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So far, generalized konane +and turning tiles have been proved to be universal partizan rulesets. +In this paper, we introduce two rulesets go on lattice and beyond the +door and prove that they are universal partizan rulesets by using game +tree preserving reduction. +1 +Introduction +Combinatorial game theory (CGT) studies two-player perfect information games +with no chance moves. We say a game is under normal play convention if the +player who moves last is the winner and a game is partizan game if the options +for both players can be different in some positions. Here, we introduce some +definitions and theorems of CGT for later discussion. For more details of CGT, +see [1, 3]. +In this theory, the two players are called Left and Right. Since the term +“game” is polysemous, we refer to each position as a game. The description of +what moves are allowed for a given position is called the ruleset. +A game is defined by Left and Right options recursively. +Definition 1. +• {|} is a game, which is called 0. +• For games GL +1 , GL +2 , . . . , GL +n, GR +1 , GR +2 , . . . , and GR +m, G = {GL +1 , GL +2 , . . . , GL +n | +GR +1 , GR +2 , . . . , GR +m} is also a game. GL +1 , GL +2 , . . . , GL +n are called left options +of G and GR +1 , GR +2 , . . . , GR +m are called right options of G. +Let G be the set of all games. +In terms of the player who has a winning strategy, G is separated into four +sets. Let L, R, N, and P be the set of positions in which Left, Right, the Next +player, and the Previous player have winning strategies, respectively. +The sets are called outcomes of the games. Every position belongs to exactly +one of the four outcomes. For a game G, let o(G) be the outcome of G. We +define the partial order of outcomes as L > P > R, L > N > R. +1 +arXiv:2301.05497v1 [math.CO] 13 Jan 2023 + +The disjunctive sum of games is an important concept in Combinatorial +Game Theory. For games G and H, a position in which a player makes a move +for one or the other on their turn is called a disjunctive sum of G and H, or +G + H. More precisely, it is as follows: +Definition 2. If the game trees of G and H are isomorphic, then we say these +games are isomorphic or G ∼= H. +Definition 3. For games G ∼= {GL +1 , GL +2 . . . GL +n | GR +1 , GR +2 , . . . , GR +m} and H ∼= +{HL +1 , HL +2 . . . , HL +n′ | HR +1 , HR +2 , . . . , HR +m′}, G + H ∼= {G + HL +1 , G + HL +2 , . . . , G + +HL +n′, GL +1 + H, GL +2 + H, . . . , GL +n + H | G + HR +1 , G + HR +2 , . . . , G + HR +m′, GR +1 + +H, GR +2 + H, . . . , GR +m + H}. +We also define equality, inequality and negative of games. +Definition 4. If for any X, o(G + X) is the same as o(H + X), then we say +G = H. +Definition 5. If o(G+H) ≥ o(H +X) holds for any X, then we say G ≥ H. On +the other hand, if o(G + H) ≤ o(H + X) holds for any X, then we say G ≤ H. +We also say G ≷ H if G ̸≥ H and G ̸≤ H. +Definition 6. For a game G ∼= {GL +1 , GL +2 , . . . , GL +n | GR +1 , GR +2 , . . . , GR +m}, let −G ∼= +{−GR +1 , −GR +2 , −GR +m | −GL +1 , −GL +2 , . . . , −GL +n}. +G + (−H) is denoted by G − H. +It is known that (G, +, =) is an abelian group and (G, ≥, =) is a partial +order. +The question arises here, will there be a ruleset in which for any game there +is a position equal to the game? If the games appearing in each ruleset are +restricted, then perhaps we should think in a narrower framework. +In fact, +however, it is known that for every game, a position equal to the game appears +in some rulesets. +1.1 +Universal partizan ruleset +Definition 7. A ruleset is universal partizan ruleset if every value in G is equal +to a position of the ruleset. +Early results showed that generalized konane and turning tiles are +universal partizan ruleset ([2, 4]). In this study, we will use the latter ruleset. +Definition 8. The ruleset of turning tiles is as follows: +• Square tiles are laid out. The front side is red or blue, and the back side +is black. +• Some pieces are on tiles. +• Each player (Left, whose color is bLue and Right, whose color is Red), +in his/her turn, take a piece and move the piece straight on the tiles of +his/her color. +2 + +• Tiles on which the piece pass over are turned over. +• The player who moves last is the winner. +Turning tiles is proved to be universal partizan ruleset even if the number +of pieces is restricted to be only one. +To distinguish this ruleset from the ruleset defined below, we will also refer +to it as blue-red turning tiles. +For games that use two colors, red and blue, corresponding to two players, +we often consider a variant that adds green, which can be used by both players. +For example, in blue-red-green hackenbush, Right can remove red or green +edges and Left can remove blue or green edges. From this point of view, we +consider a varant of turning tiles. +Definition 9. The ruleset of blue-red-green turning tiles is as follows: +• Square tiles are laid out. The front side is red, blue, or green, and the back +side is black. +• Some pieces are on tiles. +• Each player (Left, whose color is bLue and Right, whose color is Red), +in his/her turn, take a piece and move the piece straight on the tiles of +his/her color or of green. +• Tiles on which the piece pass over are turned over. +• The player who moves last is the winner. +Figure 1: Positions in blue-red turning tiles and blue-red-green turn- +ing tiles +Figure 1 is an example of positions in blue-red turning tiles and blue- +red-green turning tiles. +3 + +L +LL +RＲ +LL +RR +L +R +LLLG +R +GG +RR- +G +L +RR +R +R +G +R +G +R +L +L +R +L +R +R +R +R +RR +R +R +R +RRR +R +R +RR +R +G +G +RR +R +RR +L +7 +RR +R +一 +R +G +RR +G +R +R +P +L +R +G +G +L +R +R +LL +R +R +R +R +L +G +R +G +R +GObviously, blue-red-green turning tiles is also a universal partizan +ruleset because every position in blue-red turning tiles can be appear in +blue-red-green turning tiles. +As we have seen here, if two rulesets have an inclusion relation in terms of +the sets of positions, it can be used for proving universality of the rulesets. +Theorem 1. Let Γ and ∆ be rulesets and assume that Γ be a universal partizan +ruleset. If for every position g ∈ Γ, there is at least one position in ∆ whose +game value is the same as g, then ∆ is also a universal partizan ruleset. +Proof. This is trivial from the definition of universal partizan ruleset. +Corollary 1. Let Γ and ∆ be rulesets and assume that Γ be a universal partizan +ruleset. If for every position g ∈ Γ, there is at least one position in ∆ whose +game tree is the same as g, then ∆ is also a universal partizan ruleset. +If a ruleset is proved to be universal partizan ruleset by using Corollary 1, +we say that it is proved by game tree preserving reduction. +In the next section, we introduce two rulesets and prove that they are univer- +sal partizan ruleset by game tree preserving reduction. In Secton 3, we conclude +this study. +2 +New universal partizan rulesets +2.1 +Go on lattice +Definition 10. The rule of go on lattice is as follows: +• There is a lattice graph. There are pieces on some nodes. The edges are +colored red, blue, or dotted. +• A player, in his/her turn, chooses a piece and moves it straight on edges +colored his/her color. +• After a piece passed a node, every piece cannot get on or pass the node. +• If a player moves a piece to a node adjacent to a dotted edge, then the edge +changes to solid edge colored by the opponent’s color. +• The player who moves last is the winner. +Figure 2: Play of go on lattice +4 + +17 +MFigure 2 is a play of go on lattice. We use double line for red edges for +monochrome printing. +Theorem 2. Go on lattice is a universal partizan ruleset. +Proof. Let f be a function from a position in turning tiles to a position in +go on lattice as follows: +Let G be a position in turning tiles. In f(G) there are as many nodes +as tiles in G. The tiles in G and the nodes in f(G) are arranged exactly the +same. For each piece on a tile in G, there is a corresponding piece on the node +corresponds to the tile. For any adjacent tiles A and B in G, let A′ and B′ are +corresponding nodes in f(G). If the color of A, and B are the same, then edge +between A′ and B′ is solid and the same color as A and B. If there is a piece +on A or B, then the edge between A′ and B′ is solid and the color is the same +as the other tile. Finally, if the color of A and B are different and no piece is +on each tile, then the edge between A′ and B′ is dotted line. +● +B +B +B +B +B +R +R +R +R +● +B +R +R +B +B +B +Figure 3: Corresponding positions in turning tiles and go on lattice +. +Figure 3 shows this corresponding. Here, the game tree of G and f(G) are +isomorphic. We prove that every move in one game has a corresponding move +in the other game. +Assume that in G Left can move a piece on tile A0 to +tile An through tiles A1, A2, . . . , An−1. Then, A1, A2, . . . , An are blue tiles. Let +A′ +0, A′ +1, . . . , A′ +n be the corresponding nodes in f(G). Let (A′, B′) be the edge be- +tween A′ and B′. Then, from the definition of f, all of (A′ +1, A′ +2), (A′ +2, A′ +3), . . . , (A′ +n−1, A′ +n) +are blue edge. In addition, if A0 was a blue tile before turning, then (A′ +0, A′ +1) +is a blue edge, and if A0 was a red tile, (A′ +0, A′ +1) had been a dot edge and +after Right moved the piece to A′ +0, it changed to a blue edge. Therefore for +both case, (A′ +0, A′ +1) is a blue edge and Left can move a piece from A′ +0 to A′ +n +through A′ +1, A′ +2, . . . , A′ +n−1. Conversely, assume that in f(G), Left can move a +piece from A′ +0 to A′ +n through A′ +1, A′ +2, . . . , A′ +n−1. Then, in G, all corresponding +tiles A1, A2, . . . , An are blue tiles. Therefore, Left can move a piece from A0 to +5 + +Figure 4: Play of beyond the door. +An through A1, A2, . . . , An−1 in the corresponding position in turning tiles. +Similar proof holds for Right’s moves. +Thus, from Corollary 1, go on lattice is a universal partizan ruleset. +2.2 +Beyond the door +Definition 11. The rule of beyond the door is as follows: +• Square rooms are arranged in a grid pattern. There are doors between the +rooms. The front and back of the doors are painted red or blue. There are +pieces in several rooms. +• A player, in his/her turn, chooses a piece and moves it in a straight line. +When a piece moves beyond the door, the color of the piece’s side of the +door must be the player’s color. +• After a piece passed a room, every piece can not enter the room. +• The player who moves last is the winner. +Figure 4 shows a play of beyond the door. The red sides are masked for +monochrome printing. +Theorem 3. Beyond the door is a universal partizan ruleset. +Proof. Let f ′ be a function from a position in turning tiles to a position in +beyond the door as follows: +Let G be a position in turning tiles. In f ′(G) there are as many rooms +as tiles in G and the tiles in G and the rooms in f ′(G) are arranged exactly the +same. For each piece on a tile in G, there is a corresponding piece in the room +corresponding to the tile. For any adjacent tiles A and B in G, let A′ and B′ +are corresponding rooms in f ′(G). The color of the door between A′ and B′ is +the same as the color of A on the B′ side, and the same as the color of B on +the A′ side. +Figure 5 shows this corresponding. Here, the game tree of G and f ′(G) are +isomorphic. We prove that every move in one game has a corresponding move +in the other game. Assume that in G Left can move a piece on tile A0 to tile +An through tiles A1, A2, . . . , An−1. Then, A1, A2, . . . , An are blue tiles. Let +A′ +0, A′ +1, . . . , A′ +n be the corresponding rooms in f ′(G). Let A′ → B′ be the color +6 + +Figure 5: Corresponding positions in turning tiles and beyond the door +. +Figure 6: f and f ′ have no inverse functions. +of the door between A′ and B′ on the A′ side. Then, from the definition of f ′, +all of A′ +0 → A′ +1, A′ +1 → A′ +2, . . . , A′ +n−1 → A′ +n are blue. Therefore, Left can move +a piece from A′ +0 to A′ +n through A′ +1, A′ +2, . . . , A′ +n−1. Conversely, assume that in +f(G), Left can move a piece from A′ +0 to A′ +n through A′ +1, A′ +2, . . . , A′ +n−1. Then, +in G, all corresponding tiles A1, A2, . . . , An are blue tiles. Therefore, Left can +move a piece from A0 to An through A′ +1, A′ +2, . . . , A′ +n−1 in the corresponding +position in turning tiles. Similar proof holds for Right’s moves. +Thus, from Corollary 1, beyond the door is a universal partizan ruleset. +Note that f and f ′ have no inverse functions. For instance, Fig. 6 shows +positions in go on lattice and beyond the door. No position in turning +tiles is mapped to these positions by f and f ′ because depending on the order +of moves, both Left and Right may move pieces to the same node or the same +room in these positions. +This is somewhat interesting. +That is, even though in some ways these +rulesets are more complex than turning tiles, considering what kind of values +7 + +LLLL +L +RRR +R +R +Rcan appear in the rulesets, all of them are the same. +3 +Conclusion +In this paper, we proved go on lattice and beyond the door are universal +partizan rulesets by using game-tree preserving reduction. The method of re- +duction has been used primarily for proving complexity of problems. Since this +study shows that reduction is also effective in the proof of universality of a game, +we can expect that the knowledge accumulated in the study of computational +complexity will be utilized in the study of combinatorial game theory, and we +can expect further development of combinatorial game theory. +References +[1] M. H. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in play: An Iintro- +duction to combinatorial game theory, A K Peters, Ltd. / CRC Press(2007). +[2] A. Carvalho, C. P. Santos: A nontrivial surjective map onto the short +Conway group, Games of No Chance 5 (U. Larsson, Ed.), MSRI Book +Series 70, Cambridge University Press, pp. 271–284(2019). +[3] A. N. Siegel, Combinatorial Game Theory, American Mathematical Soci- +ety(2013). +[4] K. +Suetsugu, +Discovering +a +new +universal +partizan +ruleset, +arXiv:2201.06069 [math.CO](2022). +8 + diff --git a/0dE5T4oBgHgl3EQfOg4z/content/tmp_files/load_file.txt b/0dE5T4oBgHgl3EQfOg4z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86a4b0ed133c69912d4706c84a0195aa0ab5a080 --- /dev/null +++ b/0dE5T4oBgHgl3EQfOg4z/content/tmp_files/load_file.txt @@ -0,0 +1,288 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf,len=287 +page_content='New universal partizan rulesets Koki Suetsugu January 2023 Abstract Universal partizan ruleset is a ruleset in which every game value of partizan games can be appear as a position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' So far, generalized konane and turning tiles have been proved to be universal partizan rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In this paper, we introduce two rulesets go on lattice and beyond the door and prove that they are universal partizan rulesets by using game tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 1 Introduction Combinatorial game theory (CGT) studies two-player perfect information games with no chance moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' We say a game is under normal play convention if the player who moves last is the winner and a game is partizan game if the options for both players can be different in some positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Here, we introduce some definitions and theorems of CGT for later discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For more details of CGT, see [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In this theory, the two players are called Left and Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Since the term “game” is polysemous, we refer to each position as a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The description of what moves are allowed for a given position is called the ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' A game is defined by Left and Right options recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' {|} is a game, which is called 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For games GL 1 , GL 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GL n, GR 1 , GR 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , and GR m, G = {GL 1 , GL 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GL n | GR 1 , GR 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GR m} is also a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' GL 1 , GL 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GL n are called left options of G and GR 1 , GR 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GR m are called right options of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let G be the set of all games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In terms of the player who has a winning strategy, G is separated into four sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let L, R, N, and P be the set of positions in which Left, Right, the Next player, and the Previous player have winning strategies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The sets are called outcomes of the games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Every position belongs to exactly one of the four outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For a game G, let o(G) be the outcome of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' We define the partial order of outcomes as L > P > R, L > N > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content='05497v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content='CO] 13 Jan 2023 The disjunctive sum of games is an important concept in Combinatorial Game Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For games G and H, a position in which a player makes a move for one or the other on their turn is called a disjunctive sum of G and H, or G + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' More precisely, it is as follows: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If the game trees of G and H are isomorphic, then we say these games are isomorphic or G ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For games G ∼= {GL 1 , GL 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' GL n | GR 1 , GR 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GR m} and H ∼= {HL 1 , HL 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , HL n′ | HR 1 , HR 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , HR m′}, G + H ∼= {G + HL 1 , G + HL 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , G + HL n′, GL 1 + H, GL 2 + H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GL n + H | G + HR 1 , G + HR 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , G + HR m′, GR 1 + H, GR 2 + H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GR m + H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' We also define equality, inequality and negative of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If for any X, o(G + X) is the same as o(H + X), then we say G = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If o(G+H) ≥ o(H +X) holds for any X, then we say G ≥ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' On the other hand, if o(G + H) ≤ o(H + X) holds for any X, then we say G ≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' We also say G ≷ H if G ̸≥ H and G ̸≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For a game G ∼= {GL 1 , GL 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GL n | GR 1 , GR 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , GR m}, let −G ∼= {−GR 1 , −GR 2 , −GR m | −GL 1 , −GL 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , −GL n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' G + (−H) is denoted by G − H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' It is known that (G, +, =) is an abelian group and (G, ≥, =) is a partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The question arises here, will there be a ruleset in which for any game there is a position equal to the game?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If the games appearing in each ruleset are restricted, then perhaps we should think in a narrower framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In fact, however, it is known that for every game, a position equal to the game appears in some rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content='1 Universal partizan ruleset Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' A ruleset is universal partizan ruleset if every value in G is equal to a position of the ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Early results showed that generalized konane and turning tiles are universal partizan ruleset ([2, 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In this study, we will use the latter ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The ruleset of turning tiles is as follows: Square tiles are laid out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The front side is red or blue, and the back side is black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Some pieces are on tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Each player (Left, whose color is bLue and Right, whose color is Red), in his/her turn, take a piece and move the piece straight on the tiles of his/her color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 2 Tiles on which the piece pass over are turned over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Turning tiles is proved to be universal partizan ruleset even if the number of pieces is restricted to be only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' To distinguish this ruleset from the ruleset defined below, we will also refer to it as blue-red turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For games that use two colors, red and blue, corresponding to two players, we often consider a variant that adds green, which can be used by both players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For example, in blue-red-green hackenbush, Right can remove red or green edges and Left can remove blue or green edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' From this point of view, we consider a varant of turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The ruleset of blue-red-green turning tiles is as follows: Square tiles are laid out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The front side is red, blue, or green, and the back side is black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Some pieces are on tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Each player (Left, whose color is bLue and Right, whose color is Red), in his/her turn, take a piece and move the piece straight on the tiles of his/her color or of green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Tiles on which the piece pass over are turned over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Figure 1: Positions in blue-red turning tiles and blue-red-green turn- ing tiles Figure 1 is an example of positions in blue-red turning tiles and blue- red-green turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 3 L LL RＲ LL RR L R LLLG R GG RR- G L RR R R G R G R L L R L R R R R RR R R R RRR R R RR R G G RR R RR L 7 RR R 一 R G RR G R R P L R G G L R R LL R R R R L G R G R GObviously, blue-red-green turning tiles is also a universal partizan ruleset because every position in blue-red turning tiles can be appear in blue-red-green turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' As we have seen here, if two rulesets have an inclusion relation in terms of the sets of positions, it can be used for proving universality of the rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let Γ and ∆ be rulesets and assume that Γ be a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If for every position g ∈ Γ, there is at least one position in ∆ whose game value is the same as g, then ∆ is also a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' This is trivial from the definition of universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let Γ and ∆ be rulesets and assume that Γ be a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If for every position g ∈ Γ, there is at least one position in ∆ whose game tree is the same as g, then ∆ is also a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If a ruleset is proved to be universal partizan ruleset by using Corollary 1, we say that it is proved by game tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In the next section, we introduce two rulesets and prove that they are univer- sal partizan ruleset by game tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In Secton 3, we conclude this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 2 New universal partizan rulesets 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content='1 Go on lattice Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The rule of go on lattice is as follows: There is a lattice graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' There are pieces on some nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The edges are colored red, blue, or dotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' A player, in his/her turn, chooses a piece and moves it straight on edges colored his/her color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' After a piece passed a node, every piece cannot get on or pass the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If a player moves a piece to a node adjacent to a dotted edge, then the edge changes to solid edge colored by the opponent’s color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Figure 2: Play of go on lattice 4 17 MFigure 2 is a play of go on lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' We use double line for red edges for monochrome printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Go on lattice is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let f be a function from a position in turning tiles to a position in go on lattice as follows: Let G be a position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In f(G) there are as many nodes as tiles in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The tiles in G and the nodes in f(G) are arranged exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For each piece on a tile in G, there is a corresponding piece on the node corresponds to the tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For any adjacent tiles A and B in G, let A′ and B′ are corresponding nodes in f(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If the color of A, and B are the same, then edge between A′ and B′ is solid and the same color as A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' If there is a piece on A or B, then the edge between A′ and B′ is solid and the color is the same as the other tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Finally, if the color of A and B are different and no piece is on each tile, then the edge between A′ and B′ is dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' B B B B B R R R R B R R B B B Figure 3: Corresponding positions in turning tiles and go on lattice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Figure 3 shows this corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Here, the game tree of G and f(G) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' We prove that every move in one game has a corresponding move in the other game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Assume that in G Left can move a piece on tile A0 to tile An through tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , An−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Then, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let A′ 0, A′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n be the corresponding nodes in f(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let (A′, B′) be the edge be- tween A′ and B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Then, from the definition of f, all of (A′ 1, A′ 2), (A′ 2, A′ 3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , (A′ n−1, A′ n) are blue edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In addition, if A0 was a blue tile before turning, then (A′ 0, A′ 1) is a blue edge, and if A0 was a red tile, (A′ 0, A′ 1) had been a dot edge and after Right moved the piece to A′ 0, it changed to a blue edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Therefore for both case, (A′ 0, A′ 1) is a blue edge and Left can move a piece from A′ 0 to A′ n through A′ 1, A′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Conversely, assume that in f(G), Left can move a piece from A′ 0 to A′ n through A′ 1, A′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Then, in G, all corresponding tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Therefore, Left can move a piece from A0 to 5 Figure 4: Play of beyond the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' An through A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , An−1 in the corresponding position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Similar proof holds for Right’s moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Thus, from Corollary 1, go on lattice is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content='2 Beyond the door Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The rule of beyond the door is as follows: Square rooms are arranged in a grid pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' There are doors between the rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The front and back of the doors are painted red or blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' There are pieces in several rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' A player, in his/her turn, chooses a piece and moves it in a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' When a piece moves beyond the door, the color of the piece’s side of the door must be the player’s color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' After a piece passed a room, every piece can not enter the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Figure 4 shows a play of beyond the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The red sides are masked for monochrome printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Beyond the door is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let f ′ be a function from a position in turning tiles to a position in beyond the door as follows: Let G be a position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' In f ′(G) there are as many rooms as tiles in G and the tiles in G and the rooms in f ′(G) are arranged exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For each piece on a tile in G, there is a corresponding piece in the room corresponding to the tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For any adjacent tiles A and B in G, let A′ and B′ are corresponding rooms in f ′(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The color of the door between A′ and B′ is the same as the color of A on the B′ side, and the same as the color of B on the A′ side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Figure 5 shows this corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Here, the game tree of G and f ′(G) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' We prove that every move in one game has a corresponding move in the other game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Assume that in G Left can move a piece on tile A0 to tile An through tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , An−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Then, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let A′ 0, A′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n be the corresponding rooms in f ′(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Let A′ → B′ be the color 6 Figure 5: Corresponding positions in turning tiles and beyond the door .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Figure 6: f and f ′ have no inverse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' of the door between A′ and B′ on the A′ side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Then, from the definition of f ′, all of A′ 0 → A′ 1, A′ 1 → A′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n−1 → A′ n are blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Therefore, Left can move a piece from A′ 0 to A′ n through A′ 1, A′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Conversely, assume that in f(G), Left can move a piece from A′ 0 to A′ n through A′ 1, A′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Then, in G, all corresponding tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Therefore, Left can move a piece from A0 to An through A′ 1, A′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' , A′ n−1 in the corresponding position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Similar proof holds for Right’s moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Thus, from Corollary 1, beyond the door is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Note that f and f ′ have no inverse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' For instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 6 shows positions in go on lattice and beyond the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' No position in turning tiles is mapped to these positions by f and f ′ because depending on the order of moves, both Left and Right may move pieces to the same node or the same room in these positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' This is somewhat interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' That is, even though in some ways these rulesets are more complex than turning tiles, considering what kind of values 7 LLLL L RRR R R Rcan appear in the rulesets, all of them are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' 3 Conclusion In this paper, we proved go on lattice and beyond the door are universal partizan rulesets by using game-tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' The method of re- duction has been used primarily for proving complexity of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Since this study shows that reduction is also effective in the proof of universality of a game, we can expect that the knowledge accumulated in the study of computational complexity will be utilized in the study of combinatorial game theory, and we can expect further development of combinatorial game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' Albert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'} +page_content=' J.' 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XX, NO. XX, XXXX 2022 +1 +Detecting Severity of Diabetic Retinopathy from +Fundus Images using Ensembled Transformers +Chandranath Adak, Senior Member, IEEE, Tejas Karkera, Soumi Chattopadhyay, Member, IEEE, and +Muhammad Saqib +Abstract— Diabetic Retinopathy (DR) is considered one +of the primary concerns due to its effect on vision loss +among most people with diabetes globally. The severity of +DR is mostly comprehended manually by ophthalmologists +from fundus photography-based retina images. This paper +deals with an automated understanding of the severity +stages of DR. In the literature, researchers have focused on +this automation using traditional machine learning-based +algorithms and convolutional architectures. However, the +past works hardly focused on essential parts of the retinal +image to improve the model performance. In this paper, +we adopt transformer-based learning models to capture the +crucial features of retinal images to understand DR sever- +ity better. We work with ensembling image transformers, +where we adopt four models, namely ViT (Vision Trans- +former), BEiT (Bidirectional Encoder representation for im- +age Transformer), CaiT (Class-Attention in Image Trans- +formers), and DeiT (Data efficient image Transformers), to +infer the degree of DR severity from fundus photographs. +For experiments, we used the publicly available APTOS- +2019 blindness detection dataset, where the performances +of the transformer-based models were quite encouraging. +Index Terms— Blindness Detection, Diabetic Retinopa- +thy, Deep learning, Transformers. +I. INTRODUCTION +D +IABETES Mellitus, also known as diabetes, is a disorder +where the patient experiences increased blood sugar +levels over a long period. Diabetic Retinopathy (DR) is a mi- +crovascular complication of diabetes where the retina’s blood +vessels get damaged, which can lead to poor vision and even +blindness if untreated [1], [2]. Studies estimated that by twenty +years after diabetes onset, about 99% (or 60%) of patients +having type-I (or type-II) diabetes might have DR [1]. With +a worldwide presence of DR patients of about 126.6 million +in 2010, the current estimate is roughly around 191 million +by 2030 [3], [4]. However, about 56% of new DR cases can +be reduced by timely treatment and monitoring of the severity +[5]. The ophthalmologist analyzes fundus images for lesion- +based symptoms like microaneurysms, hard/ soft exudates, and +hemorrhages to understand the severity stages of DR [1], [2]. +The positive DR is divided into the following stages [5]: (1) +mild: the earliest stage that can contain microaneurysms, (2) +C. Adak is with Dept. of CSE, IIT Patna, India-801106, T. Karkera is +with Atharva College of Engineering, Mumbai, India-400095, S. Chat- +topadhyay is with the Dept. of CSE, IIIT Guwahati, India-781015, and +M. Saqib is with Data61, CSIRO, Australia-2122. +Corresponding author: C. Adak (e-mail: chandranath@iitp.ac.in) +negative +mild +moderate +severe +proliferative +Fig. 1. Fundus images with DR severity stages from APTOS-2019 [7]. +moderate: here, the blood vessels lose the ability to blood +transportation, (3) severe: here, blockages in blood vessels +can occur and gives a signal to grow new blood vessels, (4) +proliferative: the advanced stage where new blood vessels start +growing. Fig. 1 shows some fundus images representing the +DR severity stages. Manual examining fundus images for DR +severity stage grading may bring inconsistencies due to a high +number of patients, less number of well-trained clinicians, long +diagnosing time, unclear lesions, etc. Moreover, there may be +disagreement among ophthalmologists in choosing the correct +severity grade [6]. Therefore, computer-aided techniques have +come into the scenario for better diagnosis and broadening the +prospects of early-stage detection [2]. +Automated DR severity stage detection from fundus pho- +tographs has been performed for the last two and half decades. +Earlier, some image processing tools were used [8], [9], but +the machine learning-based DR became popular in the early +2000s. The machine learning-based techniques mostly relied +on hand-engineered features that were carefully extracted from +the fundus images and then fed to a classifier, e.g., Random +Forest (RF) [10], KNN (K-Nearest Neighbors) [11], SVM +(Support Vector Machine) [12], and ANN (Artificial Neural +Network) [13]. Although SVM and ANN-based models were +admired in the DR community, the hand-engineered feature- +based machine learning models require efficient prior feature +extraction, which may introduce errors for complex fundus +images [1], [2]. On the other hand, deep learning-based models +extract features automatically through convolution operations +[14], [15]. Besides, from 2012, deep learning architectures +rose to prominence in the computer vision community, which +also influenced the DR severity analysis from fundus images +[1]. The past deep learning-based techniques mostly employed +CNN (Convolutional Neural Network) [1], [16]. However, +the ability to give attention to certain regions/features and +fade the remaining portions hardly exists in classical CNNs. +For this reason, some contemporary methods incorporated +attention mechanism [17], [18]. Although multiple research +arXiv:2301.00973v1 [cs.CV] 3 Jan 2023 + +LOGO2 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +works are present in the literature [1], [2] and efforts were +made to detect the existence of DR in the initial stages +of its development, still there is a room for improving the +performance by incorporating higher degrees of automated +feature extraction using better deep learning models. +In this paper, we employ the transformer model for leverag- +ing its MSA (Multi-head Self-Attention) [19] to focus on the +DR revealing region of the fundus image for understanding +the severity. Moreover, the transformer model has shown high +performance in recent days [19], [20]. Initially, we adopted +ViT (Visual Transformer) [19] for detecting DR severity due to +its outperformance on image classification tasks. ViT divides +the input image into a sequence of patches and applies +global attention [19]. Moreover, since standard ViT requires +hefty amounts of data, we also adopted some other image +transformer models, such as CaiT (Class-attention in image +Transformers) [21], DeiT (Data-efficient image Transformer) +[22], and BEiT (Bidirectional Encoder representation for im- +age Transformer) [23]. CaiT is a modified version of ViT and +employs specific class-attention [21]. DeiT uses knowledge +distillation, which transfers the knowledge from one network +to another and builds a teacher-student hierarchical network +[22]. BEiT is inspired by BERT (Bidirectional Encoder Rep- +resentations from Transformers) [24] to implement masking +of image patches and to model the same for pre-training the +ViT [23]. For experiments, we used the publicly available +APTOS-2019 blindness detection dataset [7], where the in- +dividual image transformers did not perform well. Therefore, +we ensembled the image transformers to seek better predictive +performance. The ensembled image transformer obtained quite +encouraging results for DR severity stage detection. This is +one of the earliest attempts to adopt and ensemble image +transformers for DR severity stage detection, which is the main +contribution of this paper. +The rest of the paper is organized as follows. § II discusses +the relevant literature about DR and § III presents the proposed +methodology. Then § IV analyzes and discusses the experi- +mental results. Finally, § V concludes this paper. +II. RELATED WORK +This section briefly presents the literature on DR severity +detection from fundus images. The modern grading of DR +severity stages can be traced in the report by ETDRS research +group [25]. In the past, some image processing-based (e.g., +wavelet transform [8], radon transform [9]) strategies were +published. For the last two decades, machine learning and +deep learning-based approaches have shown dominance. We +broadly categorize the related works into (a) hand-engineered +feature-based models [11], [26], [27], and (b) deep feature- +based models [2], which are discussed below. +A. Hand-engineered Feature-based Models +The hand-engineered feature-based models mostly em- +ployed RF [26], KNN [28], SVM [27], ANN [29] for detecting +DR severity stages. Acharya et al. [26] employed a decision +tree with discrete wavelet/cosine transform-based features ex- +tracted from retinal images. Casanova et al. [10] introduced RF +for DR severity stage classification. In [30], RF was also used +to assess DR risk. KNN classifier was employed in [11] to +detect drusen, exudates, and cotton-wool spots for diagnosing +DR. Tang et al. [28] used KNN for retinal hemorrhage detec- +tion from fundus photographs. In [27], retinal changes due to +DR was detected by using SVM. Akram et al. [12] used SVM +and GMM (Gaussian Mixture Model) with enhanced features +such as shape, intensity, and statistics of the affected region +to identify microaneurysms for early detection of DR. ANN +was employed in [13] to classify lesions for detecting DR +severity. Osareh et al. [31] employed FCM (Fuzzy C-Means)- +based segmentation and GA (Genetic algorithm)-based feature +selection with ANN to detect exudates in DR. In [29], PSO +(Particle Swarm Optimization) was used for feature selection, +followed by ANN-based DR severity classification. +B. Deep Feature-based Models +The past deep architectures mostly used CNN for tackling +DR severity. For example, Yu et al. [16] used CNN for +detecting exudates in DR, Chudzik et al. [32] worked on +microaneurysm detection using CNN with transfer learning +and layer freezing, Gargeya and Leng [33] employed CNN- +based deep residual learning to identify fundus images with +DR. In [4], CNN was also used to identify DR severity stages +and some related eye diseases, e.g., glaucoma and AMD (Age- +related Macular Degeneration). In [34], some classical CNN +architectures (e.g., AlexNet, VGG Net, GoogLeNet, ResNet) +were employed for DR severity stage detection. Wang et al. +[17] proposed Zoom-in-Net that combined CNN, attention +mechanism, and a greedy algorithm to zoom in the region +of interest for handling DR. A modified DenseNet169 ar- +chitecture in conjunction with the attention mechanism was +used in [18] to extract refined features for DR severity +grading. In [35], a modified Xception architecture was em- +ployed for DR classification. TAN (Texture Attention Net- +work) was proposed in [36] by leveraging style (texture +features) and content (semantic and contextual features) re- +calibration mechanism. Tymchenko et al. [5] ensembled three +CNN architectures (EfficientNet-B4 [37], EfficientNet-B5, and +SE- ResNeXt50 [38]) for DR severity detection. Very recently, +a few transformer-based models have come out, e.g., CoT- +XNet [39] that combined contextual transformer and Xception +architecture, SSiT [40] that employed self-supervised image +transformers guided by saliency maps. +III. METHODOLOGY +This section first formalizes the problem statement, which +is then followed by the proposal of solution architecture. +A. Problem Formulation +In this work, we are given an image I captured by the +fundus photography, which is input to the architecture. The +task is to predict the severity stage of diabetic retinopathy (DR) +among negative, mild, moderate, severe, and proliferative, +from I. We formulate the task as a multi-class classification +problem [15]. Here, from I, features are extracted and fed to + +ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS +3 +a classifier to predict the DR severity class labels Á, where +Á = {0, 1, 2, 3, 4} corresponds to {negative, mild, moderate, +severe, proliferative}, respectively. +B. Solution Architecture +For detecting the severity stage of DR from a fundus +photograph, we adopt image transformers, i.e., ViT (Vision +Transformer) [19], BEiT (Bidirectional Encoder representation +for image Transformer) [23], CaiT (Class-attention in image +Transformers) [21], and DeiT (Data efficient image Trans- +formers) [22], and ensemble them. However, we preprocess +raw fundus images before feeding them into the transformers, +which we discuss first. +1) Preprocessing: The performance of deep learning mod- +els is susceptible to the quality and quantity of data being +passed to the model. Raw data as input can barely account for +the best achievable performance of the model due to possible +pre-existing noise and inconsistency in the images. Therefore, +a definite flow of preprocessing is essential to train the model +better [15]. +We now discuss various preprocessing and augmentation +techniques [15], [41] applied to the raw fundus photographs +for better learning. In a dataset, the fundus images may be of +various sizes; therefore, we resize the image I into 256 × 256 +sized image Iz. We perform data augmentations on training set +(DBtr), where we use centre cropping with central_fraction = +0.5, horizontal/vertical flip, random rotations within a range +of [0o, 45o], random brightness-change with max_delta = +0.95, random contrast-change in the interval [0.1, 0.9]. We +also apply CLAHE (Contrast Limited Adaptive Histogram +Equalization) [42] on 30% samples of DBtr, which ensures +over-amplification of contrast in a smaller region instead of +the entire image. +2) Transformer Networks: Deep learning models in com- +puter vision tasks have long been dominated by CNN (Convo- +lutional Neural Network) to extract high-level feature maps by +passing the image through a series of convolution operations +before feeding into the MLP (Multi-Layer Perceptron) for clas- +sification [43]. In recent days, transformer models have shown +a substantial rise in the NLP (Natural Language Processing) +domain due to its higher performances [20]. In a similar quest +to leverage high-level performance through transformers, it +has been introduced in image classification and some other +computer vision-oriented tasks [19]. Moreover, the transformer +model has lesser image-specific inductive bias than CNN [19]. +To identify the severity stages of DR from fundus images, +here we efficiently adopt and ensemble some image transform- +ers, e.g., ViT [19], BEiT [23], CaiT [21], and DeiT [22]. +Before focusing on our ensembled transformer model, we +discuss the adaptation of individual image transformers for +our task, and start with ViT. +a) Vision Transformer (ViT): The ViT model adopts the idea +of text-based transformer models [44], where the idea is to take +the input image as a series of image patches instead of textual +words, and then extract features to feed it into an MLP [19]. +The pictorial representation of ViT is presented in Fig. 2. +Here, the input image Iz is converted into a sequence of +Fig. 2. Workflow of ViT. +Fig. 3. Internal view of a transformer encoder (TE). +flattened patches xi +p (for i = 1, 2, . . . , np), each with size +wp × wp × cp, where cp denotes the number of channels +of Iz. Here, cp = 3, since Iz is an RGB fundus image. In +our task, Iz is of size 256 × 256, and empirically, we choose +wp = 64, which results np = ( 256 +64 )2 = 16. Each patch xi +p is +flattened further and mapped to a D-dimensional latent vector +(i.e., patch embedding z0) through transformer layers using a +trainable linear projection, as below. +z0 = [xclass ; x1 +p E ; x2 +p E ; . . . ; xnp +p E] + Epos +(1) +where, +E +is +the +patch +embedding +projection, +E +∈ +Rwp×wp×C×D; Epos is the position embeddings added to +patch embeddings to preserve the positional information of +patches, Epos ∈ R(np+1)×D; xclass = z0 +0 is a learnable +embedding [24]. +After mapping patch images to the embedding space with +positional information, we add a sequence of transformer +encoders [19], [45]. The internal view of a transformer encoder +can be seen in Fig. 3, which includes two blocks As and Fn. +The As and Fn contain MSA (Multi-head Self-Attention) [19] +and MLP [15] modules, respectively. LN (Layer Normaliza- +tion) [46] and residual connection [15] are employed before +and after each of these modules, respectively. This is shown +in equation 2 with general semantics. Here, the MLP module +comprises two layers having 4D and D neurons with GELU +(Gaussian Error Linear Unit) non-linear activation function + +Patch + Position +Embedding +0 +MLP Head +Softmax +* +Linear Projection of Flattened Patches +1 +Transformer Encoder (TE) +2 +3 +dn +np +pLx +LN +MSA +LN +MLP +zi +Zt4 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +similar to [19]. +z′ +l = MSA(LN(zl−1)) + zl−1; +zl = MLP(LN(z′ +l)) + z′ +l; l = 1, 2, . . . , L +(2) +where, L is the total number of transformer blocks. The core +component of the transformer encoder is MSA with h heads, +where each head includes SA (Scaled dot-product Attention) +[19], [45]. Each head i ∈ {1, 2, ..., h} of MSA calculates a +tuple comprising query, key, and value [19], i.e., (Qi, Ki, V i) +as follows. +Qi = XW i +Q ; Ki = XW i +K ; V i = XW i +V +(3) +where, X is the input embedding, and WQ, WK, WV are the +weight matrices used in the linear transformation. The tuple +(Q, K, V ) is fed to SA that computes the attention required +to pay to the input image patches, as below. +SA(Q, K, V ) = ψ +�QKT +√Dh +� +V +(4) +where, ψ is softmax function, and Dh = D/h. The outcomes +of SAs across all heads are concatenated in MSA, as follows. +MSA(Q, K, V ) = [SA1 ; SA2 ; . . . ; SAh]WL +(5) +where WL is a weight matrix. +After multiple transformer encoder blocks, the +token [24] enriches with the contextual information. The state +of the learnable embedding at the outcome of the Transformer +encoder (z0 +L) acts as the image representation y [19]. +y = LN(z0 +L) +(6) +Now, as shown in Fig. 2, we add an MLP head containing +a hidden layer with 128 neurons. To capture the non-linearity, +we use Mish [47] here. In the output layer, we keep five +neurons with softmax activation function to obtain probability +distribution s(j) in order to classify a fundus photograph into +the abovementioned five severity stages of DR. +b) Data efficient image Transformers (DeiT): For a lower +amount of training data, ViT does not generalize well. In +this scenario, DeiT can perform reasonably well and uses +lower memory [22]. DeiT adopts the ViT-specific strategy and +merges with the teacher-student scheme through knowledge +distillation [48]. The crux of DeiT is the knowledge distillation +mechanism, which is basically the knowledge transfer from +one model (teacher) to another (student) [22]. Here, we use +EfficientNet-B5 [37] as a teacher model that is trained apriori. +The student model uses a transformer, which learns from the +outcome of the teacher model through attention depending +on a distillation token [22]. In this work, we employ hard- +label distillation [22], where the hard decision of the teacher +is considered as a true label, i.e., yt = argmaxcZt(c). The +hard-label distillation objective is defined as follows. +Lhard +global = 0.5 LCE(ψ(Zs), y) + 0.5 LCE(ψ(Zs), yt) +(7) +where, LCE is the cross-entropy loss on ground-truth labels +y, ψ is the softmax function, Zs and Zt are the student and +teacher models’ logits, respectively. Using label smoothing, +Fig. 4. The distillation procedure of DeiT. +hard labels can be converted into soft ones [22]. +In Fig. 4, we present the distillation procedure of DeiT. +Here, we add the token to the transformer, +which interacts with the and tokens through +transformer encoders. The transformer encoder used here is +similar to the ViT’s one, which includes As and Fn blocks as +shown in Fig. 3. The objective of the token is to +reproduce the teacher’s predicted label instead of the ground- +truth label. The and tokens are learned +by back-propagation [15]. +A linear classifier is used in DeiT instead of the MLP head +of ViT [19], [22] to work efficiently with limited computa- +tional resources. +c) Class-attention in image Transformers (CaiT): CaiT +usually performs better than ViT and DeiT with lesser FLOPs +and learning parameters [15], when we need to increase the +depth of the transformer [21]. CaiT is basically an upgraded +version of ViT, which leverages layers with specific class- +attention and LayerScale [21]. In Fig. 5, we show the workflow +of CaiT. +LayerScale aids CaiT to work at larger depths, where we +separately multiply a diagonal matrix Mλ on the outputs of +As and Fn blocks. +z′ +l = Mλ(λl +1, . . . , λl +D) × MSA(LN(zl−1)) + zl−1; +zl = Mλ(λ′l +1, . . . , λ′l +D) × MLP(LN(z′ +l)) + z′ +l +(8) +where, λl +i and λ′l +i are learning parameters, and other symbols +denote the same as the above-mentioned ViT. +In CaiT, the transformer layers dealing with self-attention +Fig. 5. Workflow of CaiT. + +Self-attention +lass-attentionO +口ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS +5 +among patches are separated from class-attention layers that +are introduced to dedicatedly extract the content of the patches +into a vector, which can be sent to a linear classifier [21]. The + token is inserted in the latter stage, so that the initial +layers can perform the self-attention among patches devotedly. +In the class-attention stage, we alternatively use multi-head +class-attention (Ac) [21] and Fn, as shown in Fig. 5, and +update only the class embedding. +d) Bidirectional Encoder representation for image Trans- +former (BEiT): BEiT is a self-supervised model having its +root in the BERT (Bidirectional Encoder Representations from +Transformers) [23], [24]. In Fig. 6, we present the workflow +of the pre-training of BEiT. +The input image Iz is split into patches xi +p and flattened +into vectors, similar to the early-mentioned ViT. In BEiT, a +backbone transformer is engaged, for which we use ViT [19]. +On the other hand, Iz is represented as a sequence of visual +tokens vt = [vt1, vt2, . . . , vtnp] obtained by a discrete VAE +(Variational Auto-Encoder) [49]. For visual token learning, we +employ a tokenizer Tφ(vt | x) to map image pixels x to tokens +vt, and decoder Dθ(x | vt) for reconstructing input image +pixels x from vt [23]. +Here, a MIM (Masked Image Modeling) [23] task is per- +formed to pre-train the image transformers, where some image +patches are randomly masked, and the corresponding visual +tokens are then predicted. The masked patches are replaced +with a learnable embedding e[M]. We feed the corrupted image +patches xM = {xi +p : i /∈ M} �{e[M] : i ∈ M} to the +transformer encoder. Here, M is the set of indices of masked +positions. +The encoded representation hL +i +is the hidden vector of +the last transformer layer L for ith patch. For each masked +Fig. 6. Workflow of BEiT pre-training. +position, a softmax classifier ψ is used to predict the respective +visual token, i.e., pMIM(vt′ | xM) = ψ(WMhL +i + bM); where, +WM and bM contain learning parameters for linear transfor- +mation. The pre-training objective of BEiT is to maximize the +log-likelihood of the correct token vti given xM, as below: +max +� +x∈ DBtr +EM +� � +i∈M +log pMIM +� +vti | xM� +� +where, DBtr is the training dataset. The BEiT pre-training +can be perceived as VAE training [23], [49], where we follow +two stages, i.e., stage-1: minimizing loss for visual token +reconstruction, stage-2: modeling masked image, i.e., learning +prior pMIM by keeping Tφ and Dθ fixed. It can be written as +follows: +� +(xi,xM +i +) +∈ DBtr +� +� +� +�Evti∼Tφ(vt|xi) [log Dθ(xi|vti)] +� +�� +� +stage-1 ++ log pMIM +� +ˆ +vti|xM +i +� +� +�� +� +stage-2 +� +� +� +� +where, ˆ +vti = argmaxvt Tφ(vt | xi). +3) Ensembled Transformers: The abovementioned four im- +age transformers, i.e., ViT [19], DeiT [22], CaiT [21], and +BEiT [23] are pre-trained on the training set DBtr. We now +ensemble the transformers for predicting the severity stages +from fundus images of the test set DBt, since ensembling +multiple learning algorithms can achieve better performance +than the constituent algorithms alone [50]. The pictorial rep- +resentation of ensembled transformers is presented in Fig. 7. +For an image sample from DBt, we obtain the softmax +probability distribution s(j) : {P j +1 , P j +2 , . . . , P j +nc} over jth +transformer [15], for j = 1, 2, . . . , nT ; where, nc is the total +number of classes (severity stages), and nT is count of the +employed image transformers. Here, �nc +i=1 P j +i = 1, nc = 5 +(refer to § III-A), and nT = 4 since we use four separately +trained distinct image transformers, as mentioned earlier. +We obtain the severity stages/ class_labels Á|wm and Á|mv +separately using two combination methods weighted mean and +majority voting [50], respectively. +Á|wm = argmaxi P µ +i ; for i = 1, 2, . . . , nc ; +P µ +i = +�nT +j=1 αjP j +i +�nT +j=1 αj +(9) +Fig. 7. Ensembled transformers. + +Masked +Image +Patches +Original Image +latten +0 +* +Tokenizer +L +1 +BEiT Encoder +MIM Head +2 +[M] +h2 +3 +Unused at +Pre-training +Decoder +Patch + Position +Embedding +Reconstructed ImageViT +s(1) +ative +s(2) +DeiT +Mild +Moderate +Severe +CaiT +s(3) +Proliferative +c +(4) +BEiT6 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +In this task, we choose �nT +j=1 αj = 1. +Á|mv = mode +� +argmaxi(P 1 +i ), argmaxi(P 2 +i ), . . . , argmaxi(P nT +i +) +� += mode +� +argmaxi(s(1)), argmaxi(s(2)), . . . , argmaxi(s(nT )) +� +; +for i = 1, 2, . . . , nc +(10) +In this task, we use cross-entropy as the loss function [41] +in the employed image transformers. The AdamW optimizer +is used here due to its weight decay regularization effect +for tackling overfitting [51]. The training details with hyper- +parameter tuning are mentioned in Section IV-B. +IV. EXPERIMENTS AND DISCUSSIONS +In this section, we present the employed database, followed +by experimental results with discussions. +A. Database Employed +For our computational experiments, we used the publicly +available training samples of Kaggle APTOS (Asia Pacific +Tele-Ophthalmology Society) 2019 Blindness Detection dataset +[7], i.e., APTOS-2019. This database (DB) contains fundus +image samples of five severity stages of DR, i.e., negative, +mild, moderate, severe, and proliferative. Fig. 1 shows some +sample images from this dataset. In DB, a total of 3662 fundus +images are available, which we divide into training (DBtr) and +testing (DBt) datasets with a ratio of 7 : 3. As a matter of fact, +DBtr and DBt sets are disjoint. The sample counts of different +severity stages/ class_labels (Á) for DBtr and DBt are shown +in Fig. 8 individually. Here, 49.3% samples are of negative +DR (Á= 0). Among positive classes, most samples are from +the moderate stage (Á= 2). From this figure, it can be seen +DB is imbalanced due to containing a different number of +samples corresponding to various severity stages. Therefore, +we augmented the data during the training of our model as +mentioned in § III-B.1. The data augmentation also helped in +reducing the overfitting issue [15]. +Fig. 8. Count of samples in APTOS-2019 [7]. +B. Experimental Results +This section discusses the performed experiments, analyzes +the model outcome, and compares them with major state-of- +the-art methods. We begin with discussing the experimental +settings. +1) Experiment Settings: We performed the experiments on +the TensorFlow-2 framework having Python 3.7.13 over a +machine with the following configurations: Intel(R) Xeon(R) +CPU @ 2.00GHz with 52 GB RAM and Tesla T4 16 GB +GPU. All the results shown here were obtained from DBt. +The hyper-parameters of the framework were tuned and +fixed during training with respect to the performance over +some samples of DBt employed for hyper-parameter tuning. +For all the image transformers used here (i.e., ViT, DeiT, CaiT, +and BEiT), we empirically set the following hyper-parameters: +transformer_layers (L) = 12, embedding_dimension (D) = +384, num_heads (h) = 6. The following hyper-parameters +were selected for AdamW [51]: initial_learning_rate = 10−3; +exponential decay rates for 1st and 2nd moment estimates, i.e., +β1 = 0.9, β2 = 0.999; zero-denominator removal parameter +(ε) = 10−8; and weight_decay = 10−3/4. For model training, +the mini-batch size was fixed to 32. +2) Model Performance: In Table I, we present the per- +formance of our ensembled image transformer (EiT) using +the combination schemes weighted mean (wm) and majority +voting (mv), where we obtain 94.63% and 91.26% accuracy +from EiTwm and EiTmv, respectively. We also ensembled +multiple combinations of our employed transformers, and +present their performances in this table. Here, the wm scheme +performed better than mv. As evident from this table, ensem- +bling various types of transformers improved the performance. +Among single transformers (for nT = 1), CaiT performed +the best. For nT = 2 and nT = 3, “BEiT + CaiT” and +“DeiT + BEiT + CaiT” performed better than other respective +combinations. Overall, EiTwm attained the best accuracy here. +TABLE I +PERFORMANCE OVER VARIOUS ENSEMBLING OF TRANSFORMERS +nT +Ensembled Transformers +Accuracy (%) +Weighted +Majority +mean +voting +1 +ViT +82.21 +DeiT +85.65 +BEiT +86.74 +CaiT +86.91 +2 +ViT + DeiT +87.03 +86.55 +ViT + BEiT +87.48 +87.03 +ViT + CaiT +87.77 +87.21 +DeiT + BEiT +88.18 +87.69 +DeiT + CaiT +88.86 +87.93 +BEiT + CaiT +89.28 +88.12 +3 +ViT + DeiT + BEiT +90.53 +88.87 +ViT + DeiT + CaiT +91.39 +89.56 +ViT + BEiT + CaiT +92.14 +90.28 +DeiT + BEiT + CaiT +93.46 +90.91 +4 +ViT + DeiT + BEiT + CaiT +94.63 +91.26 +( EiT ) +In Fig. 10 of Appendix I, we present the coarse localization +maps generated by Grad-CAM [52] from the employed indi- +vidual image transformers to highlight the crucial regions for +understanding the severity stages. +a) Various Evaluation Metrics: Besides the accuracy, in +Table II, we present the performance of EiT with respect to +some other evaluation metrics, e.g., kappa score, precision, +recall, F1 score, specificity, balanced accuracy [53]. Here, +Cohen’s quadratic weighted kappa measures the agreement + +2000 +1800 + DBtr + DBt +1600 +541 +1400 +1200 + sampl +1000 +800 +300 +# +600 +1264 +400 +111 +699 +200 +88 +259 +135 +207 +0 +0 +1 +2 +3 +4 +severity stage/ class +label (c)ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS +7 +between human-assigned scores (i.e., DR severity stages) +and the EiT-predicted scores. Precision analyzes the true +positive samples among the total positive predictions. Recall +or sensitivity finds the true positive rate. Similarly, specificity +computes the true negative rate. F1 score is the harmonic mean +of precision and recall. Since the employed DB is imbalanced, +we also compute the balanced accuracy, which is the arithmetic +mean of sensitivity and specificity. In this table, we can see +that for both EiTwm and EiTmv, the kappa scores are greater +than 0.81, which comprehends the “almost perfect agreement” +between the human rater and EiT [53]. Here, macro means +the arithmetic mean of all per class precision/ recall/ F1 score. +TABLE II +PERFORMANCE OF EiT OVER VARIOUS EVALUATION METRICS +Metric +Weighted mean +Majority voting +(EiTwm) +(EiTmv) +Accuracy (%) +94.63 +91.26 +Kappa score +0.92 +0.87 +Macro Precision (%) +90.55 +84.65 +Macro Recall (%) +92.88 +88.81 +Macro F1-score (%) +91.67 +86.55 +Macro Specificity (%) +98.62 +97.74 +Balanced Accuracy (%) +95.75 +93.27 +b) Individual Class Performance: Table III presents the +individual performance of EiTwm and EiTmv for detecting +every severity stage of DR. From this table, we can see our +models produced the best precision and recall for negative DR +(Á= 0), and the lowest for severe DR (Á= 3). +TABLE III +PERFORMANCE OF EiT ON EVERY DR SEVERITY STAGE +class_label (Á) +0 +1 +2 +3 +4 +EiTwm +Precision (%) +98.48 +86.67 +95.00 +83.61 +89.01 +Recall (%) +95.75 +93.69 +95.00 +87.93 +92.05 +F1-score (%) +97.09 +90.04 +95.00 +85.71 +90.50 +Specificity (%) +98.56 +98.38 +98.12 +99.04 +99.01 +EiTmv +Precision (%) +96.74 +79.67 +94.14 +70.59 +82.11 +Recall (%) +93.35 +88.29 +91.00 +82.76 +88.64 +F1-score (%) +95.01 +83.76 +92.54 +76.19 +85.25 +Specificity (%) +96.95 +97.47 +97.87 +98.08 +98.32 +In each row, the best result is marked bold, second-best is italic, and lowest is underlined. +3) Comparison: In Table IV, we present a comparative +analysis with some major contemporary deep learning archi- +tectures, e.g., ResNet50 [54], InceptionV3 [55], MobileNetV2 +[56], Xception [57], DenseNet169 (Farag et al. [18]), Efficient- +Net [37], and SE-ResNeXt50 [38]. We have also compared +with recently published transformer-based models, i.e., CoT- +XNet [39], and SSiT [40]. Comparison with some major +related works [5], [35], [36] can also be seen in this table. +Our EiTwm outperformed the major state-of-the-art methods +with respect to accuracy, balanced accuracy, sensitivity, and +specificity. Our EiTmv also performed quite well in terms of +balanced accuracy. +4) Impact of Hyper-parameters: +We +tuned +the +hyper- +parameters and observed their impact on the experiment. +a) MSA Head Count: We analyzed the performance im- +pact of the number of heads (h) of MSA (Multi-head Self- +Attention) in the transformer encoder and present in Fig. 9. +As evident from this figure, the performance (accuracy) of +TABLE IV +COMPARATIVE STUDY +Method +Accuracy +Sensitivity +Specificity +Balanced +(%) +(%) +(%) +Accuracy (%) +ResNet50 [54] +74.64 +56.52 +85.71 +71.12 +InceptionV3 [55] +78.72 +63.64 +85.37 +74.51 +MobileNetV2 [56] +79.01 +76.47 +84.62 +80.55 +Xception [57] +79.59 +82.35 +86.32 +84.34 +Farag et al. [18] +82.00 +- +- +- +Kassani et al. [35] +83.09 +88.24 +87.00 +87.62 +TAN [36] +85.10 +90.30 +92.00 +- +EfficientNet-B4 [37] +90.30 +81.20 +97.60 +89.40 +EfficientNet-B5 [37] +90.70 +80.70 +97.70 +89.20 +SE-ResNeXt50 [38] +92.40 +87.10 +98.20 +92.65 +Tymchenko et al. [5] +92.90 +86.00 +98.30 +92.15 +CoT-XNet [39] +84.18 +- +95.74 +- +SSiT [40] +92.97 +- +- +- +EiTmv [ours] +91.26 +88.81 +97.74 +93.28 +EiTwm [ours] +94.63 +92.88 +98.62 +95.75 +In each column, the best result is marked bold, and the second-best is underlined. +Fig. 9. Impact of number of heads (h) in MSA on model performance. +both EiTmv and EiTwm increased with the increment of h +till h = 6, and started decreasing thereafter. +b) Weights αj of EiTwm: We tuned the weights αj (refer +to Eqn. 9) to see its impact on the performance of EiTwm. +We obtained the best accuracy of 94.63% from EiTwm for +α1 = α2 = 0.1, and α3 = α4 = 0.4. The performance of +EiTwm during tuning of αj’s is shown in Table V. +In Table VI, we also present the tuned αj’s that aided in +obtaining the best performing ensembled transformers of Table +I. +5) Ablation Study: We here present the performed ablation +study by ablating individual transformers. Our EiT is actually +an ensembling of four different image transformers, i.e., ViT, +DeiT, CaiT, and BEiT. We ablated each transformer and +observed performance degradation than EiT. For example, +considering the weighted mean scheme, when we ablated CaiT +from EiT, the accuracy dropped by 4.1%. Similarly, ablating +BEiT and CaiT deteriorated the accuracy by 7.6%. For our +task, the best individual transformer (CaiT) attained 7.72% +lower accuracy than EiTwm. More examples can be observed +in Table I. +6) Pre-training with Other Datasets: We checked the perfor- +mance of our EiT model by pre-training with some other +dataset. We took 1200 images of MESSIDOR [58] with +adjudicated grades by [59] (say, DBM). From IDRiD [60], +we also used “Disease Grading” dataset containing 516 images +(say, DBI). Here, we made four training set setups from DBM, +by taking 25%, 50%, 75%, and 100% of samples of DBM. + +96 +94.63 +EiTmv +94 +92.34 +91.92 +92 +91 +90.28 +90 +89.43 +88.59 +87.67 +88 +87 +86 +84 +82 +80 +6 +8 +108 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +TABLE V +PERFORMANCE OF EiTwm BY TUNING WEIGHTS αj +α1 +α2 +α3 +α4 +Accuracy (%) +0.25 +0.25 +0.25 +0.25 +89.53 +0.85 +0.05 +0.05 +0.05 +82.29 +0.05 +0.85 +0.05 +0.05 +85.78 +0.05 +0.05 +0.85 +0.05 +86.92 +0.05 +0.05 +0.05 +0.85 +87.05 +0.7 +0.1 +0.1 +0.1 +82.35 +0.1 +0.7 +0.1 +0.1 +85.91 +0.1 +0.1 +0.7 +0.1 +87.04 +0.1 +0.1 +0.1 +0.7 +87.20 +0.5 +0.167 +0.167 +0.166 +82.88 +0.166 +0.5 +0.167 +0.167 +86.35 +0.167 +0.166 +0.5 +0.167 +87.62 +0.167 +0.167 +0.166 +0.5 +87.74 +0.3 +0.3 +0.2 +0.2 +88.16 +0.3 +0.2 +0.3 +0.2 +89.58 +0.3 +0.2 +0.2 +0.3 +90.27 +0.2 +0.3 +0.3 +0.2 +90.85 +0.2 +0.3 +0.2 +0.3 +91.67 +0.2 +0.2 +0.3 +0.3 +92.72 +0.4 +0.4 +0.1 +0.1 +91.18 +0.4 +0.1 +0.4 +0.1 +91.49 +0.4 +0.1 +0.1 +0.4 +92.15 +0.1 +0.4 +0.4 +0.1 +92.84 +0.1 +0.4 +0.1 +0.4 +93.47 +0.1 +0.1 +0.4 +0.4 +94.63 +TABLE VI +TUNED WEIGHTS αj FOR TRANSFORMERS ENSEMBLED WITH +WEIGHTED MEAN +Transformerswm +α1 +α2 +α3 +α4 +ViT + DeiT +0.25 +0.75 +- +- +ViT + BEiT +0.4 +0.6 +- +- +ViT + CaiT +0.4 +0.6 +- +- +DeiT + BEiT +0.4 +0.6 +- +- +DeiT + CaiT +0.3 +0.7 +- +- +BEiT + CaiT +0.5 +0.5 +- +- +ViT + DeiT + BEiT +0.2 +0.3 +0.5 +- +ViT + DeiT + CaiT +0.2 +0.3 +0.5 +- +ViT + BEiT + CaiT +0.2 +0.4 +0.4 +- +DeiT + BEiT + CaiT +0.3 +0.3 +0.4 +- +ViT + DeiT + BEiT + CaiT +0.1 +0.1 +0.4 +0.4 +Similarly, four training setups were generated from DBI. As +mentioned in § IV-A, we divided APTOS-2019 database (DB) +in training (DBtr) and test (DBt) sets with a ratio of 7 : 3. In +Table VII, we present the performance of EiT on DBt, while +pre-training with DBM and DBI, and training with DBtr. +It can be observed that the performance of EiT improved +slightly when pre-trained with more data from other datasets. +TABLE VII +ACCURACY (%) OF EiT WITH PRE-TRAINING +Pre-training data +25% +50% +75% +100% +EiTwm +DBM +94.71 +94.78 +94.83 +94.88 +DBI +94.65 +94.67 +94.7 +94.79 +DBM + DBI +94.73 +94.85 +94.98 +95.13 +N.A. +94.63 +EiTmv +DBM +91.35 +91.48 +91.56 +91.61 +DBI +91.27 +91.32 +91.34 +91.35 +DBM + DBI +91.42 +91.6 +91.68 +91.75 +N.A. +91.26 +N.A.: without pre-training data +V. CONCLUSION +In this paper, we tackle the problem of automated severity +stage detection of DR from fundus images. For this purpose, +we propose two ensembled image transformers, EiTwm and +EiTmv, by using weighted mean and majority voting combi- +nation schemes, respectively. We here adopt four transformer +models, i.e., ViT, DeiT, CaiT, and BEiT. For experimentation, +we employed the publicly available APTOS-2019 blindness +detection dataset, on which EiTwm and EiTmv attained +accuracies of 94.63% and 91.26%, respectively. Although +the employed dataset was imbalanced, our models performed +quite well. Our EiTwm outperformed the major state-of-the- +art techniques. We also performed an ablation study and +observed the importance of the ensembling over the individual +transformers. +In the future, we will endeavor to improve the model perfor- +mance with some imbalanced learning techniques. 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Available: https://dx.doi.org/10.21227/H25W98 + diff --git a/0tAzT4oBgHgl3EQfDPqY/content/tmp_files/load_file.txt b/0tAzT4oBgHgl3EQfDPqY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db2f38c415533b0e33ba55a3fea04b379c738b21 --- /dev/null +++ b/0tAzT4oBgHgl3EQfDPqY/content/tmp_files/load_file.txt @@ -0,0 +1,1105 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf,len=1104 +page_content='GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, XXXX 2022 1 Detecting Severity of Diabetic Retinopathy from Fundus Images using Ensembled Transformers Chandranath Adak, Senior Member, IEEE, Tejas Karkera, Soumi Chattopadhyay, Member, IEEE, and Muhammad Saqib Abstract— Diabetic Retinopathy (DR) is considered one of the primary concerns due to its effect on vision loss among most people with diabetes globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The severity of DR is mostly comprehended manually by ophthalmologists from fundus photography-based retina images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' This paper deals with an automated understanding of the severity stages of DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In the literature, researchers have focused on this automation using traditional machine learning-based algorithms and convolutional architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' However, the past works hardly focused on essential parts of the retinal image to improve the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In this paper, we adopt transformer-based learning models to capture the crucial features of retinal images to understand DR sever- ity better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We work with ensembling image transformers, where we adopt four models, namely ViT (Vision Trans- former), BEiT (Bidirectional Encoder representation for im- age Transformer), CaiT (Class-Attention in Image Trans- formers), and DeiT (Data efficient image Transformers), to infer the degree of DR severity from fundus photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For experiments, we used the publicly available APTOS- 2019 blindness detection dataset, where the performances of the transformer-based models were quite encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Index Terms— Blindness Detection, Diabetic Retinopa- thy, Deep learning, Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' INTRODUCTION D IABETES Mellitus, also known as diabetes, is a disorder where the patient experiences increased blood sugar levels over a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Diabetic Retinopathy (DR) is a mi- crovascular complication of diabetes where the retina’s blood vessels get damaged, which can lead to poor vision and even blindness if untreated [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Studies estimated that by twenty years after diabetes onset, about 99% (or 60%) of patients having type-I (or type-II) diabetes might have DR [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' With a worldwide presence of DR patients of about 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='6 million in 2010, the current estimate is roughly around 191 million by 2030 [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' However, about 56% of new DR cases can be reduced by timely treatment and monitoring of the severity [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The ophthalmologist analyzes fundus images for lesion- based symptoms like microaneurysms, hard/ soft exudates, and hemorrhages to understand the severity stages of DR [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The positive DR is divided into the following stages [5]: (1) mild: the earliest stage that can contain microaneurysms, (2) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Adak is with Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' of CSE, IIT Patna, India-801106, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Karkera is with Atharva College of Engineering, Mumbai, India-400095, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Chat- topadhyay is with the Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' of CSE, IIIT Guwahati, India-781015, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Saqib is with Data61, CSIRO, Australia-2122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Corresponding author: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Adak (e-mail: chandranath@iitp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='in) negative mild moderate severe proliferative Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fundus images with DR severity stages from APTOS-2019 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' moderate: here, the blood vessels lose the ability to blood transportation, (3) severe: here, blockages in blood vessels can occur and gives a signal to grow new blood vessels, (4) proliferative: the advanced stage where new blood vessels start growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 1 shows some fundus images representing the DR severity stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Manual examining fundus images for DR severity stage grading may bring inconsistencies due to a high number of patients, less number of well-trained clinicians, long diagnosing time, unclear lesions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Moreover, there may be disagreement among ophthalmologists in choosing the correct severity grade [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Therefore, computer-aided techniques have come into the scenario for better diagnosis and broadening the prospects of early-stage detection [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Automated DR severity stage detection from fundus pho- tographs has been performed for the last two and half decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Earlier, some image processing tools were used [8], [9], but the machine learning-based DR became popular in the early 2000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The machine learning-based techniques mostly relied on hand-engineered features that were carefully extracted from the fundus images and then fed to a classifier, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', Random Forest (RF) [10], KNN (K-Nearest Neighbors) [11], SVM (Support Vector Machine) [12], and ANN (Artificial Neural Network) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Although SVM and ANN-based models were admired in the DR community, the hand-engineered feature- based machine learning models require efficient prior feature extraction, which may introduce errors for complex fundus images [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' On the other hand, deep learning-based models extract features automatically through convolution operations [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Besides, from 2012, deep learning architectures rose to prominence in the computer vision community, which also influenced the DR severity analysis from fundus images [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The past deep learning-based techniques mostly employed CNN (Convolutional Neural Network) [1], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' However, the ability to give attention to certain regions/features and fade the remaining portions hardly exists in classical CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For this reason, some contemporary methods incorporated attention mechanism [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Although multiple research arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00973v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='CV] 3 Jan 2023 LOGO2 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, XXXX 2022 works are present in the literature [1], [2] and efforts were made to detect the existence of DR in the initial stages of its development, still there is a room for improving the performance by incorporating higher degrees of automated feature extraction using better deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In this paper, we employ the transformer model for leverag- ing its MSA (Multi-head Self-Attention) [19] to focus on the DR revealing region of the fundus image for understanding the severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Moreover, the transformer model has shown high performance in recent days [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Initially, we adopted ViT (Visual Transformer) [19] for detecting DR severity due to its outperformance on image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' ViT divides the input image into a sequence of patches and applies global attention [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Moreover, since standard ViT requires hefty amounts of data, we also adopted some other image transformer models, such as CaiT (Class-attention in image Transformers) [21], DeiT (Data-efficient image Transformer) [22], and BEiT (Bidirectional Encoder representation for im- age Transformer) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' CaiT is a modified version of ViT and employs specific class-attention [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' DeiT uses knowledge distillation, which transfers the knowledge from one network to another and builds a teacher-student hierarchical network [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' BEiT is inspired by BERT (Bidirectional Encoder Rep- resentations from Transformers) [24] to implement masking of image patches and to model the same for pre-training the ViT [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For experiments, we used the publicly available APTOS-2019 blindness detection dataset [7], where the in- dividual image transformers did not perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Therefore, we ensembled the image transformers to seek better predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The ensembled image transformer obtained quite encouraging results for DR severity stage detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' This is one of the earliest attempts to adopt and ensemble image transformers for DR severity stage detection, which is the main contribution of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' § II discusses the relevant literature about DR and § III presents the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Then § IV analyzes and discusses the experi- mental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Finally, § V concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' RELATED WORK This section briefly presents the literature on DR severity detection from fundus images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The modern grading of DR severity stages can be traced in the report by ETDRS research group [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In the past, some image processing-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', wavelet transform [8], radon transform [9]) strategies were published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For the last two decades, machine learning and deep learning-based approaches have shown dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We broadly categorize the related works into (a) hand-engineered feature-based models [11], [26], [27], and (b) deep feature- based models [2], which are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Hand-engineered Feature-based Models The hand-engineered feature-based models mostly em- ployed RF [26], KNN [28], SVM [27], ANN [29] for detecting DR severity stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [26] employed a decision tree with discrete wavelet/cosine transform-based features ex- tracted from retinal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Casanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [10] introduced RF for DR severity stage classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In [30], RF was also used to assess DR risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' KNN classifier was employed in [11] to detect drusen, exudates, and cotton-wool spots for diagnosing DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [28] used KNN for retinal hemorrhage detec- tion from fundus photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In [27], retinal changes due to DR was detected by using SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Akram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [12] used SVM and GMM (Gaussian Mixture Model) with enhanced features such as shape, intensity, and statistics of the affected region to identify microaneurysms for early detection of DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' ANN was employed in [13] to classify lesions for detecting DR severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Osareh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [31] employed FCM (Fuzzy C-Means)- based segmentation and GA (Genetic algorithm)-based feature selection with ANN to detect exudates in DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In [29], PSO (Particle Swarm Optimization) was used for feature selection, followed by ANN-based DR severity classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Deep Feature-based Models The past deep architectures mostly used CNN for tackling DR severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For example, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [16] used CNN for detecting exudates in DR, Chudzik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [32] worked on microaneurysm detection using CNN with transfer learning and layer freezing, Gargeya and Leng [33] employed CNN- based deep residual learning to identify fundus images with DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In [4], CNN was also used to identify DR severity stages and some related eye diseases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', glaucoma and AMD (Age- related Macular Degeneration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In [34], some classical CNN architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', AlexNet, VGG Net, GoogLeNet, ResNet) were employed for DR severity stage detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [17] proposed Zoom-in-Net that combined CNN, attention mechanism, and a greedy algorithm to zoom in the region of interest for handling DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' A modified DenseNet169 ar- chitecture in conjunction with the attention mechanism was used in [18] to extract refined features for DR severity grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In [35], a modified Xception architecture was em- ployed for DR classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' TAN (Texture Attention Net- work) was proposed in [36] by leveraging style (texture features) and content (semantic and contextual features) re- calibration mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Tymchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [5] ensembled three CNN architectures (EfficientNet-B4 [37], EfficientNet-B5, and SE- ResNeXt50 [38]) for DR severity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Very recently, a few transformer-based models have come out, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', CoT- XNet [39] that combined contextual transformer and Xception architecture, SSiT [40] that employed self-supervised image transformers guided by saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' METHODOLOGY This section first formalizes the problem statement, which is then followed by the proposal of solution architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Problem Formulation In this work, we are given an image I captured by the fundus photography, which is input to the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The task is to predict the severity stage of diabetic retinopathy (DR) among negative, mild, moderate, severe, and proliferative, from I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We formulate the task as a multi-class classification problem [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, from I, features are extracted and fed to ADAK et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' : DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS 3 a classifier to predict the DR severity class labels Á, where Á = {0, 1, 2, 3, 4} corresponds to {negative, mild, moderate, severe, proliferative}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Solution Architecture For detecting the severity stage of DR from a fundus photograph, we adopt image transformers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', ViT (Vision Transformer) [19], BEiT (Bidirectional Encoder representation for image Transformer) [23], CaiT (Class-attention in image Transformers) [21], and DeiT (Data efficient image Trans- formers) [22], and ensemble them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' However, we preprocess raw fundus images before feeding them into the transformers, which we discuss first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 1) Preprocessing: The performance of deep learning mod- els is susceptible to the quality and quantity of data being passed to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Raw data as input can barely account for the best achievable performance of the model due to possible pre-existing noise and inconsistency in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Therefore, a definite flow of preprocessing is essential to train the model better [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We now discuss various preprocessing and augmentation techniques [15], [41] applied to the raw fundus photographs for better learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In a dataset, the fundus images may be of various sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' therefore, we resize the image I into 256 × 256 sized image Iz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We perform data augmentations on training set (DBtr), where we use centre cropping with central_fraction = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='5, horizontal/vertical flip, random rotations within a range of [0o, 45o], random brightness-change with max_delta = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='95, random contrast-change in the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We also apply CLAHE (Contrast Limited Adaptive Histogram Equalization) [42] on 30% samples of DBtr, which ensures over-amplification of contrast in a smaller region instead of the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 2) Transformer Networks: Deep learning models in com- puter vision tasks have long been dominated by CNN (Convo- lutional Neural Network) to extract high-level feature maps by passing the image through a series of convolution operations before feeding into the MLP (Multi-Layer Perceptron) for clas- sification [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In recent days, transformer models have shown a substantial rise in the NLP (Natural Language Processing) domain due to its higher performances [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In a similar quest to leverage high-level performance through transformers, it has been introduced in image classification and some other computer vision-oriented tasks [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Moreover, the transformer model has lesser image-specific inductive bias than CNN [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' To identify the severity stages of DR from fundus images, here we efficiently adopt and ensemble some image transform- ers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', ViT [19], BEiT [23], CaiT [21], and DeiT [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Before focusing on our ensembled transformer model, we discuss the adaptation of individual image transformers for our task, and start with ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' a) Vision Transformer (ViT): The ViT model adopts the idea of text-based transformer models [44], where the idea is to take the input image as a series of image patches instead of textual words, and then extract features to feed it into an MLP [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The pictorial representation of ViT is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, the input image Iz is converted into a sequence of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Workflow of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Internal view of a transformer encoder (TE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' flattened patches xi p (for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , np), each with size wp × wp × cp, where cp denotes the number of channels of Iz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, cp = 3, since Iz is an RGB fundus image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In our task, Iz is of size 256 × 256, and empirically, we choose wp = 64, which results np = ( 256 64 )2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Each patch xi p is flattened further and mapped to a D-dimensional latent vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', patch embedding z0) through transformer layers using a trainable linear projection, as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' z0 = [xclass ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' x1 p E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' x2 p E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' xnp p E] + Epos (1) where, E is the patch embedding projection, E ∈ Rwp×wp×C×D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Epos is the position embeddings added to patch embeddings to preserve the positional information of patches, Epos ∈ R(np+1)×D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' xclass = z0 0 is a learnable embedding [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' After mapping patch images to the embedding space with positional information, we add a sequence of transformer encoders [19], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The internal view of a transformer encoder can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 3, which includes two blocks As and Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The As and Fn contain MSA (Multi-head Self-Attention) [19] and MLP [15] modules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' LN (Layer Normaliza- tion) [46] and residual connection [15] are employed before and after each of these modules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' This is shown in equation 2 with general semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, the MLP module comprises two layers having 4D and D neurons with GELU (Gaussian Error Linear Unit) non-linear activation function Patch + Position Embedding 0 MLP Head Softmax Linear Projection of Flattened Patches 1 Transformer Encoder (TE) 2 3 dn np pLx LN MSA LN MLP zi Zt4 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, XXXX 2022 similar to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' z′ l = MSA(LN(zl−1)) + zl−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' zl = MLP(LN(z′ l)) + z′ l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , L (2) where, L is the total number of transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The core component of the transformer encoder is MSA with h heads, where each head includes SA (Scaled dot-product Attention) [19], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Each head i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', h} of MSA calculates a tuple comprising query, key, and value [19], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', (Qi, Ki, V i) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Qi = XW i Q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Ki = XW i K ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' V i = XW i V (3) where, X is the input embedding, and WQ, WK, WV are the weight matrices used in the linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The tuple (Q, K, V ) is fed to SA that computes the attention required to pay to the input image patches, as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' SA(Q, K, V ) = ψ �QKT √Dh � V (4) where, ψ is softmax function, and Dh = D/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The outcomes of SAs across all heads are concatenated in MSA, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' MSA(Q, K, V ) = [SA1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' SA2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' SAh]WL (5) where WL is a weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' After multiple transformer encoder blocks, the token [24] enriches with the contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The state of the learnable embedding at the outcome of the Transformer encoder (z0 L) acts as the image representation y [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' y = LN(z0 L) (6) Now, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 2, we add an MLP head containing a hidden layer with 128 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' To capture the non-linearity, we use Mish [47] here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In the output layer, we keep five neurons with softmax activation function to obtain probability distribution s(j) in order to classify a fundus photograph into the abovementioned five severity stages of DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' b) Data efficient image Transformers (DeiT): For a lower amount of training data, ViT does not generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In this scenario, DeiT can perform reasonably well and uses lower memory [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' DeiT adopts the ViT-specific strategy and merges with the teacher-student scheme through knowledge distillation [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The crux of DeiT is the knowledge distillation mechanism, which is basically the knowledge transfer from one model (teacher) to another (student) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, we use EfficientNet-B5 [37] as a teacher model that is trained apriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The student model uses a transformer, which learns from the outcome of the teacher model through attention depending on a distillation token [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In this work, we employ hard- label distillation [22], where the hard decision of the teacher is considered as a true label, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', yt = argmaxcZt(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The hard-label distillation objective is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Lhard global = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='5 LCE(ψ(Zs), y) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='5 LCE(ψ(Zs), yt) (7) where, LCE is the cross-entropy loss on ground-truth labels y, ψ is the softmax function, Zs and Zt are the student and teacher models’ logits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Using label smoothing, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The distillation procedure of DeiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' hard labels can be converted into soft ones [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 4, we present the distillation procedure of DeiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, we add the token to the transformer, which interacts with the and tokens through transformer encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The transformer encoder used here is similar to the ViT’s one, which includes As and Fn blocks as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The objective of the token is to reproduce the teacher’s predicted label instead of the ground- truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The and tokens are learned by back-propagation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' A linear classifier is used in DeiT instead of the MLP head of ViT [19], [22] to work efficiently with limited computa- tional resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' c) Class-attention in image Transformers (CaiT): CaiT usually performs better than ViT and DeiT with lesser FLOPs and learning parameters [15], when we need to increase the depth of the transformer [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' CaiT is basically an upgraded version of ViT, which leverages layers with specific class- attention and LayerScale [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 5, we show the workflow of CaiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' LayerScale aids CaiT to work at larger depths, where we separately multiply a diagonal matrix Mλ on the outputs of As and Fn blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' z′ l = Mλ(λl 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , λl D) × MSA(LN(zl−1)) + zl−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' zl = Mλ(λ′l 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , λ′l D) × MLP(LN(z′ l)) + z′ l (8) where, λl i and λ′l i are learning parameters, and other symbols denote the same as the above-mentioned ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In CaiT, the transformer layers dealing with self-attention Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Workflow of CaiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Self-attention lass-attentionO 口ADAK et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' : DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS 5 among patches are separated from class-attention layers that are introduced to dedicatedly extract the content of the patches into a vector, which can be sent to a linear classifier [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The token is inserted in the latter stage, so that the initial layers can perform the self-attention among patches devotedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In the class-attention stage, we alternatively use multi-head class-attention (Ac) [21] and Fn, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 5, and update only the class embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' d) Bidirectional Encoder representation for image Trans- former (BEiT): BEiT is a self-supervised model having its root in the BERT (Bidirectional Encoder Representations from Transformers) [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 6, we present the workflow of the pre-training of BEiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The input image Iz is split into patches xi p and flattened into vectors, similar to the early-mentioned ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In BEiT, a backbone transformer is engaged, for which we use ViT [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' On the other hand, Iz is represented as a sequence of visual tokens vt = [vt1, vt2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , vtnp] obtained by a discrete VAE (Variational Auto-Encoder) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For visual token learning, we employ a tokenizer Tφ(vt | x) to map image pixels x to tokens vt, and decoder Dθ(x | vt) for reconstructing input image pixels x from vt [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, a MIM (Masked Image Modeling) [23] task is per- formed to pre-train the image transformers, where some image patches are randomly masked, and the corresponding visual tokens are then predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The masked patches are replaced with a learnable embedding e[M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We feed the corrupted image patches xM = {xi p : i /∈ M} �{e[M] : i ∈ M} to the transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, M is the set of indices of masked positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The encoded representation hL i is the hidden vector of the last transformer layer L for ith patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For each masked Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Workflow of BEiT pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' position, a softmax classifier ψ is used to predict the respective visual token, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', pMIM(vt′ | xM) = ψ(WMhL i + bM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' where, WM and bM contain learning parameters for linear transfor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The pre-training objective of BEiT is to maximize the log-likelihood of the correct token vti given xM, as below: max � x∈ DBtr EM � � i∈M log pMIM � vti | xM� � where, DBtr is the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The BEiT pre-training can be perceived as VAE training [23], [49], where we follow two stages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', stage-1: minimizing loss for visual token reconstruction, stage-2: modeling masked image, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', learning prior pMIM by keeping Tφ and Dθ fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' It can be written as follows: � (xi,xM i ) ∈ DBtr � � � �Evti∼Tφ(vt|xi) [log Dθ(xi|vti)] � �� � stage-1 + log pMIM � ˆ vti|xM i � � �� � stage-2 � � � � where, ˆ vti = argmaxvt Tφ(vt | xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 3) Ensembled Transformers: The abovementioned four im- age transformers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', ViT [19], DeiT [22], CaiT [21], and BEiT [23] are pre-trained on the training set DBtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We now ensemble the transformers for predicting the severity stages from fundus images of the test set DBt, since ensembling multiple learning algorithms can achieve better performance than the constituent algorithms alone [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The pictorial rep- resentation of ensembled transformers is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For an image sample from DBt, we obtain the softmax probability distribution s(j) : {P j 1 , P j 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , P j nc} over jth transformer [15], for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , nT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' where, nc is the total number of classes (severity stages), and nT is count of the employed image transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, �nc i=1 P j i = 1, nc = 5 (refer to § III-A), and nT = 4 since we use four separately trained distinct image transformers, as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We obtain the severity stages/ class_labels Á|wm and Á|mv separately using two combination methods weighted mean and majority voting [50], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Á|wm = argmaxi P µ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , nc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' P µ i = �nT j=1 αjP j i �nT j=1 αj (9) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Ensembled transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Masked Image Patches Original Image latten 0 Tokenizer L 1 BEiT Encoder MIM Head 2 [M] h2 3 Unused at Pre-training Decoder Patch + Position Embedding Reconstructed ImageViT s(1) ative s(2) DeiT Mild Moderate Severe CaiT s(3) Proliferative c (4) BEiT6 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, XXXX 2022 In this task, we choose �nT j=1 αj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Á|mv = mode � argmaxi(P 1 i ), argmaxi(P 2 i ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , argmaxi(P nT i ) � = mode � argmaxi(s(1)), argmaxi(s(2)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , argmaxi(s(nT )) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' , nc (10) In this task, we use cross-entropy as the loss function [41] in the employed image transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The AdamW optimizer is used here due to its weight decay regularization effect for tackling overfitting [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The training details with hyper- parameter tuning are mentioned in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' EXPERIMENTS AND DISCUSSIONS In this section, we present the employed database, followed by experimental results with discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Database Employed For our computational experiments, we used the publicly available training samples of Kaggle APTOS (Asia Pacific Tele-Ophthalmology Society) 2019 Blindness Detection dataset [7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', APTOS-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' This database (DB) contains fundus image samples of five severity stages of DR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', negative, mild, moderate, severe, and proliferative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 1 shows some sample images from this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In DB, a total of 3662 fundus images are available, which we divide into training (DBtr) and testing (DBt) datasets with a ratio of 7 : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' As a matter of fact, DBtr and DBt sets are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The sample counts of different severity stages/ class_labels (Á) for DBtr and DBt are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 8 individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='3% samples are of negative DR (Á= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Among positive classes, most samples are from the moderate stage (Á= 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' From this figure, it can be seen DB is imbalanced due to containing a different number of samples corresponding to various severity stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Therefore, we augmented the data during the training of our model as mentioned in § III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The data augmentation also helped in reducing the overfitting issue [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Count of samples in APTOS-2019 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Experimental Results This section discusses the performed experiments, analyzes the model outcome, and compares them with major state-of- the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We begin with discussing the experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 1) Experiment Settings: We performed the experiments on the TensorFlow-2 framework having Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='13 over a machine with the following configurations: Intel(R) Xeon(R) CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00GHz with 52 GB RAM and Tesla T4 16 GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' All the results shown here were obtained from DBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The hyper-parameters of the framework were tuned and fixed during training with respect to the performance over some samples of DBt employed for hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For all the image transformers used here (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', ViT, DeiT, CaiT, and BEiT), we empirically set the following hyper-parameters: transformer_layers (L) = 12, embedding_dimension (D) = 384, num_heads (h) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The following hyper-parameters were selected for AdamW [51]: initial_learning_rate = 10−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' exponential decay rates for 1st and 2nd moment estimates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' zero-denominator removal parameter (ε) = 10−8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' and weight_decay = 10−3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For model training, the mini-batch size was fixed to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 2) Model Performance: In Table I, we present the per- formance of our ensembled image transformer (EiT) using the combination schemes weighted mean (wm) and majority voting (mv), where we obtain 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63% and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='26% accuracy from EiTwm and EiTmv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We also ensembled multiple combinations of our employed transformers, and present their performances in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, the wm scheme performed better than mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' As evident from this table, ensem- bling various types of transformers improved the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Among single transformers (for nT = 1), CaiT performed the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For nT = 2 and nT = 3, “BEiT + CaiT” and “DeiT + BEiT + CaiT” performed better than other respective combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Overall, EiTwm attained the best accuracy here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' TABLE I PERFORMANCE OVER VARIOUS ENSEMBLING OF TRANSFORMERS nT Ensembled Transformers Accuracy (%) Weighted Majority mean voting 1 ViT 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='21 DeiT 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='65 BEiT 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='74 CaiT 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='91 2 ViT + DeiT 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='03 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='55 ViT + BEiT 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='48 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='03 ViT + CaiT 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='77 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='21 DeiT + BEiT 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='18 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='69 DeiT + CaiT 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='86 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='93 BEiT + CaiT 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='28 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='12 3 ViT + DeiT + BEiT 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='53 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='87 ViT + DeiT + CaiT 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='39 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='56 ViT + BEiT + CaiT 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='14 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='28 DeiT + BEiT + CaiT 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='46 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='91 4 ViT + DeiT + BEiT + CaiT 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='26 ( EiT ) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 10 of Appendix I, we present the coarse localization maps generated by Grad-CAM [52] from the employed indi- vidual image transformers to highlight the crucial regions for understanding the severity stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' a) Various Evaluation Metrics: Besides the accuracy, in Table II, we present the performance of EiT with respect to some other evaluation metrics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', kappa score, precision, recall, F1 score, specificity, balanced accuracy [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, Cohen’s quadratic weighted kappa measures the agreement 2000 1800 DBtr DBt 1600 541 1400 1200 sampl 1000 800 300 # 600 1264 400 111 699 200 88 259 135 207 0 0 1 2 3 4 severity stage/ class label (c)ADAK et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' : DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS 7 between human-assigned scores (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', DR severity stages) and the EiT-predicted scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Precision analyzes the true positive samples among the total positive predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Recall or sensitivity finds the true positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Similarly, specificity computes the true negative rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' F1 score is the harmonic mean of precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Since the employed DB is imbalanced, we also compute the balanced accuracy, which is the arithmetic mean of sensitivity and specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In this table, we can see that for both EiTwm and EiTmv, the kappa scores are greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='81, which comprehends the “almost perfect agreement” between the human rater and EiT [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, macro means the arithmetic mean of all per class precision/ recall/ F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' TABLE II PERFORMANCE OF EiT OVER VARIOUS EVALUATION METRICS Metric Weighted mean Majority voting (EiTwm) (EiTmv) Accuracy (%) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='26 Kappa score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='87 Macro Precision (%) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='55 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='65 Macro Recall (%) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='88 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='81 Macro F1-score (%) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='67 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='55 Macro Specificity (%) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='62 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='74 Balanced Accuracy (%) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='75 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='27 b) Individual Class Performance: Table III presents the individual performance of EiTwm and EiTmv for detecting every severity stage of DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' From this table, we can see our models produced the best precision and recall for negative DR (Á= 0), and the lowest for severe DR (Á= 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' TABLE III PERFORMANCE OF EiT ON EVERY DR SEVERITY STAGE class_label (Á) 0 1 2 3 4 EiTwm Precision (%) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='48 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='67 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='61 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='01 Recall (%) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='75 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='69 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='93 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='05 F1-score (%) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='09 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='04 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='71 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='50 Specificity (%) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='56 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='38 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='12 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='04 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='01 EiTmv Precision (%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='74 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='67 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='14 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='59 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='11 Recall (%) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='35 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='29 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='76 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='64 F1-score (%) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='01 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='76 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='54 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='19 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='25 Specificity (%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='95 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='47 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='87 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='08 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='32 In each row, the best result is marked bold, second-best is italic, and lowest is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 3) Comparison: In Table IV, we present a comparative analysis with some major contemporary deep learning archi- tectures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', ResNet50 [54], InceptionV3 [55], MobileNetV2 [56], Xception [57], DenseNet169 (Farag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [18]), Efficient- Net [37], and SE-ResNeXt50 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We have also compared with recently published transformer-based models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', CoT- XNet [39], and SSiT [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Comparison with some major related works [5], [35], [36] can also be seen in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Our EiTwm outperformed the major state-of-the-art methods with respect to accuracy, balanced accuracy, sensitivity, and specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Our EiTmv also performed quite well in terms of balanced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 4) Impact of Hyper-parameters: We tuned the hyper- parameters and observed their impact on the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' a) MSA Head Count: We analyzed the performance im- pact of the number of heads (h) of MSA (Multi-head Self- Attention) in the transformer encoder and present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' As evident from this figure, the performance (accuracy) of TABLE IV COMPARATIVE STUDY Method Accuracy Sensitivity Specificity Balanced (%) (%) (%) Accuracy (%) ResNet50 [54] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='64 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='52 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='71 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='12 InceptionV3 [55] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='72 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='64 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='37 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='51 MobileNetV2 [56] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='01 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='47 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='62 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='55 Xception [57] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='59 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='35 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='32 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='34 Farag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [18] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 Kassani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [35] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='09 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='24 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='62 TAN [36] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='10 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='30 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 EfficientNet-B4 [37] 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='30 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='20 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='60 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='40 EfficientNet-B5 [37] 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='70 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='70 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='70 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='20 SE-ResNeXt50 [38] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='40 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='10 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='20 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='65 Tymchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' [5] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='90 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='00 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='30 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='15 CoT-XNet [39] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='18 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='74 SSiT [40] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='97 EiTmv [ours] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='26 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='81 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='74 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='28 EiTwm [ours] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='88 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='62 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='75 In each column, the best result is marked bold, and the second-best is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Impact of number of heads (h) in MSA on model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' both EiTmv and EiTwm increased with the increment of h till h = 6, and started decreasing thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' b) Weights αj of EiTwm: We tuned the weights αj (refer to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 9) to see its impact on the performance of EiTwm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We obtained the best accuracy of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63% from EiTwm for α1 = α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='1, and α3 = α4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' The performance of EiTwm during tuning of αj’s is shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In Table VI, we also present the tuned αj’s that aided in obtaining the best performing ensembled transformers of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 5) Ablation Study: We here present the performed ablation study by ablating individual transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Our EiT is actually an ensembling of four different image transformers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', ViT, DeiT, CaiT, and BEiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We ablated each transformer and observed performance degradation than EiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For example, considering the weighted mean scheme, when we ablated CaiT from EiT, the accuracy dropped by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Similarly, ablating BEiT and CaiT deteriorated the accuracy by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For our task, the best individual transformer (CaiT) attained 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='72% lower accuracy than EiTwm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' More examples can be observed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 6) Pre-training with Other Datasets: We checked the perfor- mance of our EiT model by pre-training with some other dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We took 1200 images of MESSIDOR [58] with adjudicated grades by [59] (say, DBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' From IDRiD [60], we also used “Disease Grading” dataset containing 516 images (say, DBI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Here, we made four training set setups from DBM, by taking 25%, 50%, 75%, and 100% of samples of DBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 96 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63 EiTmv 94 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='34 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='92 92 91 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='28 90 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='43 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='59 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='67 88 87 86 84 82 80 6 8 108 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' XX, XXXX 2022 TABLE V PERFORMANCE OF EiTwm BY TUNING WEIGHTS αj α1 α2 α3 α4 Accuracy (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='25 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='5 ViT + DeiT + BEiT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='5 ViT + DeiT + CaiT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='5 ViT + BEiT + CaiT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='4 DeiT + BEiT + CaiT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='4 ViT + DeiT + BEiT + CaiT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='4 Similarly, four training setups were generated from DBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' As mentioned in § IV-A, we divided APTOS-2019 database (DB) in training (DBtr) and test (DBt) sets with a ratio of 7 : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In Table VII, we present the performance of EiT on DBt, while pre-training with DBM and DBI, and training with DBtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' It can be observed that the performance of EiT improved slightly when pre-trained with more data from other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' TABLE VII ACCURACY (%) OF EiT WITH PRE-TRAINING Pre-training data 25% 50% 75% 100% EiTwm DBM 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='71 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='78 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='83 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='88 DBI 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='65 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='67 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='79 DBM + DBI 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='73 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='85 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='98 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='13 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63 EiTmv DBM 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='35 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='48 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='56 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='61 DBI 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='27 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='32 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='34 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='35 DBM + DBI 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='42 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='68 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='75 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='26 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' : without pre-training data V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' CONCLUSION In this paper, we tackle the problem of automated severity stage detection of DR from fundus images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For this purpose, we propose two ensembled image transformers, EiTwm and EiTmv, by using weighted mean and majority voting combi- nation schemes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We here adopt four transformer models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=', ViT, DeiT, CaiT, and BEiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' For experimentation, we employed the publicly available APTOS-2019 blindness detection dataset, on which EiTwm and EiTmv attained accuracies of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='63% and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='26%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Although the employed dataset was imbalanced, our models performed quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Our EiTwm outperformed the major state-of-the- art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' We also performed an ablation study and observed the importance of the ensembling over the individual transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' In the future, we will endeavor to improve the model perfor- mance with some imbalanced learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Currently, our model does not perform any lesion segmentation, which we will also attempt to explore some implicit characteristics of fundus images due to DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' APPENDIX I QUALITATIVE VISUALIZATION As mentioned in § IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content='2, we present the Grad-CAM maps of the employed individual image transformers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' negative mild moderate severe proliferative Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fundus images (1st row) with Grad-CAM maps for ViT, DeiT, BEiT, CaiT as shown in 2nd, 3rd, 4th, 5th rows, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Stolte and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' Fang, “A survey on medical image analysis in diabetic retinopathy,” Medical image analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} +page_content=' 64, p.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf'} diff --git a/0tAzT4oBgHgl3EQfDPqY/vector_store/index.pkl b/0tAzT4oBgHgl3EQfDPqY/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..5e17229ab89be832e0b99368db886b27e0c28ce9 --- /dev/null +++ b/0tAzT4oBgHgl3EQfDPqY/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cc427607bd91fe7c63af46d8869ec43535372a3a594fd7019d44feeaa3e8c4cc +size 149953 diff --git a/0tFPT4oBgHgl3EQfTjTq/content/tmp_files/2301.13054v1.pdf.txt b/0tFPT4oBgHgl3EQfTjTq/content/tmp_files/2301.13054v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bcb351b0754f84f819e3ef1c6275fa96653daadd --- /dev/null +++ b/0tFPT4oBgHgl3EQfTjTq/content/tmp_files/2301.13054v1.pdf.txt @@ -0,0 +1,1564 @@ +arXiv:2301.13054v1 [cs.FL] 30 Jan 2023 +Monadic Expressions and their Derivatives +Samira Attou1, Ludovic Mignot2, Clément Miklarz2, and Florent Nicart2 +1 Université Gustave Eiffel, +5 Boulevard Descartes — Champs s/ Marne +77454 Marne-la-Vallée Cedex 2 +2 GR2IF, +Université de Rouen Normandie, +Avenue de l’Université, +76801 Saint-Étienne-du-Rouvray, France +samira.attou@univ-eiffel.fr, +{ludovic.mignot,clement.miklarz1, florent.nicart}@univ-rouen.fr +Abstract. We propose another interpretation of well-known derivatives computations from regular expres- +sions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data +structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization +advantage, we first introduce a new derivation technique based on the graded module monad and then show an +application of this technique to generalize the parsing of expression with capture groups and back references. +We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations +(like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). +Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, +using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us +to perform this construction in a unified way, whatever the underlying monad. +Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques +of functional programming, and we provide a web interface to manipulate concrete examples. +1 +Introduction +This paper is an extended version of [2]. +Regular expressions are a classical way to represent associations between words and value sets. As an example, +classical regular expressions denote sets of words and regular expressions with multiplicities denote formal series. +From a regular expression, solving the membership test (determining whether a word belongs to the denoted +language) or the weighting test (determining the weight of a word in the denoted formal series) can be solved, +following Kleene theorems [11,17] by computing a finite automaton, such as the position automaton [9,3,5,6]. +Another family of methods to solve these tests is the family of derivative computations, that does not require the +construction of a whole automaton. The common point of these techniques is to transform the test for an arbitrary +word into the test for the empty word, which can be easily solved in a purely syntactical way (i.e. by induction over +the structure of expressions). Brzozowski [4] shows how to compute, from a regular expression E and a word w, a +regular expression dw(E) denoting the set of words w′ such that ww′ belongs to the language denoted by E. Solving +the membership test hence becomes the membership test for the empty word in the expression dw(E). Antimirov [1] +modifies this method in order to produce sets of expressions instead of expressions, i.e. defines the partial derivatives +∂w(E) as a set of expressions the sum of which denotes the same language as dw(E). If the number of derivatives +is exponential w.r.t. the length |E| of E in the worst case3, the partial derivatives produce at most a linear number +of expressions w.r.t. |E|. Lombardy and Sakarovitch [13] extends these methods to expressions with multiplicities. +Finally, Sulzmann and Lu [18] apply these derivation techniques to parse POSIX expressions. +It is well-known that these methods are based on a common operation, the quotient of languages. Furthermore, +Antimirov’s method can be interpreted as the derivation of regular expression with multiplicities in the Boolean +semiring. However, the Brzozowski computation does not produce the same expressions (i.e. equality over the syntax +trees) as the Antimirov one. +Main contributions: In this paper, we present a unification of these computations by applying notions of +category theory to the category of sets, and show how to compute categorical automata as defined in [7], by reinter- +preting the work started in [15]. We make use of classical monads to model well-known derivatives computations. +Furthermore, we deal with extended expressions in a general way: in this paper, expressions can support extended +3 as far as rules of associativity, commutativity and idempotence of the sum are considered, possibly infinite otherwise. + +operators like complement, intersection, but also any n-ary function (algebraic mean, extrema multiplications, etc.). +The main difference with [15] is that we formally state the languages and series that the expressions denote in an +inherent way w.r.t. the underlying monads. +More precisely, this paper presents: +– an extension of expressions to any n-ary function over the value set, +– a monadic generalization of expressions, +– a solution for the membership/weight test for these expressions, +– a computation of categorical derivative automata, +– a new monad that fits with the extension to n-ary functions, +– an illustration implemented in Haskell using advanced functional programming, +– an extension to capture groups and back references expressions. +Motivation: The unification of derivation techniques is a goal by itself. Moreover, the formal tools used to +achieve this unification are also useful: Monads offer both theoretical and practical advantages. Indeed, from a +theoretical point of view, these structures allow the abstraction of properties and focus on the principal mechanisms +that allow solving the membership and weight problems. Besides, the introduction of exotic monads can also facilitate +the study of finiteness of derivated terms. From a practical point of view, monads are easy to implement (even in +some other languages than Haskell) and allow us to produce compact and safe code. Finally, we can easily combine +different algebraic structures or add some technical functionalities (capture groups, logging, nondeterminism, etc.) +thanks to notions like monad transformers [10] that we consider in this paper. +This paper is structured as follows. In Section 2, we gather some preliminary material, like algebraic structures +or category theory notions. We also introduce some functions well-known to the Haskell community that can allow +us to reduce the size of our equations. We then structurally define the expressions we deal with, the associated series +and the weight test for the empty word in Section 3. In order to extend this test to any arbitrary word, we first state +in Section 4 some properties required by the monads we consider. Once this so-called support is determined, we show +in Section 5 how to compute the derivatives. The computation of derivative automata is explained in Section 6. +A new monad and its associated derivatives computation is given in Section 7. An implementation is presented +in Section 8. Finally, we show how to (alternatively to [18]) compute derivatives of capture group expressions in +Section 9 and show that as far as the same operators are concerned, the derivative formulae are the same whatever +the underlying monad is. +2 +Preliminaries +We denote by S → S′ the set of functions from a set S to a set S′. The notation λx → f(x) is an equivalent notation +for a function f. +A monoid is a set S endowed with an associative operation and a unit element. A semiring is a structure +(S, ×, +, 1, 0) such that (S, ×, 1) is a monoid, (S, +, 0) is a commutative monoid, × distributes over + and 0 is an +annihilator for ×. A starred semiring is a semiring with a unary function ⋆ such that +k⋆ = 1 + k × k⋆ = 1 + k⋆ × k. +A K-series over the free monoid (Σ∗, ·, ε) associated with an alphabet Σ, for a semiring K = (K, ×, +, 1, 0), is +a function from Σ∗ to K. The set of K-series can be endowed with the structure of semiring as follows: +1(w) = +� +1 +if w = ε, +0 +otherwise, +0(w) = 0, +(S1 + S2)(w) = S1(w) + S2(w), +(S1 × S2)(w) = +� +u·v=w +S1(u) × S2(v). +Furthermore, if S1(ε) = 0 (i.e. S1 is said to be proper), the star of S1 is the series defined by +(S1)⋆(ε) = 1, +(S1)⋆(w) = +� +n≤|w|,w=u1···un,uj̸=ε +S1(u1) × · · · × S1(un). +Finally, for any function f in Kn → K, we set: +(f(S1, . . . , Sn))(w) = f(S1(w), . . . , Sn(w)). +(1) +A functor 4 F associates with each set S a set F(S) and with each function f in S → S′ a function F(f) from +F(S) to F(S′) such that +F(id) = id, +F(f ◦ g) = F(f) ◦ F(g), +4 More precisely, a functor over a subcategory of the category of sets. + +where id is the identity function and ◦ the classical function composition. +A monad5 M is a functor endowed with two (families of) functions +– pure, from a set S to M(S), +– bind, sending any function f in S → M(S′) to M(S) → M(S′), +such that the three following conditions are satisfied: +bind(f)(pure(s)) = f(s), +bind(pure) = id, +bind(g)(bind(f)(m)) = bind(λx → bind(g)(f(x)))(m). +Example 1. The Maybe monad associates: +– any set S with the set Maybe(S) = {Just(s) | s ∈ S} ∪ {Nothing}, where Just and Nothing are two syntactic +tokens allowing us to extend a set with one value; +– any function f with the function Maybe(f) defined by +Maybe(f)(Just(s)) = Just(f(s)), +Maybe(f)(Nothing) = Nothing +– is endowed with the functions pure and bind defined by: +pure(s) = Just(s), +bind(f)(Just(s)) = f(s), +bind(f)(Nothing) = Nothing. +Example 2. The Set monad associates: +– with any set S the set 2S, +– with any function f the function Set(f) defined by Set(f)(R) = � +r∈R{f(r)}, +– is endowed with the functions pure and bind defined by: +pure(s) = {s}, +bind(f)(R) = +� +r∈R +f(r). +Example 3. The LinComb(K) monad, for K = (K, ×, +, 1, 0), associates: +– with any set S the set of K-linear combinations of elements of S, where a linear combination is a finite (formal, +commutative) sum of couples (denoted by ⊞) in K × S where (k, s) ⊞ (k′, s) = (k + k′, s), +– with any function f the function LinComb(K)(f) defined by +LinComb(K)(f)(R) = ⊞ +(k,r)∈R +(k, f(r)), +– is endowed with the functions pure and bind defined by: +pure(s) = (1, s), +bind(f)(R) = ⊞ +(k,r)∈R +k ⊗ f(r), +where k ⊗ R = ⊞ +(k′,r)∈R +(k × k′, r). +To compact equations, we use the following operators for any monad M: +f <$> s = M(f)(s), +m >>= f = bind(f)(m). +If <$> can be used to lift unary functions to the monadic level, >>= and pure can be used to lift any n-ary function +f in S1 × · · · × Sn → S, defining a function liftn sending S1 × · · · × Sn → S to M(S1) × · · · × M(Sn) → M(S) as +follows: +liftn(f)(m1, . . . , mn) =m1 >>= (λs1 → . . . +mn >>= (λsn → pure(f(s1, . . . , sn))) . . .) +Let us consider the set +1 = {⊤} with only one element. The images of this set by some previously defined monads +can be evaluated as value sets classically used to weight words in association with classical regular expressions. As +an example, Maybe(1) and Set(1) are isomorphic to the Boolean set, and any set LinComb(K)(1) can be converted +into the underlying set of K. This property allows us to extend in a coherent way classical expressions to monadic +expressions, where the type of the weights is therefore given by the ambient monad. +5 More precisely, a monad over a subcategory of the category of sets. + +3 +Monadic Expressions +As seen in the previous section, elements in M(1) can be evaluated as classical value sets for some particular +monads. Hence, we use these elements not only for the weights associated with words by expressions, but also for +the elements that act over the denoted series. +In the following, in addition to classical operators (+, · and ∗), we denote: +– the action of an element over a series by ⊙, +– the application of a function by itself. +Definition 1. Let M be a monad. An M-monadic expression E over an alphabet Σ is inductively defined as follows: +E = a, +E = ε, +E = ∅, +E = E1 + E2, +E = E1 · E2, +E = E∗ +1, +E = α ⊙ E1, +E = E1 ⊙ α, +E = f (E1, . . . , En) , +where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a +function from (M(1))n to M(1). +We denote by Exp(Σ) the set of monadic expressions over an alphabet Σ. +Example 4. As an example of functions that can be used in our extension of classical operators, one can define the +function ExtDist(x1, x2, x3) = max(x1, x2, x3) − min(x1, x2, x3) from N3 to N. +Similarly to classical regular expressions, monadic expressions associate a weight with any word. Such a relation +can be denoted via a formal series. However, before defining this notion, in order to simplify our study, we choose +to only consider proper expressions. Let us first show how to characterize them by the computation of a nullability +value. +Definition 2. Let M be a monad such that the structure (M(1), +, ×, ⋆, 1, 0) is a starred semiring. The nullability +value of an M-monadic expression E over an alphabet Σ is the element Null(E) of M(1) inductively defined as +follows: +Null(ε) = 1, +Null(∅) = 0, +Null(a) = 0, +Null(E1 + E2) = Null(E1) + Null(E2), +Null(E1 · E2) = Null(E1) × Null(E2), +Null(E∗ +1) = Null(E1)⋆, +Null(α ⊙ E1) = α × Null(E1), +Null(E1 ⊙ α) = Null(E1) × α, +Null(f(E1, . . . , En)) = f(Null(E1), . . . , Null(En)), +where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a +function from (M(1))n to M(1). +When the considered semiring is not a starred one, we restrict the nullability value computation to expressions +where a starred subexpression admits a null nullability value. In order to compute it, let us consider the Maybe +monad, allowing us to elegantly deal with such a partial function. +Definition 3. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring. The partial nullability +value of an M-monadic expression E over an alphabet Σ is the element PartNull(E) of Maybe(M(1)) defined as +follows: +PartNull(ε) = Just(1), +PartNull(∅) = Just(0), +PartNull(a) = Just(0), +PartNull(E1 + E2) = lift2(+)(PartNull(E1), PartNull(E2)), +PartNull(E1 · E2) = lift2(×)(PartNull(E1), PartNull(E2)), +PartNull(E∗ +1) = +� +Just(1) +if PartNull(E1) = Just(0), +Nothing +otherwise, +PartNull(α ⊙ E1) = (λE → α × E) <$> PartNull(E1), +PartNull(E1 ⊙ α) = (λE → E × α) <$> PartNull(E1), +PartNull(f(E1, . . . , En)) = liftn(f)(PartNull(E1), . . . , PartNull(En)), +where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a +function from (M(1))n to M(1). + +An expression E is proper if its partial nullability value is not Nothing, therefore if it is a value Just(v); in this +case, v is its nullability value, denoted by Null(E) (by abuse). +Definition 4. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring, and E be a M-monadic +proper expression over an alphabet Σ. The series S(E) associated with E is inductively defined as follows: +S(ε)(w) = +� +1 +if w = ε, +0 +otherwise, +S(∅)(w) = 0, +S(a)(w) = +� +1 +if w = a, +0 +otherwise, +S(E1 + E2) = S(E1) + S(E2), +S(E1 · E2) = S(E1) × S(E2), +S(E∗ +1) = (S(E1))⋆, +S(α ⊙ E1)(w) = α × S(E1)(w), +S(E1 ⊙ α)(w) = S(E1)(w) × α, +S(f(E1, . . . , En)) = f(S(E1), . . . , S(En)), +where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a +function from (M(1))n to M(1). +From now on, we consider the set Exp(Σ) of M-monadic expressions over Σ to be endowed with the structure of +a semiring, and two expressions denoting the same series to be equal. The weight associated with a word w in Σ∗ +by E is the value weightw(E) = S(E)(w). The nullability of a proper expression is the weight it associates with ε, +following Definition 3 and Definition 4. +Proposition 1. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring. Let E be an M-monadic +proper expression over Σ. Then: +Null(E) = weightε(E). +The previous proposition implies that the weight of the empty word can be syntactically computed (i.e. inductively +computed from a monadic expression). Now, let us show how to extend this computation by defining the computation +of derivatives for monadic expressions. +4 +Monadic Supports for Expressions +A K-left-semimodule, for a semiring K = (K, ×, +, 1, 0), is a commutative monoid (S, ±, 0) endowed with a function +⊲ from K × S to S such that: +(k × k′) ⊲ s = k ⊲ (k′ ⊲ s), +(k + k′) ⊲ s = k ⊲ s ± k′ ⊲ s, +k ⊲ (s ± s′) = k ⊲ s ± k ⊲ s′, +1 ⊲ s = s, +0 ⊲ s = k ⊲ 0 = 0. +A K-right-semimodule can be defined symmetrically. +An operad [12,14] is a structure (O, (◦j)j∈N, id) where O is a graded set (i.e. O = � +n∈N On), id is an element of +O1, ◦j is a function defined for any three integers (i, j, k)6 with 0 < j ≤ k in Ok × Oi → Ok+i−1 such that for any +elements p1 ∈ Om, p2 ∈ On, p3 ∈ Op: +∀0 < j ≤ m, id ◦1 p1 = p1 ◦j id = p1, +∀0 < j ≤ m, 0 < j′ ≤ n, p1 ◦j (p2 ◦j′ p3) = (p1 ◦j p2) ◦j+j′−1 p3, +∀0 < j′ ≤ j ≤ m, (p1 ◦j p2) ◦j′ p3 = (p1 ◦j′ p3) ◦j+p−1 p2. +Combining these compositions ◦j, one can define a composition ◦ sending Ok × Oi1 × · · · × Oik to Oi1+···+ik: for +any element (p, q1, . . . , qk) in Ok × Ok, +p ◦ (q1, . . . , qk) = (· · · ((p ◦k qk) ◦k−1 qk−1 · · · ) · · · ) ◦1 q1. +Conversely, the composition ◦ can define the compositions ◦j using the identity element: for any two elements (p, q) +in Ok × Oi, for any integer 0 < j ≤ k: +p ◦j q = p ◦ (id, . . . , id +� +�� +� +j−1 times +, q, id, . . . , id +� +�� +� +k−j times +). +As an example, the set of n-ary functions over a set, with the identity function as unit, forms an operad. +A module over an operad (O, ◦, id) is a set S endowed with a function ⋇ from On × Sn to S such that +f ⋇ (f1 ⋇ (s1,1, . . . , s1,i1), . . . , fn ⋇ (sn,1, . . . , sn,in)) += (f ◦ (f1, . . . , fn)) ⋇ (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in). +6 every couple (i, k) unambiguously defines the domain and codomain of a function ◦j + +The extension of the computation of derivatives could be performed for any monad. Indeed, any monad could +be used to define well-typed auxiliary functions that mimic the classical computations. However, some properties +should be satisfied in order to compute weights equivalently to Definition 4. Therefore, in the following we consider +a restricted kind of monads. +A monadic support is a structure (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) satisfying: +– M is a monad, +– R = (M(1), +, ×, 1, 0) is a semiring, +– M = (M(Exp(Σ)), ±, 0) is a monoid, +– (M, ⋉) is a Exp(Σ)-right-semimodule, +– (M, ⊲) is a R-left-semimodule, +– (M, ⊳) is a R-right-semimodule, +– (M(Exp(Σ)), ⋇) is a module for the operad of the functions over M(1). +An expressive support is a monadic support (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) endowed with a function toExp from +M(Exp(Σ)) to Exp(Σ) satisfying the following conditions: +weightw(toExp(m)) = m >>= weightw +(2) +toExp(m ⋉ F) = toExp(m) · F, +(3) +toExp(m ± m′) = toExp(m) + toExp(m′), +(4) +toExp(m ⊲ x) = toExp(m) ⊙ x, +(5) +toExp(x ⊳ m) = x ⊙ toExp(m), +(6) +toExp(f ⋇ (m1, . . . , mn)) = f(toExp(m1), . . . , toExp(mn)). +(7) +Let us now illustrate this notion with three expressive supports that will allow us to model well-known derivatives +computations. +Example 5 (The Maybe support). +toExp(Nothing) = 0, +toExp(Just(E)) = E, +Nothing + m = m, +m + Nothing = m, +Just(⊤) + Just(⊤) = Just(⊤), +Nothing × m = Nothing, +m × Nothing = Nothing, +Just(⊤) × Just(⊤) = Just(⊤), +Nothing ± m = m, +m ± Nothing = m, +Just(E) ± Just(E′) = Just(E + E′), +1 = Just(⊤), +0 = Nothing, +0 = Nothing, +m ⋉ F = (λE → E · F) <$> m, +m ⊲ m′ = m >>= (λx → m′), +m ⊳ m′ = m′ >>= (λx → m), +f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))). +Example 6 (The Set support). +toExp({E1, . . . , En}) = E1 + · · · + En, ++ = ∪, +× = ∩, +± = ∪, +1 = {⊤}, +0 = ∅, +0 = ∅, +m ⋉ F = (λE → E · F) <$> m, +m ⊲ m′ = m >>= (λx → m′), +m ⊳ m′ = m′ >>= (λx → m), +f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))). +Example 7 (The LinComb(K) support). +toExp((k1, E1) ⊞ · · · ⊞ (kn, En)) = k1 ⊙ E1 + · · · + kn ⊙ En, ++ = ⊞, +(k, ⊤) × (k′, ⊤) = (k × k′, ⊤), +1 = (1, ⊤), +0 = (0, ⊤), +± = ⊞, +0 = (0, ⊤), +m ⋉ F = (λE → E · F) <$> m, +m ⊲ m′ = m >>= (λx → m′), +m ⊳ k = (λE → E ⊙ k) <$> m, +f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))). + +5 +Monadic Derivatives +In the following, (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇, toExp) is an expressive support. +Definition 5. The derivative of an M-monadic expression E over Σ w.r.t. a symbol a in Σ is the element da(E) +in M(Exp(Σ)) inductively defined as follows: +da(ε) = 0, +da(∅) = 0, +da(b) = +� +pure(ε) +if a = b, +0 +otherwise, +da(E1 + E2) = da(E1) ± da(E2), +da(E∗ +1) = da(E1) ⋉ E∗ +1, +da(E1 · E2) = da(E1) ⋉ E2 ± Null(E1) ⊲ da(E2), +da(α ⊙ E1) = α ⊲ da(E1), +da(E1 ⊙ α) = da(E1) ⊳ α, +da(f(E1, . . . , En)) = f ⋇ (da(E1), . . . , da(En)) +where b is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a +function from (M(1))n to M(1). +The link between derivatives and series can be stated as follows, which is an alternative description of the classical +quotient. +Proposition 2. Let E be an M-monadic expression over an alphabet Σ, a be a symbol in Σ and w be a word in +Σ∗. Then: +weightaw(E) = da(E) >>= weightw. +Proof. Let us proceed by induction over the structure of E. All the classical cases (i.e. the function operator left +aside) can be proved following the classical methods ([1,4,13]). Therefore, let us consider this last case. +da(f(E1, . . . , En)) >>= weightw += weightw(toExp(da(f(E1, . . . , En)))) +(Eq (2)) += weightw(toExp(f ⋇ (da(E1), . . . , da(En))) +(Def 5)) += weightw(f(toExp(da(E1)), . . . , toExp(da(En)))) +(Eq (7)) += f(weightw(toExp(da(E1))), . . . , weightw(toExp(da(En)))) +(Def 4, Eq (1)) += f(da(E1) >>= weightw, . . . , da(En) >>= weightw) +(Eq (2)) += f(weightaw(E1), . . . , weightaw(En)) +(Ind. hyp.) += weightaw(f(E1, . . . , En)) +(Def 4, Eq (1)) +Let us define how to extend the derivative computation from symbols to words, using the monadic functions. +Definition 6. The derivative of an M-monadic expression E over Σ w.r.t. a word w in Σ∗ is the element dw(E) +in M(Exp(Σ)) inductively defined as follows: +dε(E) = pure(E), +da·v(E) = da(E) >>= dv, +where a is a symbol in Σ and v a word in Σ∗. +Finally, it can be easily shown, by induction over the length of the words, following Proposition 2, that the +derivatives computation can be used to define a syntactical computation of the weight of a word associated with an +expression. +Theorem 1. Let E be an M-monadic expression over an alphabet Σ and w be a word in Σ∗. Then: +weightw(E) = dw(E) >>= Null. +Notice that, restraining monadic expressions to regular ones, +– the Maybe support leads to the classical derivatives [4], +– the Set support leads to the partial derivatives [1], +– the LinComb support leads to the derivatives with multiplicities [13]. +Example 8. Let us consider the function ExtDist defined in Example 4 and the LinComb(N)-monadic expression +E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗). +da(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗) +daa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗) + +daaa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 3 ⊙ a∗) +daab(E) = ExtDist(b∗, b∗, b∗a∗) +weightaaa(E) = daaa(E) >>= Null += ExtDist(1 + 1, 1, 1 + 3) = 4 − 1 = 3 +weightaab(E) = daab(E) >>= Null = ExtDist(1, 1, 1) = 0 +In the next section, we show how to compute the derivative automaton associated with an expression. +6 +Automata Construction +A category C is defined by: +– a class ObjC of objects, +– for any two objects A and B, a set HomC(A, B) of morphisms, +– for any three objects A, B and C, an associative composition function ◦C in HomC(B, C) −→ HomC(A, B) −→ +HomC(A, C), +– for any object A, an identity morphism idA in HomC(A, A), such that for any morphisms f in HomC(A, B) and +g in HomC(B, A), f ◦C idA = f and idA ◦C g = g. +Given a category C, a C-automaton is a tuple (Σ, I, Q, F, i, δ, f) where +– Σ is a set of symbols (the alphabet), +– I is the initial object, in Obj(C), +– Q is the state object, in Obj(C), +– F is the final object, in Obj(C), +– i is the initial morphism, in HomC(I, Q), +– δ is the transition function, in Σ −→ HomC(Q, Q), +– f is the value morphism, in HomC(Q, F). +The function δ can be extended as a monoid morphism from the free monoid (Σ∗, ·, ε) to the morphism monoid +(HomC(Q, Q), ◦C, idQ), leading to the following weight definition. +The weight associated by a C-automaton A = (Σ, I, Q, F, i, δ, f) with a word w in Σ∗ is the morphism weight(w) +in HomC(I, F) defined by +weight(w) = f ◦C δ(w) ◦C i. +If the ambient category is the category of sets, and if I = +1, the weight of a word is equivalently an element of +F. Consequently, a deterministic (complete) automaton is equivalently a Set-automaton with +1 as the initial object +and B as the final object. +Given a monad M, the Kleisli composition of two morphisms f ∈ HomC(A, B) and g ∈ HomC(B, C) is the +morphism (f >=> g)(x) = f(x) >>= g in HomC(A, C). This composition defines a category, called the Kleisli +category K(M) of M, where: +– the objects are the sets, +– the morphisms between two sets A and B are the functions between A and M(B), +– the identity is the function pure. +Considering these categories: +– a deterministic automaton is equivalently a K(Maybe)-automaton, +– a nondeterministic automaton is equivalently a K(Set)-automaton, +– a weighted automaton over a semiring K is equivalently a K(LinComb(K))-automaton, +all with +1 as both the initial object and the final object. +Furthermore, for a given expression E, if i = pure(E), δ(a)(E′) = da(E′) and f = Null, we can compute +the well-known derivative automata using the three previously defined supports, and the accessible part of these +automata are finite ones as far as classical expressions are concerned [4,1,13]. +More precisely, extended expressions can lead to infinite automata, as shown in the next example. + +Example 9. Considering the computations of Example 8, it can be shown that +dan(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗). +Hence, there is not a finite number of derivated terms, that are the states in the classical derivative automaton. +This infinite automaton is represented in Figure 1, where the final weights of the states are represented by double +edges. The sink states are omitted. +ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) +ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗) +ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗) +ExtDist(b∗, b∗, b∗a∗) +ExtDist(0, 0, a∗) +ExtDist(b∗ + b∗a∗, b∗a∗b∗ + b∗, b∗a∗) +ExtDist(b∗ + b∗a∗, b∗a∗b∗ + 2 ⊙ b∗, b∗a∗) +ExtDist(a∗, a∗b∗, a∗) +ExtDist(0, b∗, 0) +ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗) +ExtDist(b∗ + b∗a∗, b∗a∗b∗ + n ⊙ b∗, b∗a∗) +1 +1 +2 +n +1 +1 +2 +1 +n +a +b +b +a +b +a +b +a +b +a +b +a +b +a +b +a +b +a +Fig. 1. The (infinite) derivative weighted automaton associated with E. +In the following section, let us show how to model a new monad in order to solve this problem. +7 +The Graded Module Monad +Let us consider an operad O = (O, ◦, id) and the association sending: +– any set S to � +n∈N On × Sn, +– any f in S → S′ to the function g in � +n∈N On × Sn → � +n∈N On × S′n: +g(o, (s1, . . . , sn)) = (o, (f(s1), . . . , f(sn))) +It can be checked that this is a functor, denoted by GradMod(O). Moreover, it forms a monad considering the two +following functions: +pure(s) = (id, s), +(o, (s1, . . . , sn)) >>= f = (o ◦ (o1, . . . , on), (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in)) +where f(sj) = (oj, sj,1, . . . , sj,ij). However, notice that GradMod(O)(1) cannot be easily evaluated as a value space. +Thus, let us compose it with another monad. As an example, let us consider a semiring K = (K, ×, +, 1, 0) and +the operad O of the n-ary functions over K. Hence, let us define the functor7 GradComb(O, K) that sends S to +GradMod(O)(LinComb(K)(S)). +7 it is folk knowledge that the composition of two functors is a functor. + +To show that this combination is a monad, let us first define a function α sending GradComb(O, K)(S) to +GradMod(O)(S). It can be easily done by converting a linear combination into an operadic combination, i.e. an +element in GradMod(O)(S), with the following function toOp: +toOp((k1, s1) ⊞ · · · ⊞ (kn, sn)) += (λ(x1, . . . , xn) → k1 × x1 + · · · + kn × xn, (s1, . . . , sn)), +α(o, (L1, . . . , Ln)) = (o ◦ (o1, . . . , on), (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in)) +where toOp(Lj) = (oj, (sj,1, . . . , sj,ij)). +Consequently, we can define the monadic functions as follows: +pure(s) = (id, (1, s)), +(o, (L1, . . . , Ln)) >>= f = α(o, (L1, . . . , Ln)) >>= f +where the second occurrence of >>= is the monadic function associated with the monad GradMod(O). +Let us finally define an expressive support for this monad: +toExp(o, (L1, . . . , Ln)) = o(toExp(L1), . . . , toExp(Ln)), +(o, (L1, . . . , Ln)) + (o′, (L′ +1, . . . , L′ +n′)) = (o + o′, (L1, . . . , Ln, L′ +1, . . . , L′ +n′)) +(o, (L1, . . . , Ln)) × (o′, (L′ +1, . . . , L′ +n′)) = (o × o′, (L1, . . . , Ln, L′ +1, . . . , L′ +n′)) +± = +, +1 = (id, (1, ⊤)), +0 = (id, (0, ⊤)), +0 = (id, (0, ⊤)), +m ⋉ F = pure(toExp(m) · F), +(o, (M1, . . . , Mk)) ⊲ (o′, (L1, . . . , Ln)) = (o(M1, . . . , Mk) × o′, (L1, . . . , Ln)), +(o, (L1, . . . , Ln)) ⊳ (o′, (M1, . . . , Mk)) = (o × o′(M1, . . . , Mk), (L1, . . . , Ln)) +f ⋇ ((o1, (L1,1, . . . , L1,i1)), . . . , (on, (Ln,1, . . . , Ln,in))) += (f ◦ (o1, . . . , on), (L1,1, . . . , L1,i1, . . . , Ln,1, . . . , Ln,in)) +where (o + o′)(x1, . . . , xn+n′) = o(x1, . . . , xn) + o′(xn+1, . . . , xn+n′) +(o × o′)(x1, . . . , xn+n′) = o(x1, . . . , xn) × o′(xn+1, . . . , xn+n′) +Example 10. Let us consider that two elements in GradComb(O, K)(Exp(Σ)) are equal if they have the same image +by toExp. Let us consider the expression E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) of Example 8. +da(E) = ExtDist ⋇ ((+, (a∗b∗, a∗)), (id, a∗b∗), (+, (a∗b∗a∗, a∗))) += (ExtDist ◦ (+, id, +), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗)) +daa(E) = (ExtDist ◦ (+, id, + ◦ (+, id)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗, a∗)) += (ExtDist ◦ (+, id, + ◦ (id, 2×)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗)) +daaa(E) = (ExtDist ◦ (+, id, + ◦ (id, 3×)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗)) +daab(E) = (ExtDist ◦ (+, id, +), (b∗, ∅, b∗, b∗a∗, ∅)) += (ExtDist, (b∗, b∗, b∗a∗)) +weightaaa(E) = daaa(E) >>= Null += ExtDist ◦ (+, id, +)(1, 1, 1, 1, 3) += ExtDist(1 + 1, 1, 1 + 3) = 4 − 1 = 3 +weightaab(E) = daab(E) >>= Null = ExtDist(1, 1, 1) = 0 +Using this monad, the number of derivated terms, that is the number of states in the associated derivative automaton, +is finite. Indeed, the computations are absorbed in the transition structure. This automaton is represented in +Figure 2. Notice that the dashed rectangle represent the functions that are composed during the traversal associated +with a word. The final weights are represented by double edges. The sink states are omitted. The state b∗ is duplicated +to simplify the representation. + +ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) +ExtDist ++ ++ +ExtDist ++ +b∗a∗b∗ ++ +a∗b∗a∗ ++ +a∗b∗ +a∗ +b∗ +b∗a∗ +b∗ +1 +1 +1 +1 +1 +1 +1 +1 +a +b +a +b +b +a +b +a +a +b +b +b +Fig. 2. The Associated Derivative Automaton of ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗). +However, notice that not every monadic expression produces a finite set of derivated terms, as shown in the next +example. +Example 11. Let us consider the expression E of Example 8 and the expression F = E · c∗. It can be shown that +dan(F) = toExp(dan(E)) · c∗ += ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗) · c∗. +The study of the necessary and sufficient conditions of monads that lead to a finite set of derivated terms is one +of the next steps of our work. +8 +Haskell Implementation +The notions described in the previous sections have been implemented in Haskell, as follows: +– The notion of monad over a sub-category of sets is a typeclass using the Constraint kind to specify a sub- +category; +– n-ary functions and their operadic structures are implemented using fixed length vectors, the size of which is +determined at compilation using type level programming; +– The notion of graded module is implemented through an existential type to deal with unknown arities: Its +monadic structure is based on an extension of heterogeneous lists, the graded vectors, typed w.r.t. the list of +the arities of the elements it contains; +– The parser and some type level functions are based on dependently typed programming with singletons [8], +allowing, for example, determining the type of the monads or the arity of the functions involved at run-time; +– An application is available here [16] illustrating the computations: +• the backend uses servant to define an API; + +• the frontend is defined using Reflex, a functional reactive programming engine and cross compiled in +JavaScript with GHCJS. +As an example, the monadic expression of the previous examples can be entered in the web application as the +input ExtDist(a*.b*+b*.a*,b*.a*.b*,a*.b*.a*). +9 +Capture Groups +Capture groups are a standard feature of POSIX regular expressions where parenthesis are used to memorize +some part of the input string being matched in order to reuse either for substitution or matching. We give here +an equivalent definition along with derivation formulae and a monadic definition. The semantic of this definition +conforms to those of POSIX expressions. Precisely, when a capture group has been involved more than one time +due to a stared subexpression, the value of the corresponding variable corresponds to the last capture. +9.1 +Syntax of Expressions with Capture Groups +A capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression for short) +is inductively defined as +E = a, +E = ε, +E = ∅, +E = F + G, +E = F · G, +E = F ∗, +E = (F)x, +E = x, +where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ. In the POSIX +syntax, capture groups are implicitly mapped with variables respectively with the order of the opening parenthesis +of a pair. Here, each capture group is associated explicitly to a variable by indexing the closing parenthesis with +the name of this variable. +9.2 +Contextual Expressions and their Contextual Languages +In order to define the contextual language and the derivation of capture-group expressions, we need to extend the +syntax of the expressions in order to attach to any capture group the current part of the input string captured +during an execution. +A contextual capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression +for short) is inductively defined as +E = a, +E = ε, +E = ∅, +E = F + G, +E = F · G, +E = F ∗, +E = (F)u +x, +E = x, +where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ. +Notice that a Σ, Γ-expression is equivalent to a contextual capture-group expression where u = ε for every +occurrence of capture group. +In the following, we consider that a context is a function from Γ to Maybe(Σ∗), modelling the possibility that +a variable was initialized (or not) during the parsing. The set of contexts is denoted by Ctxt(Γ, Σ). +Using these notions of contexts, let us now explain the semantics of contextual capture-group expressions. While +parsing, a context is built to memorize the different affectations of words to variables. Therefore, a (contextual) +language associated with an expression is a set of couples built from a language and the context that was used to +compute it. +The classic atomic cases (a symbol, the empty word or the empty set) are easy to define, preserving the context. +Another one is the case of a variable x: the context is applied here to compute the associated word (if it exists) and +is preserved. +The recursive cases are interpreted as such: +– The contextual language of a sum of two expressions is the union of their contextual languages, computed +independently. +– The contextual language of a catenation of two expressions F and G is computed in three steps. First, the +contextual language of F is computed. Secondly, for each couple (L, ctxt) of this contextual language, the +function ctxt is considered as the new context to compute the contextual language of G, leading to new couples +(L′, ctxt′). Finally, for each of these combinations, a couple (L·L′, ctxt′) is added to form the resulting contextual +language. + +– The contextual language of a starred expression is, classically, the infinite union of the powered contextual +languages, computed by iterated catenations. +– The contextual language of a captured expression (F)u +x is computed in two steps. First, the contextual language +of F is computed. Then, for each couple (L, ctxt) of it, a word w is chosen in L and the context ctxt must be +updated coherently. +More formally, the contextual language of a Σ, Γ-expression E associated with a context ctxt in Ctxt(Γ, Σ) is +the subset Lctxt(E) of 2Σ∗ × Ctxt(Γ, Σ) inductively defined as follows: +Lctxt(a) = {({a}, ctxt)}, +Lctxt(ε) = {({ε}, ctxt)}, +Lctxt(∅) = ∅, +Lctxt(x) = +� +∅ +if ctxt(x) = Nothing, +{({w}, ctxt)} +otherwise if ctxt(x) = Just(w), +Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G), +Lctxt(F · G) = +� +(L1,ctxt1)∈Lctxt(F ), +(L2,ctxt2)∈Lctxt1(G) +{(L1 · L2, ctxt2)}, +Lctxt(F ∗) = +� +n∈N +(Lctxt(F)) +n, +Lctxt((F)u +x) = +� +(L1,ctxt1)∈Lctxt(F ), +w∈L1 +{({w}, [ctxt1]x←uw)}, +where F and G are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ, u is in Σ∗, Ln is defined, for any +set L of couples (language, context) by +Ln = + + + + + + + + + + + +� +(L,ctxt)∈L +{({ε}, ctxt)} +if n = 0, +� +(L1,ctxt1)∈L, +(L2,ctxt2)∈Ln−1 +{(L1 · L2, ctxt2)} +otherwise, +and [ctxt]x←w is the context defined by +[ctxt]x←w(y) = +� +Just(w) +if x = y, +ctxt(y) +otherwise. +The contextual language of an expression E is the set of couples obtained from an uninitialised context, where +nothing is associated with any variable, that is the set +Lλ_→Nothing(E). +Finally, the language denoted by an expression E is the set of words obtained by forgetting the contexts, that is the +set +� +(L,_)∈Lλ_→Nothing(E) +L. +Example 12. Let us consider the three following expressions over the symbol alphabet {a, b, c} and the variable +alphabet {x}: +E = E1 · E2, +E1 = ((a∗)xbx)∗, +E2 = cx. +The language denoted by E2 is empty, since it is computed from the empty context, where nothing is associated +with x. However, parsing E1 allows us to compute contexts that define word values to affect to x. Let us thus show +how is defined the contextual language of E1: +– the contextual language of (a∗)x is the set� +n∈N +{({an}, λx → Just(an))} +where each word an is recorded in a context; +– the contextual language of (a∗)xbx is the set +� +n∈N +{({anban}, λx → Just(an))} +where each word an is recorded in a context applied to evaluate the variable x; +– the contextual language of E1 is the union of the two following sets S1 and S2: +S1 = {({ε}, λx → Nothing)} +S2 = {({anban | n ∈ N}∗ · {ambam}, λx → Just(am)) | m ∈ N} +where each iteration of the outermost star produces a new record for the variable x in the context; however, +notice that only the last one is recorded at the end of the process. + +Finally, the language of E is obtained by considering the contexts obtained from the parsing of E1 to evaluate the +occurrence of x in E2, leading to the set� +m∈N +({anban | n ∈ N}∗ · {ambamcam}). +Obviously, some classical equations still hold with these computations: +Lemma 1. Let E, F and G be three Σ, Γ-expressions and ctxt be a context in Ctxt(Γ, Σ). The two following +equations hold: +Lctxt(E · (F + G)) = Lctxt(E · F + E · G) +Lctxt(F ∗) = Lctxt(ε + F · F ∗) +Proof. Let us proceed by equality sequences: +Lctxt(E · (F + G)) = +� +(L1,ctxt1)∈Lctxt(E), +(L2,ctxt2)∈Lctxt1 (F +G) +{(L1 · L2, ctxt2)} += +� +(L1,ctxt1)∈Lctxt(E), +(L2,ctxt2)∈Lctxt1 (F )∪Lctxt1(G) +{(L1 · L2, ctxt2)} += +� +(L1,ctxt1)∈Lctxt(E), +(L2,ctxt2)∈Lctxt1 (F ) +{(L1 · L2, ctxt2)} +∪ +� +(L1,ctxt1)∈Lctxt(E), +(L2,ctxt2)∈Lctxt1(G) +{(L1 · L2, ctxt2)} += Lctxt(E · F) ∪ Lctxt(E · G) += Lctxt(E · F + E · G) +Lctxt(F ∗) = +� +n∈N +(Lctxt(F)) +n += (Lctxt(F)) +0 ∪ +� +n∈N,n≥1 +(Lctxt(F)) +n += (Lctxt(F)) +0 ∪ +� +n∈N +Lctxt(F) · (Lctxt(F)) +n += (Lctxt(F)) +0 ∪ Lctxt(F) · +� +n∈N +(Lctxt(F)) +n += Lctxt(ε + F · F ∗) +In order to solve the membership test for the contextual capture-group expressions, let us extend the classical +derivation method. But first, let us show how to extend the nullability predicate, needed at the end of the process. +9.3 +Nullability Computation +The nullability predicate allows us to determine whether the empty word belongs to the language denoted by +an expression. As far as capture groups are concerned, a context has to be computed. Therefore, the nullability +predicate can be represented as a set of contexts the application of which produces a language that contains the +empty word. +As we have seen, the nullability depends on the current context. Given an expression and a context ctxt, the +nullability predicate is a set in 2Ctxt(Γ,Σ), computed as follows: +Nullctxt(ε) = {ctxt} +Nullctxt(∅) = ∅ +Nullctxt(a) = ∅ +Nullctxt(x) = +� +{ctxt} +if ctxt(x) = Just(ε) +∅ +otherwise. +Nullctxt(E + F) = Nullctxt(E) ∪ Nullctxt(F) +Nullctxt(E · F) = +� +ctxt′∈Nullctxt(F ), +ctxt′′∈Nullctxt′ (G) +{ctxt′′} +Nullctxt(E∗) = {ctxt} +Nullctxt((E)u +x) = +� +ctxt′∈Nullctxt(F ) +{[ctxt′]x←u} +where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. +Example 13. Let us consider the three expressions of Example 12: +E = E1 · E2, +E1 = ((a∗)xbx)∗, +E2 = cx. +For any context ctxt, +Nullctxt(E1) = {ctxt}, +Nullctxt(E2) = ∅, +Nullctxt(E) = ∅. +The nullability predicate allows us to determine whether there exists a couple in the contextual language of an +expression such that its first component contains the empty word. + +Proposition 3. Let E be a Σ, Γ-expression and ctxt be a context in Ctxt(Γ, Σ). Then the two following conditions +are equivalent: +– Nullctxt(E) ̸= ∅, +– ∃(L, _) ∈ Lctxt(E) | ε ∈ L. +Proof. By induction over the structure of E: +– If E = a ∈ Σ or E = ∅, the property holds since Nullctxt(E) is empty and since there is no couple (L, ctxt′) in +Lctxt(E) with ε in L. +– If E = ε, the following two conditions hold, +Nullctxt(E) = {ctxt}, +Lctxt(E) = {({ε}, ctxt)}, +satisfying the stated condition. +– If E = F + G, the following two conditions hold: +Nullctxt(F + G) = Nullctxt(F) ∪ Nullctxt(G), +Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G). +Since, by induction hypothesis, the following two conditions hold +Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L, +Nullctxt(G) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(G) | ε ∈ L, +the proposition holds. +– If E = F · G, the two following conditions hold: +Nullctxt(F · G) = +� +ctxt′∈Nullctxt(F ), +ctxt′′∈Nullctxt′(G), +{ctxt′′}, +Lctxt(F · G) = +� +(L,ctxt′)∈Lctxt(F ), +(L′,ctxt′′)∈Lctxt′(G), +{(L · L′, ctxt′′)}. +Since, by induction hypothesis, the two following conditions hold, +Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L, +Nullctxt′(G) ̸= ∅ ⇔ ∃(L, ctxt′′) ∈ Lctxt′(G) | ε ∈ L, +the proposition holds. +– If E = F ∗, since the two following conditions hold +Nullctxt(F ∗) = {ctxt}, +Lctxt(F) +0 = {({ε}, ctxt)} ∈ Lctxt(F ∗), +the stated condition holds. +– If E = (F)u +x, both following conditions hold: +Nullctxt((F)u +x) = +� +ctxt′∈Nullctxt(F ) +{[ctxt′]x←u}, +Lctxt((F)u +x) = +� +(L,ctxt′)∈Lctxt(F ), +w∈L +{({w}, [ctxt′]x←uw)}. +Then, following induction hypothesis, +Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L, +the stated condition holds. +– If E = x, both following conditions hold: +Nullctxt(x) = +� +{ctxt} +if ctxt(x) = Just(ε) +∅ +otherwise, +Lctxt(x) = +� +∅ +if ctxt(x) = Nothing, +{({w}, ctxt)} +otherwise if ctxt(x) = Just(w). +Therefore, the proposition holds. +9.4 +Derivation formulae +Similarly to the nullability predicate, the derivation computation builds the context while parsing the expression. +Therefore, the derivative of an expression with respect to a context is a set of couples (expression, context), induc- +tively computed as follows, for any Σ, Γ-expression and for any context ctxt in Ctxt(Γ, Σ): +dctxt +a +(ε) = ∅ +dctxt +a +(∅) = ∅ + +dctxt +a +(b) = +� +∅ +if a ̸= b, +{(ε, ctxt)} +otherwise, +dctxt +a +(x) = +� +dctxt +a +(w) +if ctxt(x) = Just(w) +∅ +otherwise +dctxt +a +(F + G) = dctxt +a +(F) ∪ dctxt +a +(G) +dctxt +a +(F · G) = +� +(ctxt′,F ′)∈dctxt +a +(F ) +{(F ′ · G, ctxt′)} +∪ +� +ctxt′∈Nullctxt(F ) +dctxt′ +a +(G) +dctxt +a +(F ∗) = +� +(ctxt′,F ′)∈dctxt +a +(F ) +{(F ′ · F ∗, ctxt′)} +dctxt +a +((F)u +x) = +� +(ctxt′,F ′)∈dctxt +a +(F ) +{((F ′)u·a +x , ctxt′)} +where F and G are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. +Example 14. Let us consider the three expressions of Example 12: +E = E1 · E2, +E1 = ((a∗)xbx)∗, +E2 = cx. +Then, for any context ctxt, +dctxt +a +(E) = {((a∗)a +xbx((a∗)xbx)∗cx, ctxt)}, +dctxt +b +(E) = {(x((a∗)xbx)∗cx, λx → ε)}, +dctxt +c +(E) = {(x, ctxt)}. +The derivation of an expression allows us to syntactically express the computation of the quotient of the language +components in contextual languages, where the quotient w−1(L) is the set {w′ | ww′ ∈ L}. +Proposition 4. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and a be a symbol in Σ. Then: +� +(E′,ctxt′)∈dctxt +a +(E) +Lctxt′(E′) = +� +(L′,ctxt′)∈Lctxt(E) +{(a−1(L′), ctxt′)} +Proof. By induction over the structure of E, assimilating ∅ and {(∅, ctxt)} for any context ctxt. +– If E = ε or E = ∅, the property vacuously holds. +– If E = b ∈ Σ, +� +(E′,ctxt′)∈dctxt +a +(b) +Lctxt′(E′) = +� +∅ +if b ̸= a, +{({ε}, ctxt)} +otherwise, += {(a−1({b}), ctxt)} = +� +(L′,ctxt′)∈Lctxt(b) +{(a−1(L′), ctxt′)}. +– If E = F + G,� +(E′,ctxt′)∈dctxt +a +(F +G) +Lctxt′(E′) = +� +(E′,ctxt′)∈dctxt +a +(F )∪dctxt +a +(G) +Lctxt′(E′) += +� +(E′,ctxt′)∈dctxt +a +(F ) +Lctxt′(E′) ∪ +� +(E′,ctxt′)∈dctxt +a +(G) +Lctxt′(E′) += +� +(L′,ctxt′)∈Lctxt(F ) +{(a−1(L′), ctxt′)} ∪ +� +(L′,ctxt′)∈Lctxt(G) +{(a−1(L′), ctxt′)} += +� +(L′,ctxt′)∈Lctxt(F )∪Lctxt(G) +{(a−1(L′), ctxt′)} += +� +(L′,ctxt′)∈Lctxt(F +G) +{(a−1(L′), ctxt′)}. +– If E = F · G, +� +(E′,ctxt′)∈dctxt +a +(F ·G) +Lctxt′(E′) = +� +(ctxt′,F ′)∈dctxt +a +(F ) +Lctxt′(F ′ · G) ∪ +� +ctxt′∈Nullctxt(F ), +(G′,ctxt′′)∈dctxt′ +a +(G) +Lctxt′′(G′) += +� +(ctxt′,F ′)∈dctxt +a +(F ), +(L1,ctxt1)∈Lctxt(F ′), +(L2,ctxt2)∈Lctxt1(G) +{(L1 · L2, ctxt2)} ∪ +� +ctxt′∈Nullctxt(F ), +(G′,ctxt′′)∈dctxt′ +a +(G) +Lctxt′′(G′) += +� +(L1,ctxt1)∈Lctxt(F ), +(L2,ctxt2)∈Lctxt1 (G) +{(a−1(L1) · L2, ctxt2)} ∪ +� +ctxt1∈Nullctxt(F ), +(L2,ctxt2)∈Lctxt1 (G) +{(a−1(L2), ctxt2)} + += +� +(L1,ctxt1)∈Lctxt(F ), +(L2,ctxt2)∈Lctxt1 (G) +{(a−1(L1) · L2, ctxt2)} ∪ +� +∃(L,ctxt1)∈Lctxt(F )|ε∈L, +(L2,ctxt2)∈Lctxt1(G) +{(a−1(L2), ctxt2)} += +� +(L1,ctxt1)∈Lctxt(F ), +(L2,ctxt2)∈Lctxt1 (G) +{(a−1(L1) · L2, ctxt2)} ∪ +� +(L1,ctxt1)∈Lctxt(F ), +ε∈L1, +(L2,ctxt2)∈Lctxt1 (G) +{(a−1(L2), ctxt2)} += +� +(L1,ctxt1)∈Lctxt(F ), +(L2,ctxt2)∈Lctxt1 (G) +{(a−1(L1 · L2), ctxt2)} += +� +(L′,ctxt′)∈Lctxt(F ·G) +{(a−1(L′), ctxt′)}. +– If E = F ∗, +� +(E′,ctxt′)∈dctxt +a +(F ∗) +Lctxt′(E′) = +� +(ctxt′,F ′)∈dctxt +a +(F ) +Lctxt′(F ′ · F ∗) += +� +(ctxt′,F ′)∈dctxt +a +(F ), +(L1,ctxt1)∈Lctxt(F ′), +(L2,ctxt2)∈Lctxt1(F ∗) +{(L1 · L2, ctxt2)} += +� +(L1,ctxt1)∈Lctxt(F ), +(L2,ctxt2)∈Lctxt1(F ∗) +{(a−1(L1) · L2, ctxt2)} += +� +(L1,ctxt1)∈Lctxt(F ), +(L2,ctxt2)∈Lctxt1(F ∗) +{(a−1(L1 · L2), ctxt2)} += +� +(L′,ctxt′)∈Lctxt(F ·F ∗) +{(a−1(L′), ctxt′)} += +� +(L′,ctxt′)∈Lctxt(ε+F ·F ∗) +{(a−1(L′), ctxt′)} += +� +(L′,ctxt′)∈Lctxt(F ∗) +{(a−1(L′), ctxt′)} +– If E = (F)u +x, +� +(E′,ctxt′)∈dctxt +a +((F )u +x) +Lctxt′(E′) = +� +(ctxt′,F ′)∈dctxt +a +(F ) +Lctxt′((F ′)u·a +x ) += +� +(ctxt′,F ′)∈dctxt +a +(F ) +(L1,ctxt1)∈Lctxt′(F ′), +w∈L1 +{({w}, [ctxt1]x←uaw)} += +� +(L1,ctxt1)∈Lctxt(F ), +w∈a−1(L1) +{({w}, [ctxt1]x←uaw)} += +� +(L1,ctxt1)∈Lctxt(F ), +aw∈L1 +{({w}, [ctxt1]x←uaw)} += +� +(L1,ctxt1)∈Lctxt(F ), +aw∈L1 +{(a−1({aw}), [ctxt1]x←uaw)} += +� +(L1,ctxt1)∈Lctxt(F ), +w∈L1 +{(a−1({w}), [ctxt1]x←uw)} += +� +(L′,ctxt′)∈Lctxt((F )u +x) +{(a−1(L′), ctxt′)} + +– If E = x, +� +(E′,ctxt′)∈dctxt +a +(x) +Lctxt′(E′) = + + + + + +� +(E′,ctxt′)∈dctxt +a +(w) +Lctxt′(E′) +if ctxt(x) = Just(w), +∅ +otherwise, += + + + + + +� +(w,ctxt)∈dctxt +a +(aw) +Lctxt(w) +if ctxt(x) = Just(aw), +∅ +otherwise, += +� +{({w}, ctxt)} +if ctxt(x) = Just(aw), +∅ +otherwise, += +� +{(a−1({aw}), ctxt)} +if ctxt(x) = Just(aw), +∅ +otherwise, += +� +{(a−1({w}), ctxt)} +if ctxt(x) = Just(w), +∅ +otherwise, += +� +(L′,ctxt′)∈Lctxt(x) +{(a−1(L′), ctxt′)} +The derivation w.r.t. a word is, as usual, an iterated application of the derivation w.r.t. a symbol, recursively +defined as follows, for any Σ, Γ-expression E, for any context ctxt in Ctxt(Γ, Σ), for any symbol a in Σ and for +any word v in Σ∗: +dctxt +ε +(E) = {(E, ctxt)}, +dctxt +a·v (E) = +� +(E′,ctxt′)∈dctxt +a +(E) +dctxt′ +v +(E′). +Example 15. Let us consider the three expressions of Example 14: +E = E1 · E2, +E1 = ((a∗)xbx)∗, +E2 = cx. +Then, for any context ctxt, +dctxt +ab (E) = dctxt +b +((a∗)a +xbx((a∗)xbx)∗cx) += {(x((a∗)xbx)∗cx, λx → a)} +dctxt +aba (E) = dλx→a +a +(x((a∗)xbx)∗cx) += {(((a∗)xbx)∗cx, λx → a)} +dctxt +abac(E) = dλx→a +c +(((a∗)xbx)∗cx) += {(x, λx → a)} +dctxt +abaca(E) = dλx→a +a +(x) += {(ε, λx → a)} +Such an operation allows us to syntactically compute the quotient. +Proposition 5. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗. Then: +� +(E′,ctxt′)∈dctxt +w +(E) +Lctxt′(E′) = +� +(L′,ctxt′)∈Lctxt(E) +{(w−1(L′), ctxt′)} +Proof. By a direct induction over the structure of words. +Finally, the membership test of a word w can be performed as usual by first computing the derivation w.r.t. w, and +then by determining the existence of a nullable derivative, as a direct corollary of Proposition 3 and Proposition 5. +Theorem 2. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗. Then the two +following conditions are equivalent: +– ∃(L, _) ∈ Lctxt(E) | w ∈ L, +– ∃(E′, ctxt′) ∈ dctxt +w +(E) | Nullctxt′(E′) ̸= ∅. +We have shown how to compute the derivatives and solve the membership test in a classical way. Let us show how +to embed the context computation in a convenient monad, in order to generalize the definitions to other structure +than sets. + +9.5 +The StateT Monad Transformer +Monads do not compose well in general. However, ones can consider particular combinations of these objects. Among +those, well-known patterns are the monad transformers like the StateT Monad Transformer [10]. This combination +allows us to mimick the use of global variables in a functional way. In our setting, it allows us to embed the context +computation in an elegant way. +Let S be a set and M be a monad. We denote by StateT(S, M) following the mapping: +StateT(S, M)(A) = S → M(A × S). +In other terms, StateT(S, M)(A) is the set of functions from S to the monadic structure M(A×S) based on couples +in the cartesian product (A × S). +The mapping StateT(S, M) can be equipped by a structure of functor, defined for any function f from a set A +to a set B by +StateT(S, M)(f)(state)(s) = M(λ(a, s) → (f(a), s))(state(s)). +It can also be equipped with the structure of monad, defined for any function f from a set A to the set StateT(S, M)(B): +pure(a) = λs → pure(a, s) +bind(f)(state)(s) = state(s) >>= λ(a, s′) → f(a)(s′) +9.6 +Monadic Definitions +The previous definitions associated with capture-group expressions can be equivalently restated using the StateT +monad transformer specialised with the Set monad. +Let us first consider the following claims where M = StateT(Ctxt(Γ, Σ), Set), allowing us to bring closer M +and the previous notion of monadic support: +– R = (M(1), +, ×, 1, 0) is a semiring by setting: +f1 + f2 = λs → f1(s) ∪ f2(s), +f1 × f2 = f1 >>= λ_ → f2, +1 = λs → {(⊤, s)} = pure(⊤), +0 = λs → ∅, +– M = (M(Exp(Σ)), ±, 0) is a monoid by setting: +± = +, +0 = 0, +– (M, ⋉) is a Exp(Σ)-right-semimodule by setting: +f ⋉ F = λs → +� +(E,ctxt)∈f(s) +{(E · F, ctxt)}, +– (M, ⊲) is a R-left-semimodule by setting: +f1 ⊲ f2 = f1 >>= λ_ → f2. +Then, the nullable predicate formulae can be equivalently restated as an element in StateT(Ctxt(Γ, Σ), Set)(1), +which is equal by definition to Ctxt(Γ, Σ) → Set(1 × Ctxt(Γ, Σ)), isomorphic to Ctxt(Γ, Σ) → Set(Ctxt(Γ, Σ)). It +can inductively be computed as follows: +Null(ε) = 1 +Null(∅) = 0 +Null(a) = 0 +Null(E + F) = Null(E) + Null(F) +Null(E · F) = Null(E) × Null(F) +Null(E∗) = 1 +Null(x)(ctxt) = +� +pure((⊤, ctxt)) +if ctxt(x) = Just(ε), +∅ +otherwise, +Null((E)u +x)(ctxt) = Set(λ(⊤, ctxt′) → (⊤, [ctxt′]x←u))(Null(F)(ctxt)), +where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. Notice that +these formulae are the same that the ones in Definition 2 as far as classical operators are concerned, and that these +formulae can be easily generalized to other convenient monads than Set. +Moreover, the derivative of an expression is an element in StateT(Ctxt(Γ, Σ), Set)(Exp(Σ, Γ)): +da(ε) = 0 +da(∅) = 0 +da(b) = +� +0 +if a ̸= b, +pure(ε) +otherwise, +da(E + F) = da(E) ± da(F) +da(E · F) = da(E) ⋉ F + Null(E) ⊲ da(F) +da(E∗) = da(E) ⋉ E∗ +da((E)u +x) = StateT(Ctxt(Γ, Σ), Set)(λF → (F)ua +x )(da(E)) +da(x)(ctxt) = +� +pure((w, ctxt)) +if ctxt(x) = Just(aw), +∅ +otherwise, + +where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. Once again, notice +that these formulae are the same that the ones in Definition 5 as far as classical operators are concerned, and that +these formulae can be easily generalized to other convenient monads than Set. +Finally, the derivation w.r.t. a word is monadically defined as in previous sections: +dε(E) = pure(E), +dav(E) = da(E) >>= dv, +and the membership test of a word w can be equivalently rewritten as follows: +(dw(E) >>= Null)(λ_ → Nothing) ̸= ∅. +10 +Conclusion and Perspectives +In this paper, we achieved the first step of our plan to unify the derivative computation over word expressions. +Monads are indeed useful tools to abstract the underlying computation structures and thus may allow us to consider +some other functionalities, such as capture groups via the well-known StateT monad transformer [10]. We aim to +study the conditions satisfying by monads that lead to finite set of derivated terms, and to extend this method +to tree expressions using enriched categories. Finally, we plan to extend monadic derivation to other underlying +monads for capture groups, linear combinations for example. +References +1. Antimirov, V.M.: Partial derivatives of regular expressions and finite automaton constructions. Theor. Comput. Sci. +155(2) (1996) 291–319 +2. Attou, S., Mignot, L., Miklarz, C., Nicart, F.: Monadic expressions and their derivatives. In: NCMA. Volume 367 of +EPTCS (2022) 49–64 +3. Berry, G., Sethi, R.: +From regular expressions to deterministic automata. +Theoretical computer science 48 (1986) +117–126 +4. Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4) (1964) 481–494 +5. Caron, P., Flouret, M.: From glushkov wfas to k-expressions. Fundam. Informaticae 109(1) (2011) 1–25 +6. Champarnaud, J., Laugerotte, É., Ouardi, F., Ziadi, D.: From regular weighted expressions to finite automata. Int. J. +Found. Comput. Sci. 15(5) (2004) 687–700 +7. Colcombet, T., Petrisan, D.: Automata and minimization. SIGLOG News 4(2) (2017) 4–27 +8. Eisenberg, R.A., Weirich, S.: Dependently typed programming with singletons. In: Haskell, ACM (2012) 117–130 +9. Glushkov, V.M.: The abstract theory of automata. Russian Mathematical Surveys 16(5) (1961) 1 +10. Jones, M.P.: Functional programming with overloading and higher-order polymorphism. In: Adv. Func. Prog. Volume +925 of LNCS, Springer (1995) 97–136 +11. Kleene, S.: Representation of events in nerve nets and finite automata. Automata Studies Ann. Math. Studies 34 +(1956) 3–41 Princeton U. Press. +12. Loday, J.L., Vallette, B.: Algebraic operads. Volume 346. Springer Science & Business Media (2012) +13. Lombardy, S., Sakarovitch, J.: Derivatives of rational expressions with multiplicity. Theor. Comput. Sci. 332(1-3) (2005) +141–177 +14. May, J.P.: The geometry of iterated loop spaces. Volume 271. Springer (2006) +15. Mignot, L.: Une proposition d’implantation des structures d’automates, d’expressions et de leurs algorithmes associés +utilisant les catégories enrichies (in french). +Habilitation à diriger des recherches, Université de Rouen normandie +(Décembre 2020) 212 pages. +16. Mignot, L.: Monadic derivatives. https://github.com/LudovicMignot/MonadicDerivatives (2022) +17. Schützenberger, M.P.: On the definition of a family of automata. Inf. Control. 4(2-3) (1961) 245–270 +18. Sulzmann, M., Lu, K.Z.M.: POSIX regular expression parsing with derivatives. In: FLOPS. Volume 8475 of Lecture +Notes in Computer Science, Springer (2014) 203–220 + diff --git a/0tFPT4oBgHgl3EQfTjTq/vector_store/index.pkl b/0tFPT4oBgHgl3EQfTjTq/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9aca472e9ae86fcff34fe8a34c47baeeeb8682f6 --- /dev/null +++ b/0tFPT4oBgHgl3EQfTjTq/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:46a711f10f601c1825a35c4c8b3d94027a479a8a4864de43e97b9d3bcd7b1b79 +size 159172 diff --git a/29E2T4oBgHgl3EQfjQdC/content/tmp_files/2301.03966v1.pdf.txt b/29E2T4oBgHgl3EQfjQdC/content/tmp_files/2301.03966v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f37f421557b5be4f64d8181c9272ce57f1d1216c --- /dev/null +++ b/29E2T4oBgHgl3EQfjQdC/content/tmp_files/2301.03966v1.pdf.txt @@ -0,0 +1,1687 @@ +AdvBiom: Adversarial Attacks on Biometric +Matchers +Debayan Deb, Vishesh Mistry, Rahul Parthe +TECH5, +Troy, MI, USA +{debayan.deb, vishesh.mistry, rahul.parthe}@tech5-sa.com +Abstract +With the advent of deep learning models, face recognition systems have achieved +impressive recognition rates. The workhorses behind this success are Convolutional +Neural Networks (CNNs) and the availability of large training datasets. However, +we show that small human-imperceptible changes to face samples can evade most +prevailing face recognition systems. Even more alarming is the fact that the same +generator can be extended to other traits in the future. In this work, we present how +such a generator can be trained and also extended to other biometric modalities, +such as fingerprint recognition systems. +1 +Introduction +The last decade has seen a massive influx of deep learning-based technologies that have tackled +problems which were once thought to be unsolvable. Much of this progress can be attributed to +Convolutional Neural Networks (CNNs) [1, 2] which are now deployed in a plethora of applications +ranging from cancer detection to driving autonomous vehicles. Akin to the computer vision domain, +the use of CNNs have completely changed the face of biometrics due to the availability of powerful +computing devices (GPUs, TPUs) and deep architectures capable of learning rich features [3–5]. +Automated face recognition systems (AFR) have been proven to achieve accuracies as high as 99% +True Accept Rate (TAR) @ 0.1% False Accept Rate (FAR) [6], majorly owing to publicly available +large-scale face datasets. +Unfortunately, studies have shown that CNN-based networks are vulnerable to adversarial pertur- +bations1 [7–12]. It is not surprising that AFR systems too are not impervious to these attacks. +Adversarial attacks to an AFR system can be classified into two categories - (i) impersonation attack +where the hacker tries to perturb his face image to match it to a target victim, and (ii) obfuscation +attack where the hacker’s face image is perturbed to match with a random identity. Both the above at- +tacks involve the hacker adding targeted human-imperceptible perturbations to the face image. These +adversarial attacks are different from face digital manipulation that include attribute manipulation +and synthetic faces, and also from presentation attacks which involves the perpetrator wearing a +physical artifact such as a mask or replaying a photograph/video of a genuine individual which may +be conspicuous in scenarios where human operators are involved. +Let us consider, as example, the largest deployment of fingerprint recognition systems - India’s +Aadhaar Project [13], which currently has an enrolled gallery size of about 1.35 billion faces from +nearly all of its citizens. In September 2022 alone, Aadhaar received 1.3 billion authentication +requests2. In order to deny a citizen his/her rightful access to government benefits, healthcare, and +financial services, an attacker can maliciously perturb enrolled face images such that they do not +1Adversarial perturbations refer to altering an input image instance with small, human imperceptible changes +in a manner that can evade CNN models. +2https://bit.ly/3BzlpZJ +arXiv:2301.03966v1 [cs.CV] 10 Jan 2023 + +match to the genuine person during verification. In a typical AFR system, adversarial faces can +be replaced with a captured face image in order to prevent the probe face from matching to any of +its corresponding enrolled faces. Additionally, the attacker can compromise the entire gallery by +inserting adversarial faces in the enrolled gallery, where no probe face will match to the correct +identity’s gallery. +Adversarial attacks can further be categorized into two types of attacks based on how the attack vector +is trained and generated: +1. White-box attack: Attacks in which the hacker has full knowledge of the recognition system, +and iteratively perturbs every pixel by various optimization schemes are termed as white-box +attacks [14–22]. +2. Black-box attack: With no information about the parameters of the recognition system, +black-box attacks are deployed by either transferring attacks learned from an available AFR +system [23–28], or querying the the target system for score [29–31] or decision [32, 33]. +3. Semi-whitebox attack: Here, a white-box model is utilized only during training and then ad- +versarial examples are synthesized during inference without any knowledge of the deployed +AFR model. +We propose an automated adversarial synthesis method, named AdvBiom, which generates an ad- +versarial image for a probe image and satisfies all the above requirements. The contributions of the +paper are as follows: +1. GAN-based AdvBiom that learns to generate visually realistic adversarial face images that +are misclassified by state-of-the-art automated biometric systems. +2. Adversarial images generated via AdvBiom are model-agnostic and transferable, and achieve +high success rate on 5 state-of-the-art automated face recognition systems. +3. Visualizing regions where pixels are perturbed and analyzing the transferability of AdvBiom . +4. We show that AdvBiom achieves significantly higher attack success rate under current +defense mechanisms compared to baselines. +5. With the addition of the proposed Minutiae Displacement and Distortion modules, we show +thatAdvBiom can also be extended to successfully evade automated fingerprint recognition +systems. +2 +Related Work +2.1 +Adversarial Attacks +As discussed earlier, adversarial attacks are broadly classified into white-box attacks and black-box +attacks. A large number of white-box attacks are gradient-based where they analyze the gradients +during the back-propagation of an available face recognition system and perform pixel-wise per- +turbations to the target face image. While approaches such as FGSM [14] and PGD [17] exploit +the high-dimensional space of deep networks to generate adversarial attacks, C&W [18] focuses on +minimizing objective functions for optimal adversarial perturbations. However, the basic assump- +tion in white-box attacks that the target recognition system will be available is not plausible. In +real-life scenarios, the hacker will not have any information regarding the architecture, training and +deployment of the recognition system. +Black-box attacks can be classified into three major categories: transfer-based, score-based, and +decision-based attacks. Transfer-based attacks train their adversarial attack generator using readily +available recognition systems and then deploy the attacks onto a black-box target system. Dong et +al. [23] proposed the use of momentum for efficient transferability of the adversarial samples. DI2- +FGSM [24] suggested to increase input data diversity for improving transferability. Other approaches +in this category include AI-FGSM [27] and TI-FGSM [28]. Score-based attacks [29–31] query the +target system for scores and try to estimate its gradients. Decision-based attacks have the most +challenging setting wherein only the decisions from the target system are queried. Some effective +methods in this category include Evolutionary attack [32] and Boundary attack [33]. +2 + +2.2 +Adversarial Attacks on Face Recognition +Although adversarial attacks on face recognition systems have only been recently explored, there +has been a significant number of effective approaches for evading AFR systems. Attacks on face +recognition systems can be broadly categorized into physical attacks and digital attacks. Physical +attacks involve generating adversarial physical artifacts which are ’worn’ on a face. Sharif et +al. [34, 35] proposed generating adversarial eye-glass frames for attacking face recognition systems. +In [36], adversarial printed stickers placed on a hat were generated. However, methods [34–36] are +implemented in a white-box setting which is unrealistic. Additionally, Nguyen et al. [37] proposed +an adversarial light projection attack using an on-premise projector. Yin et al. [38] generated and +printed eye makeup patches to be stuck around the eyes. More recently, authors in [39] proposed an +adversarial mask for impersonation attacks in a black-box setting. However, all the above methods +suffer a major drawback of being unrealistic in an operational setting where a human operator is +present. +Digital attacks refer to manipulating and perturbing the pixels of a digital face image before being +passed through a face recognition system. Early works [9, 18, 10, 8, 40] focused on gradient-based +attacks for face recognition. However, these methods implement lp-norm perturbations to each pixel +resulting in decreased attack transferability, and vulnerability to denoising models. Cauli et al. [41] +implemented a backdoor attack where the target face recognition system’s training samples were +manipulated. Apart from the fact that gaining access to the target AFR’s training samples is highly +improbable, a thorough visual inspection of the samples can easily identify the digital artifacts. Other +works employ more stealthy attack approaches against face recognition models. Dong et al. [32] +proposed an evolutionary optimization method for generating adversarial faces in decision-based +black-box settings. However, they require a minimum of 1,000 queries to the target face recognition +system before a realistic adversarial face can be synthesized. [42] added a conditional variational +autoencoder and attention modules to generate adversarial faces in a transfer-based black-box setting. +However, they solely focused on impersonation attacks and require at least 5 image samples of the +target subject for training and inference. Zhong et al. [43] implemented dropout [44] to improve +the transferability of the adversarial examples. [38] perturbed the eye region of a face to produce +adversarial eyeshadow artifacts. However, the artifacts are visibly conspicuous under close inspection. +Deb et al. [25] used a GAN to generate minimal perturbations in salient facial regions. More recently, +[45] and [46] have focused on manipulating facial attributes for targeted adversarial attacks. +3 +Adversarial Faces +3.1 +Preliminaries +The goal of any attacker is to evade Automated Face Recognition (AFR) systems under either of the +two settings: +• Obfuscation Manipulate input face images in a manner such that they cannot be identified +as the hacker, or +• Impersonation Edit input face images such that they are identified as a target/desired +individual (victim). +While the manipulated face image evades the AFR system, a key requirement in a successful attack is +such that the input face image should appear as a legitimate face photo of the attacker. In other words, +the attacker desires an automated method of adding small and human-imperceptible changes to an +input face image such that it can evade AFR systems while appear benign to human observers. These +changes are denoted as adversarial perturbations and the manipulated image is hereby referred to as +adversarial images3. In addition, the automated method of synthesizing adversarial perturbations is +named as adversarial generator. +Formally, given an input face image, x, an adversarial generator has two requirements under the +obfuscation scenario: +• synthesize an adversarial face image, xadv = x + δ, such that AFR systems fail to match +xadv and x, and +3We interchangeably use the terms adversarial images and adversarial faces in this paper. +3 + +• limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans. +When the attack aims to impersonate a target individual, we need an image of the victim xtarget +where the identity of x and xtarget are different. Therefore, constraints under the impersonation +setting are as follows: +• synthesize an adversarial face image, xadv = x + δ, such that AFR systems erroneously +match xadv and xtarget, and +• limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans. +Obfuscation attempts (faces are perturbed such that they cannot be identified as the attacker) are gen- +erally more effective [25], computationally efficient to synthesize [14, 17], and widely adopted [47] +compared to impersonation attacks (perturbed faces can automatically match to a target subject). +Therefore, this paper focuses on crafting obfuscation attacks, however, we will still show examples +on synthesizing impersonation attacks. +3.2 +Gradient-based Attacks +In white-box attacks, the attacker is assumed to have the knowledge and access to the AFR system’s +model and parameters. Naturally, we then expect a much better attack success rate under white-box +settings since the attacker can carefully craft adversarial perturbations that necessarily evade the target +AFR system. However, these white-box manipulations of face recognition models are impractical in +real-world scenarios. For instance, assuming access to an airport’s already deployed AFR system +may be extremely difficult. +Nevertheless, it is advantageous to understand prevailing white-box methods. That is, if given access +to a CNN-based AFR system, how could one utilize all of its model parameters to launch a successful +adversarial attack? +A common approach is to utilize gradients of the whitebox AFR models. Namely the attackers modify +the image in the direction of the gradient of the loss function with respect to the input image. There +are two prevailing approaches to perform such gradient-based attacks: +• one-shot attacks, in which the attacker takes a single step in the direction of the gradient, +and +• iterative attacks where instead of a single step, several steps are taken until we obtain a +successful adversarial pattern. +3.2.1 +Fast Gradient Sign Method (FGSM) +This method computes an adversarial image by adding a pixel-wide perturbation of magnitude in the +direction of the gradient [14]. Under FGSM attack, we take a single step towards the direction of the +gradient, and therefore, FGSM is very efficient in terms of computation time. Formally, given an +input image x, we obtain an adversarial image xadv: +xadv = x + ϵ · sign (▽xJ (x, y)) +where, J is the loss function used to train the AFR system (typically, softmax cross entropy loss), and +y is the ground truth class label of x (typically, the subject ID of the identity in x). +FGSM was first proposed for the object classification domain and therefore, utilizes softmax proba- +bilities for crafting adversarial perturbations. Therefore, the number of object classes are assumed to +be known during training and testing. However, face recognition systems do not utilize the softmax +layer for classification (as the number of identities are not fixed during deployment) instead features +from the last fully connected layer are used for comparing face images. +We first modify FGSM appropriately in order to evade AFR systems rather than object classifiers. +Instead of considering the softmax cross-entropy loss as J, we craft a new loss function that models +real-world scenario4: +LfeatureMatch = 1 − Ex +� +F(x) · F(xadv) +||F(x)|| ||F(xadv)|| +� +. +4For brevity, we denote Ex ≡ Ex∈Pdata. +4 + +where, F is the matcher and F(x) is the feature representation of an input image x. The above feature +matching loss function computes the cosine distance between a pair of images and ensures that the +features between adversarial image xadv and input image x are as close as possible. Therefore, the +gradient of the above loss ensures the features do not match and hence, can be considered as an +obfuscation adversarial attack. +In Fig. 1, we show the results of launching our modified FGSM attack on a state-of-the-art AFR +system, namely ArcFace [3]. We see that with a single step and with minimal perturbations, the real +and adversarial images of Tiger Woods does not match via ArcFace while humans can easily identity +both images as pertaining to the same subject. +(a) Real Input Image +(b) Perturbation +(c) FGSM [14] +Figure 1: Adversarial face synthesized via FGSM [14]. A state-of-the-art face matcher, ArcFace [3], fails to +match the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.27, +while a score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject. +3.2.2 +Projected Gradient Descent (PGD) +An extreme case of white-box attacks is the PGD attack [17] where we assume that the attacker also +has unlimited number of attempts to try and evade the deployed AFR system. Unlike FGSM, PGD is +an iterative attack. PGD attempts to find the perturbation δ that maximises the loss of a model on a +particular input while keeping the size of the perturbation smaller than a specified amount referred +to as ϵ. We keep iterating until such a δ is obtained. Similar to FGSM, we modify the loss function +of PGD to fit the requirements of AFR system by again considering LfeatureMatch as the loss. Fig. +2 shows the results of PGD attack on ArcFace matcher. Note that due to multiple iterations, PGD +attack on AFR systems is more powerful (lower cosine similarity) but also more visible to humans as +compared to the single-step FGSM attack. +(a) Real Input Image +(b) Perturbation +(c) PGD [17] +Figure 2: Adversarial face synthesized via PGD [17]. A state-of-the-art face matcher, ArcFace [3], fails to match +the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.12, while a +score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject. +3.3 +Geometric Perturbations (GFLM) +Prior efforts in crafting adversarial faces have also tried non-linear deformations as a natural method +for evading AFR systems [48]. Non-linear deformations are applied by performing geometric warping +to the input face images. +Unlike traditional adversarial perturbations that basically add an adversarial perturbation δ, authors +in [48] propose a fast method of generating adversarial faces by altering the landmark locations of +the input images. The resulting adversarial faces completely lie on the manifold of natural images, +which makes it extremely difficult to detect any adversarial perturbations. Results of geometrically +warped adversarial faces are presented in 3. +5 + +(a) Real Input Image +(b) Perturbation +(c) GFLM +Figure 3: Adversarial face synthesized via GFLM [48]. A state-of-the-art face matcher, ArcFace [3], fails to +match the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.33, +while a score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject. +3.4 +Attribute-based Perturbations +Unlike geometric-warping and gradient-based attacks that may perturb every pixel in the image, a +few studies propose manipulating only salient regions in faces, e.g., eyes, nose, and mouth. +By restricting perturbations to only semantic regions of the face, SemanticAdv [46] generates +adversarial examples in a more controllable fashion by editing a single semantic aspect through +attribute-conditioned image editing. Fig. 4 shows results from adversarial manipulating semantic +attributes. We can see while the attacks are indeed successful, it comes at the cost of altering the +perceived identity as well as leads to degraded image quality. +(a) Real Input Image +(b) Blond +(c) Bangs +(d) Mouth Open +(e) Eyeglasses +(f) Makeup +Figure 4: Adversarial face synthesized via manipulating semantic attributes [46]. All adversarial images (b-f) +fail to match with the real image (a) via ArcFace [3]. +4 +AdvBiom: Learning to Synthesize Adversarial Attacks +We find that majority of prior efforts on crafting adversarial attacks either degrade the visual quality +where an observant human can still visually pick out the adversarial patterns. We also identify the +following challenges with prior efforts: +• Gradient-based attacks rely on white-box settings where the entire deployed CNN-based +AFR system is available to the attacker to compute its gradients. +• Geometrically-warping faces generally do not guarantee adversarial success and greatly +distort the face image. +• Semantic attribute manipulation can also degrade visual quality and may lead to greater +conspicuous changes. +Instead, we propose to train a network to “learn" the salient regions of the face that can be perturbed +to evade AFR systems in a semi-whitebox setting. These leads to the following advantages over prior +efforts: +6 + +• Perceptual Realism Given a large enough training dataset, a network can gradually learn to +synthesize adversarial face images that are perceptually realistic such that a human observer +can identify the image as a legitimate face image. +• Higher Attack Success The faces can be learned to be perturbed in a manner such that they +cannot be identified as the hacker (obfuscation at- tack) or automatically matched to a target +subject (impersonation attack) by an AFR system. +• Controllable The amount of perturbation can also be controllable by the attacker so that +they can examine the success of the learning model as a function of amount of perturbation. +• Transferability Due to the semi-whitebox setting: once the network learns to generate the +perturbed instances based on a single face recognition system, attacks can be transferred to +any black-box AFR systems. +We propose an automated adversarial biometric synthesis method, named AdvBiom, which generates +an adversarial image for a probe face image and satisfies all the above requirements. +4.1 +Methodology +Our goal is to synthesize a face image that visually appears to pertain to the target face, yet automatic +face recognition systems either incorrectly matches the synthesized image to another person or does +not match to target’s gallery images. AdvBiom comprises of a generator G, a discriminator D, and +face matcher (see Figure 5). +Probe +ℒ"#$ +Synthesized ++ +ℒ%&'()%)* +ℒ+',)-,./)%0( +Adversarial Mask +1 +ℱ +3 +Figure 5: Given a probe face image, AdvBiom automatically generates an adversarial mask that is then added to +the probe to obtain an adversarial face image. +Generator +The proposed generator takes an input face image, x ∈ X, and outputs an image, G(x). +The generator is conditioned on the input image x; for different input faces, we will get different +synthesized images. +Since our goal is to obtain an adversarial image that is metrically similar to the probe in the image +space, x, it is not desirable to perturb all the pixels in the probe image. For this reason, we treat the +output from the generator as an additive mask and the adversarial face is defined as x + G(x). If +the magnitude of the pixels in G(x) is minimal, then the adversarial image comprises mostly of the +probe x. Here, we denote G(x) as an “adversarial mask". In order to bound the magnitude of the +adversarial mask, we introduce a perturbation loss during training by minimizing the L2 norm5: +Lperturbation = Ex [max (ϵ, ∥G(x)∥2)] +(1) +where ϵ ∈ [0, ∞) is a hyperparameter that controls the minimum amount of perturbation allowed. +5For brevity, we denote Ex ≡ Ex∈X . +7 + +In order to achieve our goal of impersonating a target subject’s face or obfuscating one’s own identity, +we need a face matcher, F, to supervise the training of AdvBiom. For obfuscation attack, at each +training iteration, AdvBiom tries to minimize the cosine similarity between face embeddings of the +input probe x and the generated image x + G(x) via an identity loss function: +Lidentity = Ex[F(x, x + G(x))] +(2) +For an impersonation attack, AdvBiom maximizes the cosine similarity between the face embeddings +of a randomly chosen target’s probe, y, and the generated adversarial face x + G(x) via: +Lidentity = Ex[1 − F(y, x + G(x))] +(3) +The perturbation and identity loss functions enforce the network to learn the salient facial regions +that can be perturbed minimally in order to evade automatic face recognition systems. +Discriminator +Akin to previous works on GANs [49, 50], we introduce a discriminator in order +to encourage perceptual realism of the generated images. We use a fully-convolution network as a +patch-based discriminator [50]. Here, the discriminator, D, aims to distinguish between a probe, x, +and a generated adversarial face image x + G(x) via a GAN loss: +LGAN = +Ex [log D(x)] + +Ex[log(1 − D(x + G(x)))] +(4) +Finally, AdvBiom is trained in an end-to-end fashion with the following objectives: +min +D LD = −LGAN +(5) +min +G LG = LGAN + λiLidentity + λpLperturbation +(6) +where λi and λp are hyper-parameters controlling the relative importance of identity and perturbation +losses, respectively. Note that LGAN and Lperturbation encourage the generated images to be visually +similar to the original face images, while Lidentity optimizes for a high attack success rate. After +training, the generator G can generate an adversarial face image for any input image and can be tested +on any black-box face recognition system. +The overall algorithm describing the training procedure of AdvBiom can be found in Algorithm 1. +4.2 +Experimental Results +Evaluation Metrics +We quantify the effectiveness of the adversarial attacks generated by Ad- +vBiom and other state-of-the-art baselines via (i) attack success rate and (ii) structural similarity +(SSIM). +The attack success rate for obfuscation attack is computed as, +Attack Success Rate = (No. of Comparisons < τ) +Total No. of Comparisons +(7) +where each comparison consists of a subject’s adversarial probe and an enrollment image. Here, τ +is a pre-determined threshold computed at, say, 0.1% FAR6. Attack success rate for impersonation +attack is defined as, +Attack Success Rate = (No. of Comparisons ≥ τ) +Total No. of Comparisons +(8) +Here, a comparison comprises of an adversarial image synthesized with a target’s probe and matched +to the target’s enrolled image. We evaluate the success rate for the impersonation setting via 10-fold +cross-validation where each fold consists of a randomly chosen target. +Similar to prior studies [42], in order to measure the similarity between the adversarial example and +the input face, we compute the structural similarity index (SSIM) between the images. SSIM is a +normalized metric between −1 (completely different image pairs) to 1 (identical image pairs). +6For each face matcher, we pre-compute the threshold at 0.1% FAR on all possible image pairs in LFW. +For e.g., threshold @ 0.1% FAR for ArcFace is 0.28. +8 + +Algorithm 1 Training AdvBiom. All experiments in this work use α = 0.0001, β1 = 0.5, β2 = 0.9, +λi = 10.0, λp = 1.0, m = 32. +We set ϵ = 3.0 (obfuscation), ϵ = 8.0 (impersonation). +1: Input +2: +X +Training Dataset +3: +F +Cosine similarity between an image pair obtained by biometric matcher +4: +G +Generator with weights Gθ +5: +D +Discriminator with weights Dθ +6: +m +Batch size +7: +α +Learning rate +8: for number of training iterations do +9: +Sample a batch of probes {x(i)}m +i=1 ∼ X +10: +if impersonation attack then +11: +Sample a batch of target images y(i) ∼ X +12: +δ(i) = G((x(i), y(i)) +13: +else if obfuscation attack then +14: +δ(i) = G(x(i)) +15: +end if +16: +x(i) +adv = x(i) + δ(i) +17: +Lperturbation = 1 +m +��m +i=1 max +� +ϵ, ||δ(i)||2 +�� +18: +if impersonation attack then +19: +Lidentity = 1 +m +��m +i=1 F +� +x(i), x(i) +adv +�� +20: +else if obfuscation attack then +21: +Lidentity = 1 +m +��m +i=1 +� +1 − F +� +y(i), x(i) +adv +��� +22: +end if +23: +LG +GAN = 1 +m +��m +i=1 log +� +1 − D(x(i) +adv) +�� +24: +LD = 1 +m +�m +i=1 +� +log +� +D(x(i)) +� ++ log +� +1 − D(x(i) +adv) +�� +25: +LG = LG +GAN + λiLidentity + λpLperturbation +26: +Gθ = Adam(▽GLG, Gθ, α, β1, β2) +27: +Dθ = Adam(▽DLD, Dθ, α, β1, β2) +28: end for +Datasets +We train AdvBiom on CASIA-WebFace [51] and then test on LFW [52]7. +• CASIA-WebFace [51] is comprised of 494,414 face images belonging to 10,575 different +subjects. We removed 84 subjects that are also present in LFW and the testing images in +this paper. +• LFW [52] contains 13,233 web-collected images of 5,749 different subjects. In order to +compute the attack success rate, we only consider subjects with at least two face images. +After this filtering, 9,614 face images of 1,680 subjects are available for evaluation. +All the testing images in this paper have no identity overlap with the training set, CASIA- +WebFace [51]. +Data Preprocessing +All face images are passed through MTCNN face detector [53] to detect five +landmarks (two eyes, nose, and two mouth corners). Via similarity transformation, the face images +are aligned. After transformation, the images are resized to 160 × 160. Prior to training and testing, +each pixel in the RGB image is normalized by subtracting 127.5 and dividing by 128. +Experimental Settings +We use ADAM optimizers in Tensorflow with β1 = 0.5 and β2 = 0.9 for +the entire network. Each mini-batch consists of 32 face images. We train AdvBiom for 200,000 steps +with a fixed learning rate of 0.0001. Since our goal is to generate adversarial faces with high success +7Training on CASIA-WebFace and evaluating on LFW is a common approach in face recognition literature [3, +4] +9 + +rate, the identity loss is of utmost importance. We empirically set λi = 10.0 and λp = 1.0. We +train two separate models and set ϵ = 3.0 and ϵ = 8.0 for obfuscation and impersonation attacks, +respectively. +Gallery +Probe +Proposed AdvBiom +GFLM [48] +PGD [17] +FGSM [14] +0.68 +0.14 +0.26 +0.27 +0.04 +0.38 +0.08 +0.12 +0.21 +0.02 +(a) Obfuscation Attack +Target’s Gallery Target’s Probe +Probe +Proposed AdvBiom +A3GN [42] +FGSM [14] +0.78 +0.10 +0.30 +0.29 +0.36 +0.80 +0.15 +0.34 +0.33 +0.42 +(b) Impersonation Attack +Figure 6: Adversarial face synthesis results on LFW dataset in (a) obfuscation and (b) impersonation attack +settings (cosine similarity scores obtained from ArcFace [3] with threshold @ 0.1% FAR= 0.28). The proposed +method synthesizes adversarial faces that are seemingly inconspicuous and maintain high perceptual quality. +Architecture +Let c7s1-k be a 7 × 7 convolutional layer with k filters and stride 1. dk denotes a +4 × 4 convolutional layer with k filters and stride 2. Rk denotes a residual block that contains two +3 × 3 convolutional layers. uk denotes a 2× upsampling layer followed by a 5 × 5 convolutional +layer with k filters and stride 1. We apply Instance Normalization and Batch Normalization to the +generator and discriminator, respectively. We use Leaky ReLU with slope 0.2 in the discriminator +and ReLU activation in the generator. The architectures of the two modules are as follows: +• Generator: +c7s1-64,d128,d256,R256,R256,R256, u128, u64, c7s1-3 +• Discriminator: +d32,d64,d128,d256,d512 +A 1 × 1 convolutional layer with 3 filters and stride 1 is attached to the last convolutional layer of the +discriminator for the patch-based GAN loss LGAN. +We apply the tanh activation function on the last convolution layer of the generator to ensure +that the generated image ∈ [−1, 1]. In the paper, we denoted the output of the tanh layer as an +“adversarial mask”, G(x) ∈ [−1, 1] and x ∈ [−1, 1]. The final adversarial image is computed as +10 + +Obfuscation Attack +Proposed AdvBiom +GFLM [48] +PGD [17] +FGSM [14] +Attack Success Rate (%) @ 0.1% FAR +FaceNet [5] +99.67 +23.34 +99.70 +99.96 +SphereFace [4] +97.22 +29.49 +99.34 +98.71 +ArcFace [3] +64.53 +03.43 +33.25 +35.30 +COTS-A +82.98 +08.89 +18.74 +32.48 +COTS-B +60.71 +05.05 +01.49 +18.75 +Structural Similarity +0.95 ± 0.01 +0.82 ± 0.12 +0.29 ± 0.06 +0.25 ± 0.06 +Computation Time (s) +0.01 +3.22 +11.74 +0.03 +Impersonation Attack +Proposed AdvBiom +A3GN [42] +PGD [17] +FGSM [14] +Attack Success Rate (%) @ 0.1% FAR +FaceNet [5] +20.85 ± 0.40 +05.99 ± 0.19 +76.79 ± 0.26 +13.04 ± 0.12 +SphereFace [4] +20.19 ± 0.27 +07.94 ± 0.19 +09.03 ± 0.39 +02.34 ± 0.03 +ArcFace [3] +24.30 ± 0.44 +17.14 ± 0.29 +19.50 ± 1.95 +08.34 ± 0.21 +COTS-A +20.75 ± 0.35 +15.01 ± 0.30 +01.76 ± 0.10 +01.40 ± 0.08 +COTS-B +19.85 ± 0.28 +10.23 ± 0.50 +12.49 ± 0.24 +04.67 ± 0.16 +Structural Similarity +0.92 ± 0.02 +0.69 ± 0.04 +0.77 ± 0.04 +0.48 ± 0.75 +Computation Time (s) +0.01 +0.04 +11.74 +0.03 +White-box matcher (used for training) +Black-box matcher (never used in training) +Table 1: Attack success rates and structural similarities between probe and gallery images for obfus- +cation and impersonation attacks. Attack rates for obfuscation comprises of 484,514 comparisons and +the mean and standard deviation across 10-folds for impersonation reported. The mean and standard +deviation of the structural similarities between adversarial and probe images along with the time +taken to generate a single adversarial image (on a Quadro M6000 GPU) also reported. +xadv = 2 × clamp +� +G(x) + +� x+1 +2 +��1 +0 − 1. This ensures G(x) can either add or subtract pixels from +x when G(x) ̸= 0. When G(x) → 0, then xadv → x. +Face Recognition Systems +For all our experiments, we employ 5 state-of-the-art face matchers8. +Three of them are publicly available, namely, FaceNet [5], SphereFace [4], and ArcFace [3]. We also +report our results on two commercial-off-the-shelf (COTS) face matchers, COTS-A and COTS-B9. +We use FaceNet [5] as the white-box face recognition model, F, during training. All the testing +images in this paper are generated from the same model (trained only with FaceNet) and tested on +different matchers. +4.2.1 +Comparison with Prevailing Adversarial Face Generators +We compare our adversarial face synthesis method with state-of-the-art methods that have specifi- +cally been implemented or proposed for faces, including GFLM [48], PGD [17], FGSM [14], and +A3GN [42]10. In Table 1, we find that compared to the state-of-the-art, AdvBiom generates adversarial +faces that are similar to the probe 6. +Moreover, the adversarial images attain a high obfuscation attack success rate on 4 state-of-the-art +black-box AFR systems in both obfuscation and impersonation settings. AdvBiom learns to perturb +the salient regions of the face, unlike PGD [17] and FGSM [14], which alter every pixel in the +image. GFLM [48], on the other hand, geometrically warps the face images and thereby, results +in low structural similarity. In addition, the state-of-the-art matchers are robust to such geometric +deformation which explains the low success rate of GFLM on face matchers. A3GN, another +GAN-based method, however, fails to achieve a reasonable success rate in an impersonation setting. +8All the open-source and COTS matchers achieve 99% accuracy on LFW under LFW protocol. +9Both COTS-A and COTS-B utilize CNNs for face recognition. COTS-B is one of the top performers in the +NIST Ongoing Face Recognition Vendor Test (FRVT) [54]. +10We train the baselines using their official implementations (detailed in the supplementary material). +11 + +4.2.2 +Ablation Study +In order to analyze the importance of each module in our system, in Figure 7, we train three variants +of AdvBiom for comparison by removing the discriminator (D), perturbation loss Lperturbation, and +identity loss Lidentity, respectively. +Input +w/o D +w/o Lprt +w/o Lidt +with all +Figure 7: Variants of AdvBiom trained without the discriminator, perturbation loss, and identity loss, respectively. +Every component of AdvBiom is necessary. +The discriminator helps to ensure the visual quality of the synthesized faces are maintained. With +the generator alone, undesirable artifacts are introduced. Without the proposed perturbation loss, +perturbations in the adversarial mask are unbounded and therefore, leads to a lack in perceptual +quality. The identity loss is imperative in ensuring an adversarial image is obtained. Without the +identity loss, the synthesized image cannot evade state-of-the-art face matchers. We find that every +component of AdvBiom is necessary in order to obtain an adversarial face that is not only perceptually +realistic but can also evade state-of-the-art face matchers. +4.2.3 +What is AdvBiom Learning? +Via Lperturbation, during training, AdvBiom learns to perturb only the salient facial regions that can +evade the face matcher, F (FaceNet [5] in our case). In Figure 8, AdvBiom synthesizes the adversarial +masks corresponding to the probes. We then threshold the mask to extract pixels with perturbation +magnitudes exceeding 0.40. It can be inferred that the eyebrows, eyeballs, and nose contain highly +discriminative information that an AFR system utilizes to identify an individual. Therefore, perturbing +these salient regions are enough to evade state-of-the-art face recognition systems. +4.2.4 +Transferability of AdvBiom +In Table 1, we find that attacks synthesized by AdvBiom when trained on a white-box matcher +(FaceNet), can successfully evade 5 other face matchers that are not utilized during training in both +obfuscation and impersonation settings. In order to investigate the transferability property of AdvBiom, +we extract face embeddings of real images and their corresponding adversarial images, under the +obfuscation setting, via the white-box matcher (FaceNet) and a black-box matcher (ArcFace). In total, +we extract feature vectors from 1,456 face images of 10 subjects in the LFW dataset [52]. In Figure 9, +we plot the correlation heatmap between face features of real images, their corresponding adversarial +masks and adversarial images. First, we observe that face embeddings of real images extracted by +FaceNet and ArcFace are correlated in a similar fashion. This indicates that both matchers extract +features with related pairwise correlations. Consequently, perturbing salient features for FaceNet +can lead to high attack success rates for ArcFace as well. The similarity among the correlation +distributions of both matchers can also be observed when adversarial masks and adversarial images +are input to the matchers. That is, receptive fields for automatic face recognition systems attend to +similar regions in the face. +12 + +Probe +Adv. Mask +Visualization +Adv. Image +0.12 +0.26 +Figure 8: State-of-the-art face matchers can be evaded by slightly perturbing salient facial regions, such as +eyebrows, eyeballs, and nose (cosine similarity obtained via ArcFace [3]). +Figure 9: Correlation between face features extracted via FaceNet and ArcFace from 1,456 images belonging to +10 subjects. +To further illustrate the distributions of the embeddings of real and synthesized images, we plot +the 2D t-SNE visualization of the face embeddings for the 10 subjects in Figure 10. The identity +clusterings can be clearly observed from both real and adversarial images. In particular, the adversarial +counterpart of each subject forms a new cluster that draws closer to the adversarial clusterings of +other subjects. This shows that AdvBiom perturbs only salient pixels related to face identity while +maintaining a semantic meaning in the feature space, resulting in a similar manifold of synthesized +faces to that of real faces. +4.2.5 +Controllable Perturbation +The perturbation loss, Lperturbation is bounded by a hyper-parameter, ϵ, i.e., the L2 norm of the +adversarial mask must be at least ϵ. Without this constraint, the adversarial mask becomes a blank +image with no changes to the probe. With ϵ, we can observe a trade-off between the attack success +rate and the structural similarity between the probe and synthesized adversarial face (Fig. 11). A +higher ϵ leads to less perturbation restriction, resulting in a higher attack success rate at the cost of a +lower structural similarity. For an impersonation attack, this implies that the adversarial image may +13 + +Real Image +Adversarial Mask +Adversarial Image +0.8 +FaceNet +0.4 +0.0 +ArcFace +-0.4FaceNet +Real Image +Adversarial Image (Obfuscation) +ArcFace +Figure 10: 2D t-SNE visualization of face representations extracted via FaceNet and ArcFace from 1,456 images +belonging to 10 subjects. +contain facial features from both the hacker and the target. In our experiments, we chose ϵ = 8.0 and +ϵ = 3.0 for impersonation and obfuscation attacks, respectively. +4 +6 +8 +10 +12 +14 +16 +0.81 +0.92 +0.95 +0.76 +0.69 +5 +13 +21 +39 +52 +60 +66 +Hyper-parameter (ε) +Success Rate (%) +Structural Similarity +ε = 4.0 +ε = 8.0 +ε = 10.0 +ε = 16.0 +Figure 11: Trade-off between attack success rate and structural similarity for impersonation attacks. +4.2.6 +Attacks via AdvBiom Beyond Faces +We now show that the AdvBiommethod, coupled with the proposed Minutiae Displacement and +Distortion Modules, can be extended to effectively generate adversarial fingerprints which are visually +similar to corresponding probe fingerprints while evading two state-of-the-art COTS fingerprint +matchers as well as a deep network-based fingerprint matcher. +14 + +FS: 0.97 +FS: 0.92 +(a) Enrolled Mate +VS: 235 | FS: 0.96 +VS: 172 | FS: 0.99 +(b) Input Probe +VS: 31 | FS: 0.92 +VS: 10 | FS: 0.92 +(c) AdvBiom +VS: 134 | FS: 0.96 +VS: 104 | FS: 0.96 +(d) DeepFool [16] +VS: 139 | FS: 0.95 +VS: 104 | FS: 0.96 +(d) PGD [17] +Figure 12: Example probe and corresponding mate fingerprints along with synthesized adversarial probes. (a) +Two example mate fingerprints from NIST SD4 [55], and (b) the corresponding mates. Adversarial probe +fingerprints using different approaches are shown in: (c) proposed synthesis method, AdvBiom; (d-e) state-of- +the-art methods, DeepFool and PGD respectively. VeriFinger v11.0 match score (probe v. mate) - VS, and the +fingerprintness score (degree of similarity of a given image to a fingerprint pattern) - FS ∈ [0,1] [56], which +ranges from 1 (the highest) to 0 (the lowest), are given below each image. A VS of above 48 (at 0.01% FAR) +indicates a successful match between the probe and the mate. The proposed attack AdvBiom successfully +evades COTS and deep network-based matchers, while maintaining visual fingerprint perceptibility and high +fingerprintness scores. +Grosz et. al [57] showed that random minutiae position displacements and non-linear distortions +drastically affected the performance of COTS fingerprint matchers. AdvBiom builds upon these two +perturbations and when given a probe fingerprint, can synthesize an adversarial fingerprint image that +retains all of the original fingerprint attributes except the identity, i.e. a fingerprint recognition system +should not match the adversarial fingerprint to the probe fingerprint (obfuscation attack). +Figure 13 shows the schematic of AdvBiom conditioned for fingerprints. The following subsections +explain the major components of the approach in detail. +Minutiae Displacement Module +While the authors in [57] showed the effectiveness of random +minutiae position displacements on COTS matchers, they studied the effect of this perturbation by +directly modifying the minutiae template instead of the fingerprint image (pixel space). However, it +may be difficult to obtain the minutiae template of a given fingerprint image using COTS minutiae +extractors rather than the source fingerprint image itself. Thus, we propose a minutiae displacement +module Gdisp which, given a fingerprint image, displaces its minutiae points in random directions by +a predefined distance. To extract minutiae points from a fingerprint image, we employ a minutiae map +extractor (M) from [58]. For a fingerprint image of width w and height h, M outputs a 12 channel +heat map H ∈ Rh×w×12, where if H(i,j,c), value of the heat map at position (i,j) and channel c, is +greater than a threshold mt and is the local maximum in its 5 × 5 × 3 neighboring cube, a minutiae is +marked at (i,j). The minutiae direction θ is calculated by maximising the quadratic interpolation with +respect to: +f +� +(c − 1) × π +6 +� += H (i, j, (c − 1)%12) +(9) +f +� +c × π +6 +� += H(i, j, c) +(10) +f +� +(c + 1) × π +6 +� += H(i, j, (c + 1)%12) +(11) +Figure 14 shows a fingerprint image and its corresponding 12 channel minutiae map. Once M +extracts a minutiae map Hprobe from the input probe fingerprint x, we detect minutiae points by +applying a threshold of 0.2 on Hprobe and finding closed contours. Each detected contour, at say +15 + +𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝 +Extractor (ℳ) +𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑡𝑜𝑟 +(𝒟) +GAN Loss +ℒgan +𝑀𝑖𝑛. 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 +Module (𝒢𝑑𝑖𝑠𝑝) +Probe Fingerprint +Displaced Fingerprint +Original M.Map +𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑃𝑖𝑥. +Displacement +Target M.Map +M.Map Sim Loss +ℒmmap_sim +Pixel Loss +ℒpixel +Predicted M.Map +M.Map Dis Loss +ℒmmap_dis +𝐷𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛 +Module (𝒢𝑑𝑖𝑠𝑡) +𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝 +Extractor (ℳ) +Adversarial Fingerprint +Figure 13: Schematic of AdvBiom for generating adversarial fingerprints. Given a probe fingerprint image, it is +passed to Gdisp which randomly displaces its minutiae points. The distortion module (Gdist) identifies control +points on the displaced fingerprint and non-linearly distorts the image to output the adversarial fingerprint. The +solid black arrows show the forward pass of the network while the dotted black arrows show the propagation of +the losses. +location (i, j), is displaced by a predefined L1 distance d = |∆i|+|∆j|, giving us the target minutiae +map Htarget. +0 +𝜋/6 +𝜋/3 +𝜋/2 +2𝜋/3 +5𝜋/6 +𝜋 +7𝜋/6 +4𝜋/3 +3𝜋/2 +5𝜋/3 +11𝜋/6 +Figure 14: The 12 channel minutiae map of an example fingerprint image shown on the left. The minutiae points +(shown in red) are marked by a COTS minutiae extractor. The bright spots in each channel image indicate the +spatial location of minutiae points while the kth channel (k ∈ [0, 11]) indicate the contributions of minutiae +points to the kπ/6 orientation. +The minutiae displacement module Gdisp is essentially an autoencoder conditioned on the probe +fingerprint x and the target minutiae map Htarget. It learns to generate a displaced fingerprint xdisp +whose predicted minutiae map Hpred is as close as possible to the target minutiae map Htarget in the +pixel space. To achieve this, we have three losses that govern Gdisp: +Lmmap_sim = ||Htarget − Hpred||1 +(12) +Lmmap_dis = +1 +||Htarget − Hprobe||1 +(13) +, where Lmmap_sim is the minutiae map similarity loss which minimises the distance between the +predicted and target minutiae map, while the minutiae map dissimilarity loss Lmmap_dis maximises +16 + +G +QU +Q +Q +QQ +Q +Gthe distance between the predicted and probe minutiae map. In figure 15, we show two example +probe fingerprints and their corresponding displaced fingerprints after passing through Gdisp. +Distortion Module +One of the most noteworthy conclusions from [57] was that non-linear distor- +tions to minutiae points was one of the most successful perturbations to lower the similarity scores +between perturbed and corresponding unperturbed fingerprints. Again, the non-linear distortion +was applied to all the minutiae points in the template and not to the image. Thus, our next step in +generating adversarial fingerprints consists of a distortion module Gdist which learns to distort salient +points in a fingerprint image. +The architecture of Gdist consists of an encoder conditioned on the input probe fingerprint x and the +target minutiae map Hprobe. The output from the encoder is a predefined number of control points11 c. +The non-linear distortion model proposed in [59], learned using a thin plate spline (TPS) model [60] +from 320 already distorted fingerprint videos, was employed to calculate the displacements of the +predicted control points. The hyper-parameter σ is used to indicate the extent of the distortion. The +control points and their displacements are then fed to a differentiable warping module [61] to get the +resultant adversarial fingerprint xadv. +To limit the magnitude of non-linear distortion and to ensure that xdisp and xadv are close to the +probe fingerprint x, we introduce pixel loss between the image pairs (x, xdisp) and (x, xadv): +Lpixel = 1 +n +� +i,j +|xi,j − xdispi,j|+ 1 +n +� +i,j +|xi,j − xadvi,j| +(14) +Figure 16 shows two displaced fingerprints and their corresponding output from Gdist. +Discriminator +In order to guide the generative modules Gdisp and Gdist to synthesize realistic +fingerprint images, we introduce a fully convolutional network as a patch-based discriminator D. +The job of the discriminator is to distinguish between real fingerprint images x and the generated +adversarial fingerprint images xadv. This is accomplished through the GAN loss: +Lgan = logD(x) + log(1 − D(xadv)) +(15) +The proposed approach AdvBiom is trained in an end-to-end manner with respect to the following +objective function: +(16) +L = Lgan + λmmap_simLmmap_sim + λmmap_disLmmap_dis + λpixelLpixel +where the hyper-parameters λmmap_sim, λmmap_dis, and λpixel denote the relative importance of +their respective losses. Once trained, AdvBiom can generate an adversarial fingerprint image for +any input probe fingerprint and can be tested on any fingerprint matcher regardless of the feature +extraction method (minutiae or deep-features). +11Control points are points in an image to which non-linear distortion is applied. +Probe Fingerprint +Displaced Fingerprint +Probe Fingerprint +Displaced Fingerprint +Figure 15: Example probe fingerprints from NIST SD4 [55] and their corresponding output from the minutiae +displacement module Gdisp. The minutiae points (shown in red) are marked using a COTS minutiae extractor. +17 + +Displaced Fingerprint +Distorted Fingerprint +Displaced Fingerprint +Distorted Fingerprint +Figure 16: Fingerprints in the left column are example displaced fingerprints from Gdisp. The distortion module +Gdist predicts control points (marked in blue) and distorts the images based on their displacements (red arrows) +using the non-linear distortion model from [59]. The resultant distorted fingerprint images are shown in the right +column. +Successful +Attacks +Failed +Attacks +Original +Probe +Adv. Probe +(AdvFinge) +Mate +Original +Probe +Adv. Probe +(AdvFinge) +Mate +VS: 36 | FS: 0.82 +VS: 47 | FS: 0.88 +VS: 109 | FS: 0.87 +VS: 76 | FS: 0.89 +(AdvBiom) +(AdvBiom) +Figure 17: Example successful and failed adversarial fingerprints attack using AdvBiom on NIST SD4 [55]. The +VeriFinger matching scores (probe v. mate): VS, and fingerprintness [56] scores: FS, of adversarial probes are +shown below their respective triplet. Note that the VeriFinger matching threshold is 48 at 0.01% FAR. +Evaluation Metrics: +The requirement of a good adversarial fingerprints generator is to evade +fingerprint matchers while preserving fingerprint attributes and being model-agnostic. Thus, in order +to quantify the performance of adversarial attacks generated by AdvBiom and other state-of-the-art +baselines, we employ the following evaluation metrics: +• True Accept Rate (TAR): The extent to which an adversarial attack can evade a fingerprint +matcher is measured by the drop in TAR at an operational setting, say 0.01% False Accept +Rate (FAR). +• Fingerprintness: Soweon and Jain [56] proposed a domain-specific metric called finger- +printness to measure the degree of similarity of a given image to a fingerprint pattern. +Fingerprintness ranges from [0,1] where higher the score, higher the probability of the +pattern in the image corresponding to a fingerprint pattern. +• NFIQ 2.0: Lastly, we use NFIQ 2.0 [62] quality scores to evaluate the fingerprint quality +of adversarial fingerprint images. NFIQ scores range from [0,100] where a score of 100 +depicts the highest fingerprint quality. +Note that since non-linear distortions change the structure of the image, using the structural similarity +index (SSIM) metric is inappropriate as it essentially measures the local change in structures of the +image pairs. +Datasets: We train AdvBiom on an internal dataset of 120,000 rolled fingerprint images. Furthermore, +we evaluate the performance of the proposed fingerprint adversarial attack and other baselines on: +• 2,000 fingerprint pairs from NIST SD4 [55] +• 27,000 fingerprint pairs from NIST SD14 [63] +18 + +111@ +8 +Q +Q +QQ +Q +de +G +G母 +Q +% +d +TG +% +Q +Qd +Q +d +00 +& +QQ +% +CQ +d +3 +d +& +QQQ +@山 +G +%Q +QQ +Q +q Q +bu +G +P dppdQ +@ +d +QQ- +Q? +& +d +& +? +de +qQ +& +GQ QC +QQ +Q +Q +? +多 +& +G +lG +Q +QQ +% +Q +@ +G +QQQQ +G +G20- +Q +Q +Q +q +QQ +G +Q +G8 +a +QQ +QQ +cs +G +Q +G +QQ +0甲Accuracy +Adversarial Attacks +Original +Probes +FGSM +I-FGSM +Deep +Fool +PGD +Adv +Biom +TAR +(%) +at +0.01% +FAR +NIST +SD4 +VeriFinger +99.05 +95.20 +98.30 +95.00 +97.60 +56.25 +Innovatrics +97.00 +93.00 +95.50 +92.65 +94.75 +41.35 +DeepPrint +94.55 +36.20 +64.15 +30.40 +68.75 +46.35 +NIST +SD14 +VeriFinger +99.42 +95.20 +98.30 +95.00 +97.60 +37.67 +Innovatrics +98.24 +90.84 +95.68 +91.32 +94.01 +25.69 +DeepPrint +96.52 +48.70 +84.48 +31.44 +64.28 +69.42 +FVC +2004 +DB1 A +VeriFinger +94.89 +91.60 +91.53 +86.92 +92.69 +22.31 +Innovatrics +94.15 +87.36 +85.68 +82.32 +88.75 +5.52 +DeepPrint +75.36 +13.22 +33.31 +6.87 +27.39 +20.62 +Table 2: True Accept Rate (TAR) @ 0.01% FAR of AdvBiom along with state-of-the-art baselines attacks on +three datasets - NIST SD4 [55], NIST SD14 [63], and FVC 2004 DB1 A [64]. 2 COTS fingerprint matchers - +VeriFinger v11.0 [65] and Innovatrics v7.6.0.627 [66], and a deep network-based matcher DeepPrint [67] were +employed for the evaluation. It is observed that DeepPrint, a deep network-based matcher, is susceptible to all +types of adversarial attacks while VeriFinger and Innovatrics are more robust. +• 558 fingerprints from DB1 A of FVC 2004 [64], consisting of 1,369 genuine pairs. +Experimental Settings: AdvBiom was trained using the Adam optimizer with β1 as 0.5 and β2 as +0.9. The hyper-parameters were empirically set to λmmap_sim = 0.05, λmmap_dis = 500000, and +λpixel = 1000 for convergence. Based on the conclusions drawn in [57], d, c, and λ were set to +20, 16, and 2.0 respectively for optimal effectiveness against fingerprint matchers while ensuring +fingerprint realism. AdvBiom was trained for 16,000 steps using Tensorflow r1.14.0 on an Intel Core +i7-11700F @ 2.50GHz CPU with a RTX 3070 GPU. On the same machine, AdvBiom can synthesize +an adversarial fingerprint within 0.35 seconds. +Fingerprint Authentication Systems: Since AdvBiom is a black-box attack, we do not require any +fingerprint authentication system while training the network. However, we evaluate AdvBiom and +other baseline attacks on two COTS fingerprint matchers and one deep network-based matcher: +• VeriFinger v11.0 [65] +• Innovatrics v7.6.0.627 [66] +• DeepPrint [67] +Comparison with Prevailing Fingerprint Adversarial Generators +We show the performance +of our method AdvBiom as compared to other state-of-the-art attacks in Table 2. We observe that +the TAR of two COTS and a deep network-based fingerprint matcher for the aforementioned three +datasets. It is to note that all the baseline attacks [14, 15, 17, 16] are white-box attacks and were +trained using DeepPrint [67]. It is evident from Table 2 that AdvBiom is the most successful attack on +COTS matchers VeriFinger and Innovatrics, and is also able to effectively evade a deep network-based +fingerprint matcher, namely DeepPrint. It can also be observed that while COTS fingerprint matchers +are robust to most adversarial attacks, DeepPrint is very susceptible to the same attacks since it +heavily relies on the texture of the fingerprint which is majorly affected by adversarial attacks. +A successful adversarial attack should not only evade fingerprint matchers but should also preserve +fingerprint attributes. In order to observe the effect of adversarial attacks on fingerprint pattern +in images, we plot the fingerprintness [56] distribution of 2,000 probes from NIST SD4 [55] for +AdvBiom as well as for other baseline attacks. Since all the state-of-the-art baselines essentially add +noise to each pixel in the image, they do not change the structure of the fingerprint and thus do not +19 + +0.6 +0.7 +0.8 +0.9 +1.0 +Fingerprintedness Scores +0 +2 +4 +6 +8 +10 +12 +Probability of occurence +Original Probes ( = 0.91) +AdvFinge ( = 0.86) +FGSM ( = 0.89) +I-FGSM ( = 0.91) +PGD ( = 0.90) +DeepFool ( = 0.90) +AdvBiom +Figure 18: Fingerprintness [56] distribution of 2,000 +probes from NIST SD4 with respect to AdvBiom and +other state-of-the-art baselines attacks. +0 +20 +40 +60 +80 +100 +NFIQ Score +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +Probability of occurence +Original Probes ( = 41.70) +AdvFinge ( = 31.46) +FGSM ( = 43.30) +I-FGSM ( = 44.23) +PGD ( = 41.28) +DeepFool ( = 43.42) +AdvBiom +Figure 19: NFIQ 2.0 [62] quality scores distribution +of 2,000 probes from NIST SD4 [55] with respect to +AdvBiom and other baselines attacks. +affect fingerprintness scores. AdvBiom , on the other hand, displaces minutiae points and non-linearly +distorts the image, and still maintains a high mean fingerprintness score of µ = 0.86. +Furthermore, we also compute the NFIQ 2.0 [62] quality scores distribution (figure 19) of the original +and adversarial probes from NIST SD4 [55]. As shown in figure 12, baseline attacks tend to minutely +perturb image pixels to generate adversarial fingerprints and as a result do not have much of an effect +on the quality scores. AdvBiom , on the other hand, provides an optimal solution by successfully +attacking fingerprint matchers while maintaining high fingerprintness and NFIQ scores. +Genuine and Imposter Scores Distribution +To determine the effect of adversarial fingerprint +on both genuine and imposter pairs, we plot the genuine and imposter scores distribution of NIST +SD4 [55] in figure 20 before and after applying AdvBiom . We computed a total of 2,000 genuine +and 20,000 imposter scores for the evaluation. It can be observed that the genuine scores drastically +decrease and shift to the left of the axis as their mean drops from 183.87 to 55.55 after the attack. +However, the imposter scores remain unaffected with the mean imposter score changing by only 0.53. +0 +100 +200 +300 +400 +Matching Scores +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +Probability of Occurence +Genuine Scores | Before Attack ( = 183.88) +Genuine Scores | After Attack ( = 55.55) +Imposter Scores | Before Attack ( = 6.00) +Imposter Scores | After Attack ( = 6.52) +48: Matching Threshold at 0.01% FAR +(a) Using VeriFinger SDK [65] +(b) Using Innovatrics SDK [66] +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Matching Scores +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Probability of Occurence +Genuine Scores | Before Attack ( = 0.94) +Genuine Scores | After Attack ( = 0.78) +Imposter Scores | Before Attack ( = 0.10) +Imposter Scores | After Attack ( = 0.09) +0.837: Matching Threshold at 0.01% FAR +(c) using DeepPrint [67] +Figure 20: Genuine and imposter scores distribution of NIST SD4 [55] before and after the adversarial attack +AdvBiom using three state-of-the-art fingerprint matchers - VeriFinger v11.0 [65], Innovatrics v7.6.0.627 [66], +and DeepPrint [67]. Here, µ refers to the mean of the scores distribution. In all the three cases, the genuine +scores shift towards the left while the imposter scores do not get affected by the attack. +Is AdvBiom Biased Towards Certain Fingerprint Types? +The generated adversarial fingerprint +from AdvBiom is conditioned on the input probe fingerprint. Thus, it is essential to check if there is a +relation between the amount of perturbation applied and the fingerprint type. The confusion matrix +for the five fingerprint types (left loop, right loop, whorl, arch, tented arch) before and after applying +AdvBiom on the 2,000 probes of NIST SD4 [55] is shown in table 3. Note that we use NIST SD4 for +this evaluation since it has a uniform number of fingerprint images per each type (400 fingerprints +per type). It is evident from the table that all five fingerprint types are almost equally susceptible to +the attack, and thus the attack crafted AdvBiom is not biased towards a particular fingerprint type. +20 + +Genuine Scores LBefore Attack (u = 590.00) +Genuine Scores 1After Attack (μu = 48.72) +0.6 +Imposter Scores /Before Attack (μu = 1.34) +Imposter Scores l After Attack (μ = 1.4o) +0.5 +Probability of Occurence +40:Matching Threshold at 0.01% FAR +0.4 +0.3 +0.2 +0.1 +0.0 +200 +600 +0 +400 +800 +1000 +Matching Scores0.012 +0.010 +0.008 +0.006 +0.004 +0.002 +0.000 +0 +200 +400 +600 +800 +1000Before Attack +After Attack +TAR +L: 99.75% +R: 99.25% +W: 99.50% +T: 99.25% +A: 97.50% +FAR +L: 0% +R: 0% +W: 0% +T: 0% +A: 0% +FRR +L: 0.25% +R: 0.75% +W: 0.50% +T: 0.75% +A: 2.50% +TRR +L: 100% +R: 100% +W: 100% +T: 100% +A: 100% +TAR +L: 59.00% +R: 56.50% +W: 58.75% +T: 56.00% +A: 57.00% +FAR +L: 0% +R: 0% +W: 0% +T: 0% +A: 0% +FRR +L: 41.00% +R: 43.50% +W: 41.25% +T: 44.00% +A: 43.00% +TRR +L: 100% +R: 100% +W: 100% +T: 100% +A: 100% +Table 3: Confusion matrix for five fingerprint types (left loop: L, right loop: R, whorl: W, tented arch: T, arch: +A) from NIST SD4 [55] before and after the adversarial attack using AdvBiom. Here, TAR = True Accept Rate, +FAR = False Accept Rate, FRR = False Reject Rate, and TRR = True Reject Rate. Note that the matching +threshold was 48 at 0.01% FAR using the COTS fingerprint matcher VeriFinger. AdvBiom is not biased towards +any fingerprint type. +5 +Conclusions +We show that a new method of adversarial synthesis, namely AdvBiom, that automatically generates +adversarial face images with imperceptible perturbations evading state-of-the-art biometric matchers. +With the help of a GAN, and the proposed perturbation and identity losses, AdvBiom learns the set +of pixel locations required by face matchers for identification and only perturbs those salient facial +regions (such as eyebrows and nose). Once trained, AdvBiom generates high quality and perceptually +realistic adversarial examples that are benign to the human eye but can evade state-of-the-art black- +box face matchers, while outperforming other state-of-the-art adversarial face methods. 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Jain, “Learning a fixed-length fingerprint representation,” +IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1–1, 2019. +24 + diff --git a/29E2T4oBgHgl3EQfjQdC/content/tmp_files/load_file.txt b/29E2T4oBgHgl3EQfjQdC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1bbd8361b3136d8ff3b8690652d2c6156a849b3d --- /dev/null +++ b/29E2T4oBgHgl3EQfjQdC/content/tmp_files/load_file.txt @@ -0,0 +1,1287 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf,len=1286 +page_content='AdvBiom: Adversarial Attacks on Biometric Matchers Debayan Deb, Vishesh Mistry, Rahul Parthe TECH5, Troy, MI, USA {debayan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='deb, vishesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='mistry, rahul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='parthe}@tech5-sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='com Abstract With the advent of deep learning models, face recognition systems have achieved impressive recognition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The workhorses behind this success are Convolutional Neural Networks (CNNs) and the availability of large training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, we show that small human-imperceptible changes to face samples can evade most prevailing face recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Even more alarming is the fact that the same generator can be extended to other traits in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In this work, we present how such a generator can be trained and also extended to other biometric modalities, such as fingerprint recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 1 Introduction The last decade has seen a massive influx of deep learning-based technologies that have tackled problems which were once thought to be unsolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Much of this progress can be attributed to Convolutional Neural Networks (CNNs) [1, 2] which are now deployed in a plethora of applications ranging from cancer detection to driving autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Akin to the computer vision domain, the use of CNNs have completely changed the face of biometrics due to the availability of powerful computing devices (GPUs, TPUs) and deep architectures capable of learning rich features [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Automated face recognition systems (AFR) have been proven to achieve accuracies as high as 99% True Accept Rate (TAR) @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% False Accept Rate (FAR) [6], majorly owing to publicly available large-scale face datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Unfortunately, studies have shown that CNN-based networks are vulnerable to adversarial pertur- bations1 [7–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It is not surprising that AFR systems too are not impervious to these attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Adversarial attacks to an AFR system can be classified into two categories - (i) impersonation attack where the hacker tries to perturb his face image to match it to a target victim, and (ii) obfuscation attack where the hacker’s face image is perturbed to match with a random identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Both the above at- tacks involve the hacker adding targeted human-imperceptible perturbations to the face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' These adversarial attacks are different from face digital manipulation that include attribute manipulation and synthetic faces, and also from presentation attacks which involves the perpetrator wearing a physical artifact such as a mask or replaying a photograph/video of a genuine individual which may be conspicuous in scenarios where human operators are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Let us consider, as example, the largest deployment of fingerprint recognition systems - India’s Aadhaar Project [13], which currently has an enrolled gallery size of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='35 billion faces from nearly all of its citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In September 2022 alone, Aadhaar received 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='3 billion authentication requests2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In order to deny a citizen his/her rightful access to government benefits, healthcare, and financial services, an attacker can maliciously perturb enrolled face images such that they do not 1Adversarial perturbations refer to altering an input image instance with small, human imperceptible changes in a manner that can evade CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 2https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='ly/3BzlpZJ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='03966v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='CV] 10 Jan 2023 match to the genuine person during verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In a typical AFR system, adversarial faces can be replaced with a captured face image in order to prevent the probe face from matching to any of its corresponding enrolled faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Additionally, the attacker can compromise the entire gallery by inserting adversarial faces in the enrolled gallery, where no probe face will match to the correct identity’s gallery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Adversarial attacks can further be categorized into two types of attacks based on how the attack vector is trained and generated: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' White-box attack: Attacks in which the hacker has full knowledge of the recognition system, and iteratively perturbs every pixel by various optimization schemes are termed as white-box attacks [14–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Black-box attack: With no information about the parameters of the recognition system, black-box attacks are deployed by either transferring attacks learned from an available AFR system [23–28], or querying the the target system for score [29–31] or decision [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Semi-whitebox attack: Here, a white-box model is utilized only during training and then ad- versarial examples are synthesized during inference without any knowledge of the deployed AFR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We propose an automated adversarial synthesis method, named AdvBiom, which generates an ad- versarial image for a probe image and satisfies all the above requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The contributions of the paper are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' GAN-based AdvBiom that learns to generate visually realistic adversarial face images that are misclassified by state-of-the-art automated biometric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Adversarial images generated via AdvBiom are model-agnostic and transferable, and achieve high success rate on 5 state-of-the-art automated face recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Visualizing regions where pixels are perturbed and analyzing the transferability of AdvBiom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We show that AdvBiom achieves significantly higher attack success rate under current defense mechanisms compared to baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' With the addition of the proposed Minutiae Displacement and Distortion modules, we show thatAdvBiom can also be extended to successfully evade automated fingerprint recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1 Adversarial Attacks As discussed earlier, adversarial attacks are broadly classified into white-box attacks and black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A large number of white-box attacks are gradient-based where they analyze the gradients during the back-propagation of an available face recognition system and perform pixel-wise per- turbations to the target face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' While approaches such as FGSM [14] and PGD [17] exploit the high-dimensional space of deep networks to generate adversarial attacks, C&W [18] focuses on minimizing objective functions for optimal adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, the basic assump- tion in white-box attacks that the target recognition system will be available is not plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In real-life scenarios, the hacker will not have any information regarding the architecture, training and deployment of the recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Black-box attacks can be classified into three major categories: transfer-based, score-based, and decision-based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Transfer-based attacks train their adversarial attack generator using readily available recognition systems and then deploy the attacks onto a black-box target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [23] proposed the use of momentum for efficient transferability of the adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' DI2- FGSM [24] suggested to increase input data diversity for improving transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Other approaches in this category include AI-FGSM [27] and TI-FGSM [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Score-based attacks [29–31] query the target system for scores and try to estimate its gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Decision-based attacks have the most challenging setting wherein only the decisions from the target system are queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Some effective methods in this category include Evolutionary attack [32] and Boundary attack [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 Adversarial Attacks on Face Recognition Although adversarial attacks on face recognition systems have only been recently explored, there has been a significant number of effective approaches for evading AFR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Attacks on face recognition systems can be broadly categorized into physical attacks and digital attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Physical attacks involve generating adversarial physical artifacts which are ’worn’ on a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Sharif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [34, 35] proposed generating adversarial eye-glass frames for attacking face recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In [36], adversarial printed stickers placed on a hat were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, methods [34–36] are implemented in a white-box setting which is unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Additionally, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [37] proposed an adversarial light projection attack using an on-premise projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [38] generated and printed eye makeup patches to be stuck around the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' More recently, authors in [39] proposed an adversarial mask for impersonation attacks in a black-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, all the above methods suffer a major drawback of being unrealistic in an operational setting where a human operator is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Digital attacks refer to manipulating and perturbing the pixels of a digital face image before being passed through a face recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Early works [9, 18, 10, 8, 40] focused on gradient-based attacks for face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, these methods implement lp-norm perturbations to each pixel resulting in decreased attack transferability, and vulnerability to denoising models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Cauli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [41] implemented a backdoor attack where the target face recognition system’s training samples were manipulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Apart from the fact that gaining access to the target AFR’s training samples is highly improbable, a thorough visual inspection of the samples can easily identify the digital artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Other works employ more stealthy attack approaches against face recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [32] proposed an evolutionary optimization method for generating adversarial faces in decision-based black-box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, they require a minimum of 1,000 queries to the target face recognition system before a realistic adversarial face can be synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [42] added a conditional variational autoencoder and attention modules to generate adversarial faces in a transfer-based black-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, they solely focused on impersonation attacks and require at least 5 image samples of the target subject for training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [43] implemented dropout [44] to improve the transferability of the adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [38] perturbed the eye region of a face to produce adversarial eyeshadow artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, the artifacts are visibly conspicuous under close inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' [25] used a GAN to generate minimal perturbations in salient facial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' More recently, [45] and [46] have focused on manipulating facial attributes for targeted adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3 Adversarial Faces 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1 Preliminaries The goal of any attacker is to evade Automated Face Recognition (AFR) systems under either of the two settings: Obfuscation Manipulate input face images in a manner such that they cannot be identified as the hacker, or Impersonation Edit input face images such that they are identified as a target/desired individual (victim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' While the manipulated face image evades the AFR system, a key requirement in a successful attack is such that the input face image should appear as a legitimate face photo of the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In other words, the attacker desires an automated method of adding small and human-imperceptible changes to an input face image such that it can evade AFR systems while appear benign to human observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' These changes are denoted as adversarial perturbations and the manipulated image is hereby referred to as adversarial images3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In addition, the automated method of synthesizing adversarial perturbations is named as adversarial generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Formally, given an input face image, x, an adversarial generator has two requirements under the obfuscation scenario: synthesize an adversarial face image, xadv = x + δ, such that AFR systems fail to match xadv and x, and 3We interchangeably use the terms adversarial images and adversarial faces in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3 limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' When the attack aims to impersonate a target individual, we need an image of the victim xtarget where the identity of x and xtarget are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Therefore, constraints under the impersonation setting are as follows: synthesize an adversarial face image, xadv = x + δ, such that AFR systems erroneously match xadv and xtarget, and limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Obfuscation attempts (faces are perturbed such that they cannot be identified as the attacker) are gen- erally more effective [25], computationally efficient to synthesize [14, 17], and widely adopted [47] compared to impersonation attacks (perturbed faces can automatically match to a target subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Therefore, this paper focuses on crafting obfuscation attacks, however, we will still show examples on synthesizing impersonation attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 Gradient-based Attacks In white-box attacks, the attacker is assumed to have the knowledge and access to the AFR system’s model and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Naturally, we then expect a much better attack success rate under white-box settings since the attacker can carefully craft adversarial perturbations that necessarily evade the target AFR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, these white-box manipulations of face recognition models are impractical in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' For instance, assuming access to an airport’s already deployed AFR system may be extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Nevertheless, it is advantageous to understand prevailing white-box methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' That is, if given access to a CNN-based AFR system, how could one utilize all of its model parameters to launch a successful adversarial attack?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A common approach is to utilize gradients of the whitebox AFR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Namely the attackers modify the image in the direction of the gradient of the loss function with respect to the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' There are two prevailing approaches to perform such gradient-based attacks: one-shot attacks, in which the attacker takes a single step in the direction of the gradient, and iterative attacks where instead of a single step, several steps are taken until we obtain a successful adversarial pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1 Fast Gradient Sign Method (FGSM) This method computes an adversarial image by adding a pixel-wide perturbation of magnitude in the direction of the gradient [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Under FGSM attack, we take a single step towards the direction of the gradient, and therefore, FGSM is very efficient in terms of computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Formally, given an input image x, we obtain an adversarial image xadv: xadv = x + ϵ · sign (▽xJ (x, y)) where, J is the loss function used to train the AFR system (typically, softmax cross entropy loss), and y is the ground truth class label of x (typically, the subject ID of the identity in x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' FGSM was first proposed for the object classification domain and therefore, utilizes softmax proba- bilities for crafting adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Therefore, the number of object classes are assumed to be known during training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, face recognition systems do not utilize the softmax layer for classification (as the number of identities are not fixed during deployment) instead features from the last fully connected layer are used for comparing face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We first modify FGSM appropriately in order to evade AFR systems rather than object classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Instead of considering the softmax cross-entropy loss as J, we craft a new loss function that models real-world scenario4: LfeatureMatch = 1 − Ex � F(x) · F(xadv) ||F(x)|| ||F(xadv)|| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4For brevity, we denote Ex ≡ Ex∈Pdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4 where, F is the matcher and F(x) is the feature representation of an input image x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The above feature matching loss function computes the cosine distance between a pair of images and ensures that the features between adversarial image xadv and input image x are as close as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Therefore, the gradient of the above loss ensures the features do not match and hence, can be considered as an obfuscation adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 1, we show the results of launching our modified FGSM attack on a state-of-the-art AFR system, namely ArcFace [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We see that with a single step and with minimal perturbations, the real and adversarial images of Tiger Woods does not match via ArcFace while humans can easily identity both images as pertaining to the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' (a) Real Input Image (b) Perturbation (c) FGSM [14] Figure 1: Adversarial face synthesized via FGSM [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A state-of-the-art face matcher, ArcFace [3], fails to match the adversarial and input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Cosine similarity score (∈ [−1, 1]) between the two images is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='27, while a score above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='36 (threshold @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% False Accept Rate) indicates that two faces are of the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 Projected Gradient Descent (PGD) An extreme case of white-box attacks is the PGD attack [17] where we assume that the attacker also has unlimited number of attempts to try and evade the deployed AFR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Unlike FGSM, PGD is an iterative attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' PGD attempts to find the perturbation δ that maximises the loss of a model on a particular input while keeping the size of the perturbation smaller than a specified amount referred to as ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We keep iterating until such a δ is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Similar to FGSM, we modify the loss function of PGD to fit the requirements of AFR system by again considering LfeatureMatch as the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 2 shows the results of PGD attack on ArcFace matcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Note that due to multiple iterations, PGD attack on AFR systems is more powerful (lower cosine similarity) but also more visible to humans as compared to the single-step FGSM attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' (a) Real Input Image (b) Perturbation (c) PGD [17] Figure 2: Adversarial face synthesized via PGD [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A state-of-the-art face matcher, ArcFace [3], fails to match the adversarial and input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Cosine similarity score (∈ [−1, 1]) between the two images is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='12, while a score above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='36 (threshold @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% False Accept Rate) indicates that two faces are of the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='3 Geometric Perturbations (GFLM) Prior efforts in crafting adversarial faces have also tried non-linear deformations as a natural method for evading AFR systems [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Non-linear deformations are applied by performing geometric warping to the input face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Unlike traditional adversarial perturbations that basically add an adversarial perturbation δ, authors in [48] propose a fast method of generating adversarial faces by altering the landmark locations of the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The resulting adversarial faces completely lie on the manifold of natural images, which makes it extremely difficult to detect any adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Results of geometrically warped adversarial faces are presented in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 5 (a) Real Input Image (b) Perturbation (c) GFLM Figure 3: Adversarial face synthesized via GFLM [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A state-of-the-art face matcher, ArcFace [3], fails to match the adversarial and input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Cosine similarity score (∈ [−1, 1]) between the two images is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='33, while a score above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='36 (threshold @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% False Accept Rate) indicates that two faces are of the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='4 Attribute-based Perturbations Unlike geometric-warping and gradient-based attacks that may perturb every pixel in the image, a few studies propose manipulating only salient regions in faces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=', eyes, nose, and mouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' By restricting perturbations to only semantic regions of the face, SemanticAdv [46] generates adversarial examples in a more controllable fashion by editing a single semantic aspect through attribute-conditioned image editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4 shows results from adversarial manipulating semantic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We can see while the attacks are indeed successful, it comes at the cost of altering the perceived identity as well as leads to degraded image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' (a) Real Input Image (b) Blond (c) Bangs (d) Mouth Open (e) Eyeglasses (f) Makeup Figure 4: Adversarial face synthesized via manipulating semantic attributes [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' All adversarial images (b-f) fail to match with the real image (a) via ArcFace [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4 AdvBiom: Learning to Synthesize Adversarial Attacks We find that majority of prior efforts on crafting adversarial attacks either degrade the visual quality where an observant human can still visually pick out the adversarial patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We also identify the following challenges with prior efforts: Gradient-based attacks rely on white-box settings where the entire deployed CNN-based AFR system is available to the attacker to compute its gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Geometrically-warping faces generally do not guarantee adversarial success and greatly distort the face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Semantic attribute manipulation can also degrade visual quality and may lead to greater conspicuous changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Instead, we propose to train a network to “learn" the salient regions of the face that can be perturbed to evade AFR systems in a semi-whitebox setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' These leads to the following advantages over prior efforts: 6 Perceptual Realism Given a large enough training dataset, a network can gradually learn to synthesize adversarial face images that are perceptually realistic such that a human observer can identify the image as a legitimate face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Higher Attack Success The faces can be learned to be perturbed in a manner such that they cannot be identified as the hacker (obfuscation at- tack) or automatically matched to a target subject (impersonation attack) by an AFR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Controllable The amount of perturbation can also be controllable by the attacker so that they can examine the success of the learning model as a function of amount of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Transferability Due to the semi-whitebox setting: once the network learns to generate the perturbed instances based on a single face recognition system, attacks can be transferred to any black-box AFR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We propose an automated adversarial biometric synthesis method, named AdvBiom, which generates an adversarial image for a probe face image and satisfies all the above requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1 Methodology Our goal is to synthesize a face image that visually appears to pertain to the target face, yet automatic face recognition systems either incorrectly matches the synthesized image to another person or does not match to target’s gallery images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom comprises of a generator G, a discriminator D, and face matcher (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Probe ℒ"#$ Synthesized + ℒ%&\'()%)* ℒ+\',)-,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='/)%0( Adversarial Mask 1 ℱ 3 Figure 5: Given a probe face image, AdvBiom automatically generates an adversarial mask that is then added to the probe to obtain an adversarial face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Generator The proposed generator takes an input face image, x ∈ X, and outputs an image, G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The generator is conditioned on the input image x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' for different input faces, we will get different synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Since our goal is to obtain an adversarial image that is metrically similar to the probe in the image space, x, it is not desirable to perturb all the pixels in the probe image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' For this reason, we treat the output from the generator as an additive mask and the adversarial face is defined as x + G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' If the magnitude of the pixels in G(x) is minimal, then the adversarial image comprises mostly of the probe x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Here, we denote G(x) as an “adversarial mask".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In order to bound the magnitude of the adversarial mask, we introduce a perturbation loss during training by minimizing the L2 norm5: Lperturbation = Ex [max (ϵ, ∥G(x)∥2)] (1) where ϵ ∈ [0, ∞) is a hyperparameter that controls the minimum amount of perturbation allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 5For brevity, we denote Ex ≡ Ex∈X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 7 In order to achieve our goal of impersonating a target subject’s face or obfuscating one’s own identity, we need a face matcher, F, to supervise the training of AdvBiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' For obfuscation attack,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' at each training iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom tries to minimize the cosine similarity between face embeddings of the input probe x and the generated image x + G(x) via an identity loss function: Lidentity = Ex[F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' x + G(x))] (2) For an impersonation attack,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom maximizes the cosine similarity between the face embeddings of a randomly chosen target’s probe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' and the generated adversarial face x + G(x) via: Lidentity = Ex[1 − F(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' x + G(x))] (3) The perturbation and identity loss functions enforce the network to learn the salient facial regions that can be perturbed minimally in order to evade automatic face recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Discriminator Akin to previous works on GANs [49, 50], we introduce a discriminator in order to encourage perceptual realism of the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We use a fully-convolution network as a patch-based discriminator [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Here, the discriminator, D, aims to distinguish between a probe, x, and a generated adversarial face image x + G(x) via a GAN loss: LGAN = Ex [log D(x)] + Ex[log(1 − D(x + G(x)))] (4) Finally, AdvBiom is trained in an end-to-end fashion with the following objectives: min D LD = −LGAN (5) min G LG = LGAN + λiLidentity + λpLperturbation (6) where λi and λp are hyper-parameters controlling the relative importance of identity and perturbation losses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Note that LGAN and Lperturbation encourage the generated images to be visually similar to the original face images, while Lidentity optimizes for a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' After training, the generator G can generate an adversarial face image for any input image and can be tested on any black-box face recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The overall algorithm describing the training procedure of AdvBiom can be found in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 Experimental Results Evaluation Metrics We quantify the effectiveness of the adversarial attacks generated by Ad- vBiom and other state-of-the-art baselines via (i) attack success rate and (ii) structural similarity (SSIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The attack success rate for obfuscation attack is computed as, Attack Success Rate = (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' of Comparisons < τ) Total No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' of Comparisons (7) where each comparison consists of a subject’s adversarial probe and an enrollment image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Here, τ is a pre-determined threshold computed at, say, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% FAR6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Attack success rate for impersonation attack is defined as, Attack Success Rate = (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' of Comparisons ≥ τ) Total No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' of Comparisons (8) Here, a comparison comprises of an adversarial image synthesized with a target’s probe and matched to the target’s enrolled image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We evaluate the success rate for the impersonation setting via 10-fold cross-validation where each fold consists of a randomly chosen target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Similar to prior studies [42], in order to measure the similarity between the adversarial example and the input face, we compute the structural similarity index (SSIM) between the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' SSIM is a normalized metric between −1 (completely different image pairs) to 1 (identical image pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 6For each face matcher, we pre-compute the threshold at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% FAR on all possible image pairs in LFW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=', threshold @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% FAR for ArcFace is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 8 Algorithm 1 Training AdvBiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' All experiments in this work use α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0001, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='5, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='9, λi = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0, λp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0, m = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We set ϵ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 (obfuscation), ϵ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 (impersonation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 1: Input 2: X Training Dataset 3: F Cosine similarity between an image pair obtained by biometric matcher 4: G Generator with weights Gθ 5: D Discriminator with weights Dθ 6: m Batch size 7: α Learning rate 8: for number of training iterations do 9: Sample a batch of probes {x(i)}m i=1 ∼ X 10: if impersonation attack then 11: Sample a batch of target images y(i) ∼ X 12: δ(i) = G((x(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' y(i)) 13: else if obfuscation attack then 14: δ(i) = G(x(i)) 15: end if 16: x(i) adv = x(i) + δ(i) 17: Lperturbation = 1 m ��m i=1 max � ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' ||δ(i)||2 �� 18: if impersonation attack then 19: Lidentity = 1 m ��m i=1 F � x(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' x(i) adv �� 20: else if obfuscation attack then 21: Lidentity = 1 m ��m i=1 � 1 − F � y(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' x(i) adv ��� 22: end if 23: LG GAN = 1 m ��m i=1 log � 1 − D(x(i) adv) �� 24: LD = 1 m �m i=1 � log � D(x(i)) � + log � 1 − D(x(i) adv) �� 25: LG = LG GAN + λiLidentity + λpLperturbation 26: Gθ = Adam(▽GLG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Gθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' β2) 27: Dθ = Adam(▽DLD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' β2) 28: end for Datasets We train AdvBiom on CASIA-WebFace [51] and then test on LFW [52]7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' CASIA-WebFace [51] is comprised of 494,414 face images belonging to 10,575 different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We removed 84 subjects that are also present in LFW and the testing images in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' LFW [52] contains 13,233 web-collected images of 5,749 different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In order to compute the attack success rate, we only consider subjects with at least two face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' After this filtering, 9,614 face images of 1,680 subjects are available for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' All the testing images in this paper have no identity overlap with the training set, CASIA- WebFace [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Data Preprocessing All face images are passed through MTCNN face detector [53] to detect five landmarks (two eyes, nose, and two mouth corners).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Via similarity transformation, the face images are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' After transformation, the images are resized to 160 × 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Prior to training and testing, each pixel in the RGB image is normalized by subtracting 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='5 and dividing by 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Experimental Settings We use ADAM optimizers in Tensorflow with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='5 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='9 for the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Each mini-batch consists of 32 face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We train AdvBiom for 200,000 steps with a fixed learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Since our goal is to generate adversarial faces with high success 7Training on CASIA-WebFace and evaluating on LFW is a common approach in face recognition literature [3, 4] 9 rate, the identity loss is of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We empirically set λi = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 and λp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We train two separate models and set ϵ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 and ϵ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 for obfuscation and impersonation attacks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Gallery Probe Proposed AdvBiom GFLM [48] PGD [17] FGSM [14] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='02 (a) Obfuscation Attack Target’s Gallery Target’s Probe Probe Proposed AdvBiom A3GN [42] FGSM [14] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='42 (b) Impersonation Attack Figure 6: Adversarial face synthesis results on LFW dataset in (a) obfuscation and (b) impersonation attack settings (cosine similarity scores obtained from ArcFace [3] with threshold @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% FAR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The proposed method synthesizes adversarial faces that are seemingly inconspicuous and maintain high perceptual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Architecture Let c7s1-k be a 7 × 7 convolutional layer with k filters and stride 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' dk denotes a 4 × 4 convolutional layer with k filters and stride 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Rk denotes a residual block that contains two 3 × 3 convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' uk denotes a 2× upsampling layer followed by a 5 × 5 convolutional layer with k filters and stride 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We apply Instance Normalization and Batch Normalization to the generator and discriminator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We use Leaky ReLU with slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 in the discriminator and ReLU activation in the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The architectures of the two modules are as follows: Generator: c7s1-64,d128,d256,R256,R256,R256, u128, u64, c7s1-3 Discriminator: d32,d64,d128,d256,d512 A 1 × 1 convolutional layer with 3 filters and stride 1 is attached to the last convolutional layer of the discriminator for the patch-based GAN loss LGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We apply the tanh activation function on the last convolution layer of the generator to ensure that the generated image ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In the paper, we denoted the output of the tanh layer as an “adversarial mask”, G(x) ∈ [−1, 1] and x ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The final adversarial image is computed as 10 Obfuscation Attack Proposed AdvBiom GFLM [48] PGD [17] FGSM [14] Attack Success Rate (%) @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% FAR FaceNet [5] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='67 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='34 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='70 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='96 SphereFace [4] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='22 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='49 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='34 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='71 ArcFace [3] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='53 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='43 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='30 COTS-A 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='98 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='89 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='74 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='48 COTS-B 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='71 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='05 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='49 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 Structural Similarity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='06 Computation Time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='22 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='03 Impersonation Attack Proposed AdvBiom A3GN [42] PGD [17] FGSM [14] Attack Success Rate (%) @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1% FAR FaceNet [5] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='40 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='19 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='26 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='12 SphereFace [4] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='27 07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='19 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='39 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='03 ArcFace [3] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='44 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='29 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='95 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='21 COTS-A 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='35 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='30 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='10 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='08 COTS-B 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='28 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='24 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='16 Structural Similarity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 Computation Time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='04 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='03 White-box matcher (used for training) Black-box matcher (never used in training) Table 1: Attack success rates and structural similarities between probe and gallery images for obfus- cation and impersonation attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Attack rates for obfuscation comprises of 484,514 comparisons and the mean and standard deviation across 10-folds for impersonation reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The mean and standard deviation of the structural similarities between adversarial and probe images along with the time taken to generate a single adversarial image (on a Quadro M6000 GPU) also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' xadv = 2 × clamp � G(x) + � x+1 2 ��1 0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' This ensures G(x) can either add or subtract pixels from x when G(x) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' When G(x) → 0, then xadv → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Face Recognition Systems For all our experiments, we employ 5 state-of-the-art face matchers8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Three of them are publicly available, namely, FaceNet [5], SphereFace [4], and ArcFace [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We also report our results on two commercial-off-the-shelf (COTS) face matchers, COTS-A and COTS-B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We use FaceNet [5] as the white-box face recognition model, F, during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' All the testing images in this paper are generated from the same model (trained only with FaceNet) and tested on different matchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1 Comparison with Prevailing Adversarial Face Generators We compare our adversarial face synthesis method with state-of-the-art methods that have specifi- cally been implemented or proposed for faces, including GFLM [48], PGD [17], FGSM [14], and A3GN [42]10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In Table 1, we find that compared to the state-of-the-art, AdvBiom generates adversarial faces that are similar to the probe 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Moreover, the adversarial images attain a high obfuscation attack success rate on 4 state-of-the-art black-box AFR systems in both obfuscation and impersonation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom learns to perturb the salient regions of the face, unlike PGD [17] and FGSM [14], which alter every pixel in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' GFLM [48], on the other hand, geometrically warps the face images and thereby, results in low structural similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In addition, the state-of-the-art matchers are robust to such geometric deformation which explains the low success rate of GFLM on face matchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A3GN, another GAN-based method, however, fails to achieve a reasonable success rate in an impersonation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 8All the open-source and COTS matchers achieve 99% accuracy on LFW under LFW protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 9Both COTS-A and COTS-B utilize CNNs for face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' COTS-B is one of the top performers in the NIST Ongoing Face Recognition Vendor Test (FRVT) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 10We train the baselines using their official implementations (detailed in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 Ablation Study In order to analyze the importance of each module in our system, in Figure 7, we train three variants of AdvBiom for comparison by removing the discriminator (D), perturbation loss Lperturbation, and identity loss Lidentity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Input w/o D w/o Lprt w/o Lidt with all Figure 7: Variants of AdvBiom trained without the discriminator, perturbation loss, and identity loss, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Every component of AdvBiom is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The discriminator helps to ensure the visual quality of the synthesized faces are maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' With the generator alone, undesirable artifacts are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Without the proposed perturbation loss, perturbations in the adversarial mask are unbounded and therefore, leads to a lack in perceptual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The identity loss is imperative in ensuring an adversarial image is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Without the identity loss, the synthesized image cannot evade state-of-the-art face matchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We find that every component of AdvBiom is necessary in order to obtain an adversarial face that is not only perceptually realistic but can also evade state-of-the-art face matchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='3 What is AdvBiom Learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Via Lperturbation, during training, AdvBiom learns to perturb only the salient facial regions that can evade the face matcher, F (FaceNet [5] in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In Figure 8, AdvBiom synthesizes the adversarial masks corresponding to the probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We then threshold the mask to extract pixels with perturbation magnitudes exceeding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It can be inferred that the eyebrows, eyeballs, and nose contain highly discriminative information that an AFR system utilizes to identify an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Therefore, perturbing these salient regions are enough to evade state-of-the-art face recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='4 Transferability of AdvBiom In Table 1, we find that attacks synthesized by AdvBiom when trained on a white-box matcher (FaceNet), can successfully evade 5 other face matchers that are not utilized during training in both obfuscation and impersonation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In order to investigate the transferability property of AdvBiom, we extract face embeddings of real images and their corresponding adversarial images, under the obfuscation setting, via the white-box matcher (FaceNet) and a black-box matcher (ArcFace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In total, we extract feature vectors from 1,456 face images of 10 subjects in the LFW dataset [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In Figure 9, we plot the correlation heatmap between face features of real images, their corresponding adversarial masks and adversarial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' First, we observe that face embeddings of real images extracted by FaceNet and ArcFace are correlated in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' This indicates that both matchers extract features with related pairwise correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Consequently, perturbing salient features for FaceNet can lead to high attack success rates for ArcFace as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The similarity among the correlation distributions of both matchers can also be observed when adversarial masks and adversarial images are input to the matchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' That is, receptive fields for automatic face recognition systems attend to similar regions in the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 12 Probe Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Mask Visualization Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Image 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='26 Figure 8: State-of-the-art face matchers can be evaded by slightly perturbing salient facial regions, such as eyebrows, eyeballs, and nose (cosine similarity obtained via ArcFace [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Figure 9: Correlation between face features extracted via FaceNet and ArcFace from 1,456 images belonging to 10 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' To further illustrate the distributions of the embeddings of real and synthesized images, we plot the 2D t-SNE visualization of the face embeddings for the 10 subjects in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The identity clusterings can be clearly observed from both real and adversarial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In particular, the adversarial counterpart of each subject forms a new cluster that draws closer to the adversarial clusterings of other subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' This shows that AdvBiom perturbs only salient pixels related to face identity while maintaining a semantic meaning in the feature space, resulting in a similar manifold of synthesized faces to that of real faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='5 Controllable Perturbation The perturbation loss, Lperturbation is bounded by a hyper-parameter, ϵ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=', the L2 norm of the adversarial mask must be at least ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Without this constraint, the adversarial mask becomes a blank image with no changes to the probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' With ϵ, we can observe a trade-off between the attack success rate and the structural similarity between the probe and synthesized adversarial face (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A higher ϵ leads to less perturbation restriction, resulting in a higher attack success rate at the cost of a lower structural similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' For an impersonation attack, this implies that the adversarial image may 13 Real Image Adversarial Mask Adversarial Image 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='8 FaceNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 ArcFace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='4FaceNet Real Image Adversarial Image (Obfuscation) ArcFace Figure 10: 2D t-SNE visualization of face representations extracted via FaceNet and ArcFace from 1,456 images belonging to 10 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' contain facial features from both the hacker and the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In our experiments, we chose ϵ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 and ϵ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 for impersonation and obfuscation attacks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4 6 8 10 12 14 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='69 5 13 21 39 52 60 66 Hyper-parameter (ε) Success Rate (%) Structural Similarity ε = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 ε = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 ε = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 ε = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 Figure 11: Trade-off between attack success rate and structural similarity for impersonation attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='6 Attacks via AdvBiom Beyond Faces We now show that the AdvBiommethod, coupled with the proposed Minutiae Displacement and Distortion Modules, can be extended to effectively generate adversarial fingerprints which are visually similar to corresponding probe fingerprints while evading two state-of-the-art COTS fingerprint matchers as well as a deep network-based fingerprint matcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 14 FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='97 FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='92 (a) Enrolled Mate VS: 235 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='96 VS: 172 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='99 (b) Input Probe VS: 31 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='92 VS: 10 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='92 (c) AdvBiom VS: 134 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='96 VS: 104 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='96 (d) DeepFool [16] VS: 139 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='95 VS: 104 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='96 (d) PGD [17] Figure 12: Example probe and corresponding mate fingerprints along with synthesized adversarial probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' (a) Two example mate fingerprints from NIST SD4 [55], and (b) the corresponding mates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Adversarial probe fingerprints using different approaches are shown in: (c) proposed synthesis method, AdvBiom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' (d-e) state-of- the-art methods, DeepFool and PGD respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' VeriFinger v11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 match score (probe v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' mate) - VS, and the fingerprintness score (degree of similarity of a given image to a fingerprint pattern) - FS ∈ [0,1] [56], which ranges from 1 (the highest) to 0 (the lowest), are given below each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A VS of above 48 (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR) indicates a successful match between the probe and the mate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The proposed attack AdvBiom successfully evades COTS and deep network-based matchers, while maintaining visual fingerprint perceptibility and high fingerprintness scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Grosz et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' al [57] showed that random minutiae position displacements and non-linear distortions drastically affected the performance of COTS fingerprint matchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom builds upon these two perturbations and when given a probe fingerprint, can synthesize an adversarial fingerprint image that retains all of the original fingerprint attributes except the identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' a fingerprint recognition system should not match the adversarial fingerprint to the probe fingerprint (obfuscation attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Figure 13 shows the schematic of AdvBiom conditioned for fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The following subsections explain the major components of the approach in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Minutiae Displacement Module While the authors in [57] showed the effectiveness of random minutiae position displacements on COTS matchers, they studied the effect of this perturbation by directly modifying the minutiae template instead of the fingerprint image (pixel space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, it may be difficult to obtain the minutiae template of a given fingerprint image using COTS minutiae extractors rather than the source fingerprint image itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Thus, we propose a minutiae displacement module Gdisp which, given a fingerprint image, displaces its minutiae points in random directions by a predefined distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' To extract minutiae points from a fingerprint image, we employ a minutiae map extractor (M) from [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' For a fingerprint image of width w and height h, M outputs a 12 channel heat map H ∈ Rh×w×12, where if H(i,j,c), value of the heat map at position (i,j) and channel c, is greater than a threshold mt and is the local maximum in its 5 × 5 × 3 neighboring cube, a minutiae is marked at (i,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The minutiae direction θ is calculated by maximising the quadratic interpolation with respect to: f � (c − 1) × π 6 � = H (i, j, (c − 1)%12) (9) f � c × π 6 � = H(i, j, c) (10) f � (c + 1) × π 6 � = H(i, j, (c + 1)%12) (11) Figure 14 shows a fingerprint image and its corresponding 12 channel minutiae map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Once M extracts a minutiae map Hprobe from the input probe fingerprint x, we detect minutiae points by applying a threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 on Hprobe and finding closed contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Each detected contour, at say 15 𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝 Extractor (ℳ) 𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑡𝑜𝑟 (𝒟) GAN Loss ℒgan 𝑀𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 Module (𝒢𝑑𝑖𝑠𝑝) Probe Fingerprint Displaced Fingerprint Original M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='Map 𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑃𝑖𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Displacement Target M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='Map M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='Map Sim Loss ℒmmap_sim Pixel Loss ℒpixel Predicted M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='Map M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='Map Dis Loss ℒmmap_dis 𝐷𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛 Module (𝒢𝑑𝑖𝑠𝑡) 𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝 Extractor (ℳ) Adversarial Fingerprint Figure 13: Schematic of AdvBiom for generating adversarial fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Given a probe fingerprint image, it is passed to Gdisp which randomly displaces its minutiae points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The distortion module (Gdist) identifies control points on the displaced fingerprint and non-linearly distorts the image to output the adversarial fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The solid black arrows show the forward pass of the network while the dotted black arrows show the propagation of the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' location (i, j), is displaced by a predefined L1 distance d = |∆i|+|∆j|, giving us the target minutiae map Htarget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 0 𝜋/6 𝜋/3 𝜋/2 2𝜋/3 5𝜋/6 𝜋 7𝜋/6 4𝜋/3 3𝜋/2 5𝜋/3 11𝜋/6 Figure 14: The 12 channel minutiae map of an example fingerprint image shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The minutiae points (shown in red) are marked by a COTS minutiae extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The bright spots in each channel image indicate the spatial location of minutiae points while the kth channel (k ∈ [0, 11]) indicate the contributions of minutiae points to the kπ/6 orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The minutiae displacement module Gdisp is essentially an autoencoder conditioned on the probe fingerprint x and the target minutiae map Htarget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It learns to generate a displaced fingerprint xdisp whose predicted minutiae map Hpred is as close as possible to the target minutiae map Htarget in the pixel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' To achieve this, we have three losses that govern Gdisp: Lmmap_sim = ||Htarget − Hpred||1 (12) Lmmap_dis = 1 ||Htarget − Hprobe||1 (13) , where Lmmap_sim is the minutiae map similarity loss which minimises the distance between the predicted and target minutiae map, while the minutiae map dissimilarity loss Lmmap_dis maximises 16 G QU Q Q QQ Q Gthe distance between the predicted and probe minutiae map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In figure 15, we show two example probe fingerprints and their corresponding displaced fingerprints after passing through Gdisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Distortion Module One of the most noteworthy conclusions from [57] was that non-linear distor- tions to minutiae points was one of the most successful perturbations to lower the similarity scores between perturbed and corresponding unperturbed fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Again, the non-linear distortion was applied to all the minutiae points in the template and not to the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Thus, our next step in generating adversarial fingerprints consists of a distortion module Gdist which learns to distort salient points in a fingerprint image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The architecture of Gdist consists of an encoder conditioned on the input probe fingerprint x and the target minutiae map Hprobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The output from the encoder is a predefined number of control points11 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The non-linear distortion model proposed in [59], learned using a thin plate spline (TPS) model [60] from 320 already distorted fingerprint videos, was employed to calculate the displacements of the predicted control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The hyper-parameter σ is used to indicate the extent of the distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The control points and their displacements are then fed to a differentiable warping module [61] to get the resultant adversarial fingerprint xadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' To limit the magnitude of non-linear distortion and to ensure that xdisp and xadv are close to the probe fingerprint x, we introduce pixel loss between the image pairs (x, xdisp) and (x, xadv): Lpixel = 1 n � i,j |xi,j − xdispi,j|+ 1 n � i,j |xi,j − xadvi,j| (14) Figure 16 shows two displaced fingerprints and their corresponding output from Gdist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Discriminator In order to guide the generative modules Gdisp and Gdist to synthesize realistic fingerprint images, we introduce a fully convolutional network as a patch-based discriminator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The job of the discriminator is to distinguish between real fingerprint images x and the generated adversarial fingerprint images xadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' This is accomplished through the GAN loss: Lgan = logD(x) + log(1 − D(xadv)) (15) The proposed approach AdvBiom is trained in an end-to-end manner with respect to the following objective function: (16) L = Lgan + λmmap_simLmmap_sim + λmmap_disLmmap_dis + λpixelLpixel where the hyper-parameters λmmap_sim, λmmap_dis, and λpixel denote the relative importance of their respective losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Once trained, AdvBiom can generate an adversarial fingerprint image for any input probe fingerprint and can be tested on any fingerprint matcher regardless of the feature extraction method (minutiae or deep-features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 11Control points are points in an image to which non-linear distortion is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Probe Fingerprint Displaced Fingerprint Probe Fingerprint Displaced Fingerprint Figure 15: Example probe fingerprints from NIST SD4 [55] and their corresponding output from the minutiae displacement module Gdisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The minutiae points (shown in red) are marked using a COTS minutiae extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 17 Displaced Fingerprint Distorted Fingerprint Displaced Fingerprint Distorted Fingerprint Figure 16: Fingerprints in the left column are example displaced fingerprints from Gdisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The distortion module Gdist predicts control points (marked in blue) and distorts the images based on their displacements (red arrows) using the non-linear distortion model from [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The resultant distorted fingerprint images are shown in the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Successful Attacks Failed Attacks Original Probe Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Probe (AdvFinge) Mate Original Probe Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Probe (AdvFinge) Mate VS: 36 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='82 VS: 47 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='88 VS: 109 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='87 VS: 76 | FS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='89 (AdvBiom) (AdvBiom) Figure 17: Example successful and failed adversarial fingerprints attack using AdvBiom on NIST SD4 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The VeriFinger matching scores (probe v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' mate): VS, and fingerprintness [56] scores: FS, of adversarial probes are shown below their respective triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Note that the VeriFinger matching threshold is 48 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Evaluation Metrics: The requirement of a good adversarial fingerprints generator is to evade fingerprint matchers while preserving fingerprint attributes and being model-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Thus, in order to quantify the performance of adversarial attacks generated by AdvBiom and other state-of-the-art baselines, we employ the following evaluation metrics: True Accept Rate (TAR): The extent to which an adversarial attack can evade a fingerprint matcher is measured by the drop in TAR at an operational setting, say 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% False Accept Rate (FAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Fingerprintness: Soweon and Jain [56] proposed a domain-specific metric called finger- printness to measure the degree of similarity of a given image to a fingerprint pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Fingerprintness ranges from [0,1] where higher the score, higher the probability of the pattern in the image corresponding to a fingerprint pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' NFIQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0: Lastly, we use NFIQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 [62] quality scores to evaluate the fingerprint quality of adversarial fingerprint images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' NFIQ scores range from [0,100] where a score of 100 depicts the highest fingerprint quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Note that since non-linear distortions change the structure of the image, using the structural similarity index (SSIM) metric is inappropriate as it essentially measures the local change in structures of the image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Datasets: We train AdvBiom on an internal dataset of 120,000 rolled fingerprint images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Furthermore, we evaluate the performance of the proposed fingerprint adversarial attack and other baselines on: 2,000 fingerprint pairs from NIST SD4 [55] 27,000 fingerprint pairs from NIST SD14 [63] 18 111@ 8 Q Q QQ Q de G G母 Q % d TG % Q Qd Q d 00 & QQ % CQ d 3 d & QQQ @山 G %Q QQ Q q Q bu G P dppdQ @ d QQ- Q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' & d & ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' de qQ & GQ QC QQ Q Q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 多 & G lG Q QQ % Q @ G QQQQ G G20- Q Q Q q QQ G Q G8 a QQ QQ cs G Q G QQ 0甲Accuracy Adversarial Attacks Original Probes FGSM I-FGSM Deep Fool PGD Adv Biom TAR (%) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR NIST SD4 VeriFinger 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='05 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='20 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='30 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='60 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25 Innovatrics 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='65 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='35 DeepPrint 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='55 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='20 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='15 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='40 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='35 NIST SD14 VeriFinger 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='42 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='20 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='30 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='60 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='67 Innovatrics 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='24 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='84 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='68 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='32 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='69 DeepPrint 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='52 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='70 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='48 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='44 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='28 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='42 FVC 2004 DB1 A VeriFinger 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='89 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='60 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='53 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='92 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='69 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='31 Innovatrics 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='15 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='36 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='68 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='32 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='52 DeepPrint 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='36 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='22 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='87 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='39 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='62 Table 2: True Accept Rate (TAR) @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR of AdvBiom along with state-of-the-art baselines attacks on three datasets - NIST SD4 [55], NIST SD14 [63], and FVC 2004 DB1 A [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 2 COTS fingerprint matchers - VeriFinger v11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 [65] and Innovatrics v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='627 [66], and a deep network-based matcher DeepPrint [67] were employed for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It is observed that DeepPrint, a deep network-based matcher, is susceptible to all types of adversarial attacks while VeriFinger and Innovatrics are more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 558 fingerprints from DB1 A of FVC 2004 [64], consisting of 1,369 genuine pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Experimental Settings: AdvBiom was trained using the Adam optimizer with β1 as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='5 and β2 as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The hyper-parameters were empirically set to λmmap_sim = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='05, λmmap_dis = 500000, and λpixel = 1000 for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Based on the conclusions drawn in [57], d, c, and λ were set to 20, 16, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 respectively for optimal effectiveness against fingerprint matchers while ensuring fingerprint realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom was trained for 16,000 steps using Tensorflow r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 on an Intel Core i7-11700F @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50GHz CPU with a RTX 3070 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' On the same machine, AdvBiom can synthesize an adversarial fingerprint within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='35 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Fingerprint Authentication Systems: Since AdvBiom is a black-box attack, we do not require any fingerprint authentication system while training the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, we evaluate AdvBiom and other baseline attacks on two COTS fingerprint matchers and one deep network-based matcher: VeriFinger v11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 [65] Innovatrics v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='627 [66] DeepPrint [67] Comparison with Prevailing Fingerprint Adversarial Generators We show the performance of our method AdvBiom as compared to other state-of-the-art attacks in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We observe that the TAR of two COTS and a deep network-based fingerprint matcher for the aforementioned three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It is to note that all the baseline attacks [14, 15, 17, 16] are white-box attacks and were trained using DeepPrint [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It is evident from Table 2 that AdvBiom is the most successful attack on COTS matchers VeriFinger and Innovatrics, and is also able to effectively evade a deep network-based fingerprint matcher, namely DeepPrint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It can also be observed that while COTS fingerprint matchers are robust to most adversarial attacks, DeepPrint is very susceptible to the same attacks since it heavily relies on the texture of the fingerprint which is majorly affected by adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' A successful adversarial attack should not only evade fingerprint matchers but should also preserve fingerprint attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In order to observe the effect of adversarial attacks on fingerprint pattern in images, we plot the fingerprintness [56] distribution of 2,000 probes from NIST SD4 [55] for AdvBiom as well as for other baseline attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Since all the state-of-the-art baselines essentially add noise to each pixel in the image, they do not change the structure of the fingerprint and thus do not 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 Fingerprintedness Scores 0 2 4 6 8 10 12 Probability of occurence Original Probes ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='91) AdvFinge ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='86) FGSM ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='89) I-FGSM ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='91) PGD ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='90) DeepFool ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='90) AdvBiom Figure 18: Fingerprintness [56] distribution of 2,000 probes from NIST SD4 with respect to AdvBiom and other state-of-the-art baselines attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 0 20 40 60 80 100 NFIQ Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='025 Probability of occurence Original Probes ( = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='70) AdvFinge ( = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='46) FGSM ( = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='30) I-FGSM ( = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='23) PGD ( = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='28) DeepFool ( = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='42) AdvBiom Figure 19: NFIQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 [62] quality scores distribution of 2,000 probes from NIST SD4 [55] with respect to AdvBiom and other baselines attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' affect fingerprintness scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom , on the other hand, displaces minutiae points and non-linearly distorts the image, and still maintains a high mean fingerprintness score of µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Furthermore, we also compute the NFIQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 [62] quality scores distribution (figure 19) of the original and adversarial probes from NIST SD4 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' As shown in figure 12, baseline attacks tend to minutely perturb image pixels to generate adversarial fingerprints and as a result do not have much of an effect on the quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom , on the other hand, provides an optimal solution by successfully attacking fingerprint matchers while maintaining high fingerprintness and NFIQ scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Genuine and Imposter Scores Distribution To determine the effect of adversarial fingerprint on both genuine and imposter pairs, we plot the genuine and imposter scores distribution of NIST SD4 [55] in figure 20 before and after applying AdvBiom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' We computed a total of 2,000 genuine and 20,000 imposter scores for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It can be observed that the genuine scores drastically decrease and shift to the left of the axis as their mean drops from 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='87 to 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='55 after the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' However, the imposter scores remain unaffected with the mean imposter score changing by only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 0 100 200 300 400 Matching Scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='14 Probability of Occurence Genuine Scores | Before Attack ( = 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='88) Genuine Scores | After Attack ( = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='55) Imposter Scores | Before Attack ( = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00) Imposter Scores | After Attack ( = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='52) 48: Matching Threshold at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR (a) Using VeriFinger SDK [65] (b) Using Innovatrics SDK [66] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 Matching Scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='12 Probability of Occurence Genuine Scores | Before Attack ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='94) Genuine Scores | After Attack ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='78) Imposter Scores | Before Attack ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='10) Imposter Scores | After Attack ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='09) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='837: Matching Threshold at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR (c) using DeepPrint [67] Figure 20: Genuine and imposter scores distribution of NIST SD4 [55] before and after the adversarial attack AdvBiom using three state-of-the-art fingerprint matchers - VeriFinger v11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 [65], Innovatrics v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='627 [66], and DeepPrint [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Here, µ refers to the mean of the scores distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' In all the three cases, the genuine scores shift towards the left while the imposter scores do not get affected by the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Is AdvBiom Biased Towards Certain Fingerprint Types?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The generated adversarial fingerprint from AdvBiom is conditioned on the input probe fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Thus, it is essential to check if there is a relation between the amount of perturbation applied and the fingerprint type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' The confusion matrix for the five fingerprint types (left loop, right loop, whorl, arch, tented arch) before and after applying AdvBiom on the 2,000 probes of NIST SD4 [55] is shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Note that we use NIST SD4 for this evaluation since it has a uniform number of fingerprint images per each type (400 fingerprints per type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' It is evident from the table that all five fingerprint types are almost equally susceptible to the attack, and thus the attack crafted AdvBiom is not biased towards a particular fingerprint type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 20 Genuine Scores LBefore Attack (u = 590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00) Genuine Scores 1After Attack (μu = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='72) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='6 Imposter Scores /Before Attack (μu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='34) Imposter Scores l After Attack (μ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='4o) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='5 Probability of Occurence 40:Matching Threshold at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='0 200 600 0 400 800 1000 Matching Scores0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='000 0 200 400 600 800 1000Before Attack After Attack TAR L: 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75% R: 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25% W: 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50% T: 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25% A: 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50% FAR L: 0% R: 0% W: 0% T: 0% A: 0% FRR L: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25% R: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75% W: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50% T: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75% A: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50% TRR L: 100% R: 100% W: 100% T: 100% A: 100% TAR L: 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00% R: 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50% W: 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='75% T: 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00% A: 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00% FAR L: 0% R: 0% W: 0% T: 0% A: 0% FRR L: 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00% R: 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='50% W: 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='25% T: 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00% A: 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='00% TRR L: 100% R: 100% W: 100% T: 100% A: 100% Table 3: Confusion matrix for five fingerprint types (left loop: L, right loop: R, whorl: W, tented arch: T, arch: A) from NIST SD4 [55] before and after the adversarial attack using AdvBiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Here, TAR = True Accept Rate, FAR = False Accept Rate, FRR = False Reject Rate, and TRR = True Reject Rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Note that the matching threshold was 48 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content='01% FAR using the COTS fingerprint matcher VeriFinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' AdvBiom is not biased towards any fingerprint type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' 5 Conclusions We show that a new method of adversarial synthesis, namely AdvBiom, that automatically generates adversarial face images with imperceptible perturbations evading state-of-the-art biometric matchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' With the help of a GAN, and the proposed perturbation and identity losses, AdvBiom learns the set of pixel locations required by face matchers for identification and only perturbs those salient facial regions (such as eyebrows and nose).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Once trained, AdvBiom generates high quality and perceptually realistic adversarial examples that are benign to the human eye but can evade state-of-the-art black- box face matchers, while outperforming other state-of-the-art adversarial face methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Beyond faces, we show for the first time that such a method with the proposed Minutiae Displacement and Distortion Modules can also evade state-of-the-art automated fingerprint recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' Rumelhart and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf'} +page_content=' L.' 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+1,2027 @@ +On the motion of an electron through vacuum fluctuations +Anirudh Gundhi1, 2, ∗ and Angelo Bassi1, 2, † +1Department of Physics, University of Trieste, Strada Costiera 11, 34151 Trieste, Italy +2Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy +(Dated: January 31, 2023) +We study the effects of the electromagnetic vacuum on the motion of a non-relativistic electron. +To this end, the vacuum is treated as the environment and the electron as the system within the +framework of open quantum systems. +After tracing over the environmental degrees of freedom, +we obtain the time evolution of the reduced density matrix of the electron in the position basis. +Using the master equation, in the first part of the article we derive the equation of motion for the +expectation value of the position operator. In the presence of an external potential, the equation +turns out to be the same as its classical counterpart: the Abraham-Lorentz equation. However, in +its absence, the dynamics is free of the runaway solution. In the second part of the article we study +decoherence induced by vacuum fluctuations. We show that decoherence that appears at the level +of the reduced density matrix does not correspond to actual irreversible loss of coherence. +Numerous physical phenomena such as the Casimir ef- +fect [1–3], the Unruh effect [4–6] and the Lamb shift [7– +10] are attributed to the presence of vacuum fluctuations. +The possibility of decoherence due to vacuum fluctua- +tions, as being fundamental and unavoidable, has also +been discussed in various works [11–18] without arriving +at a general consensus. +The interaction of an electron with the vacuum fluctu- +ations can be studied within the framework of open quan- +tum systems. We use this formalism to study two spe- +cific phenomena. First, we derive the equation of motion +(EOM) for the electron in the presence of an external po- +tential that provides a quantum mechanical description +of radiation emission by an accelerated electron. +Sec- +ond, we investigate if the interactions with the vacuum +fluctuations alone can lead to spatial decoherence of the +electron. +The quantum mechanical version of the classical +Abraham-Lorentz (AL) equation, which describes the re- +coil force experienced by an accelerated electron due to +the emission of radiation [19–22], has been previously +derived, for example, in [10]. Instead of the electron’s +position, the equation was obtained for the position op- +erator and it was then argued why this operator equa- +tion is fundamentally different from the classical one. +The difficulties in making a direct connection with the +classical dynamics were attributed to the presence of the +additional transverse electric field operator of the electro- +magnetic vacuum, which is zero classically. Similar prob- +lem persists concerning the interpretation of the quantum +Langevin equation obtained in [17] for an electron inter- +acting with vacuum fluctuations. +In our work, we use the path-integral formalism to ob- +tain the explicit expression of the reduced density matrix +in the position basis. The formalism used is adopted from +[23]. +Within this framework, instead of the Langevin +∗ anirudh.gundhi@phd.units.it +† abassi@units.it +equation, we derive the master equation which yields the +EOM for the expectation value of the position operator +which provides a direct correspondence with the classical +dynamics. In the presence of an arbitrary potential, we +show that the classical EOM is the same as the one ob- +tained from the reduced quantum dynamics. Moreover, +the equation that emerges after a quantum mechanical +treatment appears to be free of the problems associated +with the AL equation: the existence of the runaway so- +lution which leads to an exponential increase of the elec- +tron’s acceleration, even in the absence of an external +potential [19–21]. +Concerning decoherence, we show that the loss of co- +herence due to vacuum fluctuations at the level of the re- +duced density matrix is only apparent and reversible. To +this end we show that by ‘switching off’ the interactions +with the EM field, the original coherence is restored at +the level of the system. Moreover, the expression for the +decoherence factor that we obtain differs from the ones +obtained in [17, 18] where the authors argue for a finite +loss of coherence for momentum superpositions, due to +vacuum fluctuations, but with different estimates for the +magnitude of decoherence. +The action. We work in the Coulomb gauge in which the +Lagrangian relevant for the dynamics of a non-relativistic +electron in the presence of an external potential and an +external radiation field is given by [24] +L(t) = 1 +2m˙r2 +e − V0(re) + +� +d3rLEM − ereE⊥(re) . +(1) +Here, re denotes the position of the electron, m the +bare mass, e the electric charge, V0(re) an arbitrary +bare external potential (acting only on the electron) and +LEM := (ϵ0/2) +� +E2 +⊥(r) − c2B2(r) +� +in which E⊥ denotes +the transverse electric field, B the magnetic field, ϵ0 the +permittivity of free space and c the speed of light. As +detailed in Appendix A, Eq. (1) is obtained from the +general Lagrangian for electrodynamics under the non- +relativistic approximation. +Following the standard prescription, the EM field is +quantized by quantizing the transverse vector potential +arXiv:2301.11946v1 [quant-ph] 27 Jan 2023 + +2 +ˆA⊥. In terms of its conjugate momentum ˆΠ (which is +not proportional to E⊥ due to the form of the interaction +term in Eq. (1), c.f. Appendix A), we define and work +with ˆΠE = − ˆΠ/ϵ0, since it appears repeatedly in the +calculations. Further, the quantized EM field is initially +assumed to be in its vacuum state. +The master equation via path integral formalism. +The +position basis representation of the full density matrix +within the path integral formalism is given by [23, 25] +⟨x′ +f| ˆρ(t) |xf⟩ = +� +D[x, x′]e +i +ℏ (S′ +T−ST)ρ(x′ +i, xi, ti) . +(2) +Eq. (2) describes the density matrix at some final time t, +starting from an initial time ti, such that xi := x(ti), +x′ +i := x′(ti), with S′ +T := ST[x′] (and similarly ST := +ST[x]) denoting the full action describing some general +dynamics along the x-axis. The path integral in Eq. (2) +is computed with the boundary conditions xf = x(t), +x′ +f = x′(t), and includes the integral over xi and x′ +i. +In our case, the quantized radiation field is treated as +the environment, initially assumed to be in its vacuum +state and the electron as the system. We are interested +in the reduced effective dynamics of the electron having +taken into account its interaction with the environment. +This is described by the reduced density matrix ˆρr which +is obtained after tracing over the environmental degrees +of freedom. After performing the trace, by assuming the +initial density matrix to be in the product state ˆρ(ti) = +ˆρS(ti) ⊗ ˆρEM(ti), ˆρr takes the form [23] (c.f. Appendix B) +ρr(x′ +f, xf, t) = +� +D[x, x′]e +i +ℏ (S′ +S−SS+SIF[x,x′])ρr(x′ +i, xi, ti) , +with, SIF = 1 +2 +� t +ti +dt1dt2xa(t1)Mab(t1; t2)xb(t2) . +(3) +Here, SS denotes the action corresponding to the sys- +tem Hamiltonian (c.f. Appendix A) and, under the Ein- +stein summation convention, we have introduced the vec- +tor notation xa(t1) = x(t1) for a = 1 and xa(t1) = x′(t1) +for a = 2 such that the matrix elements Mab are re- +lated to the two-point correlations of the canonical trans- +verse electric field operator ˆΠE (c.f. Appendix B). Since +the electron’s motion is considered to be along the x- +axis only, the two-point correlations involve only the x- +component of ˆΠE. In terms of the creation and annihi- +lation operators, and the x-component of the unit polar- +ization vector εx +k, it is given by [26] +ˆΠE(r, t) = iC +� +d3k +√ +k +� +ε +ˆaε(k)ei(k·r−ωt)εx +k + c.c , (4) +with the constant prefactor C := +� +ℏc/(2ϵ0(2π)3) +� 1 +2 . By +making a change of basis to (X(t), u(t)) with X(t) = +(x(t) + x′(t))/2 and u(t) = x′(t) − x(t), the so-called +influence functional SIF [27] takes the simplified form +SIF[X, u](t) = +� t +ti +dt1dt2 +� +iu(t1)N(t1; t2)u(t2) +2 ++ +u(t1)D(t1; t2)X(t2)] , +(5) +where the noise kernel N(t1; t2) and the dissipation ker- +nel D(t1; t2) are defined to be +N(t1; t2) := e2 +2ℏ ⟨0| {ˆΠE(t1), ˆΠE(t2)} |0⟩ , +D(t1; t2) :=ie2 +ℏ ⟨0| +� +ˆΠE(t1), ˆΠE(t2) +� +|0⟩ θ(t1 − t2) . +(6) +Here, |0⟩ is the vacuum state of the free radiation field +and θ(τ) is the Heaviside step function. As in [17, 18], +we have also used the standard non-relativistic dipole ap- +proximation in which one ignores the spatial dependence +of the EM fields. From the definitions in Eq. (6) and the +expression for ˆΠE in Eq. (4), the explicit expressions for +the noise and the dissipation kernels can be obtained. +It is important to note that the evaluation of the ker- +nels necessiates the introduction of a high frequency cut- +off in the calculations. This is due to the fact that the +expressions for the kernels, which only depend upon the +difference τ := t1 − t2, diverge at τ = 0. A cure is pro- +vided by the standard Hadamard finite part prescription +[23] which introduces the convergence factor e−k/kmax in- +side the integrals appearing in the vacuum expectation +values of the commutator and the anti-commutator. In +terms of ϵ = 1/ωmax, with ωmax = kmaxc being the high +frequency cut-off, the kernels read (c.f. Appendix C) +N(t1; t2) = N(τ) = +e2 +π2ϵ0c3 +� +ϵ4 − 6ϵ2τ 2 + τ 4� +(ϵ2 + τ 2)4 +, +(7) +D(t1; t2) = D(τ) = +e2 +3πϵ0c3 θ(τ) d3 +dτ 3 δϵ(τ) . +(8) +The function πδϵ(τ) := ϵ/(τ 2 + ϵ2) appearing in Eq. (8) +behaves like a Dirac delta for τ ≫ ϵ but is non-singular +at τ = 0 due to the finite cut-off. We refer to Appendix C +for more details. +Following [23], starting from Eq. (3) and using the ex- +plicit functional form of SIF in Eq. (5), the master equa- +tion for the reduced density matrix can be derived. Upto +second order in the interactions, we obtain its expression +to be (c.f. Appendix B for a detailed derivation) +∂tˆρr(t) = − i +ℏ +� +ˆHs, ˆρr(t) +� +− 1 +ℏ +� t−ti +0 +dτN(t; t − τ) [ˆx, [ˆxHs(−τ), ˆρr(t)]] ++ i +2ℏ +� t−ti +0 +dτD(t; t − τ) [ˆx, {ˆxHs(−τ), ˆρr(t)}] . +(9) +The first line of the master equation is the usual Liouville- +von Neuman evolution and involves only the system +Hamiltonian ˆHs. In the second and the third lines, which +encode the system’s interaction with the environment, +the operator ˆxHs(−τ) is used as a place holder for the +expression +ˆxHs(−τ) := ˆU −1 +s +(t − τ; t)ˆx ˆUs(t − τ; t) , +(10) + +3 +where ˆUs(t − τ; t) is the unitary operator that evolves +the statevector of the system from time t to t − τ via the +system Hamiltonian ˆHs only. +The operator ˆx without +the subscript is the usual Schr¨odinger operator such that +ˆxHs(0) = ˆx. +Note that due to the coupling between the position of +the electron and the transverse electric field in Eq. (1), +the system Hamiltonian receives an additional contribu- +tion such that ˆHs = ˆp2/(2m) + ˆV0(x) + ˆVEM(x), where, +having introduced a cut-off scale in the calculations and +considering the motion of the electron along the x-axis +only, ˆVEM(x) = +e2ω3 +max +3π2ϵ0c3 ˆx2 (c.f. Appendix A). We point +out that since the master equation is valid upto second +order in the interactions and since the operator ˆxHs(−τ) +appears alongside the dissipation and the noise kernels +(which are already second order in e), the time evolu- +tion governed by ˆUs(t − τ; t) in Eq. (10) is understood +to involve only ˆV0 and not ˆVEM. Therefore, upto second +order in the interactions, ˆVEM only contributes via the +Liouville-von Neuman term. +The equation of motion. Using the master equation (9), +we obtain the coupled equations for the time evolution +of ⟨ˆx⟩ and ⟨ˆp⟩. It is interesting to compare the quantum +mechanical EOM with the one derived classically. +Within classical electrodynamics, a charged spherical +shell of radius R which is accelerated by an external force +Fext, experiences an extra recoil force (radiation reaction) +due to the emission of radiation. By taking the limit R → +0 in the equation describing its dynamics, one obtains the +Abraham-Lorentz formula +mR¨x = Fext + 2ℏα +3c2 +...x , +(11) +where mR denotes the observed renormalized mass. See +for example [20, 28] and the references therein for the +derivation of the AL formula. The triple derivative term +appearing in Eq. (11) can be interpreted as the friction +term that leads to energy loss due to radiation emission. +For instance, when the external potential is taken to be +V0(x) = (1/2)mω2 +0x2, one has ...x ≈ −ω2 +0 ˙x [22]. However, +the issue with Eq. (11) is that the same triple derivative +term persists even when the external potential is switched +off, leading to an exponentially increase of the particle’s +acceleration. +A discussion of the AL formula and the +problems associated with it can be found in [19–21, 28] +and the references therein. +In the case that we are considering, the rate of change +of the expectation values is calculated from Eq. (9). The +coupled differential equations for ⟨ˆx⟩ and ⟨ˆp⟩ are given +by (c.f. Appendix E) +d +dt⟨ˆx⟩ =Tr(ˆx ˙ˆρr) = ⟨ˆp⟩ +m , +(12) +d +dt⟨ˆp⟩ = − ⟨ ˆV0,x ⟩ + Tr +� +ˆρr(t) +� t−ti +0 +dτD(τ)ˆxHs(−τ) +� +− 2e2ω3 +max⟨ˆx⟩/(3π2ϵ0c3) . +(13) +While it might not be apparent at the first glance, +Eq. (13) is actually local in time due the form of the +dissipation kernel in Eq. (8). To see this explicitly, the +integral involving the dissipation kernel needs to be eval- +uated. In order to do so, we integrate by parts such that +the derivatives acting on δϵ (which appear in the expres- +sion obtained for the dissipation kernel in Eq. (8)) are +shifted onto the adjacent function. The integral is calcu- +lated explicitly in Appendix D and the following identity +is derived +� t +0 +dτD(τ)f(τ) = − 2αℏ +3c2 f ′′′(0) − 4αℏωmax +3πc2 +f ′′(0) ++ 2e2ω3 +maxf(0)/(3π2ϵ0c3) . +(14) +Here, the prime denotes the derivative taken with respect +to τ and α = e2/(4πϵ0ℏc) the fine structure constant. +Using the identity (14), Eq. (13) becomes +d +dt⟨ˆp⟩ = − ⟨ ˆV0,x ⟩ − 4αℏωmax +3πc2 +Tr +� +ˆρr(t) d2 +dτ 2 ˆxHs(−τ) +���� +τ=0 +� +− 2αℏ +3c2 Tr +� +ˆρr(t) d3 +dτ 3 ˆxHs(−τ) +���� +τ=0 +� +. +(15) +We see that in the EOM (15) only the original bare po- +tential ˆV0 remains, because the contribution coming from +ˆVEM in the last line of Eq. (13) is canceled by the term +in the last line of the integral (14), after one introduces +the cut-off consistently throughout the calculations. For +more details we refer to Appendices A and E, or Ref. [17] +where the same cancellation was argued for. +The time derivatives of ˆxHs in Eq. (15) can be easily +computed, since from Eq. (10) we have the relation (upto +leading order in the interactions) +d +dτ ˆxHs(−τ) = − i +ℏ +� +ˆV0(x) + ˆp2 +2m, ˆxHs(−τ) +� +. +(16) +First we consider the situation when the external poten- +tial is switched off. From Eq. (16), with ˆV0(x) = 0, taking +another time derivative of ˆxHs we get +d2 +dτ 2 ˆxHs(−τ) +���� +τ=0 += +�−i +ℏ +�2 � ˆp2 +2m, +� ˆp2 +2m, ˆx +�� += 0 , +(17) +where, in Eq. (17), we have also used the relation +ˆxHs(0) = ˆx. Similarly, the third derivative term appear- +ing in Eq. (15) also vanishes. Therefore, when ˆV0(x) = 0, +Eq. (15) simply reduces to +d +dt⟨ˆp⟩ = 0 . +(18) +Unlike the AL formula in Eq. (11), we see that upto sec- +ond order in the interactions there are no solutions which +allow for an exponential increase of the particle’s accel- +eration in the absence of an external potential. +Next we consider the case when the external potential +is switched on. When the potential does not depend ex- +plicitly on time, the double and triple derivative terms + +4 +in Eq. (15) yield double and triple commutators with re- +spect to the system Hamiltonian respectively (discarding +ˆVEM upto second order). Eq. (15) can then be written as +d +dt⟨ˆp⟩ =Fext + 4αℏωmax +3πc2 +Tr +� 1 +ℏ2 ˆρr(t) +� +ˆHs, +� +ˆHs, ˆx +��� +− 2αℏ +3c2 Tr +� i +ℏ3 ˆρr(t) +� +ˆHs, +� +ˆHs, +� +ˆHs, ˆx +���� +. (19) +Here, we have defined Fext := −⟨ ˆV0(x),x ⟩. Due to the +presence of ˆV0(x), the commutators of ˆHs with ˆx no +longer vanish. To simplify the equation further, we shift +the commutators onto the density matrix using the cyclic +property Tr(ˆa · [ˆb, ˆc]) = Tr([ˆa,ˆb] · ˆc) such that +Tr +� +ˆρr +� +ˆHs, +� +ˆHs, ˆx +��� += Tr +� +ˆx +� +ˆHs, +� +ˆHs, ˆρr +��� +. +(20) +The same relationship is also obtained for the triple com- +mutator term, with an additional minus sign. Remem- +bering that the master equation is only valid upto second +order in the interaction, it is sufficient to evaluate the +trace in Eq. (19) at 0th order. This implies that within +the trace, the time dependence of the density matrix can +be evaluated only by retaining the Liouville-von Neuman +term in Eq. (9). The right hand side of Eq. (20) thus +becomes proportional to Tr(ˆx¨ˆρr). With these simplifica- +tions, Eq. (19) can be written as +mR +d2 +dt2 ⟨ˆx⟩ = Fext + 2αℏ +3c2 +d3 +dt3 ⟨ˆx⟩ . +(21) +After identifying the observed electron mass with the re- +normalized mass mR := m + (4αℏωmax)/(3πc2), Eq. (21) +reduces to the Abraham-Lorentz formula (11). The same +result is also obtained for the general case in which the +bare potential ˆV0(x, t) depends explicitly on time, as +shown in Appendix E. We remark that the equation of +motion derived quantum mechanically only reduces to +Eq. (11) in the presence of an external potential. When +the external potential is switched off, the EOM reduces +to Eq. (18) and is therefore free of the runaway solution. +Decoherence. In this final part of the article, we are inter- +ested in assessing if the spatial superposition of a charged +particle at rest can be suppressed via its interaction with +the vacuum fluctuations alone. We begin by writing the +position space representation of the master equation (9) +relevant for decoherence +∂tρr = +� +−(x′ − x)2N1(t) +ℏ +� +ρr , +(22) +where N1(τ) is defined to be N1(τ) := +� τ +0 dτ ′N(τ ′) = +−4αℏ(τ 3 − 3τϵ2)(τ 2 +ϵ2)−3(3πc2)−1 . We have set ti = 0 +and only retained the second term involving the noise ker- +nel in Eq. (9). This is because the other terms typically +give subdominant contributions when the question of in- +terest is to evaluate the rate of decay of the off-diagonal +elements of the density matrix at late times [23, 29]. We +have also used the expression of the noise kernel in Eq. (7) +inside the integral to obtain the expression for N1. Inte- +grating Eq. (22) we get +ρr(x′, x, t) = exp +� +−(x′ − x)2 +ℏ +N2(t) +� +ρr(x′, x, 0) , (23) +where N2(t) := +� t +0 dτN1(τ). The function N2(t) is in- +versely proportional to the coherence length lx(t) defined +by lx(t) := (ℏ/N2(t)) +1 +2 . +After performing the integral +over N1 the expression for the coherence length is ob- +tained to be +lx(t) = +� +3πc2 +2αω2 +max +· (t2 + ϵ2)2 +t4 + 3t2ϵ2 +t≫ϵ += +� +3π +2α +1 +kmax +. +(24) +We see that the coherence length approaches a constant +value on time scales much larger than ϵ = 1/ωmax and +that its value scales inversely with the UV cut-off. Taken +literally, if one sets kmax = 1/λdb, where λdb is the de +Broglie wavelength of the electron, one would arrive at +the conclusion that vacuum fluctuations lead to decoher- +ence with the coherence length of the charged particle +asymptotically reducing to lx ≈ 25λdb within the time +scales t ≈ λdb/c. +False Decoherence. It is clearly unsatisfactory to have +an observable effect scale explicitly with the UV cut-off, +since the precise numerical value of the cut-off is, strictly +speaking, arbitrary. A similar situation was encountered +in [30] in a different context of a harmonic oscillator cou- +pled to a massive scalar field. However, it was argued in +[30] that the reduced density matrix of the harmonic os- +cillator described false decoherence. In such a situation, +the off-diagonal elements of the density matrix are sup- +pressed simply because the state of the environment goes +into different configurations depending upon the spatial +location of the system. However, these changes in the +environmental states remain locally around the system +and are reversible. For the electron interacting with vac- +uum fluctuations, we therefore take the point of view +that if the reduced density matrix describes false deco- +herence, then after adiabatically switching off the inter- +actions with the environment (after having adiabatically +switched it on initially), the original coherence must be +fully restored at the level of the system. +To formulate the argument we consider a time depen- +dent coupling q(t) = −ef(t) such that f(t) = 1 for +most of the dynamics between the initial time t = 0 +and the final time t = T, while f(0) = f(T) = 0. +The quantity relevant for decoherence is the noise kernel +which, under the time-dependent coupling, transforms as +N → ˜ +N = f(t1)f(t2)N(t1; t2) = f(t1)f(t2)N(t1 − t2) . +The decoherence factor in the double commutator in +Eq. (9) involves replacing t2 with t1 − τ and then in- +tegrating over τ. Therefore, the function N1 transforms +as N1 → ˜ +N1, with ˜ +N1 given by +˜ +N1(t1) = f(t1) +� t1 +0 +dτf(t1 − τ)N(τ) . +(25) + +5 +From the definitions of N1 and N2 we have N1 = +(d/dτ)N, N2 = (d/dτ)N1 and N1(0) = N2(0) = 0. Using +these relations and integrating by parts, Eq. (25) becomes +˜ +N1(t1) =f(t1)N1(t1)f(0) + f(t1)N2(t1) ˙f(0) ++ f(t1) +� t1 +0 +dτN2(τ) d2 +dτ 2 f(t1 − τ) . +(26) +In the limit ϵ → 0 (taking the UV cut-off to infinity), we +see from Eq. (24) that N2 looses any time dependence. +We can therefore bring N2 outside the integral such that +˜ +N1(t1) = f(t1)N1(t1)f(0)+f(t1)N2 ˙f(0)−f(t1)N2( ˙f(0)− +˙f(t1)). The terms involving ˙f(0) cancel out and we get +˜ +N1(t1) = f(t1)N1(t1)f(0) + f(t1)N2 ˙f(t1) . +(27) +After integrating by parts Eq. (27), in order to obtain +˜ +N2(T) = +� T +0 dt1 ˜ +N1(t1), we get +˜ +N2(T) =f(0) (f(T)N2(T) − f(0)N2(0)) +− f(0)N2 +� T +0 +dt1 ˙f + N2 +2 +� T +0 +dt1 +d +dt1 +f 2 . +(28) +In the limit ϵ → 0, as we noted earlier, N2(t) takes a +constant value for any time t > 0 but is zero at t = 0 +from the way it is defined. Therefore, after completing +the remaining integrals, we get +˜ +N2(T) = N2 +2 +� +f 2(0) + f 2(T) +� +. +(29) +Since we assume that the interactions are switched off +in the very beginning and at the very end, we see that +˜ +N2(T) = 0 such that Eq. (23) becomes ˜ρr(x′, x, T) = +ρr(x′, x, 0). Therefore, by adiabatically switching off the +interactions we recover the original coherence within the +system. +This is different from standard collisional decoherence +where, for example, one originally has ∂tρr(x′, x, t) = +−Λ(x′ − x)2ρr(x′, x, t) [29]. +When in this case we +send +Λ +→ +˜Λ += +f(t)Λ, +we +get +˜ρr(x′, x, t) += +exp +� +−Λ(x′ − x)2 � t +0 dt′f(t′) +� +ρr(x′, x, 0). +The density +matrix depends on the integral of f(t) rather than its +end points and we see that coherence is indeed lost ir- +reversibly. +We interpret this result to imply that the +vacuum fluctuations alone do not lead to irreversible loss +of coherence. Moreover, our results imply that the ap- +parent decoherence cannot be due to emission of photons +as otherwise one would not be able to retrieve the coher- +ence back into the system simply by switching off the +interactions with the environment at late times. +Discussion. +We formulated the interaction of a non- +relativistic electron with the radiation field within the +framework of open quantum systems and obtained the +master equation for the reduced electron dynamics in the +position basis. We showed that the classical limit of the +quantum dynamics is free of the problems associated with +the purely classical derivation of the Abraham-Lorentz +formula. With respect to possible decoherence induced +by vacuum fluctuations alone, we showed that the ap- +parent decoherence at the level of the reduced density +matrix is reversible and is an artifact of the formalism +used. In mathematically tracing over the environment, +one traces over the degrees of freedom that physically +surround the system being observed. These degrees of +freedom must be considered part of the system being ob- +served, rather than the environment [16, 30]. We formu- +lated this interpretation by showing that one restores full +initial coherence back into the system after switching off +the interactions with the environment adiabatically. The +formulation is fairly general and might also be used in +other situations to distinguish true decoherence from a +false one. The analysis therefore brings together various +works in the literature [15–18, 30] and addresses some of +the conflicting results. +Acknowledgements. +A.G. thanks Davide Bason and +Lorenzo Di Pietro for numerous discussions. We thank +Oliviero Angeli for cross checking some of the results ob- +tained in the manuscript and Lajos Di´osi for discussions +concerning false decoherence. A.B. acknowledges finan- +cial support from the EIC Pathfinder project QuCoM +(GA no. 101046973) and the PNRR PE National Quan- +tum Science and Technology Institute (PE0000023). We +thank the University of Trieste and INFN for financial +support. +Appendix A: The Lagrangian and the Hamiltonian formulation +In the Coulomb gauge, the standard Lagrangian for electrodynamics is given by [24] +L = 1 +2m˙r2 +e − V0(re) − +� +1/2 +d3k |ρ|2 +ϵ0k2 + ϵ0 +2 +� +d3r +� +E2 +⊥(r) − c2B2(r) +� ++ +� +d3rj(r) · A⊥(r) . +(A1) +In addition to the terms that have been described in the main article, Eq. (A1) also includes the Coulomb potential +between different particles. It is given by the third term in which ρ(r) denotes the charge density and the symbol +� +1/2 means that the integral is taken over half the volume in the reciprocal space. For a single particle, it reduces to +the particle’s Coulomb self energy ECoul. After the introduction of a suitable cut-off it takes a finite value given by +ECoul = αℏωmax/π [22]. The transverse vector potential is denoted by A⊥(r, t) whose negative partial time derivative +yields the transverse electric field E⊥(r, t) while its curl gives the magnetic field B(r, t). For an electron, the current + +6 +density is given by j(r) = −e˙rδ(r−re) and the interaction term becomes −e˙reA⊥(re, t). For a non-relativistic charged +particle, the time derivative can be shifted from the position of the particle onto the transverse vector potential. This +is because in addition to a total derivative term, a term of the form erevi∂iA⊥(r, t) appears (where vi := ˙ri). After +the wave expansion of A⊥, this term is seen to be negligible with respect to ere ˙A⊥(re, t) = −ereE⊥(re, t) as long as +ωk ≫ vk or v ≪ c. Therefore, for the non-relativistic electron, the Lagrangian relevant for the dynamics reduces to +L(t) ≈ 1 +2m˙r2 +e − V0(re) + ϵ0 +2 +� +d3r +� +E2 +⊥(r) − c2B2(r) +� +− ereE⊥(re) . +(A2) +In Eq. (A2) the total derivative d/dt(reA⊥(re)) and the constant Coulomb self energy term have been omitted as +these do not affect the electron’s dynamics. +The Hamiltonian corresponding to the Lagrangian (A2) can now be obtained. In terms of the canonical variables +re, p, A⊥ and ΠE := − 1 +ϵ0 Π, it takes the form +H = HS + HEM + Hint , +(A3) +where HEM = ϵ0 +2 +� +d3r(Π2 +E(r) + c2B2(r)) is the free field Hamiltonian of the radiation field, Hint = ereΠE(re) the +interaction term and HS the system Hamiltonian given by +HS = p2 +2m + V0(re) + e2 +2ϵ0 +� +d3rriδ⊥ +im(r − re)δ⊥ +mj(r − re)rj . +(A4) +Here, the transverse Dirac delta δ⊥ +ij(r − re), which appears due to the coupling of the position of the electron with +the transverse electric field, is defined to be [22] +δ⊥ +ij(r − re) := +1 +(2π)3 +� +d3k +� +δij − kikj +k2 +� +eik·(r−re) . +(A5) +The form of HS calls for an identification of the full effective potential V (re) governing the dynamics of the electron +such that +V (re) := V0(re) + VEM(re) , +VEM(re) = e2 +2ϵ0 +� +d3rriδ⊥ +im(r − re)δ⊥ +mj(r − re)rj . +(A6) +Note that the extra term VEM(re) is not added to the bare potential by hand, but arises naturally due to the reE⊥ +coupling [17]. Although it gives a divergent contribution +e2 +2ϵ0 δ⊥ +ij(0)ri +erj +e, after regularizing the transverse delta function +on a minimum length scale rmin = 1/kmax, the contribution coming from this term scales as O( e2 +2ϵ0 r2 +ek3 +max). To be +more precise, we impose the cut-off consistently throughout the calculations by introducing the convergence factor +e−k/kmax inside the integral in the reciprocal space (c.f. Appendix C). Using this procedure, the expression for δ⊥ +ij(0) +is obtained to be +δ⊥ +ij(0) = +1 +(2π)3 +� +dkk2e−k/kmax +� +dΩ +� +δij − kikj +k2 +� +. +(A7) +First evaluating the angular integral, which gives a factor 8π +3 δij, and then the radial integral, we get +VEM(re) = e2ω3 +max +3π2ϵ0c3 r2 +e . +(A8) +Since the contribution of VEM(re) is canceled exactly by another term, as shown in the discussion around Eq. (15) of +the main text, for all practical purposes, VEM(re) has no consequences on the dynamics of the electron. +Appendix B: The master equation +The probability amplitude for a particle to be at the position xf at some final time t, starting from the position xi +at some initial time ti, is given by [25] +⟨xf| ˆU(t; ti) |xi⟩ = +� +x(t)=xf, +x(ti)=xi +D[x, p]e− i +ℏ +� t +ti dt′(HT[x,p]−p ˙x) = +� +x(t)=xf, +x(ti)=xi +D[x]e +i +ℏ ST[x] , +(B1) + +7 +where HT is the full Hamiltonian and ST is the corresponding action describing some general dynamics. From Eq. (B1) +the expression for the density matrix at time t can be written as [23] +⟨x′ +f| ˆρ(t) |xf⟩ = +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′]e +i +ℏ (ST[x′]−ST[x])ρ(x′ +i, xi, ti) , +(B2) +where the integrals over xi and x′ +i are included within the path integral. The expression analogous to Eq. (B1) also +exists for ⟨pf| ˆU(t; ti) |pi⟩ in which the boundary conditions are fixed on p(t) and the phase-space weighing function is +instead given by exp{ −i +ℏ +� t +ti dt′ (HT [x, p] + x ˙p)} such that +⟨pf| ˆU(t; ti) |pi⟩ = +� +p(t)=pf, +p(ti)=pi +D[x, p]e− i +ℏ +� t +ti dt′(HT[x,p]+x ˙p) . +(B3) +For computing the path integral over the EM field, with a slight abuse of notation, we understand exp{ i +ℏSEM} +to be simply the appropriate phase-space weighing function appearing inside the path integral with SEM := +− +� t +ti dt′d3r(HEM − Π ˙A⊥) or SEM := − +� t +ti dt′d3r(HEM + A⊥ ˙Π) depending upon the basis states between which the +transition amplitudes are calculated. We are interested in the dynamics of the electron, having taken into account its +interaction with the radiation field environment. With this distinction, the total phase-space function can be written +as ST = SS[x] + SEM[µ] + Sint[x, ΠE], where SS denotes the system action, SEM[µ] := SEM[A⊥, ΠE] the phase-space +function governing the time evolution of the free radiation field in which µ denotes its phase-space degrees of freedom +and Sint[x, ΠE] := −e +� t +ti dt′xΠE encodes the interaction between the two. The expression for the system-environment +density matrix can then be written as +� +x′ +f; Πf′ +E +�� ˆρ(t) +��xf; Πf +E +� += +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′]e +i +ℏ (SS[x′]−SS[x])ρS(x′ +i, xi, ti)× +× +� +ΠE(t)=Πf +E, +Π′ +E(t)=Πf′ +E +D[µ, µ′]e +i +ℏ (SEM[µ′]+Sint[x′,Π′ +E]−SEM[µ]−Sint[x,ΠE])ρEM(Π′ +E(ti), ΠE(ti), ti) , +(B4) +where +���Πf +E +� +denotes the basis state of the environment. Note that the precise choice of the environmental basis states +is unimportant since the reduced density matrix is obtained by tracing over the environment. In writing Eq. (B4) we +have also assumed the full density matrix ˆρ(ti) to be in the product state ˆρ(ti) = ˆρS(ti) ⊗ ˆρEM(ti) at the initial time +ti. We notice that SEM[µ] is quadratic in the environmental degrees of freedom while Sint[x, ΠE] is linear in both x +and ΠE. After tracing over the environment, that is integrating over ΠE(t) = Π′ +E(t), the term in the second line of +Eq. (B4) yields a Gaussian in x such that [23] +� +ΠE(t)=Π′ +E(t) +dΠE(t)D[µ, µ′]e +i +ℏ (SEM[µ′]+Sint[x′,Π′ +E]−SEM[µ]−Sint[x,ΠE])ρi +EM = e +i +2ℏ +�� +dt1dt2Mab(t1;t2)xa(t1)xb(t2) , +(B5) +where ρi +EM := ρEM(Π′ +E(ti), ΠE(ti), ti). We have also introduced the vector notation with the convention xa = x for +a = 1, xa = x′ for a = 2 and xa = ηabxb with ηab = diag(−1, 1). It is the matrix elements Mab which determine the +effective action of the system and contain the information about its interaction with the environment. They can be +obtained by acting with ℏ +i +δ +δxa +δ +δxb |xa=xb=0 (where xa and xb are set to zero after taking the derivatives) on Eq. (B5) +such that +M ab(t1; t2) = ie2 +ℏ +� +ΠE(t)=Π′ +E(t) +dΠE(t)D[µ, µ′]Πa +E (t1) Πb +E (t2) e +i +ℏ (SEM[µ′]−SEM[µ])ρi +EM . +(B6) +Here, in the light of the standard non-relativistic dipole approximation, we have ignored the spatial dependence of the +canonical fields (c.f. Appendix C). Depending upon the value of the indices a and b, the matrix elements correspond +to the expectation values of the time-ordered or path-ordered correlations in the Heisenberg picture [23]. For the +dynamics of the non-relativistic electron that we are considering, the expression for Mab reads +Mab(t1; t2) = ie2 +ℏ +� +� +� +˜T {ˆΠE(t1)ˆΠE(t2)} +� +0 +− +� +ˆΠE(t1)ˆΠE(t2) +� +0 +− +� +ˆΠE(t2)ˆΠE(t1) +� +0 +� +T {ˆΠE(t1)ˆΠE(t2)} +� +0 +� +� . +(B7) + +8 +The zero in the subscript denotes that the expectation values are calculated by disregarding the interaction with the +system, while T and ˜T denote the time-ordered and the anti-time ordered products respectively. It is also understood +that since the electron’s motion is considered to be along the x-axis, the canonical field operator that enters Mab is +only the x-component given by [26] +ˆΠE(r, t) = i +� +ℏc +2ϵ0(2π)3 +� 1 +2 � +d3k +√ +k +� +ε +ˆaε(k)ei(k·r−ωt)εx +k + c.c . +(B8) +In our case, the initial state of the environment is taken to be the vacuum state |0⟩ of the radiation field such that +⟨·⟩0 = ⟨0| · |0⟩. After tracing over the environment, the reduced density matrix of the electron is obtained from +Eq. (B4) to be +⟨x′ +f| ˆρr(t) |xf⟩ = +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′]e +i +ℏ (SS[x′]−SS[x]+SIF[x,x′])ρr(x′ +i, xi, ti) , +(B9) +where +SIF[x, x′] = ie2 +2ℏ +� t +ti +dt1dt2 +�� +˜T {ˆΠE(t1)ˆΠE(t2)} +� +0 x(t1)x(t2) − +� +ˆΠE(t1)ˆΠE(t2) +� +0 x(t1)x′(t2) +− +� +ˆΠE(t2)ˆΠE(t1) +� +0 x′(t1)x(t2) + +� +T {ˆΠE(t1)ˆΠE(t2)} +� +0 x′(t1)x′(t2) +� +. +(B10) +The integral +� t +ti stands for both the t1 and the t2 integrals which run from ti to t. +Alternatively, the influence +functional SIF can be written in the matrix notation as +SIF[x, x′] = 1 +2 +� t +ti +dt1dt2 +�x(t1) x′(t1)� +· +� +M11 M12 +M21 M22 +� +· +� +x(t2) +x′(t2) +� +. +(B11) +As it is more convenient, we make a change of basis to (X , u) defined by +X(t) :=(x′(t) + x(t))/2 , +u(t) = x′(t) − x(t) , +(B12) +in which the influence functional transforms as +SIF[X, u] = 1 +2 +� t +ti +dt1dt2 +�X(t1) u(t1)� +· +� ˜ +M11 +˜ +M12 +˜ +M21 +˜ +M22 +� +· +� +X(t2) +u(t2) +� +, +(B13) +where +� ˜ +M11 +˜ +M12 +˜ +M21 +˜ +M22 +� += +� +M11 + M12 + M21 + M22 +1 +2 ((M12 − M21) + (M22 − M11)) +1 +2 (−(M12 − M21) + (M22 − M11)) +1 +4((M11 + M22) − (M12 + M21)) +� +. +(B14) +From Eq. (B7) we obtain the following relations +M11 + M22 = −(M12 + M21) = ie2 +ℏ +� +{ˆΠE(t1), ˆΠE(t2)} +� +0 , +(B15) +M12 − M21 = ie2 +ℏ +�� +ˆΠE(t2), ˆΠE(t1) +�� +0 , +(B16) +M22 − M11 = ie2 +ℏ +�� +ˆΠE(t1), ˆΠE(t2) +�� +0 sgn(t1 − t2) . +(B17) +Using these relations, ˜ +M takes the simplified form +� ˜ +M11 +˜ +M12 +˜ +M21 +˜ +M22 +� += ie2 +ℏ +� +� +0 +�� +ˆΠE(t2), ˆΠE(t1) +�� +0 θ(t2 − t1) +�� +ˆΠE(t1), ˆΠE(t2) +�� +0 θ(t1 − t2) +1 +2 +� +{ˆΠE(t1), ˆΠE(t2)} +� +0 +� +� , +(B18) +where θ(t) is the Heaviside step function. Thus, in the (X, u) basis, the influence functional in Eq. (B10) takes the +compact form +SIF[X, u](t) = +� t +ti +dt1dt2 +� +iu(t1)N(t1; t2)u(t2) +2 ++ u(t1)D(t1; t2)X(t2) +� +, +(B19) + +9 +where the noise kernel N and the dissipation kernel D are defined as +N(t1; t2) := e2 +2ℏ +� +{ˆΠE(t1), ˆΠE(t2)} +� +0 , +D(t1; t2) :=ie2 +ℏ +�� +ˆΠE(t1), ˆΠE(t2) +�� +0 θ(t1 − t2) . +(B20) +Having determined the full effective action for the electron, in terms of the influence functional, we can now derive +the master equation. From Eq. (B9), it can be seen that the time derivative of the reduced density matrix will have, +in addition to the standard Liouville-von Neuman term, the contribution coming from the influence functional. In +order to compute that we need to evaluate the rate of change of SIF. It is given by +δtSIF[X, u] = u(t) +� t +ti +dt1 (iN(t; t1)u(t1) + D(t; t1)X(t1)) . +(B21) +In terms of the original (x, x′) basis, the full expression for the master equation can now be written as +∂tρr(x′ +f, xf, t) = − i +ℏ ⟨x′ +f| +� +ˆHs, ˆρr +� +|xf⟩ + i +ℏ +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′]δtSIF[x′, x]e +i +ℏ (SS[x′]−SS[x]+SIF[x,x′])ρr(x′ +i, xi, ti) +≈ − i +ℏ ⟨x′ +f| +� +ˆHs, ˆρr +� +|xf⟩ + i +ℏ +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′]δtSIF[x′, x]e +i +ℏ (SS[x′]−SS[x])ρr(x′ +i, xi, ti) +≈ − i +ℏ ⟨x′ +f| +� +ˆHs, ˆρr +� +|xf⟩ +− 1 +ℏ(x′ +f − xf) +� t +ti +dt1N(t; t1) +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′](x′(t1) − x(t1))e +i +ℏ (SS[x′]−SS[x])ρr(x′ +i, xi, ti) ++ i +2ℏ(x′ +f − xf) +� t +ti +dt1D(t; t1) +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′](x′(t1) + x(t1))e +i +ℏ (SS[x′]−SS[x])ρr(x′ +i, xi, ti) . +(B22) +The Liouville-von Neuman evolution is governed by the system Hamiltonian ˆHs alone. For the second term on the +right hand side in the second line of Eq. (B22), we have omitted SIF in the exponential. This is because SIF is second +order in the coupling constant and is already present adjacent to the exponential. Since we limit our calculations to +second order in the interactions, SIF can be neglected inside the exponential. +To simplify the master equation further, we note that the last two lines of Eq. (B22) can be written much more +compactly. This is due to the following identity [23] +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′]x′(t1)e +i +ℏ (SS[x′]−SS[x])ρr(x′ +i, xi, ti) = += +� +dx′(t1) ⟨x′ +f| ˆUs(t; t1) |x′(t1)⟩ x′(t1) ⟨x′(t1)| ˆUs(t1; ti)ˆρr(ti) ˆU −1 +s +(t; ti) |xf⟩ += ⟨x′ +f| ˆUs(t; t1)ˆx ˆUs(t1; ti)ˆρr(ti) ˆU −1 +s +(t; ti) |xf⟩ = ⟨x′ +f| ˆUs(t; t1)ˆx ˆUs(t1; ti) ˆU −1 +s +(t; ti) ˆUs(t; ti)ˆρr(ti) ˆU −1 +s +(t; ti) |xf⟩ += ⟨x′ +f| ˆUs(t; t1)ˆx ˆU −1 +s +(t; t1)ˆρr(t) |xf⟩ = ⟨x′ +f| ˆxHs(−τ)ˆρr(t) |xf⟩ , +(B23) +where +ˆxHs(−τ) := ˆU −1 +s +(t − τ; t)ˆx ˆUs(t − τ; t) , +τ := t − t1 . +(B24) +Similarly, we also have +� +x(t)=xf, +x′(t)=x′ +f +D[x, x′]x(t1)e +i +ℏ (SS[x′]−SS[x])ρr(x′ +i, xi, ti) = ⟨x′ +f| ˆρr(t)ˆxHs(−τ) |xf⟩ . +(B25) +The operator ˆxHs(−τ) is understood to be simply the placeholder for the expression that appears on the right hand +side of the first equality in Eq. (B24) such that +ˆxHs(0) = ˆx . +(B26) + +10 +Here, the operator ˆx without the subscript Hs is the usual position operator in the Schr¨odinger picture. Using these +relations, and replacing the t1 integral with the τ integral (t1 = t − τ), the master equation takes the compact form +∂tρr(x′ +f, xf, t) = − i +ℏ ⟨x′ +f| +� +ˆHs, ˆρr(t) +� +|xf⟩ +− 1 +ℏ(x′ +f − xf) +� t−ti +0 +dτN(t; t − τ) ⟨x′ +f| [ˆxHs(−τ), ˆρr(t)] |xf⟩ ++ i +2ℏ(x′ +f − xf) +� t−ti +0 +dτD(t; t − τ) ⟨x′ +f| {ˆxHs(−τ), ˆρr(t)} |xf⟩ . +(B27) +The eigenvalues outside of the integrals in Eq. (B27) can be obtained by acting with the position operator ˆx such that +⟨x′ +f| ∂tˆρr |xf⟩ = − i +ℏ ⟨x′ +f| +� +ˆHs, ˆρr(t) +� +|xf⟩ +− 1 +ℏ +� t−ti +0 +dτN(t; t − τ) ⟨x′ +f| [ˆx, [ˆxHs(−τ), ˆρr(t)]] |xf⟩ ++ i +2ℏ +� t−ti +0 +dτD(t; t − τ) ⟨x′ +f| [ˆx, {ˆxHs(−τ), ˆρr(t)}] |xf⟩ . +(B28) +The master equation in the operator form can therefore be written as +∂tˆρr = − i +ℏ +� +ˆHs, ˆρr +� +− 1 +ℏ +� t−ti +0 +dτN(t; t − τ) [ˆx, [ˆxHs(−τ), ˆρr(t)]] + i +2ℏ +� t−ti +0 +dτD(t; t − τ) [ˆx, {ˆxHs(−τ), ˆρr(t)}] . +(B29) +Appendix C: The dissipation and the noise kernels +In order to solve the master equation (B29), the kernels need to be evaluated explicitly. To achieve that, we begin +with the expression for the vacuum expectation value of the correlator +⟨0| ˆΠE(x(t1), t1)ˆΠE(x(t2), t2) |0⟩ = +−iℏc +2ϵ04π2 ˆ□ +�1 +r +� ∞ +0 +dke−ikcτ � +eikr − e−ikr�� +, +(C1) +where +r := |x(t1) − x(t2)| , +τ := t1 − t2 , +ˆ□ := − 1 +c2 ∂2 +τ + ∂2 +r . +(C2) +Here, the right hand side of Eq. (C1) is obtained with the help of the expression of the quantized canonical transverse +electric field operator in Eq. (B8). The expression in Eq. (C1) becomes convergent after resorting to the standard +Hadamard finite part prescription [23], in which the convergence factor e−ωk/ωmax is introduced inside the integral +(with ωk = kc). +Physically, this prescription cuts off the contribution coming from the modes ωk ≫ ωmax and +mathematically it is the same as using the iϵ prescription where one sends τ → τ − iϵ, with ϵ = 1/ωmax. After +completing the integral by using this prescription we get +⟨0| ˆΠE(1)ˆΠE(2) |0⟩ = +ℏc +4π2ϵ0 +ˆ□ +� +1 +r2 − c2(τ − iϵ)2 +� += +ℏc +π2ϵ0 +1 +(r2 − c2(τ − iϵ)2)2 . +(C3) +For the correlator in Eq. (C3), we ignore the spatial dependence of the fields in the spirit of the non-relativistic +approximation r ≪ cτ. In this limit, the correlator becomes +⟨0| ˆΠE(1)ˆΠE(2) |0⟩ ≈ +ℏ +π2ϵ0c3 (τ − iϵ)4 . +(C4) +Using Eq. (C4), we obtain the explicit functional form of the noise and the dissipation kernels to be +N(τ) = +e2 +π2ϵ0c3 +� +ϵ4 − 6ϵ2τ 2 + τ 4� +(ϵ2 + τ 2)4 +, +(C5) +D(τ) = +8e2 +π2ϵ0c3 +ϵτ(ϵ2 − τ 2) +(ϵ2 + τ 2)4 θ(τ) . +(C6) + +11 +With some algebraic manipulation, the dissipation kernel can be expressed more compactly as +D(τ) = +e2 +3π2ϵ0c3 θ(τ) d3 +dτ 3 +� +ϵ +τ 2 + ϵ2 +� +. +(C7) +Noticing that +ϵ +τ 2 + ϵ2 = d +dτ tan−1(τ/ϵ) = πδϵ(τ) , +(C8) +we arrive at the expression +D(τ) = +e2 +3πϵ0c3 θ(τ) d3 +dτ 3 δϵ(τ) . +(C9) +The last equality in Eq. (C8) can be understood in the limit ϵ → 0 when the function tan−1(τ/ϵ) takes the shape of +a step function. Such an expression for D would yield infinite results. For that, we keep in mind that these functions +are always well behaved for a finite ϵ and that δϵ only behaves like a Dirac delta for τ ≫ ϵ. +Appendix D: Integrals involving the dissipation kernel +In this section we derive an identity involving the integrals of the form +� +dτD(τ)f(τ). To proceed, we keep in mind +the situation where ϵ is small but finite so that all the derivatives of the smoothed Dirac delta are large but finite. +However, for times τ ≫ ϵ, we have δϵ(τ) = δ′ +ϵ(τ) = δ′′ +ϵ (τ) = 0. In addition, since the derivative of the Dirac delta is an +odd function of τ, we also have δ′ +ϵ(0) = 0. In computing the integral of D(τ) multiplying an arbitrary function f(τ), +we shift the derivatives acting on δϵ one by one onto f(τ) by integrating by parts. Since the calculations of interest +involve integrating +� t +0 dτD(τ)f(τ), where τ takes only non-negative values from 0 to t, the step function θ(τ) can be +omitted inside the integral. +The first integration by parts gives (the constant pre-factors appearing in Eq. (C9) will be plugged in at the end) +� t +0 +dτδ′′′ +ϵ (τ)f(τ) = − +� t +0 +dτδ′′ +ϵ (τ)f ′(τ) + δ′′ +ϵ (τ)f(τ)|t +0 . +(D1) +Since δ′′ +ϵ (t) = 0, only the boundary term −δ′′ +ϵ (0)f(0) survives. Further, +− +� t +0 +dτδ′′ +ϵ (τ)f ′(τ) = +� t +0 +dτδ′ +ϵ(τ)f ′′(τ) − δ′ +ϵ(τ) f ′(τ)|t +0 . +(D2) +Since δ′ +ϵ(t) = δ′ +ϵ(0) = 0 (δ′ +ϵ(τ) being an odd function of τ), both the boundary terms vanish. Proceeding further we +get +� t +0 +dτδ′ +ϵ(τ)f ′′(τ) = − +� t +0 +dτδϵ(τ)f ′′′(τ) + δϵ(τ) f ′′(τ)|t +0 . +(D3) +As before, the boundary term at τ = t is zero and only the term −δϵ(0)f ′′(0) survives. Finally, since δϵ(τ) goes to +zero much faster than a generic function f(τ) for a small ϵ, it can be treated like a Dirac delta such that +− +� t +0 +dτδϵ(τ)f ′′′(τ) = −f ′′′(0) +2 +. +(D4) +The factor of half comes because the integral is performed from 0 to t. Collecting the two boundary terms we get the +result +� t +0 +dτδ′′′ +ϵ (τ)f(τ) = −f ′′′(0) +2 +− δϵ(0)f ′′(0) − δ′′ +ϵ (0)f(0) . +(D5) +From Eq. (C8) we have δϵ(0) = 1/(πϵ) = ωmax/π and δ′′ +ϵ (0) = −2ω3 +max/π such that +� t +0 +dτD(τ)f(τ) = −2αℏ +3c2 f ′′′(0) − 4αℏωmax +3πc2 +f ′′(0) + 2e2ω3 +max +3π2ϵ0c3 f(0) . +(D6) +Here, we have now plugged in the constant prefactor appearing in Eq. (C9). + +12 +Appendix E: The Abraham-Lorentz equation as a classical limit +The rate of change of the expectation values can be obtained with the help of the master equation (B29). For the +position operator it is given by +d +dt⟨ˆx⟩ = Tr (ˆx∂tˆρr) = − i +ℏTr +� +ˆx · +� +ˆHs, ˆρr +�� ++ i +2ℏ +� t−ti +0 +dτD(t; t − τ)Tr (ˆx · [ˆx, {ˆxHs(−τ), ˆρr(t)}]) +− 1 +ℏ +� t−ti +0 +dτN(t; t − τ)Tr (ˆx · [ˆx, [ˆxHs(−τ), ˆρr(t)]]) . +(E1) +Due to the identity +Tr +� +ˆA · +� +ˆB, ˆC +�� += Tr +�� +ˆA, ˆB +� +· ˆC +� +, +(E2) +the terms involving the dissipation and the noise kernels vanish and we get +d +dt⟨ˆx⟩ = − i +ℏTr +� +ˆρr · +� +ˆx, ˆHs +�� += ⟨ˆp⟩ +m . +(E3) +Here, we remember that the system Hamiltonian ˆHs receives a contribution from ˆVEM in addition to the bare potential +ˆV0 such that (c.f. the discussion between Eqs. (A4) and (A8)) +ˆHs(t) = ˆp2 +2m + ˆV0(x, t) + e2ω3 +max +3π2ϵ0c3 ˆx2 . +(E4) +Similarly, for the momentum operator we obtain the relation +d +dt⟨ˆp⟩ = Tr (ˆp∂tˆρr) = − i +ℏTr +�� +ˆp, ˆHs +� +· ˆρr +� ++ i +2ℏ +� t−ti +0 +dτD(t; t − τ)Tr ([ˆp, ˆx] · {ˆxHs(−τ), ˆρr(t)}) +− 1 +ℏ +� t−ti +0 +dτN(t; t − τ)Tr ([ˆp, ˆx] · [ˆxHs(−τ), ˆρr(t)]) . +(E5) +Since [ˆx, ˆp] = iℏ1, the term involving the noise kernel vanishes and Eq. (E5) simplifies to +d +dt⟨ˆp⟩ = −⟨ ˆV0,x ⟩ − 2e2ω3 +max +3π2ϵ0c3 ⟨ˆx⟩ + Tr +� +ˆρr(t) +� t−ti +0 +dτD(τ)ˆxHs(−τ) +� +. +(E6) +Evaluating the integral using Eq. (D6), we see that the last term in the integral gives the contribution 2e2ω3 +max +3π2ϵ0c3 ⟨ˆx⟩ to +d +dt⟨ˆp⟩ in Eq. (E6) and cancels the contribution coming from ˆVEM. The EOM therefore reduces to +d +dt⟨ˆp⟩ = −⟨ ˆV0(x),x ⟩ − 2αℏ +3c2 Tr +� +ˆρr(t) d3 +dτ 3 ˆxHs(−τ) +���� +τ=0 +� +− 4αℏωmax +3πc2 +Tr +� +ˆρr(t) d2 +dτ 2 ˆxHs(−τ) +���� +τ=0 +� +. +(E7) +As shown in the main article, when ˆV0(x, t) = 0, the double and the triple derivatives acting on ˆxHs(−τ) vanish upto +second order in the interactions. Here, we only focus on the general case in which the external (time-dependent) +potential is switched on. To simplify the equation further, we begin by evaluating the second order derivative in +Eq. (E7). From Eq. (B24) we have +d2 +dτ 2 ˆxHs(−τ) = ˆU −1 +s +(t − τ; t)ˆx ˆU ′′ +s (t − τ; t) + 2 ˆU −1′ +s +(t − τ; t)ˆx ˆU ′ +s(t − τ; t) + ˆU −1′′ +s +(t − τ; t)ˆx ˆUs(t − τ; t) , +(E8) +where the prime denotes the derivative with respect to τ. From the Schr¨odinger equation +ˆU ′ +s(t − τ; t) = i +ℏ +ˆHs(t − τ) ˆUs(t − τ; t) , +(E9) +the derivatives acting on the unitary operator can be expressed in terms of the Hamiltonian. It is clear that taking +higher derivatives of ˆUs(t − τ; t) would result in higher powers of the Hamiltonian or the partial derivative of the +Hamiltonian with respect to τ, multiplied with only a single unitary operator on the very right. However, if in the + +13 +end τ is set to zero, the Hamiltonian and its explicit time derivatives will be evaluated at time t, and the unitary +operator on the very right disappears since ˆUs(t; t) = 1. We therefore have the following identities +ˆU (′n) +s +(t − τ; t) +��� +τ=0 = (−1)n +� dn +dtn ˆUs(t; ti) +� +ˆU −1 +s +(t; ti) , +(E10) +ˆU −1(′n) +s +(t − τ; t) +��� +τ=0 = (−1)n ˆUs(t; ti) +� dn +dtn ˆU −1 +s +(t; ti) +� +. +(E11) +The additional time parameter ti that appears in Eqs. (E10) and (E11) is only apparent. +As discussed before, +evaluating the time derivatives on the right hand side of Eq. (E10) would result in powers of ˆHs(t) and its derivatives +evaluated at t. The remaining unitary matrix ˆUs(t; ti) would be canceled by the additional ˆU −1 +s +(t; ti) on the very +right such that ti disappears from the equation. Using Eqs. (E10) and (E11) in Eq. (E8) we get +Tr +� +ˆρr(t) d2 +dτ 2 ˆxHs(−τ) +���� +τ=0 +� += Tr +��� d2 +dt2 ˆUs(t; ti) +� +ˆU −1 +s +(t; ti)ˆρr(t) ++2 +� +− d +dt +ˆUs(t; ti) +� +ˆU −1 +s +(t; ti)ˆρr(t) ˆUs(t; ti) +� +− d +dt +ˆU −1 +s +(t; ti) +� ++ˆρr(t) ˆUs(t; ti) +� d2 +dt2 ˆU −1 +s +(t; ti) +�� +ˆx +� +. +(E12) +Here, we have used the cyclic property within the trace to shift the unitary operators ˆUs and its derivatives on the +right of ˆx in Eq. (E8) onto the very left within the trace. To proceed further we note that the terms involving the +trace in Eq. (E7) are multiplied by α. It is therefore sufficient to evaluate the trace at 0th order in the interactions as +the master equation is valid only upto second order in the interactions. This implies that within the trace the time +dependence of the density matrix can be evaluated by keeping only the Liouville-von Neuman term such that +ˆρr(t) = ˆUs(t; ti)ˆρr(ti) ˆU −1 +s +(t; ti) . +(E13) +Eq. (E12) then simplifies to +Tr +� +ˆρr(t) d2 +dτ 2 ˆxHs(−τ) +���� +τ=0 +� += Tr +��� d2 +dt2 ˆUs(t; ti) +� +ˆρr(ti) ˆU −1 +s +(t; ti) + 2 +� d +dt +ˆUs(t; ti) +� +ˆρr(ti) +� d +dt +ˆU −1 +s +(t; ti) +� ++ ˆUs(t; ti)ˆρr(ti) +� d2 +dt2 ˆU −1 +s +(t; ti) +�� +ˆx +� += Tr +� d2 +dt2 ˆρr(t)ˆx +� +. +(E14) +Thus, we have the relation +Tr +� +ˆρr(t) d2 +dτ 2 ˆxHs(−τ) +���� +τ=0 +� += Tr +� d2 +dt2 ˆρr(t)ˆx +� += d2 +dt2 ⟨ˆx⟩ . +(E15) +Similar line of reasoning also leads to the identity +Tr +� +ˆρr(t) d3 +dτ 3 ˆxHs(−τ) +���� +τ=0 +� += −Tr +� d3 +dt3 ˆρr(t)ˆx +� += − d3 +dt3 ⟨ˆx⟩ . +(E16) +Using Eqs. (E15) and (E16) in Eq. (E7), the EOM for the expectation value of the position operator in the presence +of an external potential is obtained to be +mR +d2 +dt2 ⟨ˆx⟩ = −⟨ ˆV0(x),x ⟩ + 2αℏ +3c2 +d3 +dt3 ⟨ˆx⟩ , +where +mR := m + 4αℏωmax +3πc2 +. +(E17) +[1] H. B. G. Casimir, Indag. Math. 10, 261 (1948). + +14 +[2] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics +(Cambridge Univ. Press, Cambridge, UK, 1984). +[3] L. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity, Cambridge Mono- +graphs on Mathematical Physics (Cambridge University Press, 2009). +[4] W. G. Unruh, Phys. Rev. D 14, 870 (1976). +[5] S. A. Fulling, Phys. Rev. D 7, 2850 (1973). +[6] S. Takagi, Prog. Theor. Phys. Suppl. 88, 1 (1986). +[7] H. A. Bethe, Phys. Rev. 72, 339 (1947). +[8] W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241 (1947). +[9] T. A. Welton, Phys. Rev. 74, 1157 (1948). +[10] J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji, Journal de Physique 43, 1617 (1982). +[11] E. +Joos, +H. +Zeh, +D. +Giulini, +C. +Kiefer, +J. +Kupsch, +and +I. +Stamatescu, +Decoherence and the Appearance of a Classical World in Quantum Theory, +Physics +and +astronomy +online +library +(Springer, 2003). +[12] C. Kiefer, Phys. Rev. D 46, 1658 (1992). +[13] L. H. Ford, Phys. Rev. D 47, 5571 (1993). +[14] G. Baym and T. Ozawa, Proceedings of the National Academy of Sciences 106, 3035 (2009). +[15] E. Santos, Physics Letters A 188, 198 (1994). +[16] L. Di´osi, Physics Letters A 197, 183 (1995). +[17] P. M. V. B. Barone and A. O. Caldeira, Phys. Rev. A 43, 57 (1991). +[18] H.-P. Breuer and F. Petruccione, in Relativistic Quantum Measurement and Decoherence, edited by H.-P. Breuer and +F. Petruccione (Springer Berlin Heidelberg, Berlin, Heidelberg, 2000) pp. 31–65. +[19] S. Coleman, “Classical electron theory from a modern standpoint,” in Electromagnetism: Paths to Research, edited by +D. Teplitz (Springer US, Boston, MA, 1982) pp. 183–210. +[20] P. Pearle, “Classical electron models,” in Electromagnetism: Paths to Research, edited by D. Teplitz (Springer US, Boston, +MA, 1982) pp. 211–295. +[21] D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Cambridge University Press, 2017). +[22] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, “Classical electrodynamics: The fundamental equations and the +dynamical variables,” in Photons and Atoms (John Wiley and Sons, Ltd, 1997) Chap. 1, pp. 5–77. +[23] E. A. Calzetta and B.-L. B. Hu, Nonequilibrium Quantum Field Theory, Cambridge Monographs on Mathematical Physics +(Cambridge University Press, 2008). +[24] “Lagrangian and hamiltonian approach to electrodynamics, the standard lagrangian and the coulomb gauge,” in +Photons and Atoms (John Wiley & Sons, Ltd, 1997) Chap. 2, pp. 79–168. +[25] A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, 2010). +[26] C. Cohen-Tannoudji, J. Dupont-Roc, +and G. Grynberg, “Quantum electrodynamics in the coulomb gauge,” in +Photons and Atoms (John Wiley & Sons, Ltd, 1997) Chap. 3, pp. 169–252. +[27] R. Feynman and F. Vernon, Annals of Physics 24, 118 (1963). +[28] D. J. Griffiths, T. C. Proctor, and D. F. Schroeter, American Journal of Physics 78, 391 (2010). +[29] M. A. Schlosshauer, Decoherence and the Quantum-To-Classical Transition (Springer-Verlag Berlin Heidelberg, 2007). +[30] W. G. Unruh, in Relativistic Quantum Measurement and Decoherence, edited by H.-P. Breuer and F. Petruccione (Springer +Berlin Heidelberg, Berlin, Heidelberg, 2000) pp. 125–140. + diff --git a/2tFKT4oBgHgl3EQf7y43/content/tmp_files/load_file.txt b/2tFKT4oBgHgl3EQf7y43/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7abd124552b4e0d01fa1df709d56605a03091620 --- /dev/null +++ b/2tFKT4oBgHgl3EQf7y43/content/tmp_files/load_file.txt @@ -0,0 +1,644 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf,len=643 +page_content='On the motion of an electron through vacuum fluctuations Anirudh Gundhi1, 2, ∗ and Angelo Bassi1, 2, † 1Department of Physics, University of Trieste, Strada Costiera 11, 34151 Trieste, Italy 2Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy (Dated: January 31, 2023) We study the effects of the electromagnetic vacuum on the motion of a non-relativistic electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To this end, the vacuum is treated as the environment and the electron as the system within the framework of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After tracing over the environmental degrees of freedom, we obtain the time evolution of the reduced density matrix of the electron in the position basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Using the master equation, in the first part of the article we derive the equation of motion for the expectation value of the position operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In the presence of an external potential, the equation turns out to be the same as its classical counterpart: the Abraham-Lorentz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' However, in its absence, the dynamics is free of the runaway solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In the second part of the article we study decoherence induced by vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We show that decoherence that appears at the level of the reduced density matrix does not correspond to actual irreversible loss of coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Numerous physical phenomena such as the Casimir ef- fect [1–3], the Unruh effect [4–6] and the Lamb shift [7– 10] are attributed to the presence of vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The possibility of decoherence due to vacuum fluctua- tions, as being fundamental and unavoidable, has also been discussed in various works [11–18] without arriving at a general consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The interaction of an electron with the vacuum fluctu- ations can be studied within the framework of open quan- tum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We use this formalism to study two spe- cific phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' First, we derive the equation of motion (EOM) for the electron in the presence of an external po- tential that provides a quantum mechanical description of radiation emission by an accelerated electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Sec- ond, we investigate if the interactions with the vacuum fluctuations alone can lead to spatial decoherence of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The quantum mechanical version of the classical Abraham-Lorentz (AL) equation, which describes the re- coil force experienced by an accelerated electron due to the emission of radiation [19–22], has been previously derived, for example, in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Instead of the electron’s position, the equation was obtained for the position op- erator and it was then argued why this operator equa- tion is fundamentally different from the classical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The difficulties in making a direct connection with the classical dynamics were attributed to the presence of the additional transverse electric field operator of the electro- magnetic vacuum, which is zero classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Similar prob- lem persists concerning the interpretation of the quantum Langevin equation obtained in [17] for an electron inter- acting with vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In our work, we use the path-integral formalism to ob- tain the explicit expression of the reduced density matrix in the position basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The formalism used is adopted from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Within this framework, instead of the Langevin ∗ anirudh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='gundhi@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='it † abassi@units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='it equation, we derive the master equation which yields the EOM for the expectation value of the position operator which provides a direct correspondence with the classical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In the presence of an arbitrary potential, we show that the classical EOM is the same as the one ob- tained from the reduced quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Moreover, the equation that emerges after a quantum mechanical treatment appears to be free of the problems associated with the AL equation: the existence of the runaway so- lution which leads to an exponential increase of the elec- tron’s acceleration, even in the absence of an external potential [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Concerning decoherence, we show that the loss of co- herence due to vacuum fluctuations at the level of the re- duced density matrix is only apparent and reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To this end we show that by ‘switching off’ the interactions with the EM field, the original coherence is restored at the level of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Moreover, the expression for the decoherence factor that we obtain differs from the ones obtained in [17, 18] where the authors argue for a finite loss of coherence for momentum superpositions, due to vacuum fluctuations, but with different estimates for the magnitude of decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We work in the Coulomb gauge in which the Lagrangian relevant for the dynamics of a non-relativistic electron in the presence of an external potential and an external radiation field is given by [24] L(t) = 1 2m˙r2 e − V0(re) + � d3rLEM − ereE⊥(re) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (1) Here, re denotes the position of the electron, m the bare mass, e the electric charge, V0(re) an arbitrary bare external potential (acting only on the electron) and LEM := (ϵ0/2) � E2 ⊥(r) − c2B2(r) � in which E⊥ denotes the transverse electric field, B the magnetic field, ϵ0 the permittivity of free space and c the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' As detailed in Appendix A, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (1) is obtained from the general Lagrangian for electrodynamics under the non- relativistic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Following the standard prescription, the EM field is quantized by quantizing the transverse vector potential arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='11946v1 [quant-ph] 27 Jan 2023 2 ˆA⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In terms of its conjugate momentum ˆΠ (which is not proportional to E⊥ due to the form of the interaction term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (1), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix A), we define and work with ˆΠE = − ˆΠ/ϵ0, since it appears repeatedly in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Further, the quantized EM field is initially assumed to be in its vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The master equation via path integral formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The position basis representation of the full density matrix within the path integral formalism is given by [23, 25] ⟨x′ f| ˆρ(t) |xf⟩ = � D[x, x′]e i ℏ (S′ T−ST)ρ(x′ i, xi, ti) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (2) describes the density matrix at some final time t, starting from an initial time ti, such that xi := x(ti), x′ i := x′(ti), with S′ T := ST[x′] (and similarly ST := ST[x]) denoting the full action describing some general dynamics along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The path integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (2) is computed with the boundary conditions xf = x(t), x′ f = x′(t), and includes the integral over xi and x′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In our case, the quantized radiation field is treated as the environment, initially assumed to be in its vacuum state and the electron as the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We are interested in the reduced effective dynamics of the electron having taken into account its interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This is described by the reduced density matrix ˆρr which is obtained after tracing over the environmental degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After performing the trace, by assuming the initial density matrix to be in the product state ˆρ(ti) = ˆρS(ti) ⊗ ˆρEM(ti), ˆρr takes the form [23] (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix B) ρr(x′ f, xf, t) = � D[x, x′]e i ℏ (S′ S−SS+SIF[x,x′])ρr(x′ i, xi, ti) , with, SIF = 1 2 � t ti dt1dt2xa(t1)Mab(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2)xb(t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (3) Here, SS denotes the action corresponding to the sys- tem Hamiltonian (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix A) and, under the Ein- stein summation convention, we have introduced the vec- tor notation xa(t1) = x(t1) for a = 1 and xa(t1) = x′(t1) for a = 2 such that the matrix elements Mab are re- lated to the two-point correlations of the canonical trans- verse electric field operator ˆΠE (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Since the electron’s motion is considered to be along the x- axis only, the two-point correlations involve only the x- component of ˆΠE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In terms of the creation and annihi- lation operators, and the x-component of the unit polar- ization vector εx k, it is given by [26] ˆΠE(r, t) = iC � d3k √ k � ε ˆaε(k)ei(k·r−ωt)εx k + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='c , (4) with the constant prefactor C := � ℏc/(2ϵ0(2π)3) � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' By making a change of basis to (X(t), u(t)) with X(t) = (x(t) + x′(t))/2 and u(t) = x′(t) − x(t), the so-called influence functional SIF [27] takes the simplified form SIF[X, u](t) = � t ti dt1dt2 � iu(t1)N(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2)u(t2) 2 + u(t1)D(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2)X(t2)] , (5) where the noise kernel N(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) and the dissipation ker- nel D(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) are defined to be N(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) := e2 2ℏ ⟨0| {ˆΠE(t1), ˆΠE(t2)} |0⟩ , D(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) :=ie2 ℏ ⟨0| � ˆΠE(t1), ˆΠE(t2) � |0⟩ θ(t1 − t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (6) Here, |0⟩ is the vacuum state of the free radiation field and θ(τ) is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' As in [17, 18], we have also used the standard non-relativistic dipole ap- proximation in which one ignores the spatial dependence of the EM fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' From the definitions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (6) and the expression for ˆΠE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (4), the explicit expressions for the noise and the dissipation kernels can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is important to note that the evaluation of the ker- nels necessiates the introduction of a high frequency cut- off in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This is due to the fact that the expressions for the kernels, which only depend upon the difference τ := t1 − t2, diverge at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' A cure is pro- vided by the standard Hadamard finite part prescription [23] which introduces the convergence factor e−k/kmax in- side the integrals appearing in the vacuum expectation values of the commutator and the anti-commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In terms of ϵ = 1/ωmax, with ωmax = kmaxc being the high frequency cut-off, the kernels read (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix C) N(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) = N(τ) = e2 π2ϵ0c3 � ϵ4 − 6ϵ2τ 2 + τ 4� (ϵ2 + τ 2)4 , (7) D(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) = D(τ) = e2 3πϵ0c3 θ(τ) d3 dτ 3 δϵ(τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (8) The function πδϵ(τ) := ϵ/(τ 2 + ϵ2) appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (8) behaves like a Dirac delta for τ ≫ ϵ but is non-singular at τ = 0 due to the finite cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We refer to Appendix C for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Following [23], starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (3) and using the ex- plicit functional form of SIF in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (5), the master equa- tion for the reduced density matrix can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Upto second order in the interactions, we obtain its expression to be (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix B for a detailed derivation) ∂tˆρr(t) = − i ℏ � ˆHs, ˆρr(t) � − 1 ℏ � t−ti 0 dτN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) [ˆx, [ˆxHs(−τ), ˆρr(t)]] + i 2ℏ � t−ti 0 dτD(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) [ˆx, {ˆxHs(−τ), ˆρr(t)}] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (9) The first line of the master equation is the usual Liouville- von Neuman evolution and involves only the system Hamiltonian ˆHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In the second and the third lines, which encode the system’s interaction with the environment, the operator ˆxHs(−τ) is used as a place holder for the expression ˆxHs(−τ) := ˆU −1 s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t)ˆx ˆUs(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) , (10) 3 where ˆUs(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) is the unitary operator that evolves the statevector of the system from time t to t − τ via the system Hamiltonian ˆHs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The operator ˆx without the subscript is the usual Schr¨odinger operator such that ˆxHs(0) = ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Note that due to the coupling between the position of the electron and the transverse electric field in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (1), the system Hamiltonian receives an additional contribu- tion such that ˆHs = ˆp2/(2m) + ˆV0(x) + ˆVEM(x), where, having introduced a cut-off scale in the calculations and considering the motion of the electron along the x-axis only, ˆVEM(x) = e2ω3 max 3π2ϵ0c3 ˆx2 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We point out that since the master equation is valid upto second order in the interactions and since the operator ˆxHs(−τ) appears alongside the dissipation and the noise kernels (which are already second order in e), the time evolu- tion governed by ˆUs(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (10) is understood to involve only ˆV0 and not ˆVEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Therefore, upto second order in the interactions, ˆVEM only contributes via the Liouville-von Neuman term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Using the master equation (9), we obtain the coupled equations for the time evolution of ⟨ˆx⟩ and ⟨ˆp⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is interesting to compare the quantum mechanical EOM with the one derived classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Within classical electrodynamics, a charged spherical shell of radius R which is accelerated by an external force Fext, experiences an extra recoil force (radiation reaction) due to the emission of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' By taking the limit R → 0 in the equation describing its dynamics, one obtains the Abraham-Lorentz formula mR¨x = Fext + 2ℏα 3c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='x , (11) where mR denotes the observed renormalized mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' See for example [20, 28] and the references therein for the derivation of the AL formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The triple derivative term appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (11) can be interpreted as the friction term that leads to energy loss due to radiation emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For instance, when the external potential is taken to be V0(x) = (1/2)mω2 0x2, one has .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='x ≈ −ω2 0 ˙x [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' However, the issue with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (11) is that the same triple derivative term persists even when the external potential is switched off, leading to an exponentially increase of the particle’s acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' A discussion of the AL formula and the problems associated with it can be found in [19–21, 28] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In the case that we are considering, the rate of change of the expectation values is calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The coupled differential equations for ⟨ˆx⟩ and ⟨ˆp⟩ are given by (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix E) d dt⟨ˆx⟩ =Tr(ˆx ˙ˆρr) = ⟨ˆp⟩ m , (12) d dt⟨ˆp⟩ = − ⟨ ˆV0,x ⟩ + Tr � ˆρr(t) � t−ti 0 dτD(τ)ˆxHs(−τ) � − 2e2ω3 max⟨ˆx⟩/(3π2ϵ0c3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (13) While it might not be apparent at the first glance, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (13) is actually local in time due the form of the dissipation kernel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To see this explicitly, the integral involving the dissipation kernel needs to be eval- uated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In order to do so, we integrate by parts such that the derivatives acting on δϵ (which appear in the expres- sion obtained for the dissipation kernel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (8)) are shifted onto the adjacent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The integral is calcu- lated explicitly in Appendix D and the following identity is derived � t 0 dτD(τ)f(τ) = − 2αℏ 3c2 f ′′′(0) − 4αℏωmax 3πc2 f ′′(0) + 2e2ω3 maxf(0)/(3π2ϵ0c3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (14) Here, the prime denotes the derivative taken with respect to τ and α = e2/(4πϵ0ℏc) the fine structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Using the identity (14), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (13) becomes d dt⟨ˆp⟩ = − ⟨ ˆV0,x ⟩ − 4αℏωmax 3πc2 Tr � ˆρr(t) d2 dτ 2 ˆxHs(−τ) ���� τ=0 � − 2αℏ 3c2 Tr � ˆρr(t) d3 dτ 3 ˆxHs(−τ) ���� τ=0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (15) We see that in the EOM (15) only the original bare po- tential ˆV0 remains, because the contribution coming from ˆVEM in the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (13) is canceled by the term in the last line of the integral (14), after one introduces the cut-off consistently throughout the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For more details we refer to Appendices A and E, or Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' [17] where the same cancellation was argued for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The time derivatives of ˆxHs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (15) can be easily computed, since from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (10) we have the relation (upto leading order in the interactions) d dτ ˆxHs(−τ) = − i ℏ � ˆV0(x) + ˆp2 2m, ˆxHs(−τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (16) First we consider the situation when the external poten- tial is switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (16), with ˆV0(x) = 0, taking another time derivative of ˆxHs we get d2 dτ 2 ˆxHs(−τ) ���� τ=0 = �−i ℏ �2 � ˆp2 2m, � ˆp2 2m, ˆx �� = 0 , (17) where, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (17), we have also used the relation ˆxHs(0) = ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Similarly, the third derivative term appear- ing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (15) also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Therefore, when ˆV0(x) = 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (15) simply reduces to d dt⟨ˆp⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (18) Unlike the AL formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (11), we see that upto sec- ond order in the interactions there are no solutions which allow for an exponential increase of the particle’s accel- eration in the absence of an external potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Next we consider the case when the external potential is switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' When the potential does not depend ex- plicitly on time, the double and triple derivative terms 4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (15) yield double and triple commutators with re- spect to the system Hamiltonian respectively (discarding ˆVEM upto second order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (15) can then be written as d dt⟨ˆp⟩ =Fext + 4αℏωmax 3πc2 Tr � 1 ℏ2 ˆρr(t) � ˆHs, � ˆHs, ˆx ��� − 2αℏ 3c2 Tr � i ℏ3 ˆρr(t) � ˆHs, � ˆHs, � ˆHs, ˆx ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (19) Here, we have defined Fext := −⟨ ˆV0(x),x ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Due to the presence of ˆV0(x), the commutators of ˆHs with ˆx no longer vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To simplify the equation further, we shift the commutators onto the density matrix using the cyclic property Tr(ˆa · [ˆb, ˆc]) = Tr([ˆa,ˆb] · ˆc) such that Tr � ˆρr � ˆHs, � ˆHs, ˆx ��� = Tr � ˆx � ˆHs, � ˆHs, ˆρr ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (20) The same relationship is also obtained for the triple com- mutator term, with an additional minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Remem- bering that the master equation is only valid upto second order in the interaction, it is sufficient to evaluate the trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (19) at 0th order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This implies that within the trace, the time dependence of the density matrix can be evaluated only by retaining the Liouville-von Neuman term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (20) thus becomes proportional to Tr(ˆx¨ˆρr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' With these simplifica- tions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (19) can be written as mR d2 dt2 ⟨ˆx⟩ = Fext + 2αℏ 3c2 d3 dt3 ⟨ˆx⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (21) After identifying the observed electron mass with the re- normalized mass mR := m + (4αℏωmax)/(3πc2), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (21) reduces to the Abraham-Lorentz formula (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The same result is also obtained for the general case in which the bare potential ˆV0(x, t) depends explicitly on time, as shown in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We remark that the equation of motion derived quantum mechanically only reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (11) in the presence of an external potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' When the external potential is switched off, the EOM reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (18) and is therefore free of the runaway solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In this final part of the article, we are inter- ested in assessing if the spatial superposition of a charged particle at rest can be suppressed via its interaction with the vacuum fluctuations alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We begin by writing the position space representation of the master equation (9) relevant for decoherence ∂tρr = � −(x′ − x)2N1(t) ℏ � ρr , (22) where N1(τ) is defined to be N1(τ) := � τ 0 dτ ′N(τ ′) = −4αℏ(τ 3 − 3τϵ2)(τ 2 +ϵ2)−3(3πc2)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We have set ti = 0 and only retained the second term involving the noise ker- nel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This is because the other terms typically give subdominant contributions when the question of in- terest is to evaluate the rate of decay of the off-diagonal elements of the density matrix at late times [23, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We have also used the expression of the noise kernel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (7) inside the integral to obtain the expression for N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Inte- grating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (22) we get ρr(x′, x, t) = exp � −(x′ − x)2 ℏ N2(t) � ρr(x′, x, 0) , (23) where N2(t) := � t 0 dτN1(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The function N2(t) is in- versely proportional to the coherence length lx(t) defined by lx(t) := (ℏ/N2(t)) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After performing the integral over N1 the expression for the coherence length is ob- tained to be lx(t) = � 3πc2 2αω2 max (t2 + ϵ2)2 t4 + 3t2ϵ2 t≫ϵ = � 3π 2α 1 kmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (24) We see that the coherence length approaches a constant value on time scales much larger than ϵ = 1/ωmax and that its value scales inversely with the UV cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Taken literally, if one sets kmax = 1/λdb, where λdb is the de Broglie wavelength of the electron, one would arrive at the conclusion that vacuum fluctuations lead to decoher- ence with the coherence length of the charged particle asymptotically reducing to lx ≈ 25λdb within the time scales t ≈ λdb/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' False Decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is clearly unsatisfactory to have an observable effect scale explicitly with the UV cut-off, since the precise numerical value of the cut-off is, strictly speaking, arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' A similar situation was encountered in [30] in a different context of a harmonic oscillator cou- pled to a massive scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' However, it was argued in [30] that the reduced density matrix of the harmonic os- cillator described false decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In such a situation, the off-diagonal elements of the density matrix are sup- pressed simply because the state of the environment goes into different configurations depending upon the spatial location of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' However, these changes in the environmental states remain locally around the system and are reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For the electron interacting with vac- uum fluctuations, we therefore take the point of view that if the reduced density matrix describes false deco- herence, then after adiabatically switching off the inter- actions with the environment (after having adiabatically switched it on initially), the original coherence must be fully restored at the level of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To formulate the argument we consider a time depen- dent coupling q(t) = −ef(t) such that f(t) = 1 for most of the dynamics between the initial time t = 0 and the final time t = T, while f(0) = f(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The quantity relevant for decoherence is the noise kernel which, under the time-dependent coupling, transforms as N → ˜ N = f(t1)f(t2)N(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) = f(t1)f(t2)N(t1 − t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The decoherence factor in the double commutator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (9) involves replacing t2 with t1 − τ and then in- tegrating over τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Therefore, the function N1 transforms as N1 → ˜ N1, with ˜ N1 given by ˜ N1(t1) = f(t1) � t1 0 dτf(t1 − τ)N(τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (25) 5 From the definitions of N1 and N2 we have N1 = (d/dτ)N, N2 = (d/dτ)N1 and N1(0) = N2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Using these relations and integrating by parts, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (25) becomes ˜ N1(t1) =f(t1)N1(t1)f(0) + f(t1)N2(t1) ˙f(0) + f(t1) � t1 0 dτN2(τ) d2 dτ 2 f(t1 − τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (26) In the limit ϵ → 0 (taking the UV cut-off to infinity), we see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (24) that N2 looses any time dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We can therefore bring N2 outside the integral such that ˜ N1(t1) = f(t1)N1(t1)f(0)+f(t1)N2 ˙f(0)−f(t1)N2( ˙f(0)− ˙f(t1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The terms involving ˙f(0) cancel out and we get ˜ N1(t1) = f(t1)N1(t1)f(0) + f(t1)N2 ˙f(t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (27) After integrating by parts Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (27), in order to obtain ˜ N2(T) = � T 0 dt1 ˜ N1(t1), we get ˜ N2(T) =f(0) (f(T)N2(T) − f(0)N2(0)) − f(0)N2 � T 0 dt1 ˙f + N2 2 � T 0 dt1 d dt1 f 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (28) In the limit ϵ → 0, as we noted earlier, N2(t) takes a constant value for any time t > 0 but is zero at t = 0 from the way it is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Therefore, after completing the remaining integrals, we get ˜ N2(T) = N2 2 � f 2(0) + f 2(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (29) Since we assume that the interactions are switched off in the very beginning and at the very end, we see that ˜ N2(T) = 0 such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (23) becomes ˜ρr(x′, x, T) = ρr(x′, x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Therefore, by adiabatically switching off the interactions we recover the original coherence within the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This is different from standard collisional decoherence where, for example, one originally has ∂tρr(x′, x, t) = −Λ(x′ − x)2ρr(x′, x, t) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' When in this case we send Λ → ˜Λ = f(t)Λ, we get ˜ρr(x′, x, t) = exp � −Λ(x′ − x)2 � t 0 dt′f(t′) � ρr(x′, x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The density matrix depends on the integral of f(t) rather than its end points and we see that coherence is indeed lost ir- reversibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We interpret this result to imply that the vacuum fluctuations alone do not lead to irreversible loss of coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Moreover, our results imply that the ap- parent decoherence cannot be due to emission of photons as otherwise one would not be able to retrieve the coher- ence back into the system simply by switching off the interactions with the environment at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We formulated the interaction of a non- relativistic electron with the radiation field within the framework of open quantum systems and obtained the master equation for the reduced electron dynamics in the position basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We showed that the classical limit of the quantum dynamics is free of the problems associated with the purely classical derivation of the Abraham-Lorentz formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' With respect to possible decoherence induced by vacuum fluctuations alone, we showed that the ap- parent decoherence at the level of the reduced density matrix is reversible and is an artifact of the formalism used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In mathematically tracing over the environment, one traces over the degrees of freedom that physically surround the system being observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' These degrees of freedom must be considered part of the system being ob- served, rather than the environment [16, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We formu- lated this interpretation by showing that one restores full initial coherence back into the system after switching off the interactions with the environment adiabatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The formulation is fairly general and might also be used in other situations to distinguish true decoherence from a false one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The analysis therefore brings together various works in the literature [15–18, 30] and addresses some of the conflicting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' thanks Davide Bason and Lorenzo Di Pietro for numerous discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We thank Oliviero Angeli for cross checking some of the results ob- tained in the manuscript and Lajos Di´osi for discussions concerning false decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' acknowledges finan- cial support from the EIC Pathfinder project QuCoM (GA no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' 101046973) and the PNRR PE National Quan- tum Science and Technology Institute (PE0000023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We thank the University of Trieste and INFN for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix A: The Lagrangian and the Hamiltonian formulation In the Coulomb gauge, the standard Lagrangian for electrodynamics is given by [24] L = 1 2m˙r2 e − V0(re) − � 1/2 d3k |ρ|2 ϵ0k2 + ϵ0 2 � d3r � E2 ⊥(r) − c2B2(r) � + � d3rj(r) · A⊥(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A1) In addition to the terms that have been described in the main article, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A1) also includes the Coulomb potential between different particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is given by the third term in which ρ(r) denotes the charge density and the symbol � 1/2 means that the integral is taken over half the volume in the reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For a single particle, it reduces to the particle’s Coulomb self energy ECoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After the introduction of a suitable cut-off it takes a finite value given by ECoul = αℏωmax/π [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The transverse vector potential is denoted by A⊥(r, t) whose negative partial time derivative yields the transverse electric field E⊥(r, t) while its curl gives the magnetic field B(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For an electron, the current 6 density is given by j(r) = −e˙rδ(r−re) and the interaction term becomes −e˙reA⊥(re, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For a non-relativistic charged particle, the time derivative can be shifted from the position of the particle onto the transverse vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This is because in addition to a total derivative term, a term of the form erevi∂iA⊥(r, t) appears (where vi := ˙ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After the wave expansion of A⊥, this term is seen to be negligible with respect to ere ˙A⊥(re, t) = −ereE⊥(re, t) as long as ωk ≫ vk or v ≪ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Therefore, for the non-relativistic electron, the Lagrangian relevant for the dynamics reduces to L(t) ≈ 1 2m˙r2 e − V0(re) + ϵ0 2 � d3r � E2 ⊥(r) − c2B2(r) � − ereE⊥(re) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A2) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A2) the total derivative d/dt(reA⊥(re)) and the constant Coulomb self energy term have been omitted as these do not affect the electron’s dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The Hamiltonian corresponding to the Lagrangian (A2) can now be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In terms of the canonical variables re, p, A⊥ and ΠE := − 1 ϵ0 Π, it takes the form H = HS + HEM + Hint , (A3) where HEM = ϵ0 2 � d3r(Π2 E(r) + c2B2(r)) is the free field Hamiltonian of the radiation field, Hint = ereΠE(re) the interaction term and HS the system Hamiltonian given by HS = p2 2m + V0(re) + e2 2ϵ0 � d3rriδ⊥ im(r − re)δ⊥ mj(r − re)rj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A4) Here, the transverse Dirac delta δ⊥ ij(r − re), which appears due to the coupling of the position of the electron with the transverse electric field, is defined to be [22] δ⊥ ij(r − re) := 1 (2π)3 � d3k � δij − kikj k2 � eik·(r−re) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A5) The form of HS calls for an identification of the full effective potential V (re) governing the dynamics of the electron such that V (re) := V0(re) + VEM(re) , VEM(re) = e2 2ϵ0 � d3rriδ⊥ im(r − re)δ⊥ mj(r − re)rj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A6) Note that the extra term VEM(re) is not added to the bare potential by hand, but arises naturally due to the reE⊥ coupling [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Although it gives a divergent contribution e2 2ϵ0 δ⊥ ij(0)ri erj e, after regularizing the transverse delta function on a minimum length scale rmin = 1/kmax, the contribution coming from this term scales as O( e2 2ϵ0 r2 ek3 max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To be more precise, we impose the cut-off consistently throughout the calculations by introducing the convergence factor e−k/kmax inside the integral in the reciprocal space (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Using this procedure, the expression for δ⊥ ij(0) is obtained to be δ⊥ ij(0) = 1 (2π)3 � dkk2e−k/kmax � dΩ � δij − kikj k2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A7) First evaluating the angular integral, which gives a factor 8π 3 δij, and then the radial integral, we get VEM(re) = e2ω3 max 3π2ϵ0c3 r2 e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A8) Since the contribution of VEM(re) is canceled exactly by another term, as shown in the discussion around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (15) of the main text, for all practical purposes, VEM(re) has no consequences on the dynamics of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix B: The master equation The probability amplitude for a particle to be at the position xf at some final time t, starting from the position xi at some initial time ti, is given by [25] ⟨xf| ˆU(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) |xi⟩ = � x(t)=xf, x(ti)=xi D[x, p]e− i ℏ � t ti dt′(HT[x,p]−p ˙x) = � x(t)=xf, x(ti)=xi D[x]e i ℏ ST[x] , (B1) 7 where HT is the full Hamiltonian and ST is the corresponding action describing some general dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B1) the expression for the density matrix at time t can be written as [23] ⟨x′ f| ˆρ(t) |xf⟩ = � x(t)=xf, x′(t)=x′ f D[x, x′]e i ℏ (ST[x′]−ST[x])ρ(x′ i, xi, ti) , (B2) where the integrals over xi and x′ i are included within the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The expression analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B1) also exists for ⟨pf| ˆU(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) |pi⟩ in which the boundary conditions are fixed on p(t) and the phase-space weighing function is instead given by exp{ −i ℏ � t ti dt′ (HT [x, p] + x ˙p)} such that ⟨pf| ˆU(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) |pi⟩ = � p(t)=pf, p(ti)=pi D[x, p]e− i ℏ � t ti dt′(HT[x,p]+x ˙p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B3) For computing the path integral over the EM field, with a slight abuse of notation, we understand exp{ i ℏSEM} to be simply the appropriate phase-space weighing function appearing inside the path integral with SEM := − � t ti dt′d3r(HEM − Π ˙A⊥) or SEM := − � t ti dt′d3r(HEM + A⊥ ˙Π) depending upon the basis states between which the transition amplitudes are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We are interested in the dynamics of the electron, having taken into account its interaction with the radiation field environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' With this distinction, the total phase-space function can be written as ST = SS[x] + SEM[µ] + Sint[x, ΠE], where SS denotes the system action, SEM[µ] := SEM[A⊥, ΠE] the phase-space function governing the time evolution of the free radiation field in which µ denotes its phase-space degrees of freedom and Sint[x, ΠE] := −e � t ti dt′xΠE encodes the interaction between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The expression for the system-environment density matrix can then be written as � x′ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Πf′ E �� ˆρ(t) ��xf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Πf E � = � x(t)=xf, x′(t)=x′ f D[x, x′]e i ℏ (SS[x′]−SS[x])ρS(x′ i, xi, ti)× × � ΠE(t)=Πf E, Π′ E(t)=Πf′ E D[µ, µ′]e i ℏ (SEM[µ′]+Sint[x′,Π′ E]−SEM[µ]−Sint[x,ΠE])ρEM(Π′ E(ti), ΠE(ti), ti) , (B4) where ���Πf E � denotes the basis state of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Note that the precise choice of the environmental basis states is unimportant since the reduced density matrix is obtained by tracing over the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In writing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B4) we have also assumed the full density matrix ˆρ(ti) to be in the product state ˆρ(ti) = ˆρS(ti) ⊗ ˆρEM(ti) at the initial time ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We notice that SEM[µ] is quadratic in the environmental degrees of freedom while Sint[x, ΠE] is linear in both x and ΠE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After tracing over the environment, that is integrating over ΠE(t) = Π′ E(t), the term in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B4) yields a Gaussian in x such that [23] � ΠE(t)=Π′ E(t) dΠE(t)D[µ, µ′]e i ℏ (SEM[µ′]+Sint[x′,Π′ E]−SEM[µ]−Sint[x,ΠE])ρi EM = e i 2ℏ �� dt1dt2Mab(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='t2)xa(t1)xb(t2) , (B5) where ρi EM := ρEM(Π′ E(ti), ΠE(ti), ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We have also introduced the vector notation with the convention xa = x for a = 1, xa = x′ for a = 2 and xa = ηabxb with ηab = diag(−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is the matrix elements Mab which determine the effective action of the system and contain the information about its interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' They can be obtained by acting with ℏ i δ δxa δ δxb |xa=xb=0 (where xa and xb are set to zero after taking the derivatives) on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B5) such that M ab(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) = ie2 ℏ � ΠE(t)=Π′ E(t) dΠE(t)D[µ, µ′]Πa E (t1) Πb E (t2) e i ℏ (SEM[µ′]−SEM[µ])ρi EM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B6) Here, in the light of the standard non-relativistic dipole approximation, we have ignored the spatial dependence of the canonical fields (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Depending upon the value of the indices a and b, the matrix elements correspond to the expectation values of the time-ordered or path-ordered correlations in the Heisenberg picture [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For the dynamics of the non-relativistic electron that we are considering, the expression for Mab reads Mab(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) = ie2 ℏ � � � ˜T {ˆΠE(t1)ˆΠE(t2)} � 0 − � ˆΠE(t1)ˆΠE(t2) � 0 − � ˆΠE(t2)ˆΠE(t1) � 0 � T {ˆΠE(t1)ˆΠE(t2)} � 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B7) 8 The zero in the subscript denotes that the expectation values are calculated by disregarding the interaction with the system, while T and ˜T denote the time-ordered and the anti-time ordered products respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is also understood that since the electron’s motion is considered to be along the x-axis, the canonical field operator that enters Mab is only the x-component given by [26] ˆΠE(r, t) = i � ℏc 2ϵ0(2π)3 � 1 2 � d3k √ k � ε ˆaε(k)ei(k·r−ωt)εx k + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B8) In our case, the initial state of the environment is taken to be the vacuum state |0⟩ of the radiation field such that ⟨·⟩0 = ⟨0| · |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After tracing over the environment, the reduced density matrix of the electron is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B4) to be ⟨x′ f| ˆρr(t) |xf⟩ = � x(t)=xf, x′(t)=x′ f D[x, x′]e i ℏ (SS[x′]−SS[x]+SIF[x,x′])ρr(x′ i, xi, ti) , (B9) where SIF[x, x′] = ie2 2ℏ � t ti dt1dt2 �� ˜T {ˆΠE(t1)ˆΠE(t2)} � 0 x(t1)x(t2) − � ˆΠE(t1)ˆΠE(t2) � 0 x(t1)x′(t2) − � ˆΠE(t2)ˆΠE(t1) � 0 x′(t1)x(t2) + � T {ˆΠE(t1)ˆΠE(t2)} � 0 x′(t1)x′(t2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B10) The integral � t ti stands for both the t1 and the t2 integrals which run from ti to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Alternatively, the influence functional SIF can be written in the matrix notation as SIF[x, x′] = 1 2 � t ti dt1dt2 �x(t1) x′(t1)� � M11 M12 M21 M22 � � x(t2) x′(t2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B11) As it is more convenient, we make a change of basis to (X , u) defined by X(t) :=(x′(t) + x(t))/2 , u(t) = x′(t) − x(t) , (B12) in which the influence functional transforms as SIF[X, u] = 1 2 � t ti dt1dt2 �X(t1) u(t1)� � ˜ M11 ˜ M12 ˜ M21 ˜ M22 � � X(t2) u(t2) � , (B13) where � ˜ M11 ˜ M12 ˜ M21 ˜ M22 � = � M11 + M12 + M21 + M22 1 2 ((M12 − M21) + (M22 − M11)) 1 2 (−(M12 − M21) + (M22 − M11)) 1 4((M11 + M22) − (M12 + M21)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B14) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B7) we obtain the following relations M11 + M22 = −(M12 + M21) = ie2 ℏ � {ˆΠE(t1), ˆΠE(t2)} � 0 , (B15) M12 − M21 = ie2 ℏ �� ˆΠE(t2), ˆΠE(t1) �� 0 , (B16) M22 − M11 = ie2 ℏ �� ˆΠE(t1), ˆΠE(t2) �� 0 sgn(t1 − t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B17) Using these relations, ˜ M takes the simplified form � ˜ M11 ˜ M12 ˜ M21 ˜ M22 � = ie2 ℏ � � 0 �� ˆΠE(t2), ˆΠE(t1) �� 0 θ(t2 − t1) �� ˆΠE(t1), ˆΠE(t2) �� 0 θ(t1 − t2) 1 2 � {ˆΠE(t1), ˆΠE(t2)} � 0 � � , (B18) where θ(t) is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Thus, in the (X, u) basis, the influence functional in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B10) takes the compact form SIF[X, u](t) = � t ti dt1dt2 � iu(t1)N(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2)u(t2) 2 + u(t1)D(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2)X(t2) � , (B19) 9 where the noise kernel N and the dissipation kernel D are defined as N(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) := e2 2ℏ � {ˆΠE(t1), ˆΠE(t2)} � 0 , D(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t2) :=ie2 ℏ �� ˆΠE(t1), ˆΠE(t2) �� 0 θ(t1 − t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B20) Having determined the full effective action for the electron, in terms of the influence functional, we can now derive the master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B9), it can be seen that the time derivative of the reduced density matrix will have, in addition to the standard Liouville-von Neuman term, the contribution coming from the influence functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In order to compute that we need to evaluate the rate of change of SIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is given by δtSIF[X, u] = u(t) � t ti dt1 (iN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1)u(t1) + D(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1)X(t1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B21) In terms of the original (x, x′) basis, the full expression for the master equation can now be written as ∂tρr(x′ f, xf, t) = − i ℏ ⟨x′ f| � ˆHs, ˆρr � |xf⟩ + i ℏ � x(t)=xf, x′(t)=x′ f D[x, x′]δtSIF[x′, x]e i ℏ (SS[x′]−SS[x]+SIF[x,x′])ρr(x′ i, xi, ti) ≈ − i ℏ ⟨x′ f| � ˆHs, ˆρr � |xf⟩ + i ℏ � x(t)=xf, x′(t)=x′ f D[x, x′]δtSIF[x′, x]e i ℏ (SS[x′]−SS[x])ρr(x′ i, xi, ti) ≈ − i ℏ ⟨x′ f| � ˆHs, ˆρr � |xf⟩ − 1 ℏ(x′ f − xf) � t ti dt1N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1) � x(t)=xf, x′(t)=x′ f D[x, x′](x′(t1) − x(t1))e i ℏ (SS[x′]−SS[x])ρr(x′ i, xi, ti) + i 2ℏ(x′ f − xf) � t ti dt1D(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1) � x(t)=xf, x′(t)=x′ f D[x, x′](x′(t1) + x(t1))e i ℏ (SS[x′]−SS[x])ρr(x′ i, xi, ti) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B22) The Liouville-von Neuman evolution is governed by the system Hamiltonian ˆHs alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For the second term on the right hand side in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B22), we have omitted SIF in the exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This is because SIF is second order in the coupling constant and is already present adjacent to the exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Since we limit our calculations to second order in the interactions, SIF can be neglected inside the exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To simplify the master equation further, we note that the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B22) can be written much more compactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This is due to the following identity [23] � x(t)=xf, x′(t)=x′ f D[x, x′]x′(t1)e i ℏ (SS[x′]−SS[x])ρr(x′ i, xi, ti) = = � dx′(t1) ⟨x′ f| ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1) |x′(t1)⟩ x′(t1) ⟨x′(t1)| ˆUs(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti)ˆρr(ti) ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) |xf⟩ = ⟨x′ f| ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1)ˆx ˆUs(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti)ˆρr(ti) ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) |xf⟩ = ⟨x′ f| ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1)ˆx ˆUs(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti)ˆρr(ti) ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) |xf⟩ = ⟨x′ f| ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1)ˆx ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t1)ˆρr(t) |xf⟩ = ⟨x′ f| ˆxHs(−τ)ˆρr(t) |xf⟩ , (B23) where ˆxHs(−τ) := ˆU −1 s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t)ˆx ˆUs(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) , τ := t − t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B24) Similarly, we also have � x(t)=xf, x′(t)=x′ f D[x, x′]x(t1)e i ℏ (SS[x′]−SS[x])ρr(x′ i, xi, ti) = ⟨x′ f| ˆρr(t)ˆxHs(−τ) |xf⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B25) The operator ˆxHs(−τ) is understood to be simply the placeholder for the expression that appears on the right hand side of the first equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B24) such that ˆxHs(0) = ˆx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B26) 10 Here, the operator ˆx without the subscript Hs is the usual position operator in the Schr¨odinger picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Using these relations, and replacing the t1 integral with the τ integral (t1 = t − τ), the master equation takes the compact form ∂tρr(x′ f, xf, t) = − i ℏ ⟨x′ f| � ˆHs, ˆρr(t) � |xf⟩ − 1 ℏ(x′ f − xf) � t−ti 0 dτN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) ⟨x′ f| [ˆxHs(−τ), ˆρr(t)] |xf⟩ + i 2ℏ(x′ f − xf) � t−ti 0 dτD(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) ⟨x′ f| {ˆxHs(−τ), ˆρr(t)} |xf⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B27) The eigenvalues outside of the integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B27) can be obtained by acting with the position operator ˆx such that ⟨x′ f| ∂tˆρr |xf⟩ = − i ℏ ⟨x′ f| � ˆHs, ˆρr(t) � |xf⟩ − 1 ℏ � t−ti 0 dτN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) ⟨x′ f| [ˆx, [ˆxHs(−τ), ˆρr(t)]] |xf⟩ + i 2ℏ � t−ti 0 dτD(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) ⟨x′ f| [ˆx, {ˆxHs(−τ), ˆρr(t)}] |xf⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B28) The master equation in the operator form can therefore be written as ∂tˆρr = − i ℏ � ˆHs, ˆρr � − 1 ℏ � t−ti 0 dτN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) [ˆx, [ˆxHs(−τ), ˆρr(t)]] + i 2ℏ � t−ti 0 dτD(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ) [ˆx, {ˆxHs(−τ), ˆρr(t)}] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B29) Appendix C: The dissipation and the noise kernels In order to solve the master equation (B29), the kernels need to be evaluated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To achieve that, we begin with the expression for the vacuum expectation value of the correlator ⟨0| ˆΠE(x(t1), t1)ˆΠE(x(t2), t2) |0⟩ = −iℏc 2ϵ04π2 ˆ□ �1 r � ∞ 0 dke−ikcτ � eikr − e−ikr�� , (C1) where r := |x(t1) − x(t2)| , τ := t1 − t2 , ˆ□ := − 1 c2 ∂2 τ + ∂2 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C2) Here, the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C1) is obtained with the help of the expression of the quantized canonical transverse electric field operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C1) becomes convergent after resorting to the standard Hadamard finite part prescription [23], in which the convergence factor e−ωk/ωmax is introduced inside the integral (with ωk = kc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Physically, this prescription cuts off the contribution coming from the modes ωk ≫ ωmax and mathematically it is the same as using the iϵ prescription where one sends τ → τ − iϵ, with ϵ = 1/ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' After completing the integral by using this prescription we get ⟨0| ˆΠE(1)ˆΠE(2) |0⟩ = ℏc 4π2ϵ0 ˆ□ � 1 r2 − c2(τ − iϵ)2 � = ℏc π2ϵ0 1 (r2 − c2(τ − iϵ)2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C3) For the correlator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C3), we ignore the spatial dependence of the fields in the spirit of the non-relativistic approximation r ≪ cτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In this limit, the correlator becomes ⟨0| ˆΠE(1)ˆΠE(2) |0⟩ ≈ ℏ π2ϵ0c3 (τ − iϵ)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C4) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C4), we obtain the explicit functional form of the noise and the dissipation kernels to be N(τ) = e2 π2ϵ0c3 � ϵ4 − 6ϵ2τ 2 + τ 4� (ϵ2 + τ 2)4 , (C5) D(τ) = 8e2 π2ϵ0c3 ϵτ(ϵ2 − τ 2) (ϵ2 + τ 2)4 θ(τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C6) 11 With some algebraic manipulation, the dissipation kernel can be expressed more compactly as D(τ) = e2 3π2ϵ0c3 θ(τ) d3 dτ 3 � ϵ τ 2 + ϵ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C7) Noticing that ϵ τ 2 + ϵ2 = d dτ tan−1(τ/ϵ) = πδϵ(τ) , (C8) we arrive at the expression D(τ) = e2 3πϵ0c3 θ(τ) d3 dτ 3 δϵ(τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C9) The last equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C8) can be understood in the limit ϵ → 0 when the function tan−1(τ/ϵ) takes the shape of a step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Such an expression for D would yield infinite results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For that, we keep in mind that these functions are always well behaved for a finite ϵ and that δϵ only behaves like a Dirac delta for τ ≫ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Appendix D: Integrals involving the dissipation kernel In this section we derive an identity involving the integrals of the form � dτD(τ)f(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To proceed, we keep in mind the situation where ϵ is small but finite so that all the derivatives of the smoothed Dirac delta are large but finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' However, for times τ ≫ ϵ, we have δϵ(τ) = δ′ ϵ(τ) = δ′′ ϵ (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In addition, since the derivative of the Dirac delta is an odd function of τ, we also have δ′ ϵ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' In computing the integral of D(τ) multiplying an arbitrary function f(τ), we shift the derivatives acting on δϵ one by one onto f(τ) by integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Since the calculations of interest involve integrating � t 0 dτD(τ)f(τ), where τ takes only non-negative values from 0 to t, the step function θ(τ) can be omitted inside the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The first integration by parts gives (the constant pre-factors appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C9) will be plugged in at the end) � t 0 dτδ′′′ ϵ (τ)f(τ) = − � t 0 dτδ′′ ϵ (τ)f ′(τ) + δ′′ ϵ (τ)f(τ)|t 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (D1) Since δ′′ ϵ (t) = 0, only the boundary term −δ′′ ϵ (0)f(0) survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Further, − � t 0 dτδ′′ ϵ (τ)f ′(τ) = � t 0 dτδ′ ϵ(τ)f ′′(τ) − δ′ ϵ(τ) f ′(τ)|t 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (D2) Since δ′ ϵ(t) = δ′ ϵ(0) = 0 (δ′ ϵ(τ) being an odd function of τ), both the boundary terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Proceeding further we get � t 0 dτδ′ ϵ(τ)f ′′(τ) = − � t 0 dτδϵ(τ)f ′′′(τ) + δϵ(τ) f ′′(τ)|t 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (D3) As before, the boundary term at τ = t is zero and only the term −δϵ(0)f ′′(0) survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Finally, since δϵ(τ) goes to zero much faster than a generic function f(τ) for a small ϵ, it can be treated like a Dirac delta such that − � t 0 dτδϵ(τ)f ′′′(τ) = −f ′′′(0) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (D4) The factor of half comes because the integral is performed from 0 to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Collecting the two boundary terms we get the result � t 0 dτδ′′′ ϵ (τ)f(τ) = −f ′′′(0) 2 − δϵ(0)f ′′(0) − δ′′ ϵ (0)f(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (D5) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C8) we have δϵ(0) = 1/(πϵ) = ωmax/π and δ′′ ϵ (0) = −2ω3 max/π such that � t 0 dτD(τ)f(τ) = −2αℏ 3c2 f ′′′(0) − 4αℏωmax 3πc2 f ′′(0) + 2e2ω3 max 3π2ϵ0c3 f(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (D6) Here, we have now plugged in the constant prefactor appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (C9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' 12 Appendix E: The Abraham-Lorentz equation as a classical limit The rate of change of the expectation values can be obtained with the help of the master equation (B29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' For the position operator it is given by d dt⟨ˆx⟩ = Tr (ˆx∂tˆρr) = − i ℏTr � ˆx · � ˆHs, ˆρr �� + i 2ℏ � t−ti 0 dτD(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ)Tr (ˆx · [ˆx, {ˆxHs(−τ), ˆρr(t)}]) − 1 ℏ � t−ti 0 dτN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ)Tr (ˆx · [ˆx, [ˆxHs(−τ), ˆρr(t)]]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E1) Due to the identity Tr � ˆA · � ˆB, ˆC �� = Tr �� ˆA, ˆB � ˆC � , (E2) the terms involving the dissipation and the noise kernels vanish and we get d dt⟨ˆx⟩ = − i ℏTr � ˆρr · � ˆx, ˆHs �� = ⟨ˆp⟩ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E3) Here, we remember that the system Hamiltonian ˆHs receives a contribution from ˆVEM in addition to the bare potential ˆV0 such that (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' the discussion between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (A4) and (A8)) ˆHs(t) = ˆp2 2m + ˆV0(x, t) + e2ω3 max 3π2ϵ0c3 ˆx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E4) Similarly, for the momentum operator we obtain the relation d dt⟨ˆp⟩ = Tr (ˆp∂tˆρr) = − i ℏTr �� ˆp, ˆHs � ˆρr � + i 2ℏ � t−ti 0 dτD(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ)Tr ([ˆp, ˆx] · {ˆxHs(−τ), ˆρr(t)}) − 1 ℏ � t−ti 0 dτN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t − τ)Tr ([ˆp, ˆx] · [ˆxHs(−τ), ˆρr(t)]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E5) Since [ˆx, ˆp] = iℏ1, the term involving the noise kernel vanishes and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E5) simplifies to d dt⟨ˆp⟩ = −⟨ ˆV0,x ⟩ − 2e2ω3 max 3π2ϵ0c3 ⟨ˆx⟩ + Tr � ˆρr(t) � t−ti 0 dτD(τ)ˆxHs(−τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E6) Evaluating the integral using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (D6), we see that the last term in the integral gives the contribution 2e2ω3 max 3π2ϵ0c3 ⟨ˆx⟩ to d dt⟨ˆp⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E6) and cancels the contribution coming from ˆVEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The EOM therefore reduces to d dt⟨ˆp⟩ = −⟨ ˆV0(x),x ⟩ − 2αℏ 3c2 Tr � ˆρr(t) d3 dτ 3 ˆxHs(−τ) ���� τ=0 � − 4αℏωmax 3πc2 Tr � ˆρr(t) d2 dτ 2 ˆxHs(−τ) ���� τ=0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E7) As shown in the main article, when ˆV0(x, t) = 0, the double and the triple derivatives acting on ˆxHs(−τ) vanish upto second order in the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Here, we only focus on the general case in which the external (time-dependent) potential is switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To simplify the equation further, we begin by evaluating the second order derivative in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (B24) we have d2 dτ 2 ˆxHs(−τ) = ˆU −1 s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t)ˆx ˆU ′′ s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) + 2 ˆU −1′ s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t)ˆx ˆU ′ s(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) + ˆU −1′′ s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t)ˆx ˆUs(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) , (E8) where the prime denotes the derivative with respect to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' From the Schr¨odinger equation ˆU ′ s(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) = i ℏ ˆHs(t − τ) ˆUs(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) , (E9) the derivatives acting on the unitary operator can be expressed in terms of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is clear that taking higher derivatives of ˆUs(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) would result in higher powers of the Hamiltonian or the partial derivative of the Hamiltonian with respect to τ, multiplied with only a single unitary operator on the very right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' However, if in the 13 end τ is set to zero, the Hamiltonian and its explicit time derivatives will be evaluated at time t, and the unitary operator on the very right disappears since ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' We therefore have the following identities ˆU (′n) s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) ��� τ=0 = (−1)n � dn dtn ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) , (E10) ˆU −1(′n) s (t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' t) ��� τ=0 = (−1)n ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � dn dtn ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E11) The additional time parameter ti that appears in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E10) and (E11) is only apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' As discussed before, evaluating the time derivatives on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E10) would result in powers of ˆHs(t) and its derivatives evaluated at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' The remaining unitary matrix ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) would be canceled by the additional ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) on the very right such that ti disappears from the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E10) and (E11) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E8) we get Tr � ˆρr(t) d2 dτ 2 ˆxHs(−τ) ���� τ=0 � = Tr ��� d2 dt2 ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti)ˆρr(t) +2 � − d dt ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti)ˆρr(t) ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � − d dt ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � +ˆρr(t) ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � d2 dt2 ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) �� ˆx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E12) Here, we have used the cyclic property within the trace to shift the unitary operators ˆUs and its derivatives on the right of ˆx in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E8) onto the very left within the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' To proceed further we note that the terms involving the trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E7) are multiplied by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' It is therefore sufficient to evaluate the trace at 0th order in the interactions as the master equation is valid only upto second order in the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' This implies that within the trace the time dependence of the density matrix can be evaluated by keeping only the Liouville-von Neuman term such that ˆρr(t) = ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti)ˆρr(ti) ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E13) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E12) then simplifies to Tr � ˆρr(t) d2 dτ 2 ˆxHs(−τ) ���� τ=0 � = Tr ��� d2 dt2 ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � ˆρr(ti) ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) + 2 � d dt ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � ˆρr(ti) � d dt ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) � + ˆUs(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti)ˆρr(ti) � d2 dt2 ˆU −1 s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' ti) �� ˆx � = Tr � d2 dt2 ˆρr(t)ˆx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E14) Thus, we have the relation Tr � ˆρr(t) d2 dτ 2 ˆxHs(−τ) ���� τ=0 � = Tr � d2 dt2 ˆρr(t)ˆx � = d2 dt2 ⟨ˆx⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E15) Similar line of reasoning also leads to the identity Tr � ˆρr(t) d3 dτ 3 ˆxHs(−τ) ���� τ=0 � = −Tr � d3 dt3 ˆρr(t)ˆx � = − d3 dt3 ⟨ˆx⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E16) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E15) and (E16) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E7), the EOM for the expectation value of the position operator in the presence of an external potential is obtained to be mR d2 dt2 ⟨ˆx⟩ = −⟨ ˆV0(x),x ⟩ + 2αℏ 3c2 d3 dt3 ⟨ˆx⟩ , where mR := m + 4αℏωmax 3πc2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' (E17) [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Casimir, Indag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' 10, 261 (1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' 14 [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Birrell and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Press, Cambridge, UK, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' [3] L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' D 14, 870 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Fulling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' D 7, 2850 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} +page_content=' 125–140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf'} diff --git a/4tE1T4oBgHgl3EQf6QWC/content/tmp_files/2301.03521v1.pdf.txt b/4tE1T4oBgHgl3EQf6QWC/content/tmp_files/2301.03521v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1fb07e42e5de209b1c6d9d468843d58134a06dd --- /dev/null +++ b/4tE1T4oBgHgl3EQf6QWC/content/tmp_files/2301.03521v1.pdf.txt @@ -0,0 +1,1047 @@ +arXiv:2301.03521v1 [math.SP] 9 Jan 2023 +GREEN’S FUNCTIONS FOR FIRST-ORDER SYSTEMS OF +ORDINARY DIFFERENTIAL EQUATIONS WITHOUT THE +UNIQUE CONTINUATION PROPERTY +STEVEN REDOLFI AND RUDI WEIKARD +Abstract. This paper is a contribution to the spectral theory associated with +the differential equation Ju′ + qu = wf on the real interval (a, b) when J is +a constant, invertible skew-Hermitian matrix and q and w are matrices whose +entries are distributions of order zero with q Hermitian and w non-negative. +Under these hypotheses it may not be possible to uniquely continue a solution +from one point to another, thus blunting the standard tools of spectral theory. +Despite this fact we are able to describe symmetric restrictions of the maximal +relation associated with Ju′ + qu = wf and show the existence of Green’s +functions for self-adjoint relations even if unique continuation of solutions fails. +1. Introduction +This paper is a contribution to the spectral theory for the differential equation +Ju′ + qu = wf +posed on the real interval (a, b) when J is a constant, invertible, and skew-Hermitian +n × n-matrix while the entries of the matrices q and w are distributions of order +zero1 with q Hermitian and w non-negative. Ghatasheh and Weikard [7] studied this +equation under the additional hypothesis that initial value problems have unique +balanced2 solutions in the space of functions of locally bounded variation. +The equation Ju′ + qu = wf has, of course, been investigated by many people +when the coefficients q and w are locally integrable. In that situation initial value +problems always have unique solutions. This is not necessarily the case when the +measures induced by q or w have discrete components. It appears that an equation +with measure coefficients was first considered in 1952, when Krein [8] modelled a +vibrating string. In 1964 Atkinson [2] suggested to unify the treatment of differen- +tial and difference equations by writing them as systems of integral equation where +integrals were to be viewed as matrix-valued Riemann-Stieltjes integrals. Atkinson +explained that the presence of point masses may prevent the continuation of so- +lutions across such points and posed a condition avoiding that problem but more +Date: 11. May 2022. +This is a preprint of an article published in Integral Equations and Operator Theory which is +available online at https://doi.org/10.1007/s00020-022-02703-6. +©2022. +This manuscript version is made available under the CC-BY-NC-ND 4.0 license +http://creativecommons.org/licenses/by-nc-nd/4.0/. +1Recall that distributions of order 0 are distributional derivatives of functions of locally +bounded variation and hence may be thought of, on compact subintervals of (a, b), as measures. +For simplicity we might use the word measure instead of distribution of order 0 below. +2A function of locally bounded variation is called balanced, if its values at any given point are +averages of its left- and right-hand limits at that point. +1 + +2 +STEVEN REDOLFI AND RUDI WEIKARD +restrictive than the one posed in [7]. In 1999 Savchuk and Shkalikov [10] treated +Schr¨odinger equations with potentials in the Sobolev space W −1,2 +loc +. Their paper was +very influential and spurred many further developments. Nevertheless, Eckhardt +et al. [5] showed in 2013, with the help of quasi-derivatives or, equivalently, by +writing the equation as a system, that a treatment without leaving the realm of +locally integrable coefficients is possible. In the same year Eckhardt and Teschl [6] +investigated 2×2-systems with diagonal measure-valued matrices q and w requiring +essentially Atkinson’s condition. +A more thorough account of the subject’s history is given in [7]. The papers +[5] and [6], mentioned above, may also serve as excellent sources, with perhaps +different emphases, of this history. +One feature of systems of first-order equations is that, generally, they are repre- +sented by linear relations rather than linear operators. There is a well-developed +spectral theory for linear relations initiated by Arens [1], see also Orcutt [9], and +Bennewitz [3]. The most important results (for our purposes) are also surveyed in +Appendix B of [7]. +Existence or uniqueness of solutions of an initial value problem for Ju′+qu = wf +fails when, for some x ∈ (a, b), the matrices +B±(x, 0) = J ± 1 +2∆q(x) +are not invertible. Here ∆q(x) = Q+(x)−Q−(x) when Q denotes an anti-derivative +of q. Equivalently, ∆q(x) = dQ({x}) where dQ is the measure (locally) generated +by q. Assuming the unique continuation property for solutions of Ju′ + qu = wf +Ghatasheh and Weikard defined maximal and minimal relations Tmax and Tmin +associated with the differential equation Ju′ + qu = wf and showed that Tmax +is the adjoint of Tmin. They characterized the self-adjoint restrictions of Tmax, if +any, with the aid of boundary conditions and proved that resolvents are given as +integral operators, i.e., the existence of a Green’s function for any such self-adjoint +relation T . Under even more restrictive conditions they also showed the existence +of a Fourier transform diagonalizing T . +Campbell, Nguyen, and Weikard [4] defined maximal and minimal relations and +showed that Tmax = T ∗ +min without the hypothesis of unique continuation of so- +lutions. Our goal here is to advance their ideas. In particular, even though the +equation Ju′ + qu = w(λu + f) may have infinitely many linearly independent +solutions the deficiency indices, i.e., the number of linearly independent solutions +of Ju′ + qu = ±iwu of finite positive norm, is still bounded by n, the size of the +system. We show that symmetric restrictions of Tmax, in particular the self-adjoint +ones, are still given by posing boundary conditions and we show that the resolvents +of self-adjoint restrictions are integral operators by proving the existence of Green’s +functions. +We will not approach the problem of Fourier transforms and eigenfunction ex- +pansions but hope to return to it in future work. +The material in this paper is arranged as follows. In Section 2 we recall the +circumstances under which existence and uniqueness of solutions to initial value +problems does hold and investigate the sets of those x ∈ (a, b) and λ ∈ C giving +rise to trouble. +Then, in Section 3 we discuss the manifold of solutions of our +differential equation in the special case when a and b are regular endpoints. These +results are instrumental in Section 4 where we investigate the deficiency indices + +GREEN’S FUNCTIONS +3 +of the minimal relation and its symmetric extensions but without the assumption +that a and b are regular. Before we prove the existence of Green’s functions for +self-adjoint restrictions of the maximal relation in Section 6 we discuss the role +played by non-trivial solutions of zero norm in Section 5. +Let us add a few words about notation. D′0((a, b)) is the space of distributions of +order 0, i.e., the space of distributional derivatives of functions of locally bounded +variation. Any function u of locally bounded variation has left- and right-hand +limits denoted by u− and u+, respectively. Also, u is called balanced if u = u# = +(u+ + u−)/2. The space of balanced functions of bounded variation defined on +(a, b) is denoted by BV#((a, b)) while BV# +loc((a, b)) stands for the space of balanced +functions of locally bounded variation. +We use +1 to denote an identity matrix +of appropriate size and superscripts ⊤ and ∗ indicate transposition and adjoint, +respectively. The sum of two closed only trivially intersecting subspaces S and T of +some Hilbert space (i.e., their direct sum) is denoted by S ⊎ T ; if S and T are even +orthogonal we may use ⊕ instead of ⊎. The orthogonal complement of a subspace +S of a Hilbert space H is denoted by H ⊖ S or by S⊥. For c1, ..., cN ∈ Cn we +abbreviate the column vector (c⊤ +1 , ..., c⊤ +N)⊤ ∈ CnN by (c1, ..., cN)⋄. +2. Preliminaries +Throughout this paper we assume the following hypothesis to be in force. +Hypothesis 2.1. J is a constant, invertible and skew-Hermitian n × n-matrix. +Both q and w are in D′0((a, b))n×n, w is non-negative and q Hermitian. +Given that w is non-negative it gives rise to a positive measure on (a, b) and we +denote the space of functions f which satisfy +� +f ∗wf < ∞ by L2(w). This space +permits the semi-inner product ⟨f, g⟩ = +� +f ∗wg (note that ⟨f, f⟩ may be 0 without +f being 0). +Consider the differential equation +Ju′ + (q − λw)u = wf +(2.1) +where λ is a complex parameter and f an element of L2(w). The latter condition +guarantees that wf is in D′0((a, b))n. We will search for solutions in BV# +loc((a, b))n. +In this case each term in (2.1) is a distribution of order 0 so that it makes sense to +pose the equation. +The point a is called a regular endpoint for Ju′ + qu = wf, if there is a point +c ∈ (a, b) such that the left-continuous anti-derivatives Q and W of q and w are +of bounded variation on (a, c). In this case q and w may be thought of as finite +measures on (a, c). Similarly, b is called regular, if Q and W are of bounded variation +on (c, b). If an endpoint is not regular, it is called singular. Not surprisingly, the +study of our problem is less complicated when the endpoints are regular and we +will use this fact to our advantage. +Despite our earlier denigration of the existence and uniqueness theorem of so- +lutions of initial value problems it continues to play a crucial role. The following +theorem was proved in [7]. +Theorem 2.2. Suppose r ∈ D′0((a, b))n×n, g ∈ D′0((a, b))n and that the matrices +1 ± ∆r(x)/2 are invertible for all x ∈ (a, b). Let x0 be a point in (a, b). Then the +initial value problem u′ = ru + g, u(x0) = u0 ∈ Cn has a unique balanced solution +u ∈ BV# +loc((a, b))n. + +4 +STEVEN REDOLFI AND RUDI WEIKARD +If a is a regular endpoint we may pose an initial condition (for u+) at a. Simi- +larly, if b is regular we may prescribe u−(b) as the initial condition. +Suppose now that u is a solution of (2.1). Treating either side of this equation +as a measure (restricted to a compact subset of (a, b)) evaluation at a singleton {x} +shows that +J(u+(x) − u−(x)) + ∆q−λw(x)u#(x) = ∆w(x)f(x) +or, equivalently, +B+(x, λ)u+(x) − B−(x, λ)u−(x) = ∆w(x)f(x) +(2.2) +when we define +B±(x, λ) = J ± 1 +2 +� +∆q(x) − λ∆w(x) +� +. +Note that, if B+(x, λ) is not invertible, we could be in one of the following two +situations: (i) a solution given on (a, x) may fail to exist on (x, b) or (ii) there +are infinitely many ways to continue a solution on (a, x) to (x, b). An analogous +statement holds, of course, if B−(x, λ) is not invertible. +Let us now investigate the circumstances when a pair (x, λ) gives such trouble. +Define the sets Λx = {λ ∈ C : det(B+(x, λ)) det(B−(x, λ)) = 0} and Ξλ = {x ∈ +(a, b) : det(B+(x, λ)) det(B−(x, λ)) = 0}. First note, since B−(x, λ) = −B+(x, λ)∗, +we have that Ξλ = Ξλ and that each Λx is symmetric with respect to the real axis. +Also, Λx is empty unless at least one of ∆q(x) and ∆w(x) is different from 0 and +hence for all but countably many x. Next, we claim that Λx is finite as soon as it +misses one point. To see this suppose that B+(x, λ0) is invertible and that λ ̸= λ0. +Since +B+(x, λ) = (λ0 − λ)B+(x, λ0) +�1 +2B+(x, λ0)−1∆w(x) − 1/(λ − λ0) +� +we see that B+(x, λ) fails to be invertible only if 1/(λ−λ0) is an eigenvalue of some +n × n-matrix. A similar statement holds, of course, for B− proving our claim. +The really bad points x, namely those where Λx = C, are thus contained in Ξ0. +Here we wish to remove the hypothesis Ξ0 = ∅ posed in [7]. On any subinterval of +(a, b) on which q gives rise to a finite measure we find that �∞ +k=1 ∥∆q(xk)∥ must be +finite, when k �→ xk is a sequence of distinct points in that interval. It follows now +that Ξ0 is a discrete set. One shows similarly that, for any fixed complex number +λ the set Ξλ is discrete. +Lemma 2.3. Suppose [s, t] ⊂ (a, b) and (s, t)∩Ξ0 = ∅. Then we have that Λ(s,t) = +� +x∈(s,t) Λx is a discrete subset of C. +Proof. There are only finitely many points x in (s, t) where ∥J−1∆q(x)∥ > 1. Using +a Neumann series one sees that only at such points the norm of B+(x, 0)−1 can be +larger than 2∥J−1∥. Thus there is a positive number C such that ∥B+(x, 0)−1∥ ≤ C +for all x ∈ (s, t). Now suppose that B+(x, λ) is not invertible and that |λ| ≤ R. +Then 1/λ is an eigenvalue of 1 +2B+(x, 0)−1∆w(x). This requires that ∥∆w(x)∥ ≥ +2/(RC) and thus can happen only for finitely many x ∈ (s, t). Since similar argu- +ments work for B− the number of points in � +x∈(s,t) Λx which lie in a disk of radius +R centered at 0 must be finite. +□ +We remark that, when one of the anti-derivatives of q and w is only locally of +bounded variation, the set � +x∈(a,b) Λx need not be discrete even if every Λx is finite. + +GREEN’S FUNCTIONS +5 +Theorem 2.4. Suppose [s, t] ⊂ (a, b) and (s, t) ∩ Ξ0 = ∅. If u0 ∈ Cn and λ ∈ +C \ Λ(s,t), then the initial value problem Ju′ + qu = λwu, u+(s) = u0 has a unique +balanced solution in (s, t). Moreover, u(x, ·) for x ∈ (s, t) as well as u−(t, ·) are +analytic in C \ Λ(s,t) and meromorphic on C. An analogous statement holds when +the initial condition is posed at t. +Proof. The first claim is simply a consequence of Theorem 2.2. When x ∈ (s, t) +the analyticity of u(x, ·) in C \ Λ(s,t), which is an open set, was proved in Section +2.3 of [7]. If we modify q and w by setting them 0 on [t, b) we do not change the +solution on (s, t). The solution for the modified problem evaluated at t is analytic +and coincides with u−(t, ·) proving its analyticity. It remains to show that a point +λ0 ∈ Λ(s,t) can merely give rise to poles. +We know already that there are only finitely many points x in (s, t) where one of +B±(x, λ0) fails to be invertible. Suppose x′ and x′′ are two consecutive such points. +If we know the solution on (s, x′) and that u−(x′, ·) has, at worst, a pole at λ0, +then the solution in (x′, x′′) is determined by the initial value +u+(x′, λ) = B+(x′, λ)−1B−(x′, λ)u−(x′, λ) +which also has, at worst, a pole at λ0 since this is true for B+(x′, λ)−1. For x ∈ (s, t) +the claim follows now by induction. To prove that u−(t, ·) is also meromorphic we +proceed as before and modify q and w on [t, b). +□ +3. Solving the differential equation +Our goal in this section is to investigate the set of solutions of the differential +equation Ju′ + (q − λw)u = wf on (a, b) under a strengthened hypothesis. +Hypothesis 3.1. In addition to Hypothesis 2.1 we ask that a and b are regular +endpoints for Ju′ + qu = wf. +Moreover, given the partition +a = x0 < x1 < x2 < ... < xN < xN+1 = b +(3.1) +of (a, b) we require that Ξ0 ⊂ {x1, ..., xN}. We then consider only λ for which both +B+(x, λ) and B−(x, λ) are invertible unless x is in {x1, ..., xN}. +This hypothesis is in force throughout this section but later only if explicitly +mentioned. We emphasize that Ξ0 is finite when a and b are regular. Also, the set +of permissible λ, which we call Ω0, is symmetric with respect to the real axis and +avoids only a discrete set. +On each interval (xj, xj+1) we let Uj(·, λ) be a fundamental matrix of balanced +solutions of the homogeneous differential equation Ju′ + (q − λw)u = 0 such that +limx↓xj Uj(x, λ) = +1. The existence of these fundamental matrices is guaranteed by +Theorem 2.2. The general balanced solution u of the non-homogeneous equation +Ju′ + (q − λw)u = wf on (xj, xj+1) satisfies, according to Lemma 3.3 in [7], +u−(x) = U − +j (x, λ) +� +cj + J−1 +� +(xj,x) +Uj(·, λ)∗wf +� +for any cj ∈ Cn. Define +Uj(xj+1, λ) = +lim +x↑xj+1 Uj(x, λ) +and +Ij(f, λ) = +� +(xj,xj+1) +Uj(·, λ)∗wf. + +6 +STEVEN REDOLFI AND RUDI WEIKARD +Using u+(xj) = cj and u−(xj) = Uj−1(xj, λ)(cj−1 + J−1Ij−1(f, λ)) in equation +(2.2) gives +(−B−(xj, λ)Uj−1(xj, λ), B+(xj, λ)) +�cj−1 +cj +� += ∆w(xj)f(xj) + B−(xj, λ)Uj−1(xj, λ)J−1Ij−1(f, λ). +We need to consider these equations for j = 1, ..., N simultaneously. This gives rise +to the system +B(λ)˜u = F0(f, λ) +(3.2) +where ˜u = (c0, ..., cN)⋄, B(λ), to be specified presently, is in CnN×n(N+1), and +F0(f, λ) is in CnN. +The two-diagonal block-matrix structure of B suggests the +introduction of matrices E⊤ and E⊥, which, respectively, strip the first and last +n components off a vector in their domain Cn(N+1). If we also define the block- +matrices +B(λ) = diag(B+(x1, λ), ..., B+(xN, λ)), +U(λ) = diag(U0(x1, λ), ..., UN−1(xN, λ)), +and J = diag(J, ..., J) and when we note that +B(λ)∗ = diag(−B−(x1, λ), ..., −B−(xN, λ)), +we obtain +B(λ) = B(λ)∗U(λ)E⊥ + B(λ)E⊤. +(3.3) +The vector F0(f, λ) is given by +F0(f, λ) = R(f) − B(λ)∗U(λ)J −1I(f, λ) +with R(f) = ((∆wf)(x1), ..., (∆wf)(xN))⋄ and I(f, λ) = (I0(f, λ), ..., IN−1(f, λ))⋄. +We now have the following theorem. +Theorem 3.2. The differential equation Ju′ + (q − λw)u = wf has a solution u +on (a, b) if and only if ˜u = (u+(x0), ..., u+(xN))⋄ is a solution of equation (3.2). +In particular, in the homogeneous case, where f = 0, the space of solutions has +dimension n(N + 1) − rk B(λ) ≥ n. +We note that rk B(λ) = n when N = 1 so that the space of solutions of Ju′ + +(q − λw)u = 0 is then exactly n-dimensional. For N = 2, however, consider the +example (a, b) = R, J = +� 0 −1 +1 +0 +� +, q = +� 0 2 +2 0 +� +(δ1 − δ2), w = +� 2 0 +0 0 +� +(δ1 + δ2), where the +δk are Dirac point measures concentrated on {k}. It shows that the dimension of +the space of solutions of Ju′ + (q − λw)u = 0 may be strictly larger than n. +Next we investigate the connection between the right-hand limits of a solution u +of the homogeneous equation Ju′ + (q − λw)u = 0 at the points x0, ..., xN (given +by the vector ˜u) and the vector ˆu = (u(x1), ..., u(xN))⋄. We have ˆu = D(λ)˜u where +D(λ) = 1 +2(U(λ)E⊥ + E⊤) +(3.4) +is again a two-diagonal block-matrix. If N ≥ 2 we will also introduce the matrices +Bm(λ) and Dm(λ) which are obtained by deleting the first and last n columns from +B(λ) and D(λ), respectively. If N = 1 we should think of Bm(λ) and Dm(λ) as +maps from the trivial vector space to Cn. Their adjoints are the map from Cn to +{0}. With this understanding the following results hold also for N = 1 even though +they then involve “matrices” with no rows or columns. + +GREEN’S FUNCTIONS +7 +Lemma 3.3. D(λ)∗B(λ) − B(λ)∗D(λ) = diag(−J, 0, ..., 0, J) and Dm(λ)∗B(λ) − +Bm(λ)∗D(λ) = 0. +Proof. This follows since U(λ)∗J U(λ) = J which, in turn, follows from Lemma 3.2 +in [7]. +□ +Lemma 3.4. The map v �→ B(λ)v, restricted to ker D(λ), is a bijection onto +ker Dm(λ)∗. Similarly, the map v �→ D(λ)v, restricted to ker B(λ), is a bijection +onto ker Bm(λ)∗. In particular, dim ker D(λ) = dim ker Dm(λ)∗ and dim ker B(λ) = +dim ker Bm(λ)∗. +Proof. The identity Dm(λ)∗B(λ)−Bm(λ)∗D(λ) = 0 shows that B(λ) maps ker D(λ) +to ker Dm(λ)∗ as well as that D(λ) maps ker B(λ) to ker Bm(λ)∗. +If v ∈ ker B(λ) ∩ ker D(λ) one shows that E⊥v = E⊤v = 0 using the definitions +(3.3) and (3.4) of B and D and the fact that B(λ) − B(λ)∗ = 2J . This, of course, +implies that v = 0 and hence the injectivity of both B(λ)|ker D(λ) and D(λ)|ker B(λ). +Clearly, both D(λ) and Dm(λ)∗, having invertible matrices along their main +diagonal, are of full rank. +The rank-nullity theorem shows therefore that their +kernels both have dimension n. This proves surjectivity of B(λ)|ker D(λ). +Finally, assume that v ∈ ker Bm(λ)∗. Then v = D(λ)x for some x ∈ Cn(N+1) +which implies that 0 = Bm(λ)∗D(λ)x = Dm(λ)∗B(λ)x. The first part of the proof +shows that there is a y ∈ ker D(λ) such that B(λ)y = B(λ)x. Hence v = D(λ)(x−y) +where x − y ∈ ker B(λ). +□ +The following theorem establishes a connection between solutions of the differ- +ential equation Ju′ + (q − λw)u = 0 and elements of ker Bm(λ)∗. +Theorem 3.5. If u is a solution of Ju′ + (q − λw)u = 0 on (a, b), then ˆu = +(u(x1), ..., u(xN))⋄ is in ker Bm(λ)∗. +If, in addition, u+(a) = u−(b) = 0, then +ˆu ∈ ker B(λ)∗ (a subspace of ker Bm(λ)∗). +Conversely, if ˆu ∈ ker Bm(λ)∗, then Ju′ + (q − λw)u = 0 has a unique solution +u on (a, b) such that (u(x1), ..., u(xN))⋄ = ˆu. If, indeed, ˆu ∈ ker B(λ)∗, we further +have u+(a) = u−(b) = 0. +Let us emphasize that supp u ⊂ [x1, xN] when u+(a) = u−(b) = 0. +Proof. If u solves Ju′ + (q − λw)u = 0, then, by Theorem 3.2, ˜u ∈ ker B(λ). +Lemma 3.4 shows then that ˆu = D(λ)˜u is in ker Bm(λ)∗. If u+(a) = u−(b) = 0, +then Lemma 3.3 gives 0 = B(λ)∗D(λ)˜u = B(λ)∗ˆu. +Conversely, assume that ˆu ∈ ker Bm(λ)∗ = D(λ)(ker B(λ)). +Then there is a +unique vector ˜u ∈ ker B(λ) such that ˆu = D(λ)˜u, which, in turn, defines a unique +solution u of Ju′+(q−λw)u = 0 such that (u(x1), ..., u(xN))⋄ = ˆu. If ˆu ∈ ker B(λ)∗, +then, according to Lemma 3.3, diag(−J, 0, ..., 0, J)˜u = 0 which shows that u+(a) = +u−(b) = 0. +□ +Given an algebraic system Ax = b we know that there exist solutions only if +b ∈ ran A = (ker A∗)⊥. For the differential equation Ju′ + (q − λw)u = wf with +integrable coefficients q and w the unique continuation property for the solutions +gives rise to the variation of constants formula, which then guarantees the existence +of solutions for any non-homogeneity f (within reason). In the present situation, + +8 +STEVEN REDOLFI AND RUDI WEIKARD +however, the problem of existence raises its head and we now set out to give neces- +sary and sufficient conditions for f guaranteeing the existence of a solution in the +spirit of Linear Algebra. +Lemma 3.6. If ˜v ∈ ker B(λ) and ˆv = D(λ)˜v, then +˜v∗E∗ +⊥ = −ˆv∗B(λ)∗U(λ)J −1 +and +˜v∗E∗ +⊤ = ˆv∗B(λ)J −1. +Moreover, if f ∈ L2(w) and Jv′ + (q − λw)v = 0, then +� +v∗wf = ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ) = ˆv∗F0(f, λ) + ˜v∗E∗ +⊤˜I(f, λ) +where (v(x1), ..., v(xN))⋄ = ˆv = D(λ)˜v and ˜I(f, λ) = (0, ..., 0, IN(f, λ))⋄ ∈ CnN. +Proof. Using the definitions (3.3) and (3.4) of B and D and the identities B(λ) − +B(λ)∗ = 2J and U(λ)∗J U(λ) = J we obtain that B(λ)˜v = 0 implies +B(λ)D(λ)˜v = U(λ)∗−1J E⊥˜v +and +B(λ)∗D(λ)˜v = −J E⊤˜v. +Taking adjoints gives the first claim since ˆv = D(λ)˜v. +The second claim is an immediate consequence of this, since +� +v∗wf = ˆv∗R(f) + ˜v∗(I0(f, λ), ..., IN (f, λ))⋄ += ˆv∗R(f) + ˜v∗E∗ +⊥I(f, λ) + ˜v∗E∗ +⊤˜I(f, λ) += ˆv∗R(f) − ˆv∗B(λ)∗U(λ)J −1I(f, λ) + ˆv∗B(λ)J −1˜I(f, λ) += ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ). +□ +Theorem 3.7. The differential equation Ju′ + (q − λw)u = wf has a solution on +(a, b) if and only if +� +v∗wf = 0 for every solution v of Jv′ + (q − λw)v = 0 which +vanishes at a and b. +Proof. By Theorem 3.2 the solution u exists if and only if the system (3.2) has a +solution ˜u = (u+(x0), ..., u+(xN))⋄. This, in turn, happens if and only if F0(f, λ) ∈ +ran B(λ) = (ker B(λ)∗)⊥. +By Theorem 3.5 the solutions of Jv′ +(q −λw)v = 0 which vanish at a and b are +in one-to-one correspondence with elements of ker B(λ)∗. Since v+(xN) = 0 we have +˜v∗E∗ +⊤˜I(f, λ) = 0 and then, from Lemma 3.6, we obtain ˆv∗F0(f, λ) = +� +v∗wf. +□ +In the case of unique continuation of solutions the condition that v vanishes at a +or b implies, of course, that v = 0. Consequently, Ju′ + (q − λw)u = wf has then a +solution for any f ∈ L2(w). The set of all solutions is thus obtained by adding the +general solution of Ju′ +(q −λw)u = 0 whose dimension is n(N + 1)−rkB(λ) ≥ n. +Theorem 3.8. The differential equation Ju′ + (q − λw)u = wf has a solution on +(a, b) which vanishes at a and b if and only if +� +v∗wf = 0 for every solution v of +Jv′ + (q − λw)v = 0. +Proof. For u to vanish at a and b it is required that u+(x0) = 0 and u+(xN) = +−J−1IN(f, λ). The system (3.2) is therefore equivalent to +Bm(λ)(c1, ..., cN−1)⋄ = F0(f, λ) + B(λ)J −1˜I(f, λ). +The proof is now analogous to the one for Theorem 3.7. +□ + +GREEN’S FUNCTIONS +9 +We conclude this section by “counting” the solutions of Ju′ + qu = λwu which +are not compactly supported. +More precisely, we will determine the dimension +of the quotient space of all solutions of Ju′ + qu = λwu modulo the space of +compactly supported solutions. Theorem 3.5 shows that the space of all solutions of +Ju′+qu = λwu is in one-to-one correspondence with ker Bm(λ)∗ and that the space +of compactly supported solutions of Ju′+qu = λwu is in one-to-one correspondence +with ker B(λ)∗. We therefore define +˜n(λ) = dim(ker Bm(λ)∗/ ker B(λ)∗) = dim ker Bm(λ)∗ − dim ker B(λ)∗. +Lemma 3.9. ˜n(λ) + ˜n(λ) = 2n. +Proof. Since rk B(λ) = rk B(λ)∗, the rank-nullity theorem implies +dim ker B(λ) = n(N + 1) − rk B(λ)∗ = n + dim ker B(λ)∗. +Hence, using also the analogous equation for λ, +dim ker B(λ) − dim ker B(λ)∗ + dim ker B(λ) − dim ker B(λ)∗ = 2n. +Lemma 3.4 gives that dim ker B(λ) = dim ker Bm(λ)∗ yielding the claim. +□ +From Theorem 2.4 we know that the matrices Uj(xj+1, ·) are meromorphic on C +with poles at most at points in the complement of Ω0. It follows that the entries of +B are also meromorphic. Since the meromorphic functions on C form a field there +is a row-echelon matrix ˜B with meromorphic entries such that B˜u = 0 has the same +solutions as ˜B˜u = 0. Now define a set Ω as Ω0 without the set of all poles of ˜B as +well as their complex conjugates, and the set of zeros and their conjugates of any +of the pivots of ˜B. +Theorem 3.10. If λ ∈ Ω, then dim ker B(λ) = dim ker B(λ) and ˜n(λ) = n. +Proof. The construction of Ω entails that rk B(λ) = rk ˜B(λ) = rk B(λ) if λ ∈ Ω. +Since ˜n(λ) = dim ker B(λ) − dim ker B(λ)∗ = dim ker B(λ) + n − dim ker B(λ) we +obtain ˜n(λ) = n. +□ +4. Symmetric restrictions of Tmax +Given a differential equation Ju′ + qu = wf we now define associated minimal +and maximal relations. Recall that L2(w) is the space of functions f such that +� +f ∗wf < ∞. First we define +Tmax = {(u, f) ∈ L2(w) × L2(w) : u ∈ BV# +loc((a, b))n, Ju′ + qu = wf}. +Subsequently we will always tacitly assume that u ∈ BV# +loc((a, b))n, when we use +u′. Next, let +Tmin = {(u, f) ∈ Tmax : supp u is compact in (a, b)}. +Note that these are spaces of pairs of functions. To employ the power of functional +analysis we need to realize these relations in Hilbert spaces. Therefore we introduce, +as usual, the space L2(w) as the quotient of L2(w) modulo the subspace of all u ∈ +L2(w) for which ∥u∥2 = +� +u∗wu = 0. Denoting the equivalence class corresponding +to u by [u] we now set +Tmax = {([u], [f]) ∈ L2(w) × L2(w) : (u, f) ∈ Tmax} + +10 +STEVEN REDOLFI AND RUDI WEIKARD +and +Tmin = {([u], [f]) ∈ Tmax : (u, f) ∈ Tmin}. +Here (and elsewhere) we choose brevity over precision: whenever we have a pair +([u], [f]) in Tmax we choose u and f such that (u, f) ∈ Tmax. +Define the vector space +L0 = {u ∈ BV# +loc((a, b))n : Ju′ + qu = 0 and ∥u∥ = 0}. +In many cases this space is trivial and some authors restrict their attention to the +case where it is; this is then called the definiteness condition. However, we will +not do so here. Note that ∥u∥ = 0 if and only if wu is the zero distribution. The +significance of L0 stems from the following fact. Suppose ([u], [f]) ∈ Tmax and that +there are u, v ∈ [u] and f, g ∈ [f] such that Ju′ + qu = wf and Jv′ + qv = wg. +Then J(u − v)′ + q(u − v) = w(f − g) = 0 as well as w(u − v) = 0, i.e., u − v ∈ L0. +In other words, in the presence of a non-trivial space L0, the class [u] has many +representatives of locally bounded variation satisfying the differential equation for a +given class [f] (the choice of a representative of [f], on the other hand, is irrelevant). +In Section 5 we will describe a procedure to choose a representative of [u] in a +distinctive way. +In [4] it was proved that Tmin is symmetric, indeed that T ∗ +min = Tmax. In this case +it is well-known that von Neumann’s theorem holds. Setting Dλ = {([u], λ[u]) ∈ +Tmax} it states that +Tmax = Tmin ⊎ Dλ ⊎ Dλ +when Im λ ̸= 0. Moreover, when λ = ±i, these direct sums are even orthogonal. It +is also known that the dimension of Dλ does not change as λ varies in either the +upper or the lower half plane. The numbers n± = dim D±i are called deficiency +indices of Tmin and we are now setting out to investigate these. +If u is a solution of Ju′ +qu = λwu which is compactly supported then (u, λu) ∈ +Tmin and ([u], λ[u]) ∈ Tmin ∩ Dλ. If λ is not real, then Tmin ∩ Dλ is trivial and it +follows that compactly supported solutions of Ju′ + qu = λwu do not contribute to +the corresponding deficiency index. We now have, as a corollary of Theorem 3.10, +that the deficiency indices of Tmin cannot be more than n if a and b are regular +endpoints. We do not state this result separately since it is included in the next +theorem about the general case. +Thus, to emphasize, we allow in the following a and b to be either regular or +singular endpoints. Let τk, k ∈ Z, be a strictly increasing sequence in (a, b) having +a and b as its only limit points and such that all points in Ξ0 are among the +τk. Considering now only the interval Ik = (τ−k, τk) we set xj = τ−k+j for j = +0, ..., N + 1 = 2k. We can then introduce the objects from Section 3. To emphasize +their dependence on k we will add a superscript (k) to those objects. We have then, +in particular, the matrices B(k), B(k) +m and the sets Ω(k) of permissible values of λ. +We now define Ω = �∞ +k=1 Ω(k) and note that Ω is symmetric with respect to the +real axis and misses only countably many values from C. +Now fix a non-real λ ∈ Ω. If u is a solution of Ju′+qu = λwu on (a, b) we denote +its restriction to the interval Ik by u(k). We are interested in the quotient space Xk +of all solutions of Ju′ +qu = λwu on Ik modulo the compactly supported solutions. +If u is a solution of Ju′+qu = λwu on Ik we denote the associated equivalence class +in Xk by ⌊u⌋k. A compactly supported solution u of Ju′ + qu = λwu on Ik can be +extended by 0 to all of (a, b) yielding an element in Tmin ∩ Dλ. This implies, since + +GREEN’S FUNCTIONS +11 +Im λ ̸= 0, that ∥u∥2 = +� +Ik u∗wu = 0 and shows that Xk is a normed space with the +norm given by ∥u∥2 +k = +� +Ik u∗wu. According to Theorem 3.5 the quotient space Xk +is isomorphic to ker B(k) +m (λ)∗/ ker B(k)(λ)∗ and, by Theorem 3.10, its dimension is +equal to n since λ ∈ Ω ⊂ Ω(k). +Theorem 4.1. The deficiency indices of Tmin are less than or equal to n. +Proof. Fix a non-real λ ∈ Ω. Suppose u1, ..., um are solutions of Ju′ + qu = λwu +such that [u1], ..., [um] are linearly independent elements of Dλ. We will show below +that there is an interval Ip = (τ−p, τp) such that ⌊u(p) +1 ⌋p, ..., ⌊u(p) +m ⌋p are linearly +independent elements of Xp. Hence m ≤ n, the dimension of Xp. Since deficiency +indices are constant in either half-plane they cannot be larger than n. +We will now prove the existence of Ip by induction. That is we prove that, for +every k ∈ {1, ..., m}, there is an interval Iℓk such that the restrictions of u1, ..., uk +to Iℓk generate linearly independent elements ⌊u(ℓk) +1 +⌋ℓk, ..., ⌊u(ℓk) +k +⌋ℓk of Xℓk. Once +this is achieved we set p = ℓm. +Suppose k = 1 and let Iℓ1 be an interval such that ∥u(ℓ1) +1 +∥ > 0. By what we +argued above we know that u(ℓ1) +1 +is not compactly supported in Iℓ1 and thus gives +rise to a non-zero (and hence linearly independent) element of Xℓ1. +Now suppose we had already shown our claim for some k < m. If ⌊u(ℓk) +1 +⌋ℓk, ..., +⌊u(ℓk) +k+1⌋ℓk are already linearly independent as elements of Xℓk we choose ℓk+1 = ℓk +and our induction step is complete. Otherwise, there are unique complex numbers +α1, ..., αk such that +∥(α1u1 + ... + αkuk + uk+1)(ℓk)∥ℓk = 0. +However, there must be an interval Iℓk+1 ⊃ Iℓk where +∥(α1u1 + ... + αkuk + uk+1)(ℓk+1)∥ℓk+1 > 0 +on account that [u1], ..., [uk+1] are linearly independent. It follows now that, as ele- +ments of Xℓk+1 the vectors ⌊u(ℓk+1) +1 +⌋ℓk+1, ..., ⌊u(ℓk+1) +k+1 +⌋ℓk+1 are linearly independent. +This completes our induction step also in this case. +□ +Corollary 4.2. If a and b are regular, then n+ = n−. +Proof. Fix a non-real λ in Ω. Since a and b are regular, the set Ξλ = Ξλ is finite. +Thus we may assume that it is contained in Ik = (τ−k, τk) for some appropriate k. +Then dim ker B(k)(λ) is the number of linearly independent solutions of Ju′ + qu = +λwu. Theorem 3.10 shows that Ju′ + qu = λwu has the same number of linearly +independent solutions. Any of these solutions has finite norm but some may have +norm 0. Now note, that if u is a solution of Ju′+qu = λwu of norm 0, then we have +wu = 0, so that u is also a solution of Ju′ + qu = λwu. Therefore n+ = n−. +□ +As mentioned above, it is well-known, even in the case of relations, that von +Neumann’s theorem E∗ = E⊕Di⊕D−i holds when E is a closed symmetric relation +in H × H when H is a Hilbert space. In our case, when d = dim Di ⊕ D−i is finite, +as we just showed, we can use Theorem B.5 in [7] to characterize the symmetric +restriction of Tmax in terms of boundary conditions. We state that theorem here +for easy reference. The operator J appearing there is defined by J (u, f) = (f, −u) +for u, f ∈ H. + +12 +STEVEN REDOLFI AND RUDI WEIKARD +Theorem 4.3. Suppose E is a closed symmetric relation in H × H with d = +dim Di ⊕ D−i < ∞ and that m ≤ d/2 is a natural number or 0. If A : E∗ → Cd−m +is a surjective linear operator such that E ⊂ ker A and AJ A∗ has rank d−2m then +ker A is a closed symmetric restriction of E∗ for which the dimension of (ker A)⊖E +is m. Conversely, every closed symmetric restriction of E∗ is the kernel of such a +linear operator A. Finally, ker A is self-adjoint if and only if AJ A∗ = 0 (entailing +m = d/2). +A second ingredient for our next considerations is Lagrange’s identity (or Green’s +formula). If (u, f) and (v, g) are in Tmax, then v∗wf and g∗wu are finite measures. +Therefore v∗Ju′ + v′∗Ju = v∗wf − g∗wu is also a finite measure. Its antiderivative +v∗Ju is of bounded variation and thus has limits at a and b. Integration now gives +Lagrange’s identity +(v∗Ju)−(b) − (v∗Ju)+(a) = ⟨v, f⟩ − ⟨g, u⟩. +(4.1) +Note the right-hand side, and hence the left-hand side, does not change upon choos- +ing different representatives in place of u, f, v, or g. +Now, if (v, g) is an element of Di⊕D−i, then (u, f) �→ ⟨(v, g), (u, f)⟩ is a bounded +linear functional on Tmax. Conversely, since Tmax is a Hilbert space, a bounded +linear functional on Tmax is given by (u, f) �→ ⟨(v, g), (u, f)⟩ for some (v, g) ∈ Tmax. +When it is also known that Tmin is in the kernel of this functional, (v, g) may be +chosen in Di ⊕ D−i. Hence, in our situation, the operator A from Theorem 4.3 +is given by d − m linearly independent elements in Di ⊕ D−i. Lagrange’s identity +implies that the entries of the matrix AJ A∗ are then given by +(AJ A∗)k,ℓ = ⟨(vk, gk), (gℓ, −vℓ)⟩ = (g∗ +kJgℓ)−(b) − (g∗ +kJgℓ)+(a). +(4.2) +Therefore we arrive at the following theorem. +Theorem 4.4. Let d = n+ + n− and suppose that m ≤ min{n+, n−}. If (v1, g1), +..., (vd−m, gd−m) are linearly independent elements of Di⊕D−i such that the matrix +defined in (4.2) has rank d − 2m, then +T = {(u, f) ∈ Tmax : (g∗ +j Ju)−(b) − (g∗ +j Ju)+(a) = 0 for j = 1, ..., d − m} +(4.3) +is a closed symmetric restriction of Tmax. +Conversely, if T is a closed symmetric restriction of Tmax and m is the dimen- +sion of T ⊖ Tmin, then T is given by (4.3) for appropriate elements (v1, g1), ..., +(vd−m, gd−m) of Di ⊕ D−i for which the matrix defined in (4.2) has rank d − 2m. +For self-adjoint restrictions of Tmax it is hence necessary and sufficient that n+ = +n− = m = d−m and that (g∗ +kJgℓ)−(b)−(g∗ +kJgℓ)+(a) = 0 for all 1 ≤ k, ℓ ≤ m = d/2. +5. The space L0 +We mentioned earlier that the class [u] does not have a unique balanced repre- +sentative when ([u], [f]) ∈ Tmax, if the space L0 has non-trivial elements. In this +section we describe a procedure to choose a representative in a distinctive way. +To this end we assume, without loss of generality, that B+(τ0, 0) = B−(τ0, 0) = J +so that solutions of our differential equations are continuous at τ0. Define N0 = +{h(τ0) : h ∈ L0} and for each k ∈ N both Nk = {h+(τk) : h ∈ L0, supp h ⊂ [τk, b)} +and N−k = {h−(τ−k) : h ∈ L0, supp h ⊂ (a, τ−k]}. Then, for k ∈ N0, we say that a +function u ∈ BV# +loc((a, b))n satisfies condition (±k), if u±(τ±k) is perpendicular to +N±k (using always the upper sign or always the lower sign). + +GREEN’S FUNCTIONS +13 +Lemma 5.1. Suppose ([u], [f]) ∈ Tmax. Then there is a unique balanced v ∈ [u] +such that (v, f) ∈ Tmax and v satisfies condition (k) for every k ∈ Z. +Proof. First consider uniqueness. Suppose u and v are two functions satisfying the +given conditions. Then u − v ∈ L0 and hence (u − v)(τ0)∗t(τ0) = 0 for t = u and +t = v. Subtract these equations to find (u−v)(τ0) = 0, and thus u = v on (τ−1, τ1). +Moreover, h1 = (u − v)χ[τ1,b) and h−1 = (u − v)χ(a,τ−1] are in L0. Conditions (1) +and (−1) show therefore that (u−v)+(τ1) and (u−v)−(τ−1) are also 0 which proves +that u = v on (τ−2, τ2). Induction informs us now that u = v everywhere. +We now turn to existence. Pick a balanced representative u ∈ [u] such that +(u, f) ∈ Tmax. There is an element h0 ∈ L0 such that the orthogonal projection of +u(τ0) onto N0 equals h0(τ0). Thus v0 = u − h0 satisfies (v0, f) ∈ Tmax, v0 ∈ [u], +and condition (0). +Next, there is an element h1 ∈ L0 with support in [τ1, b) such that the orthogonal +projection of v+ +0 (τ1) onto N1 equals h+ +1 (τ1). We now define v1 = v0 − h1. Then +(v1, f) ∈ Tmax, v1 ∈ [u], and v1 satisfies condition (1). Notice that v1 = v0 on +(a, τ1) implying that v1 also satisfies condition (0). +Proceeding recursively, we may define, for each k ∈ N, functions hk ∈ L0 sup- +ported in [τk, b) such that vk = u−�k +j=0 hj satisfies conditions (0), ..., (k), vk ∈ [u], +and (vk, f) ∈ Tmax. +Since, for a fixed x ∈ [τ0, b), only finitely many of the numbers hk(x) are different +from 0, we find that the sequence k �→ vk converges pointwise to a function ˜v ∈ [u] +satisfying conditions (k) for all k ∈ N0 and (˜v, f) ∈ Tmax. We can now repeat +this process for negative integers starting from the function ˜v instead of u arriving +eventually at a function v ∈ [u] satisfying conditions (k) for all k ∈ Z and (v, f) ∈ +Tmax. +□ +We denote the operator which assigns the function v just constructed to a given +element ([u], [f]) ∈ Tmax by E. If Im = (τ−m, τm) we also define Em : Tmax → +BV#(Im)n by composing E with the restriction to the interval Im. +Note that +BV#(Im)n is a Banach space with the norm |||u|||m defined as the sum of the +variation of u over Im and the norm of u(τ0). +Theorem 5.2. The operator Em : Tmax → BV#(Im)n is bounded. +Proof. Due to the closed graph theorem we merely have to show that Em is a +closed operator. Thus assume that the sequence ([uj], [fj]) converges to ([u], [f]) in +Tmax and that Em([uj], [fj]) converges to v in BV#(Im)n and hence pointwise. To +simplify notation we assume that Em([uj], [fj])) and Em([u], [f]) are the restrictions +of uj and u, respectively, to the interval Im. We need to show that u = v on Im. +First note that u± +j (τ±k) ∈ N ⊥ +±k and +��u± +j (τ±k) − v±(τ±k) +�� → 0 imply that v +satisfies conditions (±k) for each k ∈ {0, ..., m − 1}. For ℓ ∈ {−m, m − 1} and +x ∈ (τℓ, τℓ+1) we have +u− +j (x) = U − +ℓ (x) +� +u+ +j (τℓ) + J−1 +� +(τℓ,x) +U ∗ +ℓ wfj +� +when Uℓ denotes the fundamental matrix of Ju′ + qu = 0 on the interval (τℓ, τℓ+1) +satisfying U + +ℓ (τℓ) = +1. Taking the limit as j → ∞ gives +v−(x) = U − +ℓ (x) +� +v+(τℓ) + J−1 +� +(τℓ,x) +U ∗ +ℓ wf +� + +14 +STEVEN REDOLFI AND RUDI WEIKARD +since the integral may be considered as a vector of scalar products which are, of +course, continuous. The variation of constants formula shows that v is a balanced +solution for Jv′ + qv = wf on (τℓ, τℓ+1). We also have +J(u+ +j (τℓ) − u− +j (τℓ)) + ∆q(τℓ)uj(τℓ) = ∆w(τℓ)fj(τℓ). +(5.1) +The fact that [fj] converges to [f] in L2(w) implies, on account of the positivity +of w, that ∆w(τℓ)fj(τℓ) converges to ∆w(τℓ)f(τℓ). +Therefore taking a limit in +(5.1) shows, in conjunction with the previous observations, that Jv′ + qv = wf on +the interval Im. Since u satisfies the same equation we have that u − v satisfies +J(u − v)′ + q(u − v) = 0 on Im. +Next we show w(u − v) = 0 on Im. Fatou’s lemma implies +0 ≤ +� +Im +(u − v)∗w(u − v) ≤ lim inf +j→∞ +� +Im +(u − uj)∗w(u − uj) = 0. +It follows that w(u − v) = 0 on Im. +Finally, a variant of Lemma 5.1 shows now that u = v. +□ +6. Green’s function +Now suppose that we have a self-adjoint restriction T of Tmax. The resolvent set +of T is the set of those λ for which T − λ : dom(T ) → L2(w) is bijective, i.e., +̺(T ) = {λ ∈ C : ker(T − λ) = {0}, ran(T − λ) = L2(w)} +which is an open set. We denote its complement, the spectrum of T , by σ(T ). +Since T is self-adjoint, σ(T ) is a subset of R. +If λ ∈ ̺(T ), then the resolvent +Rλ = (T − λ)−1 is a bounded linear operator from L2(w) to dom(T ). We now +define Rλ : L2(w) → BV# +loc((a, b))n by +Rλ[f] = E((Rλ[f], λRλ[f] + [f])). +Thus Rλ[f] is the unique solution of Ju′ + qu = w(λu + f) in L2(w) satisfying +condition (k) for every k ∈ Z. +We will now show that Rλ is an integral operator. Its kernel G is called a Green’s +function for T . +Theorem 6.1. If T is a self-adjoint restriction of Tmax, then there exists, for given +x ∈ (a, b) and λ ∈ ̺(T ), a matrix G(x, ·, λ) such that the columns of G(x, ·, λ)∗ are +in L2(w) and +(Rλ[f])(x) = +� +G(x, ·, λ)wf. +(6.1) +Proof. Fix x ∈ Im and λ ∈ ̺(T ). Consider the restriction of Rλ[f] to the interval +Im. Since Em and Rλ are bounded operators the map [f] �→ (Rλ[f])(x) is a bounded +linear map from L2(w) to Cn. Hence there are elements [g1], ..., [gn] ∈ L2(w) such +that the k-th component of (Rλ[f])(x) equals ⟨[gk], [f]⟩. Let these be the columns +of the matrix-valued function G(x, ·, λ)∗. Then we obtain (6.1). +□ +One wishes to complement this fairly abstract existence result by a more concrete +one where Green’s function is given in terms of solutions of the differential equation +as is done in the classical case, see, for instance, Zettl [11]. This was also achieved +in [7] under the assumption that Ξ0 is empty and minor generalizations of this +are certainly possible. Such an explicit construction of Green’s function, where +possible, is the cornerstone of many other results in spectral theory, in particular + +GREEN’S FUNCTIONS +15 +the development of a spectral transformation and more detailed information about +the resolvent, e.g., the compactness of the resolvent in the regular case. Due to +the difficulties posed by the absence of an existence and uniqueness theorem for +initial value problems we have, so far, not been able to obtain such a construction +in general. However, we hope to return to this issue in the future. +7. Example +In this section we treat an example where the matrices B±(x, λ) fail to be invert- +ible for infinitely many x and all λ, in other words where Ξ0 is infinite and Λx = C +for all x ∈ Ξ0 (recall that in [7] the hypothesis Ξ0 = ∅ was made causing each Λx +to be finite). The example is Ju′ + qu = wf on (a, b) = R where +J = +� +0 +−1 +1 +0 +� +, q = +� +0 +2 +2 +0 +� � +k∈Z +(δ2k − δ2k+1), and, w = +� +2 +0 +0 +0 +� � +k∈Z +δk +with δk denoting the Dirac point measure concentrated on {k}. Since we are seeking +balanced solutions we need the matrices +B−(2k − 1, λ) = +�λ +0 +2 +0 +� +and +B+(2k − 1, λ) = +�−λ +−2 +0 +0 +� +as well as +B−(2k, λ) = +� +λ +−2 +0 +0 +� +and +B+(2k, λ) = +� +−λ +0 +2 +0 +� +. +If x is not an integer we have B±(x, λ) = J. Note that f ∈ L2(w) if and only if +k �→ f1(k) is in ℓ2(Z) and any element in L2(w) is uniquely determined by these +values (here f1 denotes the first component of f). +In any interval (k, k + 1) solutions of Ju′ + qu = w(λu + f) are constant, say +(αk, βk)⊤. At x = 2k − 1 the equation +B+(2k − 1, λ)u+(2k − 1) − B−(2k − 1, λ)u−(2k − 1) = (2f1(2k − 1), 0)⊤ +implies α2k−2 = 0 and +− λα2k−1 − 2β2k−1 = 2f1(2k − 1). +(7.1) +Similarly, at x = 2k we get α2k = 0 and +− λα2k−1 + 2β2k−1 = 2f1(2k). +(7.2) +We can now describe the space Tmax. A pair (u, f) is in Tmax if and only if the +sequences k �→ f1(k) and k �→ u1(k) are in ℓ2(Z), f1(2k) = −f1(2k − 1), u1(2k) = +u1(2k − 1), and +u = +� +k∈Z +� �2u1(2k) +f1(2k) +� +χ# +(2k−1,2k) + +� 0 +β2k +� +χ# +(2k,2k+1) +� +with arbitrary numbers β2k. Note that ∥u∥2 = 4 � +k∈Z |u1(2k)|2. +Choosing here f = 0 shows that 0 is an eigenvalue of Tmax with infinite multi- +plicity. Choosing f = 0 and requiring ∥u∥ = 0 determines the space L0. Indeed, +L0 = +� � +k∈Z +� +0 +β2k +� +χ# +(2k,2k+1) : β2k ∈ C +� + +16 +STEVEN REDOLFI AND RUDI WEIKARD +which is infinite-dimensional. We now define the sequence τ setting τ0 = 1/2 and, +for k ∈ N, τk = k and τ−k = 1 − k. A solution u of Ju′ + qu = w(λu + f) always +satisfies condition (2k + 1) and it satisfies condition (2k) exactly when β2k = 0. +For f = 0 equations (7.1) and (7.2) show that no non-zero λ can be an eigenvalue +of Tmax. In particular, the deficiency indices n± are 0, i.e., Tmax is self-adjoint. Now +choose λ ̸= 0 and f arbitrary in L2(w). Then +(Rλf)(x) = − 1 +2λ +� +k∈Z +�2f1(2k − 1) + 2f1(2k) +λf1(2k − 1) − λf1(2k) +� +χ# +(2k−1,2k)(x) +(7.3) +is the unique solution of Ju′ + qu = w(λu + f) satisfying condition (k) for any +k ∈ Z. Since +∥Rλf∥2 = +� +k∈Z +2|(Rλf)1(k)|2 = +1 +|λ|2 +� +k∈Z +|f1(2k − 1) + f1(2k)|2 +(7.4) +is finite we have that C \ {0} is the resolvent set of Tmax. +We now define H = {u ∈ L2(w) : u1(2k − 1) = u1(2k)} and H∞ = {f ∈ L2(w) : +f1(2k − 1) = −f1(2k)}. These spaces are orthogonal to each other and their direct +sum is L2(w). Equation (7.4) shows that ker Rλ = H∞. Moreover, we have +Tmax = (H × {0}) ⊕ ({0} × H∞). +This is an instance of a general feature for a self-adjoint linear relation T : if H is +the closure of the domain of T , H∞ the orthogonal complement of H, and T0 = +T ∩ (H × H), then T = T0 ⊕ ({0} × H∞). The former summand is then a linear +operator densely defined in H called the operator part of T . The latter summand +is called the multi-valued part of T . +We end this example by identifying Green’s function for our example. It may +be guessed by looking at equation (7.3). In any case one can check directly that +(Rλf)(x) = +� +G(x, ·, λ)wf. Note that the second column of G is irrelevant since +the second row of w is 0. When x is not integer G(x, y, λ) is given by +� +k∈Z +� +− 1 +λ +�1 +0 +0 +0 +� ++ 1 +2 +� 0 +1 +−1 +0 +� +sgn(x − y) +� +χ# +(2k−1,2k)(x)χ# +(2k−1,2k)(y). +If x is an integer we have instead +G(2k − 1, y, λ) = 1 +2 +lim +x↓2k−1 G(x, y, λ) +and +G(2k, y, λ) = 1 +2 lim +x↑2k G(x, y, λ). +References +[1] Richard Arens. Operational calculus of linear relations. Pacific J. Math., 11:9–23, 1961. +[2] F. V. Atkinson. Discrete and continuous boundary problems. Mathematics in Science and +Engineering, Vol. 8. Academic Press, New York-London, 1964. +[3] Christer Bennewitz. Symmetric relations on a Hilbert space. Pages 212–218. Lecture Notes +in Math., Vol. 280, 1972. +[4] Kevin Campbell, Minh Nguyen, and Rudi Weikard. On the spectral theory for first-order +systems without the unique continuation property. Linear Multilinear Algebra, 69(12):2315– +2323, 2021. Published online: 04 Oct 2019. +[5] Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl. Weyl-Titchmarsh the- +ory for Sturm-Liouville operators with distributional potentials. Opuscula Math., 33(3):467– +563, 2013. +[6] Jonathan Eckhardt and Gerald Teschl. Sturm-Liouville operators with measure-valued coef- +ficients. J. Anal. Math., 120:151–224, 2013. + +GREEN’S FUNCTIONS +17 +[7] Ahmed Ghatasheh and Rudi Weikard. Spectral theory for systems of ordinary differential +equations with distributional coefficients. J. Differential Equations, 268(6):2752–2801, 2020. +[8] M. G. Kre˘ın. On a generalization of investigations of Stieltjes. Doklady Akad. Nauk SSSR +(N.S.), 87:881–884, 1952. +[9] Bruce Call Orcutt. Canonical differential equations. PhD thesis, University of Virginia, 1969. +[10] A. M. Savchuk and A. A. Shkalikov. Sturm-Liouville operators with singular potentials. Math- +ematical Notes, 66(6):741–753, 1999. Translated from Mat. Zametki, Vol. 66, pp. 897–912 +(1999). +[11] Anton Zettl. Sturm-Liouville theory, volume 121 of Mathematical Surveys and Monographs. +American Mathematical Society, Providence, RI, 2005. +Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL +35226-1170, USA +Email address: stevenre@uab.edu, weikard@uab.edu + diff --git a/4tE1T4oBgHgl3EQf6QWC/content/tmp_files/load_file.txt b/4tE1T4oBgHgl3EQf6QWC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..efbb160bc33b0a10d328d3a01fa8415a727032d7 --- /dev/null +++ b/4tE1T4oBgHgl3EQf6QWC/content/tmp_files/load_file.txt @@ -0,0 +1,705 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf,len=704 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='03521v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='SP] 9 Jan 2023 GREEN’S FUNCTIONS FOR FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS WITHOUT THE UNIQUE CONTINUATION PROPERTY STEVEN REDOLFI AND RUDI WEIKARD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This paper is a contribution to the spectral theory associated with the differential equation Ju′ + qu = wf on the real interval (a, b) when J is a constant, invertible skew-Hermitian matrix and q and w are matrices whose entries are distributions of order zero with q Hermitian and w non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with Ju′ + qu = wf and show the existence of Green’s functions for self-adjoint relations even if unique continuation of solutions fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Introduction This paper is a contribution to the spectral theory for the differential equation Ju′ + qu = wf posed on the real interval (a, b) when J is a constant, invertible, and skew-Hermitian n × n-matrix while the entries of the matrices q and w are distributions of order zero1 with q Hermitian and w non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Ghatasheh and Weikard [7] studied this equation under the additional hypothesis that initial value problems have unique balanced2 solutions in the space of functions of locally bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The equation Ju′ + qu = wf has, of course, been investigated by many people when the coefficients q and w are locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In that situation initial value problems always have unique solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This is not necessarily the case when the measures induced by q or w have discrete components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It appears that an equation with measure coefficients was first considered in 1952, when Krein [8] modelled a vibrating string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In 1964 Atkinson [2] suggested to unify the treatment of differen- tial and difference equations by writing them as systems of integral equation where integrals were to be viewed as matrix-valued Riemann-Stieltjes integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Atkinson explained that the presence of point masses may prevent the continuation of so- lutions across such points and posed a condition avoiding that problem but more Date: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This is a preprint of an article published in Integral Equations and Operator Theory which is available online at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1007/s00020-022-02703-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' ©2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This manuscript version is made available under the CC-BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='0 license http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='org/licenses/by-nc-nd/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 1Recall that distributions of order 0 are distributional derivatives of functions of locally bounded variation and hence may be thought of, on compact subintervals of (a, b), as measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For simplicity we might use the word measure instead of distribution of order 0 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 2A function of locally bounded variation is called balanced, if its values at any given point are averages of its left- and right-hand limits at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 1 2 STEVEN REDOLFI AND RUDI WEIKARD restrictive than the one posed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In 1999 Savchuk and Shkalikov [10] treated Schr¨odinger equations with potentials in the Sobolev space W −1,2 loc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Their paper was very influential and spurred many further developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Nevertheless, Eckhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' [5] showed in 2013, with the help of quasi-derivatives or, equivalently, by writing the equation as a system, that a treatment without leaving the realm of locally integrable coefficients is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In the same year Eckhardt and Teschl [6] investigated 2×2-systems with diagonal measure-valued matrices q and w requiring essentially Atkinson’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' A more thorough account of the subject’s history is given in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The papers [5] and [6], mentioned above, may also serve as excellent sources, with perhaps different emphases, of this history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' One feature of systems of first-order equations is that, generally, they are repre- sented by linear relations rather than linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' There is a well-developed spectral theory for linear relations initiated by Arens [1], see also Orcutt [9], and Bennewitz [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The most important results (for our purposes) are also surveyed in Appendix B of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Existence or uniqueness of solutions of an initial value problem for Ju′+qu = wf fails when, for some x ∈ (a, b), the matrices B±(x, 0) = J ± 1 2∆q(x) are not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Here ∆q(x) = Q+(x)−Q−(x) when Q denotes an anti-derivative of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Equivalently, ∆q(x) = dQ({x}) where dQ is the measure (locally) generated by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Assuming the unique continuation property for solutions of Ju′ + qu = wf Ghatasheh and Weikard defined maximal and minimal relations Tmax and Tmin associated with the differential equation Ju′ + qu = wf and showed that Tmax is the adjoint of Tmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' They characterized the self-adjoint restrictions of Tmax, if any, with the aid of boundary conditions and proved that resolvents are given as integral operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', the existence of a Green’s function for any such self-adjoint relation T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Under even more restrictive conditions they also showed the existence of a Fourier transform diagonalizing T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Campbell, Nguyen, and Weikard [4] defined maximal and minimal relations and showed that Tmax = T ∗ min without the hypothesis of unique continuation of so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Our goal here is to advance their ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In particular, even though the equation Ju′ + qu = w(λu + f) may have infinitely many linearly independent solutions the deficiency indices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', the number of linearly independent solutions of Ju′ + qu = ±iwu of finite positive norm, is still bounded by n, the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We show that symmetric restrictions of Tmax, in particular the self-adjoint ones, are still given by posing boundary conditions and we show that the resolvents of self-adjoint restrictions are integral operators by proving the existence of Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We will not approach the problem of Fourier transforms and eigenfunction ex- pansions but hope to return to it in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The material in this paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In Section 2 we recall the circumstances under which existence and uniqueness of solutions to initial value problems does hold and investigate the sets of those x ∈ (a, b) and λ ∈ C giving rise to trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then, in Section 3 we discuss the manifold of solutions of our differential equation in the special case when a and b are regular endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' These results are instrumental in Section 4 where we investigate the deficiency indices GREEN’S FUNCTIONS 3 of the minimal relation and its symmetric extensions but without the assumption that a and b are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Before we prove the existence of Green’s functions for self-adjoint restrictions of the maximal relation in Section 6 we discuss the role played by non-trivial solutions of zero norm in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Let us add a few words about notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' D′0((a, b)) is the space of distributions of order 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', the space of distributional derivatives of functions of locally bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Any function u of locally bounded variation has left- and right-hand limits denoted by u− and u+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Also, u is called balanced if u = u# = (u+ + u−)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The space of balanced functions of bounded variation defined on (a, b) is denoted by BV#((a, b)) while BV# loc((a, b)) stands for the space of balanced functions of locally bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We use 1 to denote an identity matrix of appropriate size and superscripts ⊤ and ∗ indicate transposition and adjoint, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The sum of two closed only trivially intersecting subspaces S and T of some Hilbert space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', their direct sum) is denoted by S ⊎ T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' if S and T are even orthogonal we may use ⊕ instead of ⊎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The orthogonal complement of a subspace S of a Hilbert space H is denoted by H ⊖ S or by S⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', cN ∈ Cn we abbreviate the column vector (c⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', c⊤ N)⊤ ∈ CnN by (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', cN)⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Preliminaries Throughout this paper we assume the following hypothesis to be in force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' J is a constant, invertible and skew-Hermitian n × n-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Both q and w are in D′0((a, b))n×n, w is non-negative and q Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Given that w is non-negative it gives rise to a positive measure on (a, b) and we denote the space of functions f which satisfy � f ∗wf < ∞ by L2(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This space permits the semi-inner product ⟨f, g⟩ = � f ∗wg (note that ⟨f, f⟩ may be 0 without f being 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Consider the differential equation Ju′ + (q − λw)u = wf (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) where λ is a complex parameter and f an element of L2(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The latter condition guarantees that wf is in D′0((a, b))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We will search for solutions in BV# loc((a, b))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In this case each term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) is a distribution of order 0 so that it makes sense to pose the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The point a is called a regular endpoint for Ju′ + qu = wf, if there is a point c ∈ (a, b) such that the left-continuous anti-derivatives Q and W of q and w are of bounded variation on (a, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In this case q and w may be thought of as finite measures on (a, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Similarly, b is called regular, if Q and W are of bounded variation on (c, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If an endpoint is not regular, it is called singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Not surprisingly, the study of our problem is less complicated when the endpoints are regular and we will use this fact to our advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Despite our earlier denigration of the existence and uniqueness theorem of so- lutions of initial value problems it continues to play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The following theorem was proved in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose r ∈ D′0((a, b))n×n, g ∈ D′0((a, b))n and that the matrices 1 ± ∆r(x)/2 are invertible for all x ∈ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Let x0 be a point in (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then the initial value problem u′ = ru + g, u(x0) = u0 ∈ Cn has a unique balanced solution u ∈ BV# loc((a, b))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 4 STEVEN REDOLFI AND RUDI WEIKARD If a is a regular endpoint we may pose an initial condition (for u+) at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Simi- larly, if b is regular we may prescribe u−(b) as the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose now that u is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Treating either side of this equation as a measure (restricted to a compact subset of (a, b)) evaluation at a singleton {x} shows that J(u+(x) − u−(x)) + ∆q−λw(x)u#(x) = ∆w(x)f(x) or, equivalently, B+(x, λ)u+(x) − B−(x, λ)u−(x) = ∆w(x)f(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) when we define B±(x, λ) = J ± 1 2 � ∆q(x) − λ∆w(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Note that, if B+(x, λ) is not invertible, we could be in one of the following two situations: (i) a solution given on (a, x) may fail to exist on (x, b) or (ii) there are infinitely many ways to continue a solution on (a, x) to (x, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' An analogous statement holds, of course, if B−(x, λ) is not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Let us now investigate the circumstances when a pair (x, λ) gives such trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Define the sets Λx = {λ ∈ C : det(B+(x, λ)) det(B−(x, λ)) = 0} and Ξλ = {x ∈ (a, b) : det(B+(x, λ)) det(B−(x, λ)) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' First note, since B−(x, λ) = −B+(x, λ)∗, we have that Ξλ = Ξλ and that each Λx is symmetric with respect to the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Also, Λx is empty unless at least one of ∆q(x) and ∆w(x) is different from 0 and hence for all but countably many x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Next, we claim that Λx is finite as soon as it misses one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' To see this suppose that B+(x, λ0) is invertible and that λ ̸= λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since B+(x, λ) = (λ0 − λ)B+(x, λ0) �1 2B+(x, λ0)−1∆w(x) − 1/(λ − λ0) � we see that B+(x, λ) fails to be invertible only if 1/(λ−λ0) is an eigenvalue of some n × n-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' A similar statement holds, of course, for B− proving our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The really bad points x, namely those where Λx = C, are thus contained in Ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Here we wish to remove the hypothesis Ξ0 = ∅ posed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' On any subinterval of (a, b) on which q gives rise to a finite measure we find that �∞ k=1 ∥∆q(xk)∥ must be finite, when k �→ xk is a sequence of distinct points in that interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It follows now that Ξ0 is a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' One shows similarly that, for any fixed complex number λ the set Ξλ is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose [s, t] ⊂ (a, b) and (s, t)∩Ξ0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then we have that Λ(s,t) = � x∈(s,t) Λx is a discrete subset of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' There are only finitely many points x in (s, t) where ∥J−1∆q(x)∥ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Using a Neumann series one sees that only at such points the norm of B+(x, 0)−1 can be larger than 2∥J−1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Thus there is a positive number C such that ∥B+(x, 0)−1∥ ≤ C for all x ∈ (s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Now suppose that B+(x, λ) is not invertible and that |λ| ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then 1/λ is an eigenvalue of 1 2B+(x, 0)−1∆w(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This requires that ∥∆w(x)∥ ≥ 2/(RC) and thus can happen only for finitely many x ∈ (s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since similar argu- ments work for B− the number of points in � x∈(s,t) Λx which lie in a disk of radius R centered at 0 must be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ We remark that, when one of the anti-derivatives of q and w is only locally of bounded variation, the set � x∈(a,b) Λx need not be discrete even if every Λx is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' GREEN’S FUNCTIONS 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose [s, t] ⊂ (a, b) and (s, t) ∩ Ξ0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If u0 ∈ Cn and λ ∈ C \\ Λ(s,t), then the initial value problem Ju′ + qu = λwu, u+(s) = u0 has a unique balanced solution in (s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Moreover, u(x, ·) for x ∈ (s, t) as well as u−(t, ·) are analytic in C \\ Λ(s,t) and meromorphic on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' An analogous statement holds when the initial condition is posed at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The first claim is simply a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' When x ∈ (s, t) the analyticity of u(x, ·) in C \\ Λ(s,t), which is an open set, was proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3 of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If we modify q and w by setting them 0 on [t, b) we do not change the solution on (s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The solution for the modified problem evaluated at t is analytic and coincides with u−(t, ·) proving its analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It remains to show that a point λ0 ∈ Λ(s,t) can merely give rise to poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We know already that there are only finitely many points x in (s, t) where one of B±(x, λ0) fails to be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose x′ and x′′ are two consecutive such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If we know the solution on (s, x′) and that u−(x′, ·) has, at worst, a pole at λ0, then the solution in (x′, x′′) is determined by the initial value u+(x′, λ) = B+(x′, λ)−1B−(x′, λ)u−(x′, λ) which also has, at worst, a pole at λ0 since this is true for B+(x′, λ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For x ∈ (s, t) the claim follows now by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' To prove that u−(t, ·) is also meromorphic we proceed as before and modify q and w on [t, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Solving the differential equation Our goal in this section is to investigate the set of solutions of the differential equation Ju′ + (q − λw)u = wf on (a, b) under a strengthened hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Hypothesis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In addition to Hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1 we ask that a and b are regular endpoints for Ju′ + qu = wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Moreover, given the partition a = x0 < x1 < x2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' < xN < xN+1 = b (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) of (a, b) we require that Ξ0 ⊂ {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', xN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We then consider only λ for which both B+(x, λ) and B−(x, λ) are invertible unless x is in {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', xN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This hypothesis is in force throughout this section but later only if explicitly mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We emphasize that Ξ0 is finite when a and b are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Also, the set of permissible λ, which we call Ω0, is symmetric with respect to the real axis and avoids only a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' On each interval (xj, xj+1) we let Uj(·, λ) be a fundamental matrix of balanced solutions of the homogeneous differential equation Ju′ + (q − λw)u = 0 such that limx↓xj Uj(x, λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The existence of these fundamental matrices is guaranteed by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The general balanced solution u of the non-homogeneous equation Ju′ + (q − λw)u = wf on (xj, xj+1) satisfies, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3 in [7], u−(x) = U − j (x, λ) � cj + J−1 � (xj,x) Uj(·, λ)∗wf � for any cj ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Define Uj(xj+1, λ) = lim x↑xj+1 Uj(x, λ) and Ij(f, λ) = � (xj,xj+1) Uj(·, λ)∗wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 6 STEVEN REDOLFI AND RUDI WEIKARD Using u+(xj) = cj and u−(xj) = Uj−1(xj, λ)(cj−1 + J−1Ij−1(f, λ)) in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) gives (−B−(xj, λ)Uj−1(xj, λ), B+(xj, λ)) �cj−1 cj � = ∆w(xj)f(xj) + B−(xj, λ)Uj−1(xj, λ)J−1Ij−1(f, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We need to consider these equations for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', N simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This gives rise to the system B(λ)˜u = F0(f, λ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) where ˜u = (c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', cN)⋄, B(λ), to be specified presently, is in CnN×n(N+1), and F0(f, λ) is in CnN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The two-diagonal block-matrix structure of B suggests the introduction of matrices E⊤ and E⊥, which, respectively, strip the first and last n components off a vector in their domain Cn(N+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If we also define the block- matrices B(λ) = diag(B+(x1, λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', B+(xN, λ)), U(λ) = diag(U0(x1, λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', UN−1(xN, λ)), and J = diag(J, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', J) and when we note that B(λ)∗ = diag(−B−(x1, λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', −B−(xN, λ)), we obtain B(λ) = B(λ)∗U(λ)E⊥ + B(λ)E⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3) The vector F0(f, λ) is given by F0(f, λ) = R(f) − B(λ)∗U(λ)J −1I(f, λ) with R(f) = ((∆wf)(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', (∆wf)(xN))⋄ and I(f, λ) = (I0(f, λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', IN−1(f, λ))⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The differential equation Ju′ + (q − λw)u = wf has a solution u on (a, b) if and only if ˜u = (u+(x0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', u+(xN))⋄ is a solution of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In particular, in the homogeneous case, where f = 0, the space of solutions has dimension n(N + 1) − rk B(λ) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We note that rk B(λ) = n when N = 1 so that the space of solutions of Ju′ + (q − λw)u = 0 is then exactly n-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For N = 2, however, consider the example (a, b) = R, J = � 0 −1 1 0 � , q = � 0 2 2 0 � (δ1 − δ2), w = � 2 0 0 0 � (δ1 + δ2), where the δk are Dirac point measures concentrated on {k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It shows that the dimension of the space of solutions of Ju′ + (q − λw)u = 0 may be strictly larger than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Next we investigate the connection between the right-hand limits of a solution u of the homogeneous equation Ju′ + (q − λw)u = 0 at the points x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', xN (given by the vector ˜u) and the vector ˆu = (u(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', u(xN))⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We have ˆu = D(λ)˜u where D(λ) = 1 2(U(λ)E⊥ + E⊤) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4) is again a two-diagonal block-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If N ≥ 2 we will also introduce the matrices Bm(λ) and Dm(λ) which are obtained by deleting the first and last n columns from B(λ) and D(λ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If N = 1 we should think of Bm(λ) and Dm(λ) as maps from the trivial vector space to Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Their adjoints are the map from Cn to {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' With this understanding the following results hold also for N = 1 even though they then involve “matrices” with no rows or columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' GREEN’S FUNCTIONS 7 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' D(λ)∗B(λ) − B(λ)∗D(λ) = diag(−J, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', 0, J) and Dm(λ)∗B(λ) − Bm(λ)∗D(λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This follows since U(λ)∗J U(λ) = J which, in turn, follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The map v �→ B(λ)v, restricted to ker D(λ), is a bijection onto ker Dm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Similarly, the map v �→ D(λ)v, restricted to ker B(λ), is a bijection onto ker Bm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In particular, dim ker D(λ) = dim ker Dm(λ)∗ and dim ker B(λ) = dim ker Bm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The identity Dm(λ)∗B(λ)−Bm(λ)∗D(λ) = 0 shows that B(λ) maps ker D(λ) to ker Dm(λ)∗ as well as that D(λ) maps ker B(λ) to ker Bm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If v ∈ ker B(λ) ∩ ker D(λ) one shows that E⊥v = E⊤v = 0 using the definitions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4) of B and D and the fact that B(λ) − B(λ)∗ = 2J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This, of course, implies that v = 0 and hence the injectivity of both B(λ)|ker D(λ) and D(λ)|ker B(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Clearly, both D(λ) and Dm(λ)∗, having invertible matrices along their main diagonal, are of full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The rank-nullity theorem shows therefore that their kernels both have dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This proves surjectivity of B(λ)|ker D(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Finally, assume that v ∈ ker Bm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then v = D(λ)x for some x ∈ Cn(N+1) which implies that 0 = Bm(λ)∗D(λ)x = Dm(λ)∗B(λ)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The first part of the proof shows that there is a y ∈ ker D(λ) such that B(λ)y = B(λ)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Hence v = D(λ)(x−y) where x − y ∈ ker B(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ The following theorem establishes a connection between solutions of the differ- ential equation Ju′ + (q − λw)u = 0 and elements of ker Bm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If u is a solution of Ju′ + (q − λw)u = 0 on (a, b), then ˆu = (u(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', u(xN))⋄ is in ker Bm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If, in addition, u+(a) = u−(b) = 0, then ˆu ∈ ker B(λ)∗ (a subspace of ker Bm(λ)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Conversely, if ˆu ∈ ker Bm(λ)∗, then Ju′ + (q − λw)u = 0 has a unique solution u on (a, b) such that (u(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', u(xN))⋄ = ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If, indeed, ˆu ∈ ker B(λ)∗, we further have u+(a) = u−(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Let us emphasize that supp u ⊂ [x1, xN] when u+(a) = u−(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If u solves Ju′ + (q − λw)u = 0, then, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2, ˜u ∈ ker B(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4 shows then that ˆu = D(λ)˜u is in ker Bm(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If u+(a) = u−(b) = 0, then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3 gives 0 = B(λ)∗D(λ)˜u = B(λ)∗ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Conversely, assume that ˆu ∈ ker Bm(λ)∗ = D(λ)(ker B(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then there is a unique vector ˜u ∈ ker B(λ) such that ˆu = D(λ)˜u, which, in turn, defines a unique solution u of Ju′+(q−λw)u = 0 such that (u(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', u(xN))⋄ = ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If ˆu ∈ ker B(λ)∗, then, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3, diag(−J, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', 0, J)˜u = 0 which shows that u+(a) = u−(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ Given an algebraic system Ax = b we know that there exist solutions only if b ∈ ran A = (ker A∗)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For the differential equation Ju′ + (q − λw)u = wf with integrable coefficients q and w the unique continuation property for the solutions gives rise to the variation of constants formula, which then guarantees the existence of solutions for any non-homogeneity f (within reason).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In the present situation, 8 STEVEN REDOLFI AND RUDI WEIKARD however, the problem of existence raises its head and we now set out to give neces- sary and sufficient conditions for f guaranteeing the existence of a solution in the spirit of Linear Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If ˜v ∈ ker B(λ) and ˆv = D(λ)˜v, then ˜v∗E∗ ⊥ = −ˆv∗B(λ)∗U(λ)J −1 and ˜v∗E∗ ⊤ = ˆv∗B(λ)J −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Moreover, if f ∈ L2(w) and Jv′ + (q − λw)v = 0, then � v∗wf = ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ) = ˆv∗F0(f, λ) + ˜v∗E∗ ⊤˜I(f, λ) where (v(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', v(xN))⋄ = ˆv = D(λ)˜v and ˜I(f, λ) = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', 0, IN(f, λ))⋄ ∈ CnN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Using the definitions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4) of B and D and the identities B(λ) − B(λ)∗ = 2J and U(λ)∗J U(λ) = J we obtain that B(λ)˜v = 0 implies B(λ)D(λ)˜v = U(λ)∗−1J E⊥˜v and B(λ)∗D(λ)˜v = −J E⊤˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Taking adjoints gives the first claim since ˆv = D(λ)˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The second claim is an immediate consequence of this, since � v∗wf = ˆv∗R(f) + ˜v∗(I0(f, λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', IN (f, λ))⋄ = ˆv∗R(f) + ˜v∗E∗ ⊥I(f, λ) + ˜v∗E∗ ⊤˜I(f, λ) = ˆv∗R(f) − ˆv∗B(λ)∗U(λ)J −1I(f, λ) + ˆv∗B(λ)J −1˜I(f, λ) = ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The differential equation Ju′ + (q − λw)u = wf has a solution on (a, b) if and only if � v∗wf = 0 for every solution v of Jv′ + (q − λw)v = 0 which vanishes at a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2 the solution u exists if and only if the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) has a solution ˜u = (u+(x0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', u+(xN))⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This, in turn, happens if and only if F0(f, λ) ∈ ran B(λ) = (ker B(λ)∗)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='5 the solutions of Jv′ +(q −λw)v = 0 which vanish at a and b are in one-to-one correspondence with elements of ker B(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since v+(xN) = 0 we have ˜v∗E∗ ⊤˜I(f, λ) = 0 and then, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='6, we obtain ˆv∗F0(f, λ) = � v∗wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ In the case of unique continuation of solutions the condition that v vanishes at a or b implies, of course, that v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Consequently, Ju′ + (q − λw)u = wf has then a solution for any f ∈ L2(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The set of all solutions is thus obtained by adding the general solution of Ju′ +(q −λw)u = 0 whose dimension is n(N + 1)−rkB(λ) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The differential equation Ju′ + (q − λw)u = wf has a solution on (a, b) which vanishes at a and b if and only if � v∗wf = 0 for every solution v of Jv′ + (q − λw)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For u to vanish at a and b it is required that u+(x0) = 0 and u+(xN) = −J−1IN(f, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) is therefore equivalent to Bm(λ)(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', cN−1)⋄ = F0(f, λ) + B(λ)J −1˜I(f, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The proof is now analogous to the one for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ GREEN’S FUNCTIONS 9 We conclude this section by “counting” the solutions of Ju′ + qu = λwu which are not compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' More precisely, we will determine the dimension of the quotient space of all solutions of Ju′ + qu = λwu modulo the space of compactly supported solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='5 shows that the space of all solutions of Ju′+qu = λwu is in one-to-one correspondence with ker Bm(λ)∗ and that the space of compactly supported solutions of Ju′+qu = λwu is in one-to-one correspondence with ker B(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We therefore define ˜n(λ) = dim(ker Bm(λ)∗/ ker B(λ)∗) = dim ker Bm(λ)∗ − dim ker B(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' ˜n(λ) + ˜n(λ) = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since rk B(λ) = rk B(λ)∗, the rank-nullity theorem implies dim ker B(λ) = n(N + 1) − rk B(λ)∗ = n + dim ker B(λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Hence, using also the analogous equation for λ, dim ker B(λ) − dim ker B(λ)∗ + dim ker B(λ) − dim ker B(λ)∗ = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4 gives that dim ker B(λ) = dim ker Bm(λ)∗ yielding the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4 we know that the matrices Uj(xj+1, ·) are meromorphic on C with poles at most at points in the complement of Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It follows that the entries of B are also meromorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since the meromorphic functions on C form a field there is a row-echelon matrix ˜B with meromorphic entries such that B˜u = 0 has the same solutions as ˜B˜u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Now define a set Ω as Ω0 without the set of all poles of ˜B as well as their complex conjugates, and the set of zeros and their conjugates of any of the pivots of ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If λ ∈ Ω, then dim ker B(λ) = dim ker B(λ) and ˜n(λ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The construction of Ω entails that rk B(λ) = rk ˜B(λ) = rk B(λ) if λ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since ˜n(λ) = dim ker B(λ) − dim ker B(λ)∗ = dim ker B(λ) + n − dim ker B(λ) we obtain ˜n(λ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Symmetric restrictions of Tmax Given a differential equation Ju′ + qu = wf we now define associated minimal and maximal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Recall that L2(w) is the space of functions f such that � f ∗wf < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' First we define Tmax = {(u, f) ∈ L2(w) × L2(w) : u ∈ BV# loc((a, b))n, Ju′ + qu = wf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Subsequently we will always tacitly assume that u ∈ BV# loc((a, b))n, when we use u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Next, let Tmin = {(u, f) ∈ Tmax : supp u is compact in (a, b)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Note that these are spaces of pairs of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' To employ the power of functional analysis we need to realize these relations in Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Therefore we introduce, as usual, the space L2(w) as the quotient of L2(w) modulo the subspace of all u ∈ L2(w) for which ∥u∥2 = � u∗wu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Denoting the equivalence class corresponding to u by [u] we now set Tmax = {([u], [f]) ∈ L2(w) × L2(w) : (u, f) ∈ Tmax} 10 STEVEN REDOLFI AND RUDI WEIKARD and Tmin = {([u], [f]) ∈ Tmax : (u, f) ∈ Tmin}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Here (and elsewhere) we choose brevity over precision: whenever we have a pair ([u], [f]) in Tmax we choose u and f such that (u, f) ∈ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Define the vector space L0 = {u ∈ BV# loc((a, b))n : Ju′ + qu = 0 and ∥u∥ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In many cases this space is trivial and some authors restrict their attention to the case where it is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' this is then called the definiteness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' However, we will not do so here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Note that ∥u∥ = 0 if and only if wu is the zero distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The significance of L0 stems from the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose ([u], [f]) ∈ Tmax and that there are u, v ∈ [u] and f, g ∈ [f] such that Ju′ + qu = wf and Jv′ + qv = wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then J(u − v)′ + q(u − v) = w(f − g) = 0 as well as w(u − v) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', u − v ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In other words, in the presence of a non-trivial space L0, the class [u] has many representatives of locally bounded variation satisfying the differential equation for a given class [f] (the choice of a representative of [f], on the other hand, is irrelevant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In Section 5 we will describe a procedure to choose a representative of [u] in a distinctive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In [4] it was proved that Tmin is symmetric, indeed that T ∗ min = Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In this case it is well-known that von Neumann’s theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Setting Dλ = {([u], λ[u]) ∈ Tmax} it states that Tmax = Tmin ⊎ Dλ ⊎ Dλ when Im λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Moreover, when λ = ±i, these direct sums are even orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It is also known that the dimension of Dλ does not change as λ varies in either the upper or the lower half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The numbers n± = dim D±i are called deficiency indices of Tmin and we are now setting out to investigate these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If u is a solution of Ju′ +qu = λwu which is compactly supported then (u, λu) ∈ Tmin and ([u], λ[u]) ∈ Tmin ∩ Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If λ is not real, then Tmin ∩ Dλ is trivial and it follows that compactly supported solutions of Ju′ + qu = λwu do not contribute to the corresponding deficiency index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now have, as a corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='10, that the deficiency indices of Tmin cannot be more than n if a and b are regular endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We do not state this result separately since it is included in the next theorem about the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Thus, to emphasize, we allow in the following a and b to be either regular or singular endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Let τk, k ∈ Z, be a strictly increasing sequence in (a, b) having a and b as its only limit points and such that all points in Ξ0 are among the τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Considering now only the interval Ik = (τ−k, τk) we set xj = τ−k+j for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', N + 1 = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We can then introduce the objects from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' To emphasize their dependence on k we will add a superscript (k) to those objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We have then, in particular, the matrices B(k), B(k) m and the sets Ω(k) of permissible values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now define Ω = �∞ k=1 Ω(k) and note that Ω is symmetric with respect to the real axis and misses only countably many values from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Now fix a non-real λ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If u is a solution of Ju′+qu = λwu on (a, b) we denote its restriction to the interval Ik by u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We are interested in the quotient space Xk of all solutions of Ju′ +qu = λwu on Ik modulo the compactly supported solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If u is a solution of Ju′+qu = λwu on Ik we denote the associated equivalence class in Xk by ⌊u⌋k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' A compactly supported solution u of Ju′ + qu = λwu on Ik can be extended by 0 to all of (a, b) yielding an element in Tmin ∩ Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This implies, since GREEN’S FUNCTIONS 11 Im λ ̸= 0, that ∥u∥2 = � Ik u∗wu = 0 and shows that Xk is a normed space with the norm given by ∥u∥2 k = � Ik u∗wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='5 the quotient space Xk is isomorphic to ker B(k) m (λ)∗/ ker B(k)(λ)∗ and, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='10, its dimension is equal to n since λ ∈ Ω ⊂ Ω(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The deficiency indices of Tmin are less than or equal to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Fix a non-real λ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', um are solutions of Ju′ + qu = λwu such that [u1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', [um] are linearly independent elements of Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We will show below that there is an interval Ip = (τ−p, τp) such that ⌊u(p) 1 ⌋p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', ⌊u(p) m ⌋p are linearly independent elements of Xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Hence m ≤ n, the dimension of Xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since deficiency indices are constant in either half-plane they cannot be larger than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We will now prove the existence of Ip by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' That is we prove that, for every k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', m}, there is an interval Iℓk such that the restrictions of u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', uk to Iℓk generate linearly independent elements ⌊u(ℓk) 1 ⌋ℓk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', ⌊u(ℓk) k ⌋ℓk of Xℓk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Once this is achieved we set p = ℓm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose k = 1 and let Iℓ1 be an interval such that ∥u(ℓ1) 1 ∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' By what we argued above we know that u(ℓ1) 1 is not compactly supported in Iℓ1 and thus gives rise to a non-zero (and hence linearly independent) element of Xℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Now suppose we had already shown our claim for some k < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If ⌊u(ℓk) 1 ⌋ℓk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', ⌊u(ℓk) k+1⌋ℓk are already linearly independent as elements of Xℓk we choose ℓk+1 = ℓk and our induction step is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Otherwise, there are unique complex numbers α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', αk such that ∥(α1u1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' + αkuk + uk+1)(ℓk)∥ℓk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' However, there must be an interval Iℓk+1 ⊃ Iℓk where ∥(α1u1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' + αkuk + uk+1)(ℓk+1)∥ℓk+1 > 0 on account that [u1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', [uk+1] are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It follows now that, as ele- ments of Xℓk+1 the vectors ⌊u(ℓk+1) 1 ⌋ℓk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', ⌊u(ℓk+1) k+1 ⌋ℓk+1 are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This completes our induction step also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If a and b are regular, then n+ = n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Fix a non-real λ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since a and b are regular, the set Ξλ = Ξλ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Thus we may assume that it is contained in Ik = (τ−k, τk) for some appropriate k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then dim ker B(k)(λ) is the number of linearly independent solutions of Ju′ + qu = λwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='10 shows that Ju′ + qu = λwu has the same number of linearly independent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Any of these solutions has finite norm but some may have norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Now note, that if u is a solution of Ju′+qu = λwu of norm 0, then we have wu = 0, so that u is also a solution of Ju′ + qu = λwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Therefore n+ = n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ As mentioned above, it is well-known, even in the case of relations, that von Neumann’s theorem E∗ = E⊕Di⊕D−i holds when E is a closed symmetric relation in H × H when H is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In our case, when d = dim Di ⊕ D−i is finite, as we just showed, we can use Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='5 in [7] to characterize the symmetric restriction of Tmax in terms of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We state that theorem here for easy reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The operator J appearing there is defined by J (u, f) = (f, −u) for u, f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 12 STEVEN REDOLFI AND RUDI WEIKARD Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose E is a closed symmetric relation in H × H with d = dim Di ⊕ D−i < ∞ and that m ≤ d/2 is a natural number or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If A : E∗ → Cd−m is a surjective linear operator such that E ⊂ ker A and AJ A∗ has rank d−2m then ker A is a closed symmetric restriction of E∗ for which the dimension of (ker A)⊖E is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Conversely, every closed symmetric restriction of E∗ is the kernel of such a linear operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Finally, ker A is self-adjoint if and only if AJ A∗ = 0 (entailing m = d/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' A second ingredient for our next considerations is Lagrange’s identity (or Green’s formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If (u, f) and (v, g) are in Tmax, then v∗wf and g∗wu are finite measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Therefore v∗Ju′ + v′∗Ju = v∗wf − g∗wu is also a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Its antiderivative v∗Ju is of bounded variation and thus has limits at a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Integration now gives Lagrange’s identity (v∗Ju)−(b) − (v∗Ju)+(a) = ⟨v, f⟩ − ⟨g, u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) Note the right-hand side, and hence the left-hand side, does not change upon choos- ing different representatives in place of u, f, v, or g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Now, if (v, g) is an element of Di⊕D−i, then (u, f) �→ ⟨(v, g), (u, f)⟩ is a bounded linear functional on Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Conversely, since Tmax is a Hilbert space, a bounded linear functional on Tmax is given by (u, f) �→ ⟨(v, g), (u, f)⟩ for some (v, g) ∈ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' When it is also known that Tmin is in the kernel of this functional, (v, g) may be chosen in Di ⊕ D−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Hence, in our situation, the operator A from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3 is given by d − m linearly independent elements in Di ⊕ D−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Lagrange’s identity implies that the entries of the matrix AJ A∗ are then given by (AJ A∗)k,ℓ = ⟨(vk, gk), (gℓ, −vℓ)⟩ = (g∗ kJgℓ)−(b) − (g∗ kJgℓ)+(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) Therefore we arrive at the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Let d = n+ + n− and suppose that m ≤ min{n+, n−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If (v1, g1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', (vd−m, gd−m) are linearly independent elements of Di⊕D−i such that the matrix defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) has rank d − 2m, then T = {(u, f) ∈ Tmax : (g∗ j Ju)−(b) − (g∗ j Ju)+(a) = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', d − m} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3) is a closed symmetric restriction of Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Conversely, if T is a closed symmetric restriction of Tmax and m is the dimen- sion of T ⊖ Tmin, then T is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3) for appropriate elements (v1, g1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', (vd−m, gd−m) of Di ⊕ D−i for which the matrix defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) has rank d − 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For self-adjoint restrictions of Tmax it is hence necessary and sufficient that n+ = n− = m = d−m and that (g∗ kJgℓ)−(b)−(g∗ kJgℓ)+(a) = 0 for all 1 ≤ k, ℓ ≤ m = d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The space L0 We mentioned earlier that the class [u] does not have a unique balanced repre- sentative when ([u], [f]) ∈ Tmax, if the space L0 has non-trivial elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In this section we describe a procedure to choose a representative in a distinctive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' To this end we assume, without loss of generality, that B+(τ0, 0) = B−(τ0, 0) = J so that solutions of our differential equations are continuous at τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Define N0 = {h(τ0) : h ∈ L0} and for each k ∈ N both Nk = {h+(τk) : h ∈ L0, supp h ⊂ [τk, b)} and N−k = {h−(τ−k) : h ∈ L0, supp h ⊂ (a, τ−k]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then, for k ∈ N0, we say that a function u ∈ BV# loc((a, b))n satisfies condition (±k), if u±(τ±k) is perpendicular to N±k (using always the upper sign or always the lower sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' GREEN’S FUNCTIONS 13 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose ([u], [f]) ∈ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then there is a unique balanced v ∈ [u] such that (v, f) ∈ Tmax and v satisfies condition (k) for every k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' First consider uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Suppose u and v are two functions satisfying the given conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then u − v ∈ L0 and hence (u − v)(τ0)∗t(τ0) = 0 for t = u and t = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Subtract these equations to find (u−v)(τ0) = 0, and thus u = v on (τ−1, τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Moreover, h1 = (u − v)χ[τ1,b) and h−1 = (u − v)χ(a,τ−1] are in L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Conditions (1) and (−1) show therefore that (u−v)+(τ1) and (u−v)−(τ−1) are also 0 which proves that u = v on (τ−2, τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Induction informs us now that u = v everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now turn to existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Pick a balanced representative u ∈ [u] such that (u, f) ∈ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' There is an element h0 ∈ L0 such that the orthogonal projection of u(τ0) onto N0 equals h0(τ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Thus v0 = u − h0 satisfies (v0, f) ∈ Tmax, v0 ∈ [u], and condition (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Next, there is an element h1 ∈ L0 with support in [τ1, b) such that the orthogonal projection of v+ 0 (τ1) onto N1 equals h+ 1 (τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now define v1 = v0 − h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then (v1, f) ∈ Tmax, v1 ∈ [u], and v1 satisfies condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Notice that v1 = v0 on (a, τ1) implying that v1 also satisfies condition (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proceeding recursively, we may define, for each k ∈ N, functions hk ∈ L0 sup- ported in [τk, b) such that vk = u−�k j=0 hj satisfies conditions (0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', (k), vk ∈ [u], and (vk, f) ∈ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since, for a fixed x ∈ [τ0, b), only finitely many of the numbers hk(x) are different from 0, we find that the sequence k �→ vk converges pointwise to a function ˜v ∈ [u] satisfying conditions (k) for all k ∈ N0 and (˜v, f) ∈ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We can now repeat this process for negative integers starting from the function ˜v instead of u arriving eventually at a function v ∈ [u] satisfying conditions (k) for all k ∈ Z and (v, f) ∈ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ We denote the operator which assigns the function v just constructed to a given element ([u], [f]) ∈ Tmax by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If Im = (τ−m, τm) we also define Em : Tmax → BV#(Im)n by composing E with the restriction to the interval Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Note that BV#(Im)n is a Banach space with the norm |||u|||m defined as the sum of the variation of u over Im and the norm of u(τ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The operator Em : Tmax → BV#(Im)n is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Due to the closed graph theorem we merely have to show that Em is a closed operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Thus assume that the sequence ([uj], [fj]) converges to ([u], [f]) in Tmax and that Em([uj], [fj]) converges to v in BV#(Im)n and hence pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' To simplify notation we assume that Em([uj], [fj])) and Em([u], [f]) are the restrictions of uj and u, respectively, to the interval Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We need to show that u = v on Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' First note that u± j (τ±k) ∈ N ⊥ ±k and ��u± j (τ±k) − v±(τ±k) �� → 0 imply that v satisfies conditions (±k) for each k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For ℓ ∈ {−m, m − 1} and x ∈ (τℓ, τℓ+1) we have u− j (x) = U − ℓ (x) � u+ j (τℓ) + J−1 � (τℓ,x) U ∗ ℓ wfj � when Uℓ denotes the fundamental matrix of Ju′ + qu = 0 on the interval (τℓ, τℓ+1) satisfying U + ℓ (τℓ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Taking the limit as j → ∞ gives v−(x) = U − ℓ (x) � v+(τℓ) + J−1 � (τℓ,x) U ∗ ℓ wf � 14 STEVEN REDOLFI AND RUDI WEIKARD since the integral may be considered as a vector of scalar products which are, of course, continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The variation of constants formula shows that v is a balanced solution for Jv′ + qv = wf on (τℓ, τℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We also have J(u+ j (τℓ) − u− j (τℓ)) + ∆q(τℓ)uj(τℓ) = ∆w(τℓ)fj(τℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) The fact that [fj] converges to [f] in L2(w) implies, on account of the positivity of w, that ∆w(τℓ)fj(τℓ) converges to ∆w(τℓ)f(τℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Therefore taking a limit in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) shows, in conjunction with the previous observations, that Jv′ + qv = wf on the interval Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since u satisfies the same equation we have that u − v satisfies J(u − v)′ + q(u − v) = 0 on Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Next we show w(u − v) = 0 on Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Fatou’s lemma implies 0 ≤ � Im (u − v)∗w(u − v) ≤ lim inf j→∞ � Im (u − uj)∗w(u − uj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It follows that w(u − v) = 0 on Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Finally, a variant of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1 shows now that u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Green’s function Now suppose that we have a self-adjoint restriction T of Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The resolvent set of T is the set of those λ for which T − λ : dom(T ) → L2(w) is bijective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', ̺(T ) = {λ ∈ C : ker(T − λ) = {0}, ran(T − λ) = L2(w)} which is an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We denote its complement, the spectrum of T , by σ(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since T is self-adjoint, σ(T ) is a subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If λ ∈ ̺(T ), then the resolvent Rλ = (T − λ)−1 is a bounded linear operator from L2(w) to dom(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now define Rλ : L2(w) → BV# loc((a, b))n by Rλ[f] = E((Rλ[f], λRλ[f] + [f])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Thus Rλ[f] is the unique solution of Ju′ + qu = w(λu + f) in L2(w) satisfying condition (k) for every k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We will now show that Rλ is an integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Its kernel G is called a Green’s function for T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If T is a self-adjoint restriction of Tmax, then there exists, for given x ∈ (a, b) and λ ∈ ̺(T ), a matrix G(x, ·, λ) such that the columns of G(x, ·, λ)∗ are in L2(w) and (Rλ[f])(x) = � G(x, ·, λ)wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Fix x ∈ Im and λ ∈ ̺(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Consider the restriction of Rλ[f] to the interval Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since Em and Rλ are bounded operators the map [f] �→ (Rλ[f])(x) is a bounded linear map from L2(w) to Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Hence there are elements [g1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', [gn] ∈ L2(w) such that the k-th component of (Rλ[f])(x) equals ⟨[gk], [f]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Let these be the columns of the matrix-valued function G(x, ·, λ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then we obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' □ One wishes to complement this fairly abstract existence result by a more concrete one where Green’s function is given in terms of solutions of the differential equation as is done in the classical case, see, for instance, Zettl [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This was also achieved in [7] under the assumption that Ξ0 is empty and minor generalizations of this are certainly possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Such an explicit construction of Green’s function, where possible, is the cornerstone of many other results in spectral theory, in particular GREEN’S FUNCTIONS 15 the development of a spectral transformation and more detailed information about the resolvent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', the compactness of the resolvent in the regular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Due to the difficulties posed by the absence of an existence and uniqueness theorem for initial value problems we have, so far, not been able to obtain such a construction in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' However, we hope to return to this issue in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Example In this section we treat an example where the matrices B±(x, λ) fail to be invert- ible for infinitely many x and all λ, in other words where Ξ0 is infinite and Λx = C for all x ∈ Ξ0 (recall that in [7] the hypothesis Ξ0 = ∅ was made causing each Λx to be finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The example is Ju′ + qu = wf on (a, b) = R where J = � 0 −1 1 0 � , q = � 0 2 2 0 � � k∈Z (δ2k − δ2k+1), and, w = � 2 0 0 0 � � k∈Z δk with δk denoting the Dirac point measure concentrated on {k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since we are seeking balanced solutions we need the matrices B−(2k − 1, λ) = �λ 0 2 0 � and B+(2k − 1, λ) = �−λ −2 0 0 � as well as B−(2k, λ) = � λ −2 0 0 � and B+(2k, λ) = � −λ 0 2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If x is not an integer we have B±(x, λ) = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Note that f ∈ L2(w) if and only if k �→ f1(k) is in ℓ2(Z) and any element in L2(w) is uniquely determined by these values (here f1 denotes the first component of f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In any interval (k, k + 1) solutions of Ju′ + qu = w(λu + f) are constant, say (αk, βk)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' At x = 2k − 1 the equation B+(2k − 1, λ)u+(2k − 1) − B−(2k − 1, λ)u−(2k − 1) = (2f1(2k − 1), 0)⊤ implies α2k−2 = 0 and − λα2k−1 − 2β2k−1 = 2f1(2k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) Similarly, at x = 2k we get α2k = 0 and − λα2k−1 + 2β2k−1 = 2f1(2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) We can now describe the space Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' A pair (u, f) is in Tmax if and only if the sequences k �→ f1(k) and k �→ u1(k) are in ℓ2(Z), f1(2k) = −f1(2k − 1), u1(2k) = u1(2k − 1), and u = � k∈Z � �2u1(2k) f1(2k) � χ# (2k−1,2k) + � 0 β2k � χ# (2k,2k+1) � with arbitrary numbers β2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Note that ∥u∥2 = 4 � k∈Z |u1(2k)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Choosing here f = 0 shows that 0 is an eigenvalue of Tmax with infinite multi- plicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Choosing f = 0 and requiring ∥u∥ = 0 determines the space L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Indeed, L0 = � � k∈Z � 0 β2k � χ# (2k,2k+1) : β2k ∈ C � 16 STEVEN REDOLFI AND RUDI WEIKARD which is infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now define the sequence τ setting τ0 = 1/2 and, for k ∈ N, τk = k and τ−k = 1 − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' A solution u of Ju′ + qu = w(λu + f) always satisfies condition (2k + 1) and it satisfies condition (2k) exactly when β2k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' For f = 0 equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='2) show that no non-zero λ can be an eigenvalue of Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In particular, the deficiency indices n± are 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', Tmax is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Now choose λ ̸= 0 and f arbitrary in L2(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Then (Rλf)(x) = − 1 2λ � k∈Z �2f1(2k − 1) + 2f1(2k) λf1(2k − 1) − λf1(2k) � χ# (2k−1,2k)(x) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3) is the unique solution of Ju′ + qu = w(λu + f) satisfying condition (k) for any k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Since ∥Rλf∥2 = � k∈Z 2|(Rλf)1(k)|2 = 1 |λ|2 � k∈Z |f1(2k − 1) + f1(2k)|2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4) is finite we have that C \\ {0} is the resolvent set of Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We now define H = {u ∈ L2(w) : u1(2k − 1) = u1(2k)} and H∞ = {f ∈ L2(w) : f1(2k − 1) = −f1(2k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' These spaces are orthogonal to each other and their direct sum is L2(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='4) shows that ker Rλ = H∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Moreover, we have Tmax = (H × {0}) ⊕ ({0} × H∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' This is an instance of a general feature for a self-adjoint linear relation T : if H is the closure of the domain of T , H∞ the orthogonal complement of H, and T0 = T ∩ (H × H), then T = T0 ⊕ ({0} × H∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The former summand is then a linear operator densely defined in H called the operator part of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' The latter summand is called the multi-valued part of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' We end this example by identifying Green’s function for our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' It may be guessed by looking at equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' In any case one can check directly that (Rλf)(x) = � G(x, ·, λ)wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Note that the second column of G is irrelevant since the second row of w is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' When x is not integer G(x, y, λ) is given by � k∈Z � − 1 λ �1 0 0 0 � + 1 2 � 0 1 −1 0 � sgn(x − y) � χ# (2k−1,2k)(x)χ# (2k−1,2k)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' If x is an integer we have instead G(2k − 1, y, λ) = 1 2 lim x↓2k−1 G(x, y, λ) and G(2k, y, λ) = 1 2 lim x↑2k G(x, y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' References [1] Richard Arens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Operational calculus of linear relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Academic Press, New York-London, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' [3] Christer Bennewitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Symmetric relations on a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Pages 212–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', Vol.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Sturm-Liouville operators with measure-valued coef- ficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=', 120:151–224, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' GREEN’S FUNCTIONS 17 [7] Ahmed Ghatasheh and Rudi Weikard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Spectral theory for systems of ordinary differential equations with distributional coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Differential Equations, 268(6):2752–2801, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Kre˘ın.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' On a generalization of investigations of Stieltjes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Doklady Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Nauk SSSR (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' ), 87:881–884, 1952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' [9] Bruce Call Orcutt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Canonical differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' PhD thesis, University of Virginia, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Savchuk and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Shkalikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Sturm-Liouville operators with singular potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Math- ematical Notes, 66(6):741–753, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Translated from Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Zametki, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 66, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' 897–912 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' [11] Anton Zettl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Sturm-Liouville theory, volume 121 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content=' Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35226-1170, USA Email address: stevenre@uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='edu, weikard@uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQf6QWC/content/2301.03521v1.pdf'} diff --git a/7dE4T4oBgHgl3EQf2Q2c/content/tmp_files/load_file.txt b/7dE4T4oBgHgl3EQf2Q2c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..90afade6ff0db426bd04798c0ff5c5b44f6fed33 --- /dev/null +++ b/7dE4T4oBgHgl3EQf2Q2c/content/tmp_files/load_file.txt @@ -0,0 +1,735 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf,len=734 +page_content='Towards Dependable Autonomous Systems Based on Bayesian Deep Learning Components Fabio Arnez∗, Huascar Espinoza†, Ansgar Radermacher∗ and Franc¸ois Terrier∗ ∗Universit´e Paris-Saclay, CEA, List, F-91120, Palaiseau, France {name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='lastname}@cea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='fr †KDT JU, TO 56 05/16, B-1049 Brussels, Belgium huascar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='espinoza@kdt-ju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='eu Abstract—As autonomous systems increasingly rely on Deep Neural Networks (DNN) to implement the navigation pipeline functions, uncertainty estimation methods have be- come paramount for estimating confidence in DNN predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Bayesian Deep Learning (BDL) offers a principled approach to model uncertainties in DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However, in DNN-based systems, not all the components use uncertainty estimation methods and typically ignore the uncertainty propagation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' This paper provides a method that considers the uncertainty and the interaction between BDL components to capture the overall system uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' We study the effect of uncertainty propagation in a BDL-based system for autonomous aerial navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Experiments show that our approach allows us to capture useful uncertainty estimates while slightly improving the system’s performance in its final task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In addition, we discuss the benefits, challenges, and implications of adopting BDL to build dependable autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Index Terms—Bayesian Deep Learning, Uncertainty Propaga- tion, Unmanned Aerial Vehicle, Navigation, Dynamic Depend- ability I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' INTRODUCTION Navigation in complex environments still represents a big challenge for autonomous systems (AS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Particular instances of this problem are autonomous driving and autonomous aerial navigation in the context of self-driving cars and Unmanned Aerial Vehicles (UAVs), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In both cases, the naviga- tion task is addressed by first acquiring rich and complex raw sensory information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', from camera, radar, LiDAR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' ), which is then processed to drive the autonomous agent towards its goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Usually, this process is done in sequence, where tasks and specific software components are linked together in the so- called perception-planning-control software pipeline [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Over the last decade, Deep Neural Networks (DNNs) have become a popular choice to implement navigation pipeline components thanks to their effectiveness in processing com- plex sensory inputs, and their powerful representation learning that surpasses the performance of traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Cur- rently, three main paradigms exist to develop and train navi- gation components based on DNNs: Modular (isolated), End- to-End (E2E) learning, and mixed or hybrid approaches [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Preprint version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Accepted and presented at the 18th European Depend- able Computing Conference (EDCC), Zaragoza, Spain, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Digital Object Identifier (DOI) is available in the preprint description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' UAV BDL-based Aerial Navigation Pipeline: The downstream control component gets predictions of the previous perception component as input and must take their uncertainty into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Despite the remarkable progress in representation learning, DNNs should also represent the confidence in their predictions to deploy them in safety-critical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' McAllister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' [2] proposed using Bayesian Deep Learning (BDL) to implement the components from navigation pipelines or stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Bayesian methods offer a principled framework to model and capture system uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However, if the Bayesian approach is followed, all the components in the system pipeline should use BDL to enable uncertainty propagation in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Hence, BDL components should admit uncertainty information as an input to account for the uncertainty from the outputs of preceding BDL components (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In recent years, a large body of literature has employed uncertainty estimation methods in robotic tasks thanks to its potential to improve the safety of automated functions [3], and the capacity to increase the task performance [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However, uncertainty is captured partially in navigation pipelines that utilize DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' BDL methods are used mainly in perception tasks, and downstream components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', planning and control) usually ignore the uncertainty from the preceding components or do not capture uncertainty in their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Although some works propagate downstream perceptual uncertainty from intermediate representations [6]–[8], the overall system output does not take into account all the uncertainty sources from DNN components in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Moreover, proposed frameworks for dynamic dependability management that use uncertainty information focus only on DNN-based perception tasks [9], [10], ignoring uncertainty propagation through the system pipeline, the interactions be- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='05297v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='RO] 12 Jan 2023 Perception Controltween uncertainty-aware components, and the potential impact on system performance and safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Quantifying uncertainty in a BDL-based system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', a pipeline of BDL components) still remains a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Uncertainties from BDL components must be assembled in a principled way to provide a reliable measure of overall system uncertainty, based on which safe decisions can be made [2], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In this paper, we propose to capture the uncertainty along a pipeline of BDL components and study the impact of uncertainty propagation on the aerial navigation task in a UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In addition, we propose an uncertainty-centric dynamic dependability management framework to cope with the challenges that arise from propagating uncertainty through BDL-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Neural Network Uncertainty Estimation Bayesian neural networks (BNN) have been widely used to represent the confidence in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' A proper con- fidence representation in DNN predictions can be achieved by modeling two sources of uncertainty: aleatoric (data) and epistemic (model) uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For epistemic uncertainty, Bayesian inference is used to estimate the posterior predic- tive distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In practice, approximate Bayesian inference methods are often used [12]–[15] since the posterior on the model parameters p(θ | D) is intractable in DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To model data uncertainty, [14], [16] propose to incorporate additional outputs to represent the parameters (mean and vari- ance) of a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Loquercio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' [17] forward propagate sensor noise through the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' This approach does not require retraining, however, it assumes a fixed uncertainty value for the sensor noise at the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Another family of methods aim to capture complex stochastic patterns such as multimodality or heteroscedasticity (aleatoric uncertainty) using latent variables (LV) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' When BNNs are used with LV (BNN+LV), both types of uncertainty can be captured [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In this approach, a BNN receives an input combined with a random disturbance coming from an LV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', features are partially stochastic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In contrast, this paper considers that a BNN can receive a complete stochastic features at the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Uncertainty in DNN-Based Navigation In an autonomous driving context, perception uncertainty is captured from implicit [8] and explicit representations [7] and used downstream for scene motion forecasting and trajectory planning respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In reinforcement learning, input uncer- tainty has been employed for model-based [20] and model-free control policies [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the former case, a collision predictor uncertainty is passed to a model predictive controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the latter, perception uncertainty is mapped to the control policy uncertainty using heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the context of aerial navigation, a few works have considered uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' [17] uses a fixed uncertainty value for sensors as an input to a control policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' [6] extends the work from [22] to use the uncertainty from perception noisy representations downstream in a BNN control policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Although these approaches use perception uncertainty in downstream components, not all the DNN components in the pipeline employ uncertainty estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Uncertainty-based Dependability Frameworks For the deployment of dependable autonomous systems that use machine learning (ML) components, Trapp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' [23] and Henne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' [9] conceptualized the use and runtime monitoring of perception uncertainty to ensure safe behavior on AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To model system behavior, probabilistic graphical models (PGMs) and, in particular, Bayesian Networks (BNs) have been used in dependability research for safety and reliability analyses and risk assessment applications [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' BNs allow incorporating ex- pert domain knowledge, model complex relationships between components, and enable decision-making under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the context of autonomous aviation systems, [10] proposes a method for quantifying system assurance using perception component uncertainty and dynamic BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For autonomous vehicles, [25] offers a framework for dynamic risk assessment, using BNs to predict the behavior intents of other traffic par- ticipants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Unlike these works, this paper considers uncertainty from Bayesian deep learning components beyond perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' SYSTEM TASK FORMULATION In this paper, we address the problem of autonomous aerial navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The goal of the autonomous agent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', UAV) is to navigate through a set of gates with unknown locations disposed in a circular track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Following prior work from [6], [22], the navigation architecture consists of two DNN-based components: one for perception and the other for control (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Both DNNs are trained following the hybrid paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To achieve the agent goal, the navigation task is formulated as a sequential-decision making problem, where a sequence of control actions are produced given environ- ment observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In this regard, the simulation environment provides at each time step an observation comprised of an RGB image x acquired from a front-facing camera on the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The perception component defines an encoder function qφ : X → Z that maps the input image x to a rich low dimensional representation z ∈ R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Next, a control policy πw : Z → Y maps the compact representation z to control commands y = [ ˙x, ˙y, ˙z, ˙ψ] ∈ R4, corresponding to linear and angular (yaw) velocities in the UAV body frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the perception component, a cross-modal variational autoencoder (CMVAE) [22], [26] is used to learn a rich and robust compact representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' A CMVAE is a variant of the traditional variational autoencoder (VAE) [27] that learns a single latent representation for multiple data modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In this case, the perception dataset Dp has two data modalities: the RGB images and the pose of the gate relative to the UAV body-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' During training, the CMVAE encoder qφ maps an input image x to a noisy representation with mean µφ(x) and variance σ2 φ(x) in the latent space, from where latent vectors z are sampled, z ∼ N(µφ, σ2 φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Next, a latent vector z is used to reconstruct the input image and estimate the gate pose (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', recover the two data modalities) using two DNNs, a decoder and a feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The CMVAE encoder qφ is based Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' System architecture for aerial navigation on the Dronet architecture [28], and additional constraints on the latent space are imposed through the loss function to promote the learning of robust disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Once the perception component is trained, the downstream control task (control policy π) uses a feed-forward network to operate on the latent vectors z at the output of the CMVAE encoder qφ to predict UAV velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To this end, the control policy network is added at the output of the perception encoder qφ, forming the navigation pipeline DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The control component π uses a control imitation learning dataset (Dc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' During training, we freeze the perception encoder qφ to update only the control policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For more information about the general architecture for aerial navigation, datasets, and training procedures, we refer the reader to [6], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Uncertainty from Perception Representations Although the CMVAE encoder qφ employs Bayesian in- ference to obtain latent vectors z, CMVAE does not capture epistemic uncertainty since the encoder lacks a distribution over parameters φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To capture uncertainty in the perception encoder we follow prior work from [29], [30] that attempts to capture epistemic uncertainty in VAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' We adapt the CMVAE to capture the posterior qΦ(z | x, Dp) as shown in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' qΦ(z | x, Dp) = � q(z | x, φ)p(φ | Dp)dφ (1) To approximate (1), we take a set Φ = {φm}M m of encoder parameters samples φm ∼ p(φ | Dp), to obtain a set of latent samples {zm}M m=1 ∼ qΦ(z | x, Dp) at the output of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In practice, we modify CMVAE by adding a dropout layer in the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Then, we use Monte Carlo Dropout (MCD) [12] to approximate the posterior on the encoder weights p(φ | Dp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Finally, for a given input image x we perform M stochastic forward passes (with dropout “turned on”) to compute a set of M latent vector samples z at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Input Uncertainty for Control In BDL, downstream uncertainty propagation assumes that a neural network component is able to handle or admit uncer- tainty at the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In our navigation case, this implies that the DNN-based controller is able to handle the uncertainty coming from the perception encoder qΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To capture the navigation model uncertainty (overall system uncertainty at the output of the controller), we use the Bayesian approach to compute the posterior predictive distribution for target variable y∗ associated with a new input image x∗, as shown in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' p(y∗ | x∗, Dc, Dp) = �� π(y | z, w)p(w | Dc)qΦ(z | x∗, Dp)dwdz (2) The integrals from (2) are intractable, and we rely on approximations to obtain an estimation of the predictive dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The posterior p(w | Dc) is difficult to evaluate, thus we can approximate the inner integral using an ensemble of neural networks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In practice, we train an ensemble of N probabilistic control policies {πn(y | z, wn)}N n=1, with weights {wn}N n=1 ∼ p(w|D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Each control policy πn in the ensemble predicts the mean µ and variance σ2 for each velocity command, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', yµ = [µ ˙x, µ ˙y, µ ˙z, µ ˙ψ] and yσ2 = [σ2 ˙x, σ2 ˙y, σ2 ˙z, σ2 ˙ψ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For training the control policy we use imitation learning and the heteroscedastic loss function, as suggested by [14], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The outer integral is approximated by taking a set of samples from the perception component latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In [6] latent samples are drawn using the encoder mean and variance z ∼ N(µφ, σ2 φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For the sake of simplicity, we directly use the samples obtained in the perception component {zm}M m ∼ qΦ(z | x, Dp) to take into account the epistemic uncertainty from the previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Finally, the predictions that we get from passing each latent vector z through each ensemble member are used to estimate the posterior predictive distribution in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' From the control policy perspective, using multiple latent samples z can be seen as taking a better “picture” of the latent space (perception representation) to gather more information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' EXPERIMENTS & DISCUSSION For our experiments, we seek to study the impact of uncertainty propagation in the navigation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In par- ticular, we seek to answer the following research questions: RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' How does uncertainty from perception representations affect downstream component uncertainty estimation quality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Can uncertainty propagation improve system perfor- mance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Could uncertainty-aware components in the pipeline help detect challenging scenes that can threaten the system mission?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To answer these questions we perform a quantitative and qualitative comparison between uncertainty- aware aerial navigation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Experimental setup 1) Navigation Model Baselines: All the navigation archi- tectures are based on [22] and are implemented using PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Table I shows the uncertainty-aware navigation architectures used in our experiments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' detailing the type of perception component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' the number of latent variable samples (LVS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' the type of control policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' and the number of control prediction TrainingOnly Ensemble Probabilistic NeuralNetworks 2 CMVAE Yμl 2 元1 Yol Z1 Yμ3 Z2 : 元3 Yo3 p(y* I x*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Dc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Dp) 2 Yμ5 Yo5 元5 q(z / x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Dp) [Tn(y I z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Wn))N Perception ControlTABLE I UNCERTAINTY-AWARE NAVIGATION MODELS IN THE EXPERIMENTS Model Perception (qφ) LVS Control Policy (π) CPS M0 MCD-CMVAE 32 Ensemble (N = 5) Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 160 M1 CMVAE 32 Ensemble (N = 5) Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 160 M2 CMVAE 1 Ensemble (N = 5) Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 5 M3 CMVAE 32 Deterministic 32 M4 CMVAE 1 Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 1 samples (CPS) at the output of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For instance, M0 represents our Bayesian navigation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' M0 perception component captures epistemic uncertainty using MCD with 32 forward passes for each input to get 32 latent variable pre- dictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For the sake of simplicity, perception predictions are directly used as latent variable samples in downstream control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The control component uses an ensemble of 5 probabilistic control policies obtaining 160 control prediction samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' M1 to M4 partially capture uncertainty in the pipeline since they use a deterministic perception component (CMVAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For the control component, M1 and M2 take 32 and 1 latent variable samples (LVS) respectively, and use the samples later with an ensemble of 5 probabilistic control policies capturing epistemic and aleatoric uncertainty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' M3 uses 32 LVS, and the control component is completely deterministic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' M4 uses 1 LVS with a probabilistic control policy to capture aleatoric uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For UAV control, we use the expected value of the predicted velocities means at the output of the control component [14], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', ˆyµ = E([µ ˙x, µ ˙y, µ ˙z, µ ˙ψ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 2) Datasets: We use two independent datasets for each component in the navigation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The perception CMVAE uses a dataset (Dp) of 300k images where a gate is visible and gate-pose annotations area available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The control component uses a dataset (Dc) of 17k images with UAV velocity anno- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Dc is collected by flying the UAV in a circular track with gates, using traditional methods for trajectory planning and control (see [22] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The perception dataset is divided into 80% for training, and the remaining 20% for validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The control dataset uses a split of 90% for training and the remaining for validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In both cases the image size is 64x64 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In addition, using the validation data from Dp and Dc, we generate refined validation sub-datasets with images that have: exactly one visible gate (ideal situation), no visible gate in front, and multiple gates visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The last two types of images represent situations that can pose a risk to the system task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Each sub-dataset contains 200 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Experiments In the context of RQ2, we use the validation dataset from the control component to measure the regression Expected Cali- bration Error (ECE) [31] to compare the quality of uncertainty estimates from navigation models at the output of the system, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', the control component output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In order to answer RQ2, we evaluate our navigation archi- tecture under controlled simulations using the AirSim simu- (a) Circular track view without noise (left) and with noise (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' (b) UAV mission scenes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' UAV Mission: Navigation tracks and scenes from birds-eye view, and view from UAV perspective lation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The UAV mission resembles the scenario and the conditions observed in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Therefore, we use a circular track with eight equally spaced gates posi- tioned initially in a radius of 8m and constant height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To assess the system performance to perturbations in the environment, we generate new tracks adding random noise to each gate radius and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the context of the AirSim [32] simulation environment, a track is entirely defined by a set of gates, their poses in three- dimensional space, and the expected navigation direction of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For perception-based navigation, the complexity of a track resides in the “gate-visibility” difficulty [33], [34], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', how well the camera Field-of-View (FoV) captures the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' A natural way to increase track complexity is by adding a random displacement to the position of each gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' A track without random displacement in the gates has a circular fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Gate position randomness alters the shape of the track, affecting the gate visibility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', gates are: not visible, partially visible, or multiple gates can be captured in the UAV FoV as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The images from these scenarios are challenging given its potential impact on system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To measure the system performance we take the average number of gates passed in all generated tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For track generation we use a random seed to produce circular tracks with two levels of noise in the gates offset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', each random seed generates two (reproducible) noisy tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In total, we use 6 random seeds to produce 12 tracks, 6 tracks per noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The two noise levels are a combination of Gate Radius Noise (GRN) and Gate Height Noise (GHN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Finally, all navigation models are tested in the same generated tracks for a fair comparison, and each model has 3 trials per track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To address RQ3, we perform a qualitative comparison of the component predicted densities using scenes (images) from challenging situations during the UAV mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To this end, we first use the images from the generated sub-datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Next, we use the scenes from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 3b as an input to the Bayesian navigation model M0 to analyze the effect uncertainty prop- agation under specific situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 口口 口 口 口TABLE II UNCERTAINTY-AWARE NAVIGATION MODELS: ECE & AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' NUMBER OF GATES PASSED Model ECE (↓) Performance with Track Gate Noise (↑) GRN ∼ U[−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0) GHN ∼ U[0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0) GRN ∼ U[−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='5) GHN ∼ U[0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0) M0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='00700 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='77 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='22 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='00129 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 M2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='00136 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 M3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='05709 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='33 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 M4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='00050 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Results Table II summarizes the ECE for all the navigation models using the validation dataset from the control component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' M4 has the best uncertainty quality since the model learned to predict the noise from the data using the heteroscedastic loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' On the contrary, M2 has the worst calibration results caused by the deterministic control choice and its inability to learn the data uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' M1 and M2 have similar values since both receive the one noisy encoding from perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However, M1 takes multiple samples from the noisy perception encoding which causes a reduction of the ECE value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Finally, M0 shows a higher ECE value compared to the previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' This is caused by applying MCD in the perception CMVAE and the dispersion of the latent codes at the output of the perception encoder qΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The uncertainty quality of the downstream control is slightly affected because the control component did not see the same perception encod- ing dispersion (uncertainty) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For RQ2, Table II presents the navigation performance results for all the navigation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In general, learning to predict uncertainty in the control component can boost the performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However, for M3, sampling from a noisy perception representation adds sufficient diversity to the downstream control predictions, resulting in better decisions than M2 in tracks with higher noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In M4, the good performance suggests that the track noise observed at test time (lower noise level), resembles the data noise observed during the training of the single probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In case of M0, the diversity from perception prediction samples improves the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Interestingly, the perfor- mance difference with other models is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' This situation can make us wonder if an uncertainty estimation is needed along the whole pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Nonetheless, we believe that performance similarity is rooted in how we use our model predictions and uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The control output is computed by taking the mean and variance of the policy ensemble mixture, and only the mean values are passed to the UAV control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However, the multimodal predictions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 5 show that admitting perception uncertainty (samples) at the input of the control component permits the representation of ambiguity in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Hence, a proper use of predictions and associated uncertainties is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For example, in a bi-modal predictive distribution at the output, we can use the modes (a) Visible gate sub-dataset (b) No visible gate sub-dataset (c) Multiple gates visible sub-dataset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Navigation model standard deviation (ˆσ) prediction comparison (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', distribution peaks) instead of the expected value to avoid sub-optimal control decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', near distribution valleys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the context of RQ3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 4 shows the estimated uncer- tainty densities (ˆσ) for each velocity command at the output of the system, using the images from the generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In this case, M0 allows higher uncertainty estimates while reducing the dispersion in the sub-datasets from each situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 5 shows M0 predictions at the output of the perception (z) and control (ˆµ, ˆσ) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Predictions are made using the three sample images from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 3b, using the LVS and CPS to estimate the densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' M0 perception and control outputs show high uncertainty (dispersion) values when a gate is not visible (mid-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The ˆµ ˙y density suggests that the UAV control predictions will follow the training dataset (Dc) bias, rotating clockwise and moving to the right when no gate is in-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Interestingly, the predicted densities in the bottom plots show that M0 is able to represent the ambiguity in the input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' sample image DoubleorMultipleVisibleGatesSubdataset:PredictedStandardDeviationDensities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='2 Navigation Model Velocity (m/s) or (deg/s) Mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 Mi M2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 ox ModelPredictionVisibleGate Subdataset:Predicted Standard DeviationDensities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='2 Navigation Model Mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 Mi M2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 ox Mode/PredictionNoVisibleGate Subdataset:Predicted Standard DeviationDensities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='2 Navigation Model Velocity (m/s) or (deg/s) Mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 M1 M2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 x ModelPrediction(a) Single gate prediction densities (b) No visible gate prediction densities (c) Double gate prediction densities Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Bayesian navigation model M0: Perception qΦ predictions z density (left);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Predicted velocity ˆµ density (mid);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Predicted velocity ˆσ (right) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' with two gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The predicted densities have a multimodal distribution (two peaks) for ˆµ ˙y and ˆσ ˙y commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Further, the predicted densities for the latent vector z show that the uncertainty from perception outputs is different for each type of sample, which is suitable for the early detection of anomalies based on uncertainty information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In addition, detecting multi-modality in prediction distributions can help expressing situations where decisions must be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Dynamic Dependability Management using Uncertainty from DNN-Based Systems Based on the results and observations in the previous sub-sections, uncertainty propagation through a DNN-based can impact downstream component predictions and their per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Thus, using uncertainty information to improve system dependability or safety can be a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For example, building monitoring functions based on uncertainty information is no simple task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The uncertainty intervals we ob- served for different situations present overlaps that can lead to false-positive or false-negative verdicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Moreover, the multi- modal nature of some predictions under specific conditions or scenes demands knowledge of multiple intervals around the monitored uncertainty value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Therefore dependable and safe automated systems require more than a simple composition of predicates around some confidence measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Towards building dependable autonomous systems, we pro- pose to align with previous frameworks that leverage percep- tion uncertainty (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' subsection II-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However existing frame- works for system dependability do not consider the impact of uncertainty propagation in uncertainty-aware systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To overcome these new challenges, we propose to capture and use uncertainty beyond perception and consider as well the uncertainty from downstream components along the navigation pipeline, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='6 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Our approach for dynamic dependability management takes inspiration from [35] and focuses on safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Therefore, we propose an architecture for dynamic risk assessment and management where we devise three functional blocks, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 6 2 : Monitoring functions, risk estimation and behavior arbitration modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 1) Monitoring Functions: Monitoring is a widely-known dependability technique for runtime verification intended to track system variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' component inputs and outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the automotive domain, SOTIF and ISO26262 suggest the use of monitoring functions as a solution for error detection in hardware and software components [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Monitoring functions are designed using a set of rules, based on a model of the system and its environment, and the properties they should guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Hence, monitors basically perform a binary classi- fication task to check if a property holds or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Designing monitoring functions for ML components is different given the probabilistic nature of the outputs and the difficulty in specifying the component behavior at design time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' For ML-based components in general, typical monitoring PerceptionCMVAEEncodergoPredictionDensities value iable vari .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='atent Zo Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 LatentvectorzvariablesControl EnsembleMixtureVelocity μPredictionDensities Mo Prediction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='75 μx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='50 py 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='0 Predicted velocity (m/s)or (deg/s)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Runtime risk assessment & management framework function tasks include Out-of-Distribution (OoD) detection or Out-of-Boundary (OoB) detection and can be implemented with rules, data-driven methods or a mix of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 2) Probabilistic Inference for Risk Assessment: To enable dynamic uncertainty-aware reasoning and provide context to risk estimates, we propose to use Bayesian networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Fol- lowing the methodology described in [24], BNs for risk assessment and safety can be constructed using a combination of expert domain knowledge and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The experts provide a model of causal relations and can have support from traditional dependability analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=', fault tree analysis) to build the BN structure while system data is used to provide the conditional probabilities between random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In our framework, the BN of the system can receive the predictions from components in the pipeline (probability distributions) and the verdicts from monitoring functions ap- plied to system sensors, component predictions, and relevant environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The output of the BN is represented by all the critical events identified by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Hence, during inference, the BN estimates the probability of a critical event, which is used along with its severity to compute the system’s risk at runtime [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Though we focus on risk assessment, in a general way the output of BNs can be any assurance measure variables linked to dependability attributes [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Further, the BN should handle uncertain evidence [38] to preserve the probabilistic nature of component and monitor predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' 3) Behavior Arbitration: The last building block in our framework aims at keeping the system in a safe state by taking or discarding navigation pipeline predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Safe decisions must be made in the presence of high-risk values in a given context caused by erroneous component predictions or associated uncertainties and external environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' To this end, we propose using Behavior Trees (BTs) to adopt different system behaviors while facing high-risk situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' BTs are sophisticated modular decision-making engines for reactive and fault-tolerant task execution [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Compositions of BTs can preserve safety and robustness properties [40] and are widely adopted tools in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In the context of our system, we can have a dedicated behavior to search for a gate when we detect that there are no gates in the UAV FoV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' This behavior will put the system back into a state where the levels of uncertainty do not represent a risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' CONCLUSION We presented a method to capture and propagate uncertainty along a navigation pipeline implemented with Bayesian deep learning components for UAV aerial navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' We analyzed the effect of uncertainty propagation regarding system com- ponent predictions and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' Our experiments show that our approach to capturing and propagating uncertainty along the system can provide valuable predictions for decision- making and identifying situations that are critical for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' However, proper use and management of component predictions and uncertainty estimates are needed to create dependable and highly-performant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In this sense and based on our observations, we also proposed a framework for system dependability management using system uncertainty and focused on safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' In future work, we aim to implement our proposed dependability framework and explore sampling- free methods [41] for uncertainty estimation to reduce the computational budget and memory footprint in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' ACKNOWLEDGMENT This work has received funding from the COMP4DRONES project, under ECSEL Joint Undertaking (JU) grant agreement N°826610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf'} +page_content=' The ECSEL JU receives support from the European Union’s Horizon 2020 research and innovation programme 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Member, IEEE, Masood Parvania Senior Member, +IEEE, Pramod P. Khargonekar Fellow, IEEE +Abstract—This paper proposes a two-level hierarchical match- +ing framework for Integrated Hybrid Resources (IHRs) with grid +constraints. An IHR is a collection of Renewable Energy Sources +(RES) and flexible customers within a certain power system zone, +endowed with an agent to match. The key idea is to pick the +IHR zones so that the power loss effects within the IHRs can +be neglected. This simplifies the overall matching problem into +independent IHR-level matching problems, and an upper-level +optimal power flow problem to meet the IHR-level upstream +flow requirements while respecting the grid constraints. Within +each IHR, the agent employs a scalable Deep Reinforcement +Learning algorithm to identify matching solutions such that +the customer’s service constraints are met. The central agent +then solves an optimal power flow problem with the IHRs as +the nodes, with their active power flow and reactive power +limits, and grid constraints to determine the final flows such +that matched power can be delivered to the extent the grid +constraints are satisfied. The proposed framework is implemented +on a test power distribution system, and multiple case studies are +presented to substantiate the welfare efficiency of the proposed +solution and the satisfaction of the grid and customers’ servicing +constraints. +Index Terms—Hierarchical dynamic matching, integrated hy- +brid resources, deep reinforcement learning, uncertainty. +I. INTRODUCTION +D +RIVEN by the advances in communication technologies +and supporting policies, Distributed Energy Resources +(DERs) and flexible loads are going to be highly penetrated in +power grids. Federal Energy Regulatory Commission (FERC) +order 2222 requires power system operators to facilitate the +participation of demand-side resources in the electricity mar- +kets, reflecting their significant potential to provide energy +flexibility [1]. The DERs and flexible loads, if coordinated and +controlled carefully, can make the grid flexible and energy- +efficient [2], [3]. However, a large number of DERs and +flexible loads might challenge the structure and capacity of +power distribution grids. Hence, it is essential to develop +intelligent energy management solutions that can manage +different sorts of DERs and flexible loads in a scalable manner +without compromising the power grid’s stability. +In recent years, several efforts have been made to develop +energy management solutions for the coordination of DERs +in power distribution systems. One promising solution is +This work is supported in part by the National Science Foundation under +Grant ECCS-1839429. +M. Majidi, and M. Parvania are with the Department of Electrical and +Computer Engineering, the University of Utah, Salt Lake City, UT 84112 USA +(e-mails: {majid.majidi, masood.parvania,}@utah.edu). Deepan Muthirayan +and Pramod P. Khargonekar are with the Department of Electrical Engineering +and Computer Science, University of California, Irvine, CA 92697 USA (e- +mails: {deepan.m, pramod.khargonekar,}@uci.edu). +matching, which is a Peer-to-Peer (P2P) solution. Unlike +traditional energy management solutions based on pooling re- +sources, matching offers optimal use of energy flexibility while +accounting for the energy preferences of flexible customers. +This is the key feature that makes matching very promising to +future power grids. However, developing a matching solution +for power grids still has several challenges: (i) the solution +has to be online and capable of adapting the matching strategy +with the state of the whole grid, (ii) it has to cope with a large +number of DERs and flexible loads, and (iii) it has to satisfy +the security constraints of the grid. Although [4], [5] propose +online solutions for dynamic matching, the proposed solutions +are heuristics and therefore can be sub-optimal. Alternatively, +data-driven approaches like Reinforcement Learning (RL) can +be used to discover high-performing dynamic matching poli- +cies. However, RL approaches have severe limitations when +it comes to learning policies for power grids with constraints +such as power flow limits. For instance, [6] proposes a Deep +Reinforcement Learning (DRL) solution for dynamic matching +but it fails to account for the grid constraints. Moreover, a +central matching structure might not be efficient or feasible +for managing a large number of DERs and flexible loads in +the power grid. Hence, an efficient learning-based hierarchical +matching model with Integrated Hybrid Resources (IHRs) is of +interest in addressing dynamic matching in power grids with +grid constraints. +IHRs present a viable solution to facilitate efficient energy +management and control of uncertain DERs in various appli- +cations [7]–[10]. In a power grid, different types of renewable +and non-renewable DERs and flexible can be combined and +operated as an IHR to supply distributed energy flexibility. +The key feature of an IHR is that the resources within the IHR +can be treated as an integrated set of resources with a single +interconnection point. Therefore, from a matching solution +point of view, each IHR can be treated as a separate matching +market with a single interconnection to the distribution grid. +This then enables the use of an RL-type algorithm to determine +the optimal way to manage the DERs within each IHR, +where the RL solution does not need to take into account +the grid constraints. Once the IHR-level matching results are +determined, a central controller can re-dispatch the IHRs using +a reduced-dimension Optimal Power Flow (OPF) model with +each IHR as a node to balance the excesses while ensuring that +the grid constraints across the distribution system are satisfied. +This paper proposes a hierarchical framework for dynamic +matching markets in power distribution systems composed of +IHRs. The schematic of the proposed framework is shown +in Fig. 1. An IHR consists of an agent that employs DRL +arXiv:2301.13796v1 [eess.SY] 31 Jan 2023 + +2 +to locally match the distributed Renewable Energy Sources +(RES) and flexible customers in each IHR while satisfying +the quality of service constraints of customers, i.e., critical- +ity, servicing deadline, etc. Such a learning-based approach +allows for developing a very effective online matching pol- +icy, which is otherwise very difficult to design. Once the +IHR-level matching results are determined, each IHR agent +communicates the net active power flow (i.e., net active +power consumption/generation), as well as the reactive power +limits of the IHR to a central agent. The central agent then +formulates a reduced-dimension OPF model with the IHRs as +the nodes to determine the final flows (set-points) such that +the grid constraints are met. In this stage, the central agent +can curtail the IHR-level matching decisions and control the +reactive power flow from/into each IHR in order to make sure +that power flows in the grid and voltage levels across the +distribution nodes are not violated. +A. Related Works and Contributions +Several works in the literature have explored the control +and management of DERs in distribution systems. P2P energy +trading markets of different types are developed in [11]– +[20]. The authors in [12] studied the operation and benefits +of centralized and decentralized battery energy storage under +different P2P market designs. A P2P market design based +on bilateral contracts is proposed in [13] for energy trading +between multiple DERs and flexible loads, with the objective +of minimizing peak load in the low-voltage power distribution +system. A decentralized P2P optimization framework is pre- +sented in [14] to enable local energy sharing between DER +agents in the low-voltage distribution systems. A local energy +sharing framework is proposed for prosumers in low-voltage +distribution systems in [15], where the voltage regulation capa- +bility of the proposed energy sharing framework is highlighted. +An iterative sequential approach is implemented in [16] to +enable P2P energy and reserve sharing between prosumers in +power distribution systems with grid constraints. A negotiation +algorithm is proposed to facilitate energy sharing between +interconnected DER owners in [17]. The use of game theory +to determine the interaction strategy of DERs in P2P energy +sharing is investigated in [18]–[20]. +The application of data-driven approaches to energy man- +agement of DERs is studied in [21]–[25]. The authors in [21] +implemented a multi-agent learning framework to determine +real-time local energy trading strategies for DER owners in +regional microgrids. A multi-energy sharing model based on +RL is proposed in [22], [23] for local heat and power sharing +in energy microgrids equipped with DERs. In [24], [25], the +authors proposed a specific price-based market framework for +coordinating the prosumers in the market to minimize the +peak load. In [26], a hierarchical energy management model +based on DRL is proposed for local energy management of +energy storage systems to improve the resilience of the power +distribution system. +Although the works reviewed here study the management +and coordination of DERs in distribution systems, their P2P +solutions address specific scenarios, and many do not account +for the grid constraints. Moreover, the existing hierarchical +energy management models for DERs in power distribution +systems neglect the preference and dynamic characteristics +of the DERs and flexible loads, which impacts the energy +flexibility available to the grid. In contrast, this paper develops +a broadly applicable online matching solution that (i) is de- +signed to maximize the integration of local RES and the overall +welfare in a very generic real-time operating scenario that can +be riddled with uncertainties, (ii) takes into consideration the +flexible loads’ servicing deadline, as well as their dynamic +criticality, and willingness to pay for a unit of energy, and (iii) +at the same time accounts for the power grid constraints. The +key contributions of the paper can be summarized as follows: +• A hierarchical dynamic matching framework for power +distribution systems constituted by IHRs. +• An efficient and scalable learning-based solution to match +the flexible loads and uncertain RES within each IHR +with no need for prior experience or expert supervision +or elaborate design. An independent DRL algorithm for +each IHR agent that matches the flexible loads and supply +sources to improve the utilization of RES and therefore +maximize the social welfare in each IHR such that the +quality of service constraints of the loads are satisfied +prior to their servicing deadline, and their dynamic will- +ingness to pay for a unit of energy is taken into account. +• An upper-level optimization model to fix the excesses or +imbalances within the IHRs. Once the matching results in +each IHR are determined, a central agent runs a reduced- +dimension OPF problem to fix the possible imbalances +within the IHRs and, at the same time, ensures the +power flow limits and voltage security constraints of the +distribution system are met. +The remaining of the paper is categorized as follows: Hi- +erarchical matching market framework is presented in Section +II. The proposed learning-based solution approach is explained +in Section III. The simulation results are presented in Section +IV, and the paper is concluded in Section V. +II. HIERARCHICAL MATCHING MARKET +The proposed hierarchical matching framework is composed +of (i) a market operator or central agent, and (ii) multiple +IHRs, with each IHR operating as a separate matching market +and the central agent acting as the coordinating agent between +the IHRs. In the proposed framework, the grid is divided into +multiple IHRs, where each IHR is an integrated unit of several +types of DERs that are treated as a single resource with a single +interconnection point to the grid. This is feasible to do when +the region of the grid representing the IHR is such that the +voltage variation within each IHR is within a small δ as [26]: +|Vit − Vjt| < δ, +∀t, ∀b, b′, ∈ Bh, ∀h ∈ H, +(1) +where H is the set of all IHRs and b, b′, ∈ Bh represent any +pair of buses in IHR h ∈ H. Hence, each IHR is treated as a +regular matching market with no power flow constraints, and +the rest of the grid as the upstream supply source, to balance +any imbalances or excesses in the IHRs. + +3 +IHR Agent +Matching +Policy +Matching +Policy +Matching Market +Constraints +Supply-Demand Balance +RES Generation Availability +Objective +Maximize Social Welfare +Customers Deadline +IHR Matching Problem +Active Customers +RES Availability +Customers Criticality +Wait +State +Match +Action +Match +Reward +Social Welfare +Constraints +Optimal Power Flow +Integrated Hybrid Resource +Objective +Minimize System Energy Cost +System-Level Power Flow Constraints +- Net Active Power +- Min & Max Reactive Power Capacity +Fixed Mismatches +Fig. 1. Structure of hierarchical dynamic matching model in power distribu- +tion systems using deep reinforcement learning. +The matching market within each IHR is an online market, +with customers and RES that are characterized by uncertain +arrivals and generation over time. In the proposed model, +each IHR is endowed with an agent which can adapt its +decision according to the state of the local IHR market, which +includes the history of customers and renewable generation. +This ensures that the decisions can be optimized with respect +to the underlying state and the expected future conditioned +on the current state. But, such state-dependent solutions are +difficult to characterize for an online market. Alternatively, +data-driven approaches like RL can be used to derive state- +dependent solutions for systems like electricity markets, which +are dynamic and uncertain. Given this, the IHR agents are +endowed with DRL algorithms to discover state-dependent +matching policies from their respective operational data. +The design of a DRL model for a matching market has many +practical limitations like scalability because of the size of the +action space, which can grow exponentially with the number +of customers, in addition to the servicing constraints that the +matching outputs from the DRL model are required to satisfy. +In this study, the scalable DRL-based solution proposed in +our prior work [6] is adopted as the DRL model for each +of the IHRs. This model is designed specifically to address +the scalability and convergence of DRL applied to matching +markets. The DRL model is discussed in the next section. +The central agent plays the role of managing the whole +grid, coordinating the upstream demand of the IHRs such +that the grid constraints are satisfied. At any moment, after +the IHR-level matching decisions are determined, the IHR +agents send their net active power and reactive power limits +to the central agent. The central agent then solves a reduced- +dimension OPF problem with IHR as the nodes to compute the +active and reactive power flows for the nodes such that the grid +constraints (i.e., overall power flow constraints and voltage +boundary limits derived from (1) are met. The voltage bound- +ary constraints ensure that the condition in (1) is satisfied, +and therefore the matched power is delivered without much +loss. The inclusion of the power flow constraints ensures that +the overall flow delivers the matched power to the extent the +constraints can be satisfied. Now, a single centralized market +can perform matching, and be adaptable to the changing +market condition, but it is typically hard to compute a solution +with DRL that satisfies stability constraints, i.e., power flow +constraints. This is the key benefit of the proposed hierarchical +approach, which uses DRL to identify state-dependent policies +and optimization to ensure the feasibility of the policies for +grid operation. The rest of the section describes the supply +and customer models, the market state, the matching market +formulation, and finally the IHR-level matching and upper- +level OPF problems. +A. Supply Model +Lets denote the time within a day by t. The matching market +comprises two sources of supply: 1) grid supply (type g), +which it can be drawn from its interconnection to the grid +and 2) RES (type s). The upstream grid supply at a time t, +denoted by gt ∈ R, is priced at the retail price of electricity, +c/kWh, while the unit cost of RES generation, rt ∈ R, is +assumed to be zero. +B. Flexible Customer Model +Each customer (or load) is characterized by three param- +eters, {ai, pi, di}, where ai ∈ N is the arrival time of the +customer, pi ∈ R is the load requested by the customer, and +di ∈ N is the servicing deadline by which the customer is +to be served. Moreover, each customer has a criticality rate +bi, at which its willingness to pay decreases from ai until di. +The heterogeneity of customers lies in the differing deadlines +and their criticality. Hence, the utility function of customer i +representing its willingness to pay for a unit of energy can be +defined as follows: +πi +t = c − bi(t − ai), +πi +t ≥ 0, ai ≤ t ≤ di, +bi = ϕc/(di − ai), +(2) +where ϕ ∈ [0, 1] determines the reduction rate in customers’ +willingness to pay for a unit of energy. The utility function +for different values of criticality rate is shown in Fig. 2. In +addition to the flexible customers, the market can also have +non-flexible loads. +In Fig. 2, customer’s willingness to pay is less than or equal +to the grid supply price c/kWh. This is reasonable considering +that the grid supply is available at this price at all times. A +customer with ϕ = 1 will only be willing to pay zero if it +is served at its deadline. On the other hand, a customer with +ϕ = 0 can be served at any time without any change to the + +4 +πt +t +b > 0 +a +d +c +πt +t +b = 0 +a +d +c +flexible load c at bus i in time t, the state equation of queuing +system can be expressed as: +˙Qc +t,i = Ac +t,i − P c +t,i +(4) +Qc +t,i = Qc +t−∆t,i + +� +Ac +t,i − P c +t,i +� +∆t +(5) +According to the state equations expressed in In (4)-(5), the +flexible load queue backlog in each time interval t is equal to +customer i at time k by qj (k). Then, the function χk is given +by +χk(j, i, Z) = qi +j(k), χk(j, st, Z) = qst +j (k) +(8) +Given these definitions, the matching problem for the distri- +bution system is given by the following optimization problem +Fig. 2. Illustration of utility function for different values of b. +willingness to pay. This model captures a variety of customers +in the market, where customers’ willingness to pay can remain +fixed or decay with time and at distinct rates. As the number +of customers is finite in real markets, the number of customers +arriving on the platform at any time t, nt ∈ N, is assumed to +be upper bound by a constant n. +C. IHR Market State +Let zt +:= +[a⊤ +t , p⊤ +t , b⊤ +t , d⊤ +t , rt]1 be the vector of state +parameters, where at ∈ Nn is the vector of the arrival times of +the customers which arrive at time t, pt ∈ Nn is the vector of +their respective requested loads, bt ∈ [0, 1] is the criticality rate +of customer at time t, dt ∈ Nn is the vector of their respective +deadlines, and rt ∈ R is the amount of RES generation at time +t. The scenario at time t is given by +Z⊤ +t = [z⊤ +1 , z⊤ +2 , ..., z⊤ +t−1, z⊤ +t ]. +The probability that zt = z is given by the stochastic process +modeled by P (zt = z|Zt−1). This process is not known to +the market operator. Let xt := [a⊤ +t , p⊤ +t , b⊤ +t , d⊤ +t , p⊤ +u,t, b⊤ +u,t, rt], +where pu,t denotes the vector of the portion of the requested +load that has not been served to the customers who arrived at +t and expressed the criticality b⊤ +u,t. Let denote the set of all +possible states at time t by Ωt and the state of the market by +Xt. Then Xt is given by +X⊤ +t = [x⊤ +1 , x⊤ +2 , ..., x⊤ +t−1, x⊤ +t ]. +Note that the state Xt depends on the scenario Zt and the +matching decisions till time t − 1. Given that the state of the +market evolves, the matching solution will have to be able to +adapt to the changing market state. +D. Matching Market for Integrated Hybrid Resources +This part formulates the IHR-level dynamic matching mar- +ket problem for a duration of T, divided into time periods +spaced equally at an interval ∆t. The IHR-level dynamic +matching market problem aims to match the load request +of flexible and inflexible customers to maximize the social +welfare in IHR h subject to satisfying the supply-demand +balance constraint for non-flexible loads and the quality of +service constraints for flexible loads arriving sequentially. Let +define Ah,t as the set of all active customers at time t and IHR +h and define Sh,t = {g, s} as the set of supply types. The IHR +agent, at each time, can decide to match and supply or skip the +load requests. Let define pi +h,t as the skipped and unsupplied +1[.]⊤ denotes the transpose operation. +load request of the customer i and Mh,t(j, i, Xt) ∈ R define +the amount of supply of type j matched to customer i at time +t and IHR h, at the unit cost of ch,t. The matching market +problem can be then stated as: +sup +T +� +t=1 +� +i∈Ah,t +� +j∈Sh,t +(πi +h,t − ch,j)Mh,t(j, i, Xt), s.t. +(3) +� +j∈Sh,t +Mh,t(j, i, Xt) ≤ pi +h,t, ∀h ∈ H, ∀i ∈ Ah +t , t ̸= di +h, +(4) +� +j∈Sh,t +Mh,t(j, i, Xt) = pi +h,t, ∀h ∈ H, ∀i ∈ Ah,t, t = di +h, (5) +� +i∈Ah,t +Mh,t(r, i, Xt) ≤ rp +h,t, ∀h ∈ H, ∀t. +(6) +pNet +h,t = +� +i∈Ah,t +Mh,t(g, i, Xt), ∀h ∈ H, ∀t, +(7) +where the dependency of Mt on Xt accounts for the depen- +dency of the matching decision on the full state information in +each IHR. Here, the power balance constraint for the flexible +loads is given in (4). The power balance for the non-flexible +loads and the critical flexible loads at their departure (t=di +h) +is given in (5). The constraint (6) limits the matching power +from RES to the active power output of RES rp +h,t. Finally, +the net active power flow exchanged between the IHR and +upstream grid, pNet +h,t , is given by (7). +The output of the above problem is a matching policy M1:T +for the entire duration of a day. Because of the interdependence +across time, the optimal policy Mt is dependent on the load +arrivals and RES generation for the full day. This makes +the computation of the optimal policy through the above +approach infeasible in real-time operation. There are also no +known explicit characterizations for Mt. This is what makes +approaches like DRL very appealing, since they are general- +purpose methods that can be used to discover state-dependent +policies, such as Mt, from just operational data. Therefore, we +use a DRL algorithm to compute the matching decisions. The +proposed DRL model for a specific IHR is designed to output +a matching decision at any point of time depending on the +market state of the IHR. The DRL framework for the IHRs +matching is described in the next section. +Once the matching decisions are computed by the IHRs, +the agent determines the net active power flow, i.e., the active +power to be taken or injected from and to the upstream grid, +as well as the reactive power limits of its zone to the central +agent. The reactive power limits are utilized by the central +agent to adjust the nodal reactive power demands such that +the constraint (1) is satisfied. The reactive power limits for +each IHR, denoted by qNet +h,t +and qNet +h,t , are obtained through +(8)-(10): +rq +h,t=− +� +rs +h,t +2−rp +h,t +2 , +rq +h,t= +� +rs +h,t +2−rp +h,t +2, ∀h∈H,∀t, (8) +qNet +h,t = +� +i∈Ah,t +qi +h,t − rq +h,t, +∀h ∈ H, ∀t, +(9) +qNet +h,t = +� +i∈Ah,t +qi, +h,t − rq +h,t, +∀h ∈ H, ∀t, +(10) + +5 +where rq +h,t, rq +h,t are the minimum and maximum reactive +power output of RES, rs +h,t is the available RES generation and +the term qi +h,t represents the reactive power load of the IHR, +determined based on the non-flexible reactive power load and +matched power to flexible loads. +E. Reduced-Dimension Optimal Power Flow +This section describes the reduced-dimension OPF problem +solved by the central agent to determine the final flows for the +nodes in the network to deliver the matched power to the extent +it does not violate the grid constraints. The agent specifically +solves a quadratic optimization model with the IHRs as the +nodes, where the constraints are the power flow constraints +with the active power flow demand and reactive power limits +of the IHRs, and the voltage limit constraints, defined based on +(1). The voltage limit constraints ensure that the final flows are +consistent with the matched power in each of the IHRs. The +central agent also curtails the active power flow demand by ph +C +to the extent that the flow constraints are satisfied. The central +agent’s objective function, which is the distribution system +cost, is given in (11): +min +� +λRT P G− +H +� +h=1 +λCpC +h +� +, +(11) +where P G is the active power taken from the transmission +system at the real-time market price λRT and pC +h is the active +load request curtailment with the unit cost of λC. +1) Power Balance Constraints: The active and reactive +energy balance equations for the slack buses in the distribution +system are given in (12)-(13), where P1h, Q1h are the active +and reactive power flows from the substation bus to the IHR +h, V sq +1 +is the squared voltage on the substation bus, g1, b1 are +the shunt conductance and susceptance at the substation bus +and L is the set of lines in the distribution grid. +P G = +� +1h∈L +P1h + g1V sq +1 , +(12) +QG = +� +1h∈L +Q1h + b1V sq +1 . +(13) +The energy balance constraints for the IHR nodes are +presented in (14)-(15), where pNet +h +is the net active power +flow submitted by the IHR, Phh′′, Ph′h and Qhh′′, Qh′h are +the active and reactive power flows in lines hh′′ and h′h, V sq +h +is the squared voltage on IHR node h, Isq +h′h is the squared +current flow in line h′h, and rh′h, xh′h are the resistance and +reactance of the line h′h and gh, bh are the shunt conductance +and susceptance at the IHR node h. In the active power balance +constraint, the curtailment pC +h ensures that the final net active +power flow is consistent with the power flow constraints. Here, +the curtailment is only made to the extent that the constraints +are satisfied. The reactive power flow of the IHRs, denoted by +qNet +h +, is limited to the reactive power limits of IHRs in (16). +Phh′′ +pNet +h +−pC +h = +� +h′h∈L +(Ph′h−rh′hIsq +h′h)+ghV sq +h , ∀h∈B, (14) +Qhh′′ + qNet +h += +� +h′h∈L +(Qh′h− xh′hIsq +h′h)+ bhV sq +h , ∀h ∈ B, (15) +qNet +h +≤ qNet +h +≤ qNet +h +, ∀h. +(16) +2) Voltage and Power Flow Limits: The voltage drop across +the grid is given by (17). The limits on the squared voltage +level and the limits on the current flow are given in Eq. (18) +and Eq. (19), where V sq +h , V +sq +h are the minimum and maximum +squared voltage boundaries, defined based on the nominal node +voltage and δ in (1) and I +sq +h′h is the squared current flow limit. +Finally, the complex power flow constraint is given in (20). +V sq +h − V sq +h′ = −2 (rh′hPh′h + xh′hQh′h) ++ +� +r2 +h′h + x2 +h′h +� +Isq +h′h, +∀(h′h) ∈ L, +(17) +V sq +h ≤ V sq +h +≤ V +sq +h , +∀h ∈ B, +(18) +Isq +h′h ≤ I +sq +h′h, +∀(h′h) ∈ L, +(19) +V sq +h,tIsq +h′h ≥ P 2 +h′h + Q2 +h′h, +∀(h′h) ∈ L. +(20) +Any feasible solution to the online OPF problem in (11)- +(20) ensures the matched power in each of the IHRs is +delivered to the extent the flow and voltage constraints are +met. In case a solution is feasible without any curtailment, +then the matched power will be delivered to the customers. +III. DEEP REINFORCEMENT LEARNING FOR IHRS +In the proposed hierarchical framework, each IHR agent +is endowed with a trainable policy that outputs a probability +distribution over the set of matching decisions for the flexible +loads and RES with the IHR. A policy gradient RL algorithm +is applied to train the matching policy given the load and +generation data for multiple instances of the market. This +algorithm does not require supervision or expert knowledge as +it measures its own performance for the training process. The +following subsections briefly discuss the matching policy’s +structure for an IHR and then the learning algorithm. Note +that the expectation with respect to all sources of randomness +is denoted by E[.]. It is implicit that all the descriptions in this +section are confined to a single IHR. +A. General Discrete Matching Policy +Each IHR agent in the proposed study learns an online +matching policy given by χ = {χ1, χ2, χ3, ..., χT }. The +discrete matching policy for time t, denoted by χt, indicates +whether a customer is to be matched to a supply or not, +regardless of the amount of matching. Let define Mt as the +space of discrete matching at time t. Each component in this +set m ∈ Mt, is a feasible discrete matching that specifies +whether a customer is matched or not (i.e., mi,k ∈ {0, 1} with +one indicating “matched” and zero indicating “not matched”). +Hence, the general matching policy χt can be given by: +χt : Ωt → Mt. +Aside from the fact that the matching problem is an +online decision-making procedure with a future ridden with +uncertainties, there are still several general challenges from +an RL point of view. Firstly, the action space of the matching +problem is large and specifically exponential in the number +of customers. For example, if there are m supplies and n +customers, then there are mn ways of matching; thus it is + +6 +exponential in the number of active customers. Secondly, +not all the actions from this space are feasible as supply +unavailability might limit the matching decisions. There might +also be some restrictions enforced by the customers servicing +constraints. Hence, some actions are infeasible, and their +infeasibility is state-dependent. Thirdly, RL algorithms can +converge to a local optimum, a general challenge that applies +to the matching problem. Therefore, the goal is to develop +a framework based on RL that is simple and efficient to +learn, simultaneously satisfies the action constraints, and can +converge to a good solution. The proposed framework in this +paper simplifies the output of the policy to be trained by RL +to just “match” or “not to match” for each active customer, +regardless of the supply type and action feasibility. Thus, the +action space of the output of the component that is trained +is linear in the number of active consumers. Further details +regarding the proposed matching policy are given below. +B. Proposed Matching Policy +The proposed discrete matching policy is characterized by +a learnable and fixed component [6]. The first component, de- +noted by µt, determines the probability of matching customers, +and the latter makes sure that the customers are matched before +their deadline. Let PMt be the space of probability measures +over the set Mt. Then, the policy µt can be defined: +µt : Ωt → PMt. +Let mt ∈ Mt be given by mt ∼ µt. The component of mt +corresponding to the customer i is defined by mi,t, where +mi,t ∈ {0, 1}. The output mi is input to a second function, ϕ. +The function ϕ matches the customers with mi,t = 1 to the +available RES in each IHR. When total matching implied by +the discrete matching is in excess of the RES, the remaining +customers with mi,t = 1 are matched to the grid supply. When +total matching implied by the discrete matching is less than the +available RES, the excess RES generation is assigned to the +remaining customers. Denote the component of ϕ that specifies +whether customer i is matched to supply type j by ϕj,i. +The output ϕj,i is input to a third function, ν, that overturns +the matching decision for the customers with an immediate +deadline and ensures that the flexible customers in IHRs are +served by their deadline: +νj,i = +� +� +� +1 +if di = t, i is active, ϕs,i = 0 ∀s +and j = g, +ϕj,i +otherwise. +Thus, the overall discrete matching policy for time t, χt, is +given by: +χt = ν ◦ ϕ ◦ mt, mt ∼ µt. +(21) +The proposed policy is parameterized by θt, where the pa- +rameterization is denoted by µt(.; θt). The learning algorithm, +presented next, uses the observations from load and generation +data of the IHR and trains θt for every time step t by evaluating +its own performance. +C. Policy Gradient Learning Algorithm +This part describes the proposed policy gradient learning +algorithm. EXt∼Pt(.) is used as a shorthand for expectation +over Xt ∼ P(.|Xt−1, χt−1), where P(.|Xt−1, χt−1) denotes +the transition probability from state Xt−1 under the pol- +icy χt−1. Let mt:T += {mt, ml+1, ..., mT } and µl:T += +{µt, µl+1, ..., µT }. Denoting Em∼µ as a shortened form of +Emt+1:T ∼µt+1:T , the market welfare can be defined as: +V χ +t+(Xt+1):=Em∼µ +T +� +l=t+1 +� +j +� +k∈At +(πi +l − cj)χt,j,i(Xt). +(22) +Let: +vχ +t := +� +j +� +i∈At +(πi +l − cj)χt,j,i(Xt), +(23) +V χ +t (Xt|mt) := vχ +t + EXt+1∼Pt+1(.) +� +V χ +t+(Xt+1) +� +. +(24) +Then, from the definitions of Vχ and V χ +t (Xt), the gradient +of the value function with respect to the policy parameter θt +can be calculated as follows: +∂Vχ +∂θt += EXt +�∂V χ +t (Xt) +∂θt +� +, +(25) +∂Vχ +∂θt += EXt +� +mt∈Ht +∂µt(mt; θt) +∂θt +[vχ +t ++EXt+1∼Pt+1(.)V χ +t+(Xt+1) +� +. +(26) +The gradient of the value function with respect to the policy +parameter derived in (25)-(26) can be written as follows: +∂Vχ +∂θt += EXt,mt∼µt +�∂ log µt(mt; θt) +∂θt +V χ +t (Xt|mt) +� +, +(27) +where an unbiased estimate of this relationship can defined as +follows: +δθ +t = +�∂ log µt(mt; θt) +∂θt +V χ +t (Xt|mt) +� +. +(28) +Since the term V χ +t (Xt|mt) is unknown, the gradient in (28) +is not computable. Therefore, this term is replaced with the +social welfare from t to T for a sample epoch under policy χ: +δθ +t,r = ∂ log µt(mt; θt) +∂θt +� T +� +l=t +vχ +l +� +, +(29) +where the gradient is computable using the data from a sample +epoch (Z = {Z1, Z2, ..., ZT }) and matching decisions under +the policy χ for the same epoch. Furthermore, the gradient +estimate is unbiased as +∂Vχ +∂θt += E[δθ +t,r]. The vanilla policy +gradient learning algorithm learns the policy parameter θt for +each time step using the following stochastic gradient ascent +algorithm: +θt+1 ← θt + γθδθ +t,r, +(30) +where θt is updated using the computed gradient δθ +t,r for +multiple sample epochs in every update step. +In addition to the vanilla policy gradient learning algorithm +described above, the actor-critic algorithm AC−k is also +proposed for the dynamic matching of IHRs. This algorithm +learns both the matching policy µ and an approximation of + +7 +value function V χ +t (Xt). This function, which is also called +the critic function, is parameterized by φk and expressed by +V χ +k (Xk; φk). Hence, the approximate policy gradient for the +actor-critic algorithm AC−k can be defined as follows: +δθ +t,k = ∂ log µt(mt; θt) +∂θt +�t+k−1 +� +l=t +vχ +l + V χ +t+k(Xt+k; φt+k) +� +, +(31) +where the policy parameters are learned using the following +stochastic gradient ascent algorithm in (32) and similarly the +parameter φk of the critic function is learned by stochastic +gradient descent for its least-squares error in (33). +θt+1 ← θt + γθδθ +t,k, +(32) +φk+1 ← φk − γφ +� +V χ +k (.; φk) − +T +� +l=k +vχ +l +� +. +(33) +Further details regarding the actor-critic algorithm are given +in Algorithm 1, where the ADAM gradient algorithm of the +gradient updates in (32) and (33) is implemented. +Algorithm 1 Actor-Critic (AC−k) Policy Gradient Learn- +ing Algorithm for an IHR +1) Initialize D = ∅, j = 0 +2) Initialize θk ∀ k ∈ [1, ..., T]. N: number of epochs +3) for i = 1 : N +a) j = j + 1 +b) Set Di = {{Xk, mk, vχ +k } ∀ k ∈ [1, ..., T]} +c) Include Di into D +d) if j == M +Update θk by ADAM of Eq. (32) using D +Update φk by ADAM of Eq. (33) using D +j = 0; D = ∅ +end +end +4) end +The matching policy in the proposed study is trained +with the Temporal Convolution Network (TCN), denoted +by TCNµ. Let +˜Xt denote the input sequence to the TCN +at each time step, where +˜X⊤ +t += [˜x⊤ +1 , ˜x⊤ +2 , ˜x⊤ +3 , ..., ˜x⊤ +t ], and +˜xt = [a⊤ +t , p⊤ +t , b⊤ +t , d⊤ +t , p⊤ +u,t, b⊤ +u,t, rt]. The vector of matching +probabilities P m +µ +∈ [0, 1]n×T as the output of TCN can be +determined as P m +µ = TCNµ( ˜Xt). The output of TCN in the +proposed study is fixed and capped at the maximum number +of active customers at any time, n × T. Let P m +µ,i denote the +probability of matching for the active customer i at time step +t. Hence, the distribution µt is constructed as: +P(mt,i = 1) = P m +µ,i, P(mt,i = 0) = 1 − P m +µ,i. +IV. SIMULATIONS AND RESULTS +The proposed hierarchical matching framework is imple- +mented on the IEEE 33-bus test distribution system using the +30-minute real-time California Independent System Operator +(CAISO) load and solar generation data from January 1, 2021 +to September 28, 2021. The distribution system is divided +into 5 IHRs, each consisting of a learning agent to control +and match DERs with flexible loads. The structure of the +distribution system with IHRs is shown in Fig. 3, where +the electric vehicle (EV) charging stations supply charging +requests of 6.6 kWh to 24 EVs in IHR 1, 30 EVs in IHR +2, 8 EVs in IHR 3 and 30 EVs in each one of IHRs 4 and +5. The CAISO solar power data is scaled to the inverter’s +nominal capacity of 105 kW in IHR 1, 150 kW in IHR 2, 45 +kW in IHR 3, and 150 kW in each one of IHRs 4 and 5. The +distribution system active and reactive loads are scaled to 50 % +of their nominal rates, 3715 kW and 2300 kVAr, respectively. +The electricity tariff is assumed to be 120 $/MWh, and the +curtailment penalty for the central agent is assumed to be 500 +$/MWh. To validate the efficiency of the proposed hierarchical +framework, the following scenarios are considered: +• Scenario 1: This is a scenario where the EVs are char- +acterized by earlier arrival times and longer departure +times. In this scenario, waiting to match will fetch higher +welfare. This scenario tests the capability of the IHR +agents to learn to let the customers wait in the market +and not match them immediately upon their arrival. +• Scenario 2: This is a scenario where the EVs are charac- +terized by moderate arrival and longer departure times. In +this scenario, waiting may not result in improved welfare. +Here, a strategy that partially waits and partially matches +upon arrival might be needed. This scenario tests the +capability of the algorithm to learn such hybrid strategies. +For illustration, two matching algorithms are considered, +one is the Learning Algorithm (LA) described in Section III, +and the other is the standard matching heuristic, Matching on +Arrival (MA). These matching algorithms are implemented in +scenarios 1 and 2 under the following market models: +• Centralized Model: In this model, a single agent manages +the matching of the whole distribution system. +• Decentralized Model: In this model, the distribution sys- +tem is divided into multiple IHRs as described earlier, +with each IHR employing a separate matching algorithm. +The central agent solves the reduced-dimension OPF +model described earlier to meet the respective IHRs flow +requirements and ensure that the grid constraints are met. +This model with the learning algorithm is the proposed +hierarchical framework. +The best hyper-parameters for the TCN model were iden- +tified to be 3 for the number of blocks, 4 for the number +of filters, 3 for the filter size, 0.1 for the dropout factor, +and 4 for the dilation factor. Sigmoid function is utilized +as the activation function for each output of TCN and the +following values are considered as the parameters of the +ADAM algorithm: α = [0.25, 0.99], β1 = 0.9, β2 = 0.999, +ϵ = 10−8, where α is the learning rate and β1, β2 are +exponential decay rates for the moment estimates. The best +batch size for the LA is 20. +A. Numerical Results +The average social welfare achieved in scenarios 1 and 2 +for the centralized and decentralized models is summarized in +Table I. In scenario 1, it can be seen that the MA algorithm +achieves a welfare of 17.98$ and 16.17$ in both the centralized + +8 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 +12 13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +26 +27 28 +29 +30 +31 +32 +33 +24 +25 +L1 +L2 +L3 +L4 +L5 +L6 +L7 +L8 +L9 +L10 L11 +L12 +L13 +L14 +L15 +L16 +L17 +L23 +L24 +L26 +L27 L28 +L29 L30 +L31 +L32 +L18 +L19 +L20 +L21 +L22 +L25 +IHR 4 +IHR 5 +IHR 1 +IHR 3 +IHR 2 +Fig. 3. +Structure of the 33-bus power distribution system, divided into 5 +integrated hybrid resources. +TABLE I +AVERAGE SOCIAL WELFARE ($) +Algorithm +Scenario +Centralized Model +Decentralized Model +LA +Scenario1 +218.59 +232.70 +Scenario2 +866.33 +893.52 +Average +542.46 +563.11 +MA +Scenario1 +17.98 +16.17 +Scenario2 +742.75 +808.20 +Average +380.365 +412.185 +and decentralized models, while the LA algorithm achieves +a higher welfare of 218.59$ and 232.7$ in the centralized +and decentralized models respectively. This shows that the LA +algorithm leverages the flexibility better to match the excess +RES during the middle of the day. In scenario 2, the results +reveal that the optimal matching policy is not to match all +the loads on their arrival but to only match the critical ones +on their arrival, so as to efficiently utilize the RES that is +available during the middle of the day. In this scenario, the +LA algorithm is the top-performing, achieving a social welfare +of 866.33$ and 893.52$ in the centralized and decentralized +models, followed by the MA algorithm, which achieves a +social welfare of 742.75$ and 808.2$ in the centralized and +decentralized models, respectively. +Comparing the performance of LA and MA algorithms in +the centralized and decentralized models, it can be seen that +the decentralized model with the learning algorithm is the best +performing, substantiating the efficacy of our approach. Table +II summarizes the social welfare achieved by LA and MA +in the decentralized matching model. Comparing the social +welfare, it can be found that the LA algorithm outperforms the +MA algorithm in each of the IHRs, showing the superiority +of the learning-based approach. +TABLE II +AVERAGE SOCIAL WELFARE IN THE DECENTRALIZED MODEL ($) +Algorithm +Model +IHR1 +IHR2 +IHR3 +IHR4 +IHR5 +LA +Scenario1 +41.65 +55.06 +12.3 +60.02 +63.67 +Scenario2 +184.15 +216.67 +49.47 +221.36 +221.87 +Average +112.9 +135.865 +30.885 +140.69 +142.77 +MA +Scenario1 +3.06 +3.89 +1.02 +4.164 +4.04 +Scenario2 +163.08 +202.4 +43.53 +199.46 +199.73 +Average +83.07 +103.145 +22.275 +101.81 +101.885 +B. Matching Market Analysis +This section analyzes the performance of the matching +algorithms under the centralized and decentralized matching +markets in scenarios 1 and 2. +1) Scenario 1: EVs with Earlier Arrival and Longer Depar- +ture Times: In this scenario, the flexible loads are character- +ized by earlier arrival and longer departure (deadline) times, +the RES generation is available during the middle of the day. +Thus, the market operator (agent) can queue the load requests +and match them to the RES available during the middle of +the day. The matching by the LA and MA of IHR 2 in this +scenario is shown in Fig. 4. The results clearly show that MA +fails to wait to avail the RES during the middle of the day, and +instead matches the loads to the grid supply. On the contrary, +the LA learns to queue the non-critical loads and shift them +to the periods where RES generation is available. +-40 +-20 +0 +20 +40 +60 +80 +100 +1 +21 +41 +61 +81 +101 +121 +141 +161 +181 +201 +221 +Social Welfare ($) +Epoch +Average: LA +Average: MA +Actual: LA +Actual: MA +Fig. 4. Average and actual social welfare of LA and MA for IHR 2 under +scenario 1. +Figure 5 shows the matching market results for the LA of +IHR 2 for a representative epoch of scenario 1. In Fig. 5, +the initial load request of critical loads is supplied using the +grid power, while a significant portion of non-critical loads +is shifted to the middle of the day and matched to the RES +generation, indicating the efficacy of the fixed and trainable +components of the matching policy to match flexible loads +with RES, while satisfying the quality of service constraints of +the loads. This is evident in Fig. 5, where all the requested load +is supplied without any curtailment and the RES generation +is efficiently allocated to supply the queued flexible loads and +the non-flexible loads when the flexible loads are unavailable. +0 +30 +60 +90 +120 +150 +180 +210 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +Matching Results (kW) +Time (h) +RES Generation +Matched Flexible Load to RES +Matched Flexible Load to Grid +Curtailed Flexible Load +Matched Non-Flexible Load to RES +Curtailed RES Generation +Requested Flexible Load +Total Supplied Flexible Load +Fig. 5. Matching market results for LA of IHR 2 for a representative epoch +of scenario 1. + +9 +The performance of the decentralized and centralized mod- +els with LA is compared in Fig. 6. As shown, the centralized +model obtains higher welfare in the initial epochs, but the +decentralized model achieves a higher average social welfare +with experience. +-30 +0 +30 +60 +90 +120 +150 +180 +210 +240 +1 +21 +41 +61 +81 +101 +121 +141 +161 +181 +201 +221 +Average Social Welfare ($) +Epoch +IHR 1 +IHR 2 +IHR 3 +IHR 4 +IHR 5 +Centralized +Fig. 6. Average social welfare of decentralized and centralized models with +LA in scenario 1. +2) Scenario 2: EVs with Moderate Arrival and Longer +Departure Times: In this scenario, the flexible loads are +characterized by moderate arrival and longer departure times +and the RES generation is available during the middle of the +day. The social welfare achieved by the LA and MA of IHR 5 +in this scenario is shown in Fig. 7. The results show that both +the LA and MA achieve a similar performance. However, as +shown, the LA is superior to the MA in that it doesn’t match +all the flexible loads on their arrival. This is evident in Fig. 8, +where a portion of the load request is shifted from their arrival +and matched to the RES generation during the middle of the +day. As in the previous scenario, the LA is able to meet the +quality of service constraints of the loads and utilize the RES +generation, while ensuring that the outcomes are economically +efficient. +25 +75 +125 +175 +225 +275 +325 +375 +1 +21 +41 +61 +81 +101 +121 +141 +161 +181 +Social Welfare ($) +Epoch +Average: LA +Average: MA +Actual: LA +Actual: MA +Fig. 7. Average and actual social welfare of LA and MA of IHR 5 in scenario +2. +C. Distribution System Constraints +In the proposed hierarchical matching framework, the cen- +tral agent solves a reduced-dimension OPF model with the +IHRs as the nodes to deliver the IHR flow requirements while +ensuring that the grid constraints are met. The agent can +also curtail the flow to each IHR to the extent that the grid +constraints are not violated. Figure 9 shows the voltage profiles +of IHRs in the decentralized model and scenario 1. +0 +100 +200 +300 +400 +500 +600 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +Matching Results (kW) +Time (h) +RES Generation +Matched Flexible Load to RES +Matched Flexible Load to Grid +Matched Non-Flexible Load to RES +Curtailed Flexible Load +Curtailed RES Generation +Requested Flexible Load +Total Supplied Flexible Load +Fig. 8. +Matching results for LA of IHR 5 for a representative epoch in +scenario 2. +12.5 +12.55 +12.6 +12.65 +12.7 +1 +5 +9 +13 +17 +21 +25 +29 +33 +37 +41 +45 +Voltage Level (kV) +Time (h) +Substation Bus +IHR 1 +IHR 2 +IHR 3 +IHR 4 +IHR 5 +Fig. 9. Voltage profiles of IHR nodes in the decentralized model, scenario 1. +In this epoch, the lower and upper voltage boundaries of +IHRs are respectively V h = [12.37, 12.2, 12.05, 12.08, 12.22] +and V h =[12.948, 13.11, 13.25, 13.22, 13.09]. As shown, the +voltage level of all IHRs is within the safe lower-bound and +upper-bound limits in all the time periods. The extent to which +the matching decisions are met in each IHR depends on the +power flow in the grid operation, which can be curtailed by +the central agent to ensure that the grid constraints are met. +Figure 10 shows the matching curtailment of different IHRs +in the decentralized model and scenario 1. The results show +that the initial IHR-level matching decisions are not curtailed +in most of the epochs, though the matching decisions in some +initial epochs are curtailed to ensure the safe operation of the +power grid. +0 +100 +200 +300 +400 +500 +600 +1 +21 +41 +61 +81 +101 +121 +141 +161 +181 +201 +221 +Load Curtailment (kWh) +Epoch +IHR 1 +IHR 2 +IHR 3 +IHR 4 +IHR 5 +Fig. 10. Matching curtailment of the central agent in the decentralized model, +scenario 1. + +10 +V. CONCLUSIONS +This paper proposes a learning-based hierarchical frame- +work for dynamic matching in power distribution systems. +In the proposed framework, the power distribution system is +divided into multiple IHRs, each consisting of flexible loads +and RES. The IHR agents employ DRL to output an efficient +and scalable online matching policy to match the available +RES and active customers as the day progresses such that +their quality of service constraints are not violated. Once +the IHR-level matching decisions are determined, a central +agent uses the net active power flow, as well as the reactive +power limits of each IHR to formulate a reduced-dimension +OPF model to determine the final flows such that the flow +requirements of the IHRs are met and the grid constraints are +not violated. The hierarchical approach offers a very effective +way to combine the ability of DRL to learn state-dependent (or +online) matching policies and that of optimization to ensure +safe grid operation. The proposed hierarchical framework was +implemented on the IEEE 33-bus test distribution system and +tested on multiple scenarios with different matching algo- +rithms, including the proposed learning algorithm. The results +show that the hierarchical framework utilizes the flexible loads +better, resulting in higher social welfare compared to the +centralized approach that matches across the whole distribution +system. +REFERENCES +[1] “Ferc +order +no. +2222: +A +new +day +for +distributed +energy +resources,” +2020. +[Online]. +Available: +https://www.ferc.gov/media/ +ferc-order-no-2222-fact-sheet +[2] K. Oikonomou, M. Parvania, and R. 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Parvania, “Hierarchical intelligent operation of +energy storage systems in power distribution grids,” IEEE Transactions +on Sustainable Energy, 2022. + diff --git a/99FJT4oBgHgl3EQfpCw2/content/2301.11598v1.pdf b/99FJT4oBgHgl3EQfpCw2/content/2301.11598v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c001c7e2d8e9f182ea41354e275958a8f2bf0684 --- /dev/null +++ b/99FJT4oBgHgl3EQfpCw2/content/2301.11598v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3909851f657db119e0c1d90718d9b4ae49d9a8602af2ce63c86108b8784323f5 +size 1654502 diff --git a/99FJT4oBgHgl3EQfpCw2/content/tmp_files/2301.11598v1.pdf.txt b/99FJT4oBgHgl3EQfpCw2/content/tmp_files/2301.11598v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ee26ae45f6c50a5b8a228221636b564bf9b5146 --- /dev/null +++ b/99FJT4oBgHgl3EQfpCw2/content/tmp_files/2301.11598v1.pdf.txt @@ -0,0 +1,1960 @@ +arXiv:2301.11598v1 [math.NA] 27 Jan 2023 +Practical Sketching Algorithms for Low-Rank +Tucker Approximation of Large Tensors +Wandi Dong1, Gaohang Yu1*, Liqun Qi2,1,3 and Xiaohao Cai4 +1Department of Mathematics, Hangzhou Dianzi University, +Hangzhou, 310018, China. +2Huawei Theory Research Lab, Hong Kong, China. +3Department of Applied Mathematics, Hongkong Polytechnic +University, Hong Kong, China. +4School of Electronics and Computer Science, University of +Southampton, Southampton, SO17 1BJ, UK. +*Corresponding author(s). E-mail(s): maghyu@163.com; +Contributing authors: 15560159213@163.com; +liqun.qi@polyu.edu.hk; x.cai@soton.ac.uk; +Abstract +Low-rank approximation of tensors has been widely used in high- +dimensional data analysis. It usually involves singular value decom- +position (SVD) of large-scale matrices with high computational com- +plexity. Sketching is an effective data compression and dimension- +ality reduction technique applied to the low-rank approximation of +large matrices. This paper presents two practical randomized algo- +rithms for low-rank Tucker approximation of large tensors based +on sketching and power scheme, with a rigorous error-bound analy- +sis. Numerical experiments on synthetic and real-world tensor data +demonstrate the competitive performance of the proposed algorithms. +Keywords: tensor sketching, randomized algorithm, Tucker decomposition, +subspace power iteration, high-dimensional data +MSC Classification: 68W20 , 15A18 , 15A69 +1 + +2 +Sketching Algorithms for Low-Rank Tucker Approximation +1 Introduction +In practical applications, high-dimensional data, such as color images, hyper- +spectral images and videos, often exhibit a low-rank structure. Low-rank +approximation of tensors has become a general tool for compressing and +approximating high-dimensional data and has been widely used in scientific +computing, machine learning, signal/image processing, data mining, and many +other fields [1]. The classical low-rank tensor factorization models include, +e.g., Canonical Polyadic decomposition (CP) [2, 3], Tucker decomposition [4– +6], Hierarchical Tucker (HT) [7, 8], and Tensor Train decomposition (TT) +[9]. This paper focuses on low-rank Tucker decomposition, also known as +the low multilinear rank approximation of tensors. When the target rank +of Tucker decomposition is much smaller than the original dimensions, it +will have good compression performance. For a given Nth-order tensor X ∈ +RI1×I2×...×IN , the low-rank Tucker decomposition can be formulated as the +following optimization problem, i.e., +min +Y ∥X − Y∥2 +F , +(1) +where Y ∈ RI1×I2×...×IN , with rank(Y(n)) ≤ rn for n = 1, 2, . . ., N, Y(n) is the +mode-n unfolding matrix of Y, and rn is the rank of the mode-n unfolding +matrix of X. +For the Tucker approximation of higher-order tensors, the most fre- +quently used non-iterative algorithms are the improved algorithms for the +higher-order singular value decomposition (HOSVD) [5], the truncated higher- +order SVD (THOSVD) [10] and the sequentially truncated higher-order SVD +(STHOSVD) [11]. Although the results of THOSVD and STHOSVD are usu- +ally sub-optimal, they can use as reasonable initial solutions for iterative +methods such as higher-order orthogonal iteration (HOOI) [10]. However, both +algorithms rely directly on SVD when computing the singular vectors of inter- +mediate matrices, requiring large memory and high computational complexity +when the size of tensors is large. +Strikingly, randomized algorithms can reduce the communication among +different levels of memories and are parallelizable. In recent years, many schol- +ars have become increasingly interested in randomized algorithms for finding +approximation Tucker decomposition of large-scale data tensors [12–17, 19, 20]. +For example, Zhou et al. [12] proposed a randomized version of the HOOI +algorithm for Tucker decomposition. Che and Wei [13] proposed an adaptive +randomized algorithm to solve the multilinear rank of tensors. Minster et al. +[14] designed randomized versions of the THOSVD and STHOSVD algorithms, +i.e., R-STHOSVD. Sun et al. [17] presented a single-pass randomized algorithm +to compute the low-rank Tucker approximation of tensors based on a practical +matrix sketching algorithm for streaming data, see also [18] for more details. +Regarding more randomized algorithms proposed for Tucker decomposition, +please refer to [15, 16, 19, 20] for a detailed review of randomized algorithms + +Sketching Algorithms for Low-Rank Tucker Approximation +3 +for solving Tucker decomposition of tensors in recent years involving, e.g., ran- +dom projection, sampling, count-sketch, random least-squares, single-pass, and +multi-pass algorithms. +This paper presents two efficient randomized algorithms for finding the +low-rank Tucker approximation of tensors, i.e., Sketch-STHOSVD and sub- +Sketch-STHOSVD summarized in Algorithms 6 and 8, respectively. The main +contributions of this paper are threefold. Firstly, we propose a new one-pass +sketching algorithm (i.e., Algorithm 6) for low-rank Tucker approximation, +which can significantly improve the computational efficiency of STHOSVD. +Secondly, we present a new matrix sketching algorithm (i.e., Algorithm 7) by +combining the two-sided sketching algorithm proposed by Tropp et al. [18] +with subspace power iteration. Algorithm 7 can accurately and efficiently com- +pute the low-rank approximation of large-scale matrices. Thirdly, the proposed +Algorithm 8 can deliver a more accurate Tucker approximation than sim- +pler randomized algorithms by combining the subspace power iteration. More +importantly, sub-Sketch-STHOSVD can converge quickly for any data tensors +and independently of singular value gaps. +The rest of this paper is organized as follows. Section 2 briefly introduces +some basic notations, definitions, and tensor-matrix operations used in this +paper and recalls some classical algorithms, including THOSVD, STHOSVD, +and R-STHOSVD, for low-rank Tucker approximation. Our proposed two- +sided sketching algorithm for STHOSVD is given in Section 3. In Section 4, +we present an improved algorithm with subspace power iteration. The effec- +tiveness of the proposed algorithms is validated thoroughly in Section 5 by +numerical experiments on synthetic and real-world data tensors. We conclude +in Section 6. +2 Preliminary +2.1 Notations and basic operations +Some common symbols used in this paper are shown in the following Table 1. +Table 1 Common symbols used in this paper. +Symbols +Notations +a +scalar +A +matrix +X +tensor +X(n) +mode-n unfolding matrix of X +×n +mode-n product of tensor and matrix +In +identity matrix with size n × n +σi(A) +the ith largest singular value of A +A⊤ +transpose of A +A† +pseudo-inverse of A + +4 +Sketching Algorithms for Low-Rank Tucker Approximation +We denote an Nth-order tensor X ∈ RI1×I2×...×IN with entries given by +xi1,i2,...,iN, 1 ≤ in ≤ In, n = 1, 2, ..., N. The Frobenius norm of X is defined as +∥X∥F = +� +� +� +� +I1,I2,...,IN +� +i1,i2,...,iN +x2 +i1,i2,...,iN . +The mode-n tensor-matrix multiplication is a frequently encountered operation +in tensor computation. The mode-n product of a tensor X ∈ RI1×I2×...×IN +by a matrix A ∈ RK×In (with entries ak,in) is denoted as Y = X ×n A ∈ +RI1×...×In−1×K×In+1×...×IN, with entries +yi1,...,in−1,k,in+1,...,iN = +In +� +in=1 +xi1,...,in−1,in,in+1,...,iNak,in. +The mode-n matricization of higher-order tensors is the reordering of ten- +sor elements into a matrix. The columns of mode-n unfolding matrix X(n) ∈ +RIn×(� +N̸=n IN ) are the mode-n fibers of X. More specifically, a element +(i1, i2, ..., iN) of X is maps on a element (in, j) of X(n), where +j = 1 + +N +� +k=1,k̸=n +[(ik − 1) +k−1 +� +m=1,m̸=n +Im]. +Let the rank of mode-n unfolding matrix X(n) is rn, the integer array +(r1, r2, ..., rN) is Tucker-rank of Nth-order tensor X, also known as the mul- +tilinear rank. The Tucker decomposition of X with rank (r1, r2, ..., rN) is +expressed as +X = G ×1 U (1) ×2 U (2) . . . ×N U (N), +(2) +where G ∈ Rr1×r2×...×rN is the core tensor, and {U (n)}N +n=1 with U (n) ∈ RIn×rn +is the mode-n factor matrices. The graphical illustration of Tucker decom- +position for a third-order tensor shows in Figure 1. We denote an optimal +rank-(r1, r2, ..., rN) approximation of a tensor X as ˆ +Xopt, which is the optimal +Tucker approximation by solving the minimization problem in (1). Below we +Fig. 1 Tucker decomposition of a third-order tensor. +present the definitions of some concepts used in this paper. + +B +9 +3 +2 +ASketching Algorithms for Low-Rank Tucker Approximation +5 +Definition 1 (Kronecker products) The Kronecker product of matrices A ∈ Rm×n +and B ∈ Rk×l is defined as +A ⊗ B = + + +a11B +a12B +... a1nB +a21B +a22B +... a2nB +: +: +... +: +am1B am2B ... amnB + + ∈ Rmk×nl. +The Kronecker product helps express Tucker decomposition. The Tucker +decomposition in (2) implies +X(n) = U (n)G(n)(U (N) ⊗ ... ⊗ U (n+1) ⊗ U (n−1) ⊗ ... ⊗ U (1))⊤. +Definition 2 (Standard normal matrix) The elements of a standard normal matrix +follow the real standard normal distribution (i.e., Gaussian with mean zero and +variance one) form an independent family of standard normal random variables. +Definition 3 (Standard Gaussian tensor) The elements of a standard Gaussian +tensor follow the standard Gaussian distribution. +Definition 4 (Tail energy) The jth tail energy of a matrix X is defined as +τ 2 +j (X) := +min +rank(Y ) 0 is the oversampling parameter satisfying r+p ≤ min{m, n}. +Algorithm 3 is an efficient randomized algorithm for computing rank-r +approximations to matrices. Minster et al. [14] applied Algorithm 3 directly +to the STHOSVD algorithm and then presented a randomized version of +STHOSVD (i.e., R-STHOSVD), see Algorithm 4. +Algorithm 4 R-STHOSVD +Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing +order sp : {i1, i2, . . . , iN}, and oversampling parameter p ≥ 0 +Ensure: Tucker approximation ˆ +X = G ×1 U (1) ×2 U (2) . . . ×N U (N) +1: G ← X +2: for n = i1, i2, . . . , iN do +3: +( ˆU, ˆS, ˆV ⊤) ← R-SVD(G(n), rn, p) (cf. Algorithm 3) +4: +U (n) ← ˆU +5: +G ← foldn( ˆS ˆV ⊤) +6: end for +3 Sketching algorithm for STHOSVD +A drawback of R-SVD algorithm is that when both dimensions of the inter- +mediate matrices are enormous, the computational cost can still be high. To +resolve this problem, we could resort to the two-sided sketching algorithm for +low-rank matrix approximation proposed by Joel A. Tropp et al. [22]. The +preprocessing of sketching algorithm needs two sketch matrices to contain +information regarding the rows and columns of input matrix A ∈ Rm×n. Thus +we should choose two sketch size parameters k and l, s.t. , r ≤ k ≤ min{l, n}, +0 < l ≤ m. The random matrices Ω ∈ Rn×k and Ψ ∈ Rl×m are fixed indepen- +dent standard normal matrices. Then we can multiply matrix A left and right +respectively to obtain random sketch matrices Y ∈ +Rm×k and W ∈ Rl×n, +which collect sufficient data about the input matrix to compute the low-rank +approximation. The dimensionality and distribution of the random sketch +matrices determine the approximation’s potential accuracy, with larger values +of k and l resulting in better approximations but also requiring more storage +and computational cost. +The sketching algorithm for low-rank approximation is given in Algorithm +5. Function orth(A) in Step 2 produces an orthonormal basis of A. Using +orthogonalization matrices will achieve smaller errors and better numerical +stability than directly using the randomly generated Gaussian matrices. In +particular, when A is dense, the arithmetic cost of Algorithm 5 is O((k + +l)mn + kl(m + n)) flops. Algorithm 5 is simple, practical, and possesses the +sub-optimal error-bound as stated in the following Theorem 3. In Theorem 3, + +10 +Sketching Algorithms for Low-Rank Tucker Approximation +Algorithm 5 Sketch for low-rank approximation +Require: matrix A ∈ Rm×n, and sketch size parameters k, l +Ensure: rank-k approximation ˆA = QX of A +1: Ω ← randn(n, k), Ψ ← randn(l, m) +2: Ω ← orth(Ω), Ψ⊤ ← orth(Ψ⊤) +3: Y ← AΩ +4: W ← ΨA +5: (Q, ∼) ← thinQR(Y ) +6: X ← (ΨQ)†W +function f(s, t) := s/(t − s − 1)(t > s + 1 > 1). The minimum in Theorem +3 reveals that the low rank approximation of given matrix A automatically +exploits the decay of tail energy. +Theorem 3 ([22], Theorem 4.3) Assume that the sketch size parameters satisfy +l > k + 1, and draw random test matrices Ω ∈ Rn×k and Ψ∈ Rl×m independently +forming the standard normal distribution. Then the rank-k approximation ˆA obtained +from Algorithm 5 satisfies +E ∥ A − ˆA ∥2 +F ≤ (1 + f(k, l)) · min +̺ 0 +Ensure: rank-k approximation ˆA = QX of A +1: Ω ← randn(n, k), Ψ ← randn(l, m) +2: Ω ← orth(Ω), Ψ⊤ ← orth(Ψ⊤) +3: Y = AΩ, W = ΨA +4: Q0 ← thinQR(Y ) +5: for j = 1, . . . , q do +6: +ˆYj = A⊤Qj−1 +7: +( ˆQj, ∼) ← thinQR( ˆYj) +8: +Yj = A ˆQj +9: +(Qj, ∼) ← thinQR(Yj) +10: end for +11: Q = Qq +12: X ← (ΨQ)†W +Although power iteration can improve the accuracy of Algorithm 5 to some +extent, it still suffers from a problem, i.e., during the execution with power +iteration, the rounding errors will eliminate all information about the singular +modes associated with the singular values. To address this issue, we propose an + +Sketching Algorithms for Low-Rank Tucker Approximation +13 +improved sketching algorithm by orthonormalizing the columns of the sample +matrix between each application of A and A⊤, see Algorithm 7. When A is +dense, the arithmetic cost of Algorithm 7 is O((q + 1)(k + l)mn + kl(m + n)) +flops. Numerical experiments show that a good approximation can achieve +with a choice of 1 or 2 for subspace power iteration parameter [21]. +Algorithm 8 sub-Sketch-STHOSVD +Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing +order sp : {i1, i2, . . . , iN}, sketch size parameters {l1, l2, ..., lN}, and integer +q > 0 +Ensure: Tucker approximation ˆ +X = G ×1 U (1) ×2 U (2) . . . ×N U (N) +1: G ← X +2: for n = i1, i2, . . . , iN do +3: +(Q, X) ← sub-Sketch(G(n), rn, ln, q) (cf. Algorithm 7) +4: +U (n) ← Q +5: +G ← foldn(X) +6: end for +Using Algorithm 7 to compute the low-rank approximations of intermedi- +ate matrices, we can obtain an improved sketching algorithm for STHOSVD, +called sub-Sketch-STHOSVD, see Algorithm 8. The error-bound for Algorithm +8 states in the following Theorem 5. Its proof is deferred in Appendix. +Theorem 5 Let ˆ +X = G ×1 U(1) ×2 U(2) . . . ×N U(N) be the Tucker approximation +of a tensor X ∈ RI1×I2×...×IN obtained by the sub-Sketch-STHOSVD algorithm +(i.e., Algorithm 8) with target rank rn < In, n = 1, 2, ..., N, sketch size parameters +{l1, l2, ..., lN} and processing order p : {1, 2, . . . , N}. Let ̟k ≡ +σk+1 +σk +denote the +singular value gap, then +E{Ωj}N +j=1∥X − � +X ∥2 +F ≤ +N +� +n=1 +(1 + f(rn, ln)) · +min +̺n k + 1. Draw random +test matrices Ω ∈ Rn×k and Ψ∈ Rl×m independently from the standard normal +distribution. Then the rank-k approximation ˆA obtained from Algorithm 7 satisfies +E ∥ A − ˆA ∥2 +F ≤ (1 + f(k, l)) · min +̺ 0 and δ ∈ [0, 1] imply +stronger privacy guarantees. +Although significant advances have been made recently in understanding the utility-privacy +trade-offs in canonical statistical tasks, several important questions remain open. We provide a +survey in App. A. Consider a canonical statistical task of linear regression with n i.i.d. samples, +{(xi ∈ Rd, yi ∈ R)}n +i=1, drawn from xi ∼ N(0, Σ), yi = x⊤ +i w∗ + zi, and zi ∼ N(0, σ2) for some +true parameter w ∈ Rd. The error is measured in ∥ ˆw − w∗∥Σ := ∥Σ1/2( ˆw − w∗)∥, which correctly +accounts for the signal-to-noise ratio in each direction; in the direction of large eigenvalue of Σ, +we have larger signal in xi but the noise zi remains the same; we expect smaller error in those +directions, which is accounted for in ∥ ˆw − w∗∥Σ. +When computational complexity is not concerned, the best known algorithm is introduced by +Liu et al. (2022b), called High-dimensional Propose-Test-Release (HPTR). For linear regression, +n = ˜O(d/α2 + d/(εα)) samples are sufficient for HPTR to achieve an error of (1/σ)∥ ˆw − w∗∥Σ = α +with high probability. After a series of work surveyed in App. A, Varshney et al. (2022) achieve the +∗Paul Allen School of Computer Science & Engineering, University of Washington, xiyangl@cs.washington.edu +†Google Research, prajain@google.com +‡Google Research, weihaokong@google.com +§Paul Allen School of Computer Science & Engineering, University of Washington, and Google Research, +sewoong@cs.washington.edu +¶Google Research, arunss@google.com +1 +arXiv:2301.13273v1 [cs.LG] 30 Jan 2023 + +best known sample complexity for an efficient algorithm: n = ˜O(κ2d/ε + d/α2 + κd/(εα)). The last +term has an extra factor of κ, the condition number of the covariance Σ of the covariates, and the +first term is unnecessary. +In this work, we propose a novel method (Algorithm 1) that uses full-batch gradient descent +with adaptive clipping. Furthermore, using a intuitive but intricate analysis, we improve this sample +complexity. +Theorem 1 (informal version of Thm. 4 with no adversary). Alg. 1 is (ε, δ)-DP. Under the +(Σ, σ2, w∗, K, a)-model in Assumption 1, n = ˜O(d/α2 + κ1/2d/(εα)) samples are sufficient for Alg. 1 +to achieve an error rate of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α), where κ := λmax(Σ)/λmin(Σ). +That is, we can get rid of the first unnecessary term in alaysis of Varshney et al. (2022), while +also improving dependency on κ term which is quite critical for practical applications. +Perhaps surprisingly, we show that the same algorithm is also robust against label-corruption, +where an adversary selects arbitrary αcorrupt fraction of the data points and changes their response +variables arbitrarily. When computational complexity is not concerned, the best known algorithm +is again HPTR that also provides optimal robustness and (ε, δ)-DP simultaneously, i.e., n = +˜O(d/α2 + d/(εα)) samples are sufficient for HPTR to achieve an error of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α) +for any corruption bounded by αcorrupt ≤ α. Note that this is a stronger adversary than the +label-corruption we study in this paper; this adversary can corrupt both the covariate, xi, and the +response variable, yi. Currently, there is no efficient algorithm that can guarantee both privacy +and robustness for linear regression. Under a weaker adversary that only corrupts yi’s, we provide +the first efficient algorithm that achieves both privacy and robustness with a near-optimal sample +complexity. +Theorem 2 (informal version of Thm. 4 with adversarial label corruption). Alg. 1 is (ε, δ)- +DP. Under the hypotheses of Thm. 1 and under αcorrupt-corruption model of Assumption 2, if +αcorrupt ≤ α then n = ˜O(d/α2 + κ1/2d/(εα)) samples are sufficient for Alg. 1 to achieve an error +rate of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α) , where κ := λmax(Σ)/λmin(Σ). +We focus on sub-Weibull distributions in the main text. A similar algorithm can be applied to +the case where the noise in the samples is heavy-tailed, i.e., k-th moment bounded for k ≥ 4. This +results in an increased sample complexity of n = ˜O(d/α2k/(k−1) + κ1/2d/(εαk/(k−1))) to achieve the +same level of error. We explain the heavy-tailed setting, provide detailed analysis and a proof, and +discuss the results in App. L. +Theorem 3 (informal version of Coro. L.4). Alg. 4 is (ε, δ)-DP. Under (Σ, σ2, w∗, K, a, κ2, k)- +model of Assumption 3 and αcorrupt-corruption of Assumption 4, if 1.2αcorrupt ≤ αk/(k−1), then +n = ˜O(κ1/2d/(εαk/(k−1)) + d/α2k/(k−1))) samples are sufficient for Algorithm 4 to achieve an error +rate of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α), where κ := λmax(Σ)/λmin(Σ). +Contributions. +For a canonical problem of private linear regression under sub-Gaussian +distributions, we provide a novel algorithm that achieves the state-of-the-art sample complexity and +computational efficiency, improving upon the sample complexity of the prior efficient algorithms +Varshney et al. (2022); Cai et al. (2019) and nearly matching that of an exponential-time algorithm +Liu et al. (2022b). For the same problem, we show that the same algorithm is the first to achieve +robustness against adversarial corruption of the response variables. Under a heavy-tailed distribution +of the noise, we provide the first computationally efficient algorithm, to the best of our knowledge, +that achieves a sample complexity close to that of an exponential-time algorithm of Liu et al. (2022b). +This algorithm is also the first to achieve robustness against adversarial corruption of the response +variables, under heavy-tailed noise. +2 + +We start with the formal description of the setting in Sec. 2, where we present the prior work of +Varshney et al. (2022) and explain our main technical contributions. Varshney et al. (2022) propose +a streaming version of DP-SGD with adaptive clipping. Streaming algorithm ensures independence +between the current iterate and the current gradient, simplifying the analysis. Adaptive clipping +finds the appropriate threshold to clip the norm of the gradients, which is an appropriate technique +when there is no adversarial corruption. However, these two algorithmic choices are sub-optimal. +First, streaming DP-SGD can only use O(n/κ) samples at each round, which increases the +sensitivity and leads to an extra κ1/2 factor in the sample complexity. Instead, we propose using +a full-batch gradient descent and overcome the challenges in the analysis by relying on resilience +(explained in Sec. 6). Together with the novel analysis technique we explain in Sec. 2.1, this results +in the gain of κ1/2. +Next, the gradient-norm clipping is vulnerable against label corruption. Recall that a gradient +is a product of the residual, (w⊤ +t xi − yi), and the covariate, xi. An adversary can target those +samples with small covariates and make big changes to the residuals, while managing to evade the +clipping by the norm. Instead, we propose clipping the residual and the covariate separately. With +adaptively estimated clipping thresholds, this provides robustness against label corruption. +We present our main algorithm (Alg. 1) in Sec. 3 with theoretical analyses and justification of +the assumptions. We propose a novel adaptive clipping in Sec. 4 We present numerical experiments +on synthetic data that demonstrates the sample efficiency of our approach in Sec. 5. We end with a +sketch of our main proof ideas in Sec. 6. +2 +Problem formulation and background +In linear regression without corruption, the following assumption is standard for the uncorrupted +dataset Sgood, except for the fact that we assume a more general family of (K, a)-sub-Weibull +distributions that recovers the standard sub-Gaussian family as a special case when a = 0.5. +Assumption 1 ((Σ, σ2, w∗, K, a)-model). A multiset Sgood = {(xi ∈ Rd, yi ∈ R)}n +i=1 of n i.i.d. sam- +ples is from a linear model yi = ⟨xi, w∗⟩ + zi, where the input vector xi is zero mean, E[xi] = 0, +with a positive definite covariance Σ := E[xix⊤ +i ] ≻ 0, and the (input dependent) label noise zi is zero +mean, E[zi] = 0, with variance σ2 := E[z2 +i ]. We further assume E[xizi] = 0, which is equivalent to +assuming that the true parameter w∗ = Σ−1E[yixi]. We assume that the marginal distribution of xi +is (K, a)-sub-Weibull and that of zi is also (K, a)-sub-Weibull, as defined below. +Sub-Weibull distributions provide Gaussian-like tail bounds determining the resilience of the +dataset in Lemma J.7, which our analysis critically relies on and whose necessity is justified in +Sec. 3.3. +Definition 2.1 (sub-Weibull distribution (Kuchibhotla & Chakrabortty, 2018) ). For some K, a > 0, +we say a random vector x ∈ Rd is from a (K, a)-sub-Weibull distribution if for all v ∈ Rd, +E +� +exp +�� +⟨v,x⟩2 +K2E[⟨v,x⟩2] +�1/(2a)�� +≤ 2. +Our goal is to estimate the unknown parameter w∗, given upper bounds on the sub-Weibull +parameters, (K, a), and a corrupted dataset under the the standard definition of label corruption in +(Bhatia et al., 2015). There are variations in literature on the definition, which we survey in App. A. +Assumption 2 (αcorrupt-corruption). Given a dataset Sgood = {(xi, yi)}n +i=1, an adversary inspects +all the data points, selects αcorruptn data points denoted as Sr, and replaces the labels with arbitrary +labels while keeping the covariates unchanged. We let Sbad denote this set of αcorruptn newly labelled +3 + +examples by the adversary. Let the resulting set be S := Sgood ∪Sbad \Sr. We further assume that the +corruption rate is bounded by αcorrupt ≤ ¯α, where ¯α is a known positive constant satisfying ¯α ≤ 1/10, +72C2 K2 ¯α log2a(1/(6¯α)) log(κ) ≤ 1/2, and 2C2K2 log2a(1/(2¯α)) ≥ 1 for the (K, a)-sub-Weibull +distribution of interest and a positive constant C2 defined in Lemma J.7 that only depends on (K, a). +Notations. A vector x ∈ Rd has the Euclidean norm ∥x∥. For a matrix M, we use ∥M∥2 to denote +the spectral norm. The error is measured in ∥ ˆw − w∗∥Σ := ∥Σ1/2( ˆw − w∗)∥ for some PSD matrix Σ. +The identity matrix is denoted by Id ∈ Rd×d. Let [n] = {1, 2, . . . , n}. ˜O(·) hides some constants +terms, K, a = Θ(1), and poly-logarithmic terms in n, d, 1/ε, log(1/δ), 1/ζ, and 1/αcorrupt. For a +vector x ∈ Rd, we define clipa(x) := x · min{1, +a +∥x∥}. +Background on DP. Differential Privacy is a standard measure of privacy leakage when data is +accessed via queries, introduced by Dwork et al. (2006). Two datasets S and S′ are said to be +neighbors if they differ at most by one entry, which is denoted by S ∼ S′. A stochastic query q is said +to be (ε, δ)-differentially private for some ε > 0 and δ ∈ [0, 1], if P(q(S) ∈ A) ≤ eεP(q(S) ∈ A) + δ, +for all neighboring datasets S ∼ S′ and all subset A of the range of the query. We build upon two +widely used DP primitives, the Gaussian mechanism and the private histogram. A central concept +in DP mechanism design is the sensitivity of a query, defined as ∆q := supS∼S′ ∥q(S) − q(S′)∥. We +describe Gaussian mechanism and private histogram in App. B. +2.1 +Comparisons with the prior work +When there is no adversarial corruption, the state-of-the-art approach introduced by Varshney et al. +(2022) is based on stochastic gradient descent, where privacy is ensured by gradient norm clipping +and the Gaussian mechanism to ensure privacy. There are two main components in this approach: +adaptive clipping and streaming SGD. Adaptive clipping with an appropriate threshold θt ensures +that no data point is clipped while providing a bound on the sensitivity of the average mini-batch +gradient, which ensures we do not add too much noise. The streaming approach, where each data +point is only touched once and discarded, ensures independence between the past iterate wt and +the gradients at round t + 1, which the analysis critically relies on. For T = ˜Θ(κ) iterations where +κ is the condition number of the covariance Σ of the covariates, the dataset S = {(xi, yi)}n +i=1 is +partitioned into {Bt}T−1 +t=0 subsets of equal size: |Bt| = ˜Θ(n/κ). At each round t < T, the gradients +are clipped and averaged with additive Gaussian noise chosen to satisfy (ε, δ)-DP: +wt+1 ← wt − η +� 1 +|Bt| +� +i∈Bt +clipθt(xi(w⊤ +t xi − yi)) + θt +� +2 log(1.25/δ) +ε|Bt| +νt +� +, +(1) +where νt ∼ N(0, Id). +In Varshney et al. (2022), a variation of this streaming SGD requires +n = ˜O(κ2d/ε + d/α2 + κd/(εα)) to achieve an error of ∥wT − w∗∥2 +Σ = O(σ2α2). +Our technical innovations. Our approach builds upon such gradient based methods but +makes several important innovations. First, we use full-batch gradient descent, as opposed to the +streaming SGD above. Using all n samples reduces the sensitivity of the per-round gradient average, +since n > |Bt| = ˜Θ(n/κ). This improves the sample complexity to n = ˜O(d/α2 + κ1/2d/(εα)) to +achieve an error of ∥wT − w∗∥2 +Σ = O(σ2α2). However, full-batch GD loses the independence that the +streaming SGD enjoyed between wt and the samples used in the round t+1. This dependence makes +the analysis more challenging. We instead propose using the resilience property of sub-Weibull +distributions to precisely track the bias and variance of the (dependent) full-batch gradient average. +Resilience is a central concept in robust statistics which we explain in Sec. 6. +4 + +Next, one critical component in achieving this improved sample complexity is the new analysis +technique we introduce for tracking the end-to-end gradient updates. Since our gradient descent +algorithm is not guaranteed to make progress every step, we can not use the vanilla one-step +ahead analysis. Taking the full end-to-end analysis by expanding the whole gradient trajectory will +introduce too many correlated cross-terms which are very hard to control. Therefore, we leverage +an every κ-step analysis and show that the objective function at least decreases geometrically +every κ steps. To be more specific, our analysis technique in App. H steps 3 and 4 opens up the +iterative updates from beginning to end, and exploits the fact that λmax((ηΣ)1/2(1 − ηΣ)i(ηΣ)1/2) +is upper bounded by 1/(i + 1) when ∥ηΣ∥ ≤ 1. This technique is critical in achieving the near- +optimal dependence in κ. This might be of independent interest to other analysis of gradient-based +algorithms. We refer to the beginning of step 3 in App. H for a detailed explanation. +Finally, we propose a novel clipping method that separately clips xi and (w⊤ +t xi − yi) in the +gradient. This is critical in achieving robustness to label-corruption, as we explain in detail in +Sec. 3.1. +3 +Robust and DP linear regression +We introduce a gradient descent approach for linear regression with a novel adaptive clipping that +ensures robustness against label-corruption. This achieves a near-optimal sample complexity and, +for the special case of private linear regression without adversarial corruption, improves upon the +state-of-the-art algorithm. +3.1 +Algorithm +The skeleton of our approach in Alg. 1 is the general DP-SGD (Abadi et al., 2016; Song et al., 2013) +with adaptive clipping (Andrew et al., 2021). However, the standard adaptive clipping is not robust +against label-corruption under the more general (K, a)-sub-Weibull assumption. In particular, it is +possible under sub-Weibull distribution that a positive fraction of the covariates are close to the +origin, which is not possible under Gaussian data due to concentration. In this case, the adversary +can choose to corrupt those points with small norm, ∥xi∥, making large changes in the residual, +(yi −w⊤ +t xi), while evading the standard clipping (by the norm of the gradient), since the norm of the +gradient, ∥xi(yi − w⊤ +t xi)∥ = ∥xi∥ |yi − w⊤ +t xi|, can remain under the threshold. This is problematic, +since the bias due to the corrupted samples in the gradient scales proportional to the magnitude of +the residual (after clipping). To this end, we propose clipping the norm and the residual separately: +clipΘ(xi)clipθt +� +w⊤ +t xi − yi +� +. This keeps the sensitivity of gradient average bounded by Θθt, and the +subsequent Gaussian mechanism in line 11 ensures (ε0, δ0)-DP at each round. Applying advanced +composition in Lemma B.5 of T rounds, this ensures end-to-end (ε, δ)-DP. +Novel adaptive clipping. When clipping with clipΘ(xi), the only purpose of clipping the +covariate by its norm, ∥xi∥, is to bound the sensitivity of the resulting clipped gradient. In particular, +we do not need to make it robust as there is no corruption in the covariates. Ideally, we want to +select the smallest threshold Θ that does not clip any of the covariates. Since the norm of a covariate +is upper bounded by ∥xi∥2 ≤ K2Tr(Σ) log2a(1/ζ) with probability 1 − ζ (Lemma J.3), we estimate +the unknown Tr(Σ) using Private Norm Estimator in Alg. 3 in App. F and set the norm threshold +Θ = K +√ +2Γ loga(n/ζ) (Alg. 1 line 4). The n in the logarithm ensures that the union bound holds. +When clipping with clipθt(w⊤ +t xi − yi), the purpose of clipping the residual by its magnitude, +|yi − w⊤ +t xi| = |(w∗ − wt)⊤xi + zi|, is to bound the sensitivity of the gradient and also to provide +robustness against label-corruption. We want to choose a threshold that only clips corrupt data +points and at most a few clean data points. In order to achieve an error (1/σ)∥wT − w∗∥Σ = O(α), +5 + +we know that any set of (1 − α) fraction of the clean data points is sufficient to get a good +estimate of the average gradient, and we can find such a large enough set of points that satisfy +|(w∗ − wt)⊤xi + zi|2 ≤ (∥wt − w∗∥2 +Σ + σ2)CK2 log2a(1/(2α)) from Lemma J.3. +At the same +time, this threshold on the residual is small enough to guarantee robustness against the label- +corrupted samples. We introduce the robust and private Distance Estimator in Alg. 2 to estimate +the unknown (squared and shifted) distance, ∥wt − w∗∥2 +Σ + σ2, and set the distance threshold +θt = 2√2γt +� +9C2K2 log2a(1/(2α)) (Alg. 1 line 7). Both norm and distance estimation rely on +private histogram (Lemma B.2), but over a set of statistics computed on partitioned datasets, which +we explain in detail in Sec. 4. +Algorithm 1: Robust and Private Linear Regression +Input: S = {(xi, yi)}3n +i=1, DP parameters (ε, δ), T, learning rate η, failure probability ζ, +target error α, distribution parameter (K, a) +1 Partition dataset S into three equal sized disjoint subsets S = S1 ∪ S2 ∪ S3. +2 δ0 ← +δ +2T , ε0 ← +ε +4√ +T log(1/δ0), ζ0 ← ζ +3, w0 ← 0 +3 Γ ← PrivateNormEstimator(S1, ε0, δ0, ζ0) /* using Algorithm 3, Appendix F +*/ +4 Θ ← K +√ +2Γ loga(n/ζ0) +5 for t = 0, 1, 2, . . . , T − 1 do +6 +γt ← RobustPrivateDistanceEstimator(S2, wt, ε0, δ0, α, ζ0) /* using Algorithm 2 +*/ +7 +θt ← 2√2γt · +� +9C2K2 log2a(1/(2α)). +8 +Sample νt ∼ N (0, Id) +9 +˜g(t) +i +← clipΘ(xi)clipθt(x⊤ +i wt − yi) +10 +φt = ( +� +2 log(1.25/δ0)Θθt)/(ε0n) +11 +wt+1 ← wt − η +� +1 +n +� +i∈S3 ˜g(t) +i ++ φtνt +� +12 Return wT +3.2 +Analysis +We show that Algorithm 1 achieves a near-optimal sample complexity. We provide a proof in +Appendix H and a sketch of the proof in Section 6. We address the necessity of the assumptions in +Sec. 3.3, along with some lower bounds. +Theorem 4. Algorithm 1 is (ε, δ)-DP. Under (Σ, σ2, w∗, K, a)-model of Assumption 1 and αcorrupt- +corruption of Assumption 2 and for any failure probability ζ ∈ (0, 1) and target error rate α ≥ αcorrupt, +if the sample size is large enough such that +n = ˜O +� +K2d log2a+1 �1 +ζ +� ++ d + log(1/ζ) +α2 ++ +K2dT 1/2 log( 1 +δ) loga( 1 +ζ ) +εα +� +, +(2) +with a large enough constant where ˜O hides poly-logarithmic terms in d, n, and κ, then the choices +of a step size η = 1/(Cλmax(Σ)) for any C ≥ 1.1 and the number of iterations, T = ˜Θ (κ log (∥w∗∥)) +for a condition number of the covariance κ := λmax(Σ)/λmin(Σ), ensures that, with probability 1 − ζ, +Algorithm 1 achieves +Eν1,··· ,νt∼N(0,Id) +� +∥wT − w∗∥2 +Σ +� += ˜O +� +K4σ2α2 log4a � 1 +α +� � +, +(3) +6 + +where the expectation is taken over the noise added for DP, and ˜Θ(·) hides logarithmic terms in +K, σ, d, n, 1/ε, log(1/δ), 1/α, and κ. +Optimality. Omitting some constant and logarithmic terms, Alg. 1 requires +n += +˜O +� d +α2 + κ1/2d log(1/δ) +εα +� +, +(4) +samples to ensure an error rate of E[∥wT − w∗∥2 +Σ] = ˜O(σ2α2) for any α ≥ αcorrupt. The lower bound +on the achievable error of σ2α2 ≥ σ2α2 +corrupt is due to the label-corruption and cannot be improved, +as it matches an information theoretic lower bound we provide in Proposition 3.1. In the special case +when the covariate follows a sub-Gaussian distribution, that is (K, 1/2)-sub-Weibull for a constant +K, there is an n = Ω(d/α2 + d/(εα)) lower bound (Cai et al. (2019), Theorem 4.1), and our upper +bound matches this lower bound up to a factor of κ1/2 in the second term and other logarithmic +factors. Eq. (4) is the best known rate among all efficient private linear regression algorithms, +strictly improving upon existing methods when log(1/δ) = ˜O(1). We discuss some exponential time +algorithms that closes the κ1/2 gap in Sec. 3.3. +Comparisons with the state-of-the-art. The best existing efficient algorithm by Varshney +et al. (2022) can only handle the case where there is no adversarial corruption, and requires +n = ˜O(κ2d +� +log(1/δ)/ε + d/α2 + κd +� +log(1/δ)/(εα)) to achieve an error rate of σ2α2. Compared +to Eq. (4), the first term dominates in its dependence in κ, which is a factor of κ larger than Eq. (4). +The third term is larger by a factor of κ1/2 but smaller by a factor of log1/2(1/δ), compared to the +second term in Eq. (4). +In the non-private case, when ε = ∞, a recent line of work has developed algorithms for linear +regression that are robust to label corruptions (Bhatia et al., 2015, 2017; Suggala et al., 2019; +Dalalyan & Thompson, 2019). Of these, Bhatia et al. (2015); Dalalyan & Thompson (2019) are +relevant to our work as they consider the same adversary model as Assumption 2. When xi’s and +zi’s are sampled from N(0, Σ) and N(0, σ2), Dalalyan & Thompson (2019) proposed a Huber loss +based estimator that achieves error rate of σ2α2 log2(n/δ) when n = ˜O +� +κ2d/α2� +. Under the same +setting, Bhatia et al. (2015) propoased a hard thresholding based estimator that achieves σ2α2 +error rate with ˜O +� +d/α2� +sample complexity. Our results in Theorem 4 match these rates, except +for the sub-optimal dependence on log4a(1/α). Another line of work considered both label and +covariate corruptions and developed optimal algorithms for parameter recovery (Diakonikolas et al., +2019c,b; Prasad et al., 2018; Pensia et al., 2020; Cherapanamjeri et al., 2020; Jambulapati et al., +2020; Klivans et al., 2018; Bakshi & Prasad, 2021; Zhu et al., 2019; Depersin, 2020). The best +existing efficient algorithm , e.g. (Pensia et al., 2020), achieves error rate of σ2α2 log(1/α) when +n = ˜O +� +d/α2� +, and the xi and zi are sampled from N(0, I) and N(0, σ2). +Under both privacy requirements and adversarial corruption, the only algorithm with a provable +guarantee is an exponential time approach, known as High-dimensional Propose-Test-Release +(HPTR), of Liu et al. (2022b, Corollary C.2), which achieves a sample complexity of n = O(d/α2 + +(d + log(1/δ))/(εα)). Notice that there is no dependence on κ and the log(1/δ) term scales as +1/(εα) as opposed to κd1/2/(εα) of Eq. (4). It remains an open question if computationally efficient +private linear regression algorithms can achieve such a κ-independent sample complexity. Further, +HPTR is robust against a stronger adversary who corrupts the covariates also and not just the +labels. Under this stronger adversary, it remains open if there is an efficient algorithm that achieves +n = O(d/α2 + d/(εα)) sample complexity even for constant κ and δ. +3.3 +Lower bounds +Necessity of our assumptions. A tail assumption on the covariate xi such as Assumption 1 is +7 + +necessary to achieve n = O(d) sample complexity in Eq. (4). Even when the covariance Σ is close +to identity, without further assumptions on the tail of covariate x, the result in (Bassily et al., 2014) +implies that for δ < 1/n and sufficiently large n, no (ε, δ)-DP estimator can achieve excess risk +∥ ˆw − w∗∥2 +Σ better than Ω(d3/(ε2n2)) (see Eq. (3) in (Wang, 2018)). Note that this lower bound is a +factor d larger than our upper bound that benefits from the additional tail assumption. +A tail assumption on the noise zi such as Assumption 1 is necessary to achieve n = O(d/(εα)) +dependence on the sample complexity in Eq. (4). For heavy-tailed noise, such as k-th moment +bounded noise, the dependence can be significantly larger. Liu et al. (2022b, Proposition C.5) +implies that for δ = e−Θ(d) and 4-th moment bounded xi and zi, any (ε, δ)-DP estimator requires +n = Ω(d/(εα2)), which is a factor of 1/α larger, to achieve excess risk E[∥ ˆw − w∗∥2 +Σ] = ˜O(σ2α2). +The assumption that only label is corrupted is critical for Algorithm 1. The average of the +clipped gradients can be significantly more biased, if the adversary can place the covariates of the +corrupted samples in the same direction. In particular, the bound on the bias of our gradient step +in Eq. (41) in App. H would no longer hold. Against such strong attacks, one requires additional +steps to estimate the mean of the gradients robustly and privately, similar to those used in robust +private mean estimation (Liu et al., 2021; Kothari et al., 2021; Hopkins et al., 2022a; Ashtiani & +Liaw, 2022). This is outside the scope of this paper. +Lower bounds under label corruption. Under the αcorrupt label corruption setting (As- +sumption 2), even with infinite data and without privacy constraints, no algorithm is able to learn +w∗ with ℓ2 error better than αcorrupt. We provide a formal derivation for completeness. +Proposition 3.1. Let DΣ,σ2,w∗,K,a be a class of joint distributions on (xi, yi) from (Σ, σ2, w∗, K, a)- +model in Assumption 1. Let Sn,α be an α-corrupted dataset of n i.i.d. samples from some distribution +D ∈ DΣ,σ2,w∗,K,a under Assumption 2. Let M be a class of estimators that are functions over the +datasets Sn,α. Then there exists a positive constant c such that +min +n, ˆw∈M +max +Sn,α,D∈DΣ,σ2,w∗,K,a,w∗,K,a, E[∥ ˆw − w∗∥2 +Σ] ≥ c α2 σ2 +. +A proof is provided in Appendix I.1. A similar lower bound can be found in Bakshi & Prasad +(2021, Theorem 6.1). +4 +Adaptive clipping for the gradient norm +In the ideal clipping thresholds for the norm and the residual, there are unknown terms which we +need to estimate adaptively, (∥wt − w∗∥2 +Σ + σ2) and Tr(Σ), up to constant multiplicative errors. We +privately estimate the (squared and shifted) distance to optimum, (∥wt − w∗∥2 +Σ + σ2), with Alg. 2 +and privately estimate the average input norm, E[∥xi∥2] = Tr(Σ), with Alg. 3 in App. F. These +are used to get the clipping thresholds in Alg. 1. We propose a trimmed mean approach below for +distance estimation. The norm estimator is similar and is provided in App. F. +Private distance estimation using private trimmed mean. The goal is to estimate the +(shifted) distance to optimum, ∥wt − w∗∥2 +Σ + σ2, up to some constant multiplicative error. Note +that this is precisely the task of estimating the variance of the residual bi = yi − w⊤ +t xi. When there +is no adversarial corruption and no privacy constraint, we can simply use the empirical variance +estimator (1/n) � +i∈[n](yi − w⊤ +t xi)2 to obtain a good estimate. However, the empirical variance +estimator is not robust against adversarial corruptions since one outlier can make the estimate +arbitrarily large. A classical idea is using the trimmed estimator from (Tukey & McLaughlin, 1963), +8 + +which throws away the 2α fraction of residuals bi with the largest magnitude. For datasets with +resilience property as assumed in this paper, this will guarantee an accurate estimate of the distance +to optimum in the presence of α fraction of corruptions. +To make the estimator private, it is tempting to simply add a Laplacian noise to the estimate. +However, the sensitivity of the trimmed estimator is unknown and depends on the distance to +the optimum that we aim to estimate; this makes it challenging to determine the variance of the +Laplacian noise we add. Instead, we propose to partition the dataset into k batches, compute +an estimate for each batch, and form a histogram with over those k estimates. Using a private +histogram mechanism with geometrically increasing bin sizes, we propose using the bin with the +most estimates to guarantee a constant factor approximation of the distance to the optimum. We +describe the algorithm as follows. +Algorithm 2: Robust and Private Distance Estimator +Input: S2 = {(xi, yi)}n +i=1, current wt, (ε0, δ0), ¯α, ζ +1 Let bi ← (yi − w⊤ +t xi)2, ∀i ∈ [n] and ˜S ← {bi}n +i=1. +2 Partition ˜S into k = ⌈C1 log(1/(δ0ζ))/ε0⌉ subsets of equal size and let Gj be the j-th +partition. +3 For j ∈ [k], denote ψj as the (1 − 3¯α)-quantile of Gj and φj ← +1 +|Gj| +� +i∈Gj bi1{bi ≤ ψj}. +4 Partition [0, ∞) into geometrically increasing intervals +Ω := +� +. . . , +� +2−1, 1 +� +, [1, 2) , +� +2, 22� +, . . . +� +∪ {[0, 0]} +5 Run (ε0, δ0)-DP histogram of Lemma B.2 on {φj}k +j=1 over Ω +6 if all the bins are empty then Return ⊥ +7 Let [ℓ, r] be a non-empty bin that contains the maximum number of points in the DP +histogram +8 return ℓ +This algorithm gives an estimate of the distance up to a constant multiplicative error as we +show in the following theorem. We provide a proof in App. D. +Theorem 5. Algorithm 2 is (ε0, δ0)-DP. For an αcorrupt-corrupted dataset S2 and an upper bound +¯α on αcorrupt that satisfy Assumption 1 and 37C2K2 · ¯α log2a(1/(6¯α)) ≤ 1/4 and any ζ ∈ (0, 1), if +n = O +�(d + log((log(1/(δ0ζ)))/ε0ζ))(log(1/(δ0ζ))) +¯α2ε0 +� +, +(5) +with a large enough constant then, with probability 1 − ζ, Algorithm 2 returns ℓ such that 1 +4(∥wt − +w∗∥2 +Σ + σ2) ≤ ℓ ≤ 4(∥wt − w∗∥2 +Σ + σ2). +Note that in Theorem 5, we only need to estimate distance up to a constant multiplicative error, +as opposed to an error that depends on our final end-to-end desired level α. Consequently, we +require smaller sample complexity (that doesn’t depend on α) than other parts of our approach. +Remark 4.1. While DP-STAT (Algorithm 3 in Varshney et al. (2022)) can also be used to estimate +∥wt − w∗∥Σ + σ (and it would not change the ultimate sample complexity in its dependence on κ, d, +ε, and n), there are three important improvements we make: (i) DP-STAT requires the knowledge +of ∥w∗∥Σ + σ; (ii) our utility guarantee has improved dependence in K and log2a(n); and (iii) +Algorithm 2 is robust against label corruption. +9 + +Figure 1: Performance of various techniques on DP linear regression. d = 10 in all the experiments. +n = 107, κ = 1 in the 2nd experiment. n = 107, σ = 1 in the 3rd experiment, where κ is the condition +number of Σ and σ2 is the variance of the label noise zi. +Upper bound on clipped good data points. Using the above estimated distance to the +optimum in selecting a threshold θt, we also need to ensure that we do not clip too many clean +data points. The tolerance in our algorithm to reach the desired level of accuracy is clipping O(α) +fraction of clean data points. This is ensured by the following lemma, and we provide a proof in +Appendix E. +Lemma 4.2. Under Assumption 1 and for all t ∈ [T], if θt ≥ +� +9C2K2 log2a(1/(2α))·(∥w∗ − wt∥Σ + σ) +then +��� +i ∈ S3 ∩ Sgood : +��w⊤ +t xi − yi +�� ≥ θt +��� ≤ αn. +5 +Experimental results +5.1 +DP Linear Regression +We present experimental results comparing our proposed technique (DP-RobGD) with other +baselines. We consider non-corrupted regression in this section and defer corrupted regression to +the App. K. We begin by describing the problem setup and the baseline algorithms first. +Experiment Setup. We generate data for all the experiments using the following generative model. +The parameter vector w∗ is uniformly sampled from the surface of a unit sphere. The covariates +{xi}n +i=1 are first sampled from N(0, Σ) and then projected to unit sphere. We consider diagonal +covariances Σ of the following form: Σ[0, 0] = κ, and Σ[i, i] = 1 for all i ≥ 1. Here κ ≥ 1 is the +condition number of Σ. We generate noise zi from uniform distribution over [−σ, σ]. Finally, the +response variables are generated as follows yi = x⊤ +i w∗ + zi. All the experiments presented below +are repeated 5 times and the averaged results are presented. We set the DP parameters (ϵ, δ) as +ϵ = 1, δ = min(10−6, n−2). Experiments for ϵ = 0.1 can be found in Fig. 2 in the App. K. +Baseline Algorithms. We compare our estimator with the following baseline algorithms: +• Non private algorithms: ordinary least squares (OLS), one-pass stochastic gradient descent with +tail-averaging (SGD). For SGD, we use a constant step-size of 1/(2λmax) with n/T minibatch +size, where T = 3κ log n. +• Private algorithms: sufficient statistics perturbation (DP-SSP) (Foulds et al., 2016; Wang, +2018), differentially private stochastic gradient descent (DP-AMBSSGD) (Varshney et al., 2022). +DP-SSP had the best empirical performance among numerous techniques studied by Wang (2018), +and DP-AMBSSGD has the best known theoretical guarantees. The DP-SSP algorithm involves +releasing XT X and XT y differentially privately and computing (� +XT X)−1 � +XT y. DP-AMBSSGD +is a private version of SGD where the DP noise is set adaptively according to the excess error +in each iteration. For both the algorithms, we use the hyper-parameters recommended in their +10 + +d=10.0=1,K=1,E=1 +10- +[] +10- +105 +106 +107 +NumberofSamplesE=1 +Estimation Eror +[0] +10 +Parameter +10-6 +10] +10-4 +10-3 +10-2 +10-1 +100 +aE=1 +Parameter Estimation Error +10' +100 +101 +KNon Private OLS +DP-SSP +Non Private SGD +DP-AMBSSGD +DP-RobGD +DP-RobGD*respective papers. To improve the performance of DP-AMBSSGD, we reduce the clipping +threshold recommended by the theory by a constant factor. +DP-RobGD. We implement Algorithm 1 with the following key changes. Instead of relying on +PrivateNormEstimator to estimate Γ, we set it to its true value Tr(Σ). This is done for a fair +comparison with DP-AMBSSGD which assumes the knowledge of Tr(Σ). Next, we use 20% of the +samples to compute γt in line 5 (instead of the 50% stated in Algorithm 1). In our experiments +we also present results for a variant of our algorithm called DP-RobGD* which outputs the best +iterate based on γt, instead of the last iterate. One could also perform tail-averaging instead of +picking the best iterate. Both these modifications are primarily used to reduce the variance in the +output of Algorithm 1 and achieved similar performance in our experiments. +Results. Figure 1 presents the performance of various algorithms as we vary n, κ, σ. It can be +seen that DP-RobGD outperforms DP-AMBSSGD in almost all the settings (and DP-RobGD* +outperforms DP-RobGD in all cases). DP-SSP has poor performance when the noise σ is low, +but performs slightly better than DP-RobGD in other settings. A major drawback of DP-SSP +is its computational complexity which scales as O(nd2 + dω). In contrast, the computational +complexity of DP-RobGD has smaller dependence on d and scales as ˜O(ndκ). Thus the latter is +more computationally efficient for high-dimensional problems. More experimental results on both +robust and private linear regression can be found in the App. K. +6 +Sketch of the main ideas in the analysis +We provide the main ideas behind the proof of Theorem 4. The privacy proof is straightforward +since no matter what clipping threshold we use the noise we add is always proportionally to the +clipping threshold which guarantees privacy. In the remainder, we focus on the utility analysis. +The proof of the utility heavily relies on the resilience (Steinhardt et al., 2017) (also known as +stability (Diakonikolas & Kane, 2019)), which states that given a large enough sample set S, various +statistics (for example, sample mean and sample variance) of any large enough subset of S will be +close to each other. We define resilience in App. C. +The main effort for proving Theorem 4 lies in the analysis of the gradient descent algorithm. +Without clipping and adding noise for differential privacy, convergence property of gradient descent +for linear regression is well known. The convergence proof of noisy gradient descent is also relatively +straightforward. However, our algorithm requires both clipping and adding noise for robustness and +privacy purposes. The key difference between our setting and the classical setting is the existence of +adversarial bias and random noise in the gradient. We give an overview of the proof of our robust +and private gradient descent as follows. +First, we introduce some notations. Let g(t) +i +:= (x⊤ +i wt − yi)xi be the raw gradient. Note that +when the data follows from our distributional assumption, uncorrupted samples are not clipped: +clipΘ(xi) = xi for i ∈ Sgood. Let G := Sgood ∩ S3 = S3 \ Sbad denote the clean data that remains in +the input dataset. We can write down one step of gradient update as follows: +wt+1 − w∗ += +wt − η +� +1 +n +� +i∈S +˜g(t) +i ++ φtνt +� +− w∗ += +� +I − η +n +� +i∈G +xix⊤ +i +� +(wt − w∗) + η +n +� +i∈G +xizi + η +n +� +i∈G +(g(t) +i +− ˜g(t) +i ) − η +n +� +i∈Sbad +˜g(t) +i +− ηφtνt . +11 + +In the above equation, the first term is a contraction, meaning wt is moving toward w∗. The second +term captures the noise from the randomness in the samples. The third term captures the bias +introduced by the clipping operation, the fourth term (η/n) � +i∈Sbad ˜g(t) +i +captures the bias introduced +by the adversarial datapoints, and the fifth term captures the added Gaussian noise for privacy. +The second term is standard and relatively easy to control, and our main focus is on the last three +terms. +The third term (η/n) � +i∈G(g(t) +i +− ˜g(t) +i ) can be controlled using the resilience property. We +prove that with our estimated threshold, the clipping will only affect a small amount of datapoints, +whose contribution to the gradient is small collectively. The fourth term (η/n) � +i∈Sbad ˜g(t) +i += +(η/n) � +i∈Sbad clipθt(x⊤ +i wt − yi)xi can be controlled since there is only a small amount data points +whose label is corrupted, the clipθt(x⊤ +i wt − yi) is controlled by the clipping threshold and the xi +part satisfies resilience property which implies a small, say Sbad, must have small ∥ � +i∈Sbad xi∥. +Now we have controlled the deterministic bias. Then, we upper bound the fifth term, which +is the noise introduced by the Gaussian noise for the purpose of privacy, and show the expected +prediction error decrease in every gradient step. The difficulty is that, since our clipping threshold +is adaptive, the decrease of the estimation error depends on the estimation error of all the previous +steps. This causes that in some iterations, the estimation error actually increase. In order to get +around this, we split the iterations into length κ chunks, and argue that the maximum estimation +error in a chunk must be a constant factor smaller than the previous chunk. This implies we will +reach the desired error within ˜O(κ) steps. +7 +Discussion +We provide a novel variant of DP-SGD algorithm for differentially private linear regression under +label corruption. We show the first near-optimal rate that achieves privacy and robustness to label +corruptions simultaneously. When there is no label corruption, our result also improves upon the +state-of-the-art method (Varshney et al., 2022) in terms of the condition number κ. Compared +to (Varshney et al., 2022), our algorithm has three innovations: 1) we introduce a novel adaptive +clipping, which is critical in achieving robustness against label corruptions; and 2) we use full +batch gradient descent and a novel convergence analysis to get the near-optimal sample complexity. +Compared to the lower bound and upper bound from a computationally inefficient algorithm in +(Liu et al., 2022b), our sample complexities ˜O(d/α2 + κ1/2d/(εα)) has additional κ1/2 factor in the +privacy term. It remains an open question if there is an efficient algorithm to achieve the optimal +rate without the κ dependence. +Acknowledgement +We thank Abhradeep Guha Thakurta for helpful discussions while working on this paper. 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In International Conference on Artificial Intelligence and Statistics, pp. 4782–4817. +PMLR, 2022. +18 + +Appendix +A +Related work +Differentially private optimization. There is a long line of work at the intersection of differ- +entially privacy and optimization (Chaudhuri et al., 2011; Kifer et al., 2012; Bassily et al., 2014; +Song et al., 2013; Bassily et al., 2019; Wu et al., 2017; Andrew et al., 2021; Feldman et al., 2020; +Song et al., 2020; Asi et al., 2021; Kulkarni et al., 2021; Kamath et al., 2021; Zhang et al., 2022). +As one of the most well-studied problem in differentially privacy, DP Empirical Risk Minimization +(DP-ERM) aims to minimize the empirical risk (1/n) � +i∈S ℓ(xi; w) privately. The optimal excess +empirical risk for approximate DP (i.e., δ > 0) is known to be GD · +√ +d/(εn), where the loss ℓ is +convex and G-Lipschitz with respect to the data, and D is the diameter of the convex parameter +space (Bassily et al., 2014). This bound can be achieved by several DP-SGD methods, e.g., (Song +et al., 2013; Bassily et al., 2014), with different computational complexities. Differentially private +stochastic convex optimization considers minimizing the population risk Ex∼D[ℓ(x, w)], where data +is drawn i.i.d. from some unknown distribution D. Using some variations of DP-SGD, Bassily et al. +(2019) and Feldman et al. (2020) achieves a population risk of GD(1/√n + +√ +d/(εn)). +DP linear regression. Applying above results for the linear model, by observing that G = O(d) +if D = O(1), the sample complexity required for achieving generalization error is n = d2. Existing +works for DP linear regression, for example (Vu & Slavkovic, 2009; Kifer et al., 2012; Mir, 2013; +Dimitrakakis et al., 2014; Wang et al., 2015; Foulds et al., 2016; Minami et al., 2016; Wang, 2018; +Sheffet, 2019; Agarwal et al., 2019) typically consider deterministic data. Under the i.i.d. Gaussian +data setting, this translates into a sample complexity of n = d3/2/(εα), where the extra d1/2 due to +the fact that no statistical assumptions are made. For i.i.d. sub-Weibull data, recent work (Varshney +et al., 2022) achieved nearly optimal excess population risk d/n + d2/(ε2n2) using DP-SGD with +adaptive clipping, up to extra factors on the condition number. This is closest to our work and +we provide detailed comparisons in Sections 2.1 and 3.2. Under Gaussian assumptions, Milionis +et al. (2022) analyze linear regression algorithm with sub-optimal guarantees. (Dwork & Lei, 2009; +Amin et al., 2022; Alabi et al., 2020; Liu et al., 2022b) also consider using robust statistics like +Tukey median (Tukey, 1975) or Theil–Sen estimator (Theil, 1950) for differentially private regression. +However, (Dwork & Lei, 2009) and (Amin et al., 2022) lack utility guarantees and (Alabi et al., +2020) is restricted to one-dimensional data. Liu et al. (2022b) achieves optimal sample complexity +but takes exponential time. +Robust linear regression. Robust mean estimation and linear regression have been studied +for a long time in the statistics community (Tukey & McLaughlin, 1963; Huber, 1992; Tukey, +1975). However, for high dimensional data, these estimators generalizing the notion of median to +higher dimensions are typically computationally intractable. Recent advances in the filter-based +algorithms, e.g., (Diakonikolas et al., 2017, 2020, 2019a, 2018; Cheng et al., 2019; Dong et al., 2019), +achieve nearly optimal guarantees for mean estimation in time linear in the dimension of the dataset. +Motivated by the filter algorithms, Diakonikolas et al. (2019c,b); Prasad et al. (2018); Pensia et al. +(2020); Cherapanamjeri et al. (2020); Jambulapati et al. (2020) achieved nearly optimal rate with d +samples for robust linear regression, where both data xi and label yi are corrupted. Another type of +efficient methods that achieve similar rates and sample complexity in polynomial time is based on +sum-of-square proofs (Klivans et al., 2018; Bakshi & Prasad, 2021), which can be computationally +expensive in practice. Zhu et al. (2019) and Liu et al. (2022b) achieves nearly optimal rates using +d samples but require exponential time complexities. An important special case of adversarial +corruption is when the adversary only corrupts the response variable in supervised learning (Khetan +19 + +et al., 2018) and also in unsupervised learning (Thekumparampil et al., 2018). For linear regression, +when there is only label corruptions, (Bhatia et al., 2015; Dalalyan & Thompson, 2019; Kong et al., +2022) achieve nearly optimal rates with O(d) samples. Under the oblivious label corruption model, +i.e., the adversary only corrupts a fraction of labels in complete ignorance of the data, (Bhatia et al., +2017; Suggala et al., 2019) provide consistent estimator ˆwn such that limn→∞ E [ �wn − w∗]2 = 0 with +O(d) samples. +Robust and private linear regression. Under the settings of both DP and data corrup- +tions, the only algorithm by Liu et al. (2022b) achieves nearly optimal rates α log(1/α)σ with +optimal sample complexities of d/α2 + d/(εα). However, their algorithm requires exponential time +complexities. +Robust and private mean estimation Based on sum-of-square proofs, recent works (Hopkins +et al., 2022b; Alabi et al., 2022) are able to achieve nearly optimal rates α log(1/α) with ˜O(d) +samples for sub-Gaussian data with known covariance. +B +Preliminary on differential privacy +Our algorithm builds upon two DP primitive: Gaussian mechanism and private histogram. The +Gaussian mechanism is one examples of a larger family of mechanisms known as output perturbation +mechanisms. In practice, it is possible to get better utility trade-off for a output perturbation +mechanism by carefully designing the noise, such as the stair-case mechanism which are shown to +achieve optimal utility in the variance (Geng et al., 2015) and also in hypothesis testing (Kairouz +et al., 2014). However, the gain is only by constant factors, which we do not try to optimize in this +paper. We provide a reference for the Gaussian mechanism and private histogram below. +Lemma B.1 (Gaussian mechanism (Dwork & Roth, 2014)). For a query q with sensitivity ∆q, the +Gaussian mechanism outputs q(S) + N(0, (∆q +� +2 log(1.25/δ)/ε)2Id) and achieves (ε, δ)-DP. +Lemma B.2 (Stability-based histogram (Karwa & Vadhan, 2017, Lemma 2.3)). For every K ∈ +N ∪ ∞, domain Ω, for every collection of disjoint bins B1, . . . , BK defined on Ω, n ∈ N, ε ≥ 0, δ ∈ +(0, 1/n), β > 0 and α ∈ (0, 1) there exists an (ε, δ)-differentially private algorithm M : Ωn → RK +such that for any set of data X1, . . . , Xn ∈ Ωn +1. ˆpk = 1 +n +� +Xi∈Bk 1 +2. (˜p1, . . . , ˜pK) ← M(X1, . . . , Xn), and +3. +n ≥ min +� 8 +εβ log(2K/α), 8 +εβ log(4/αδ) +� +then, +P(|˜pk − ˆpk| ≤ β) ≥ 1 − α +When the databse is accessed multiple times, we use the following composition theorems to +account for the end-to-end privacy leakage. +Lemma B.3 (Parallel composition McSherry (2009)). Consider a sequence of interactive queries +{qk}K +k=1 each operating on a subset Sk of the database and each satisfying (ε, δ)-DP. If Sk’s are +disjoint then the composition (q1(S1), q2(S2), . . . , qK(SK)) is (ε, δ)-DP. +20 + +Lemma B.4 (Serial composition Dwork & Roth (2014)). If a database is accessed with an (ε1, δ1)- +DP mechanism and then with an (ε2, δ2)-DP mechanism, then the end-to-end privacy guarantee is +(ε1 + ε2, δ1 + δ2)-DP. +In most modern privacy analysis of iterative processes, advanced composition theorem from +Kairouz et al. (2015) gives tight accountant for the end-to-end privacy budget. It can be improved +for specific mechanisms using tighter accountants, e.g., in Mironov (2017); Girgis et al. (2021); Wang +et al. (2019); Zhu et al. (2022); Gopi et al. (2021). +Lemma B.5 (Advanced composition Kairouz et al. (2015)). For ε ≤ 0.9, an end-to-end guar- +antee of (ε, δ)-differential privacy is satisfied if a database is accessed k times, each with a +(ε/(2 +� +2k log(2/δ)), δ/(2k))-differential private mechanism. +C +Definition of resilience +Definition C.1 ((Liu et al., 2022b, Definition 23)). For some α ∈ (0, 1), ρ1 ∈ R+, ρ2 ∈ R+, and +ρ3 ∈ R+, ρ4 ∈ R+, we say dataset Sgood = {(xi ∈ Rd, yi ∈ R)}n +i=1 is (α, ρ1, ρ2, ρ3, ρ4)-resilient +with respect to (w∗, Σ, σ) for some w∗ ∈ Rd, positive definite Σ ≻ 0 ∈ Rd×d, and σ > 0 if for any +T ⊂ Sgood of size |T| ≥ (1 − α)n, the following holds for all v ∈ Rd: +��� 1 +|T| +� +(xi,yi)∈T +⟨v, xi⟩(yi − x⊤ +i w∗) +��� ≤ ρ1 +√ +v⊤Σv σ , +(6) +��� 1 +|T| +� +xi∈T +⟨v, xi⟩2 − v⊤Σv +��� ≤ ρ2v⊤Σv , +(7) +��� 1 +|T| +� +(xi,yi)∈T +(yi − x⊤ +i w∗)2 − σ2��� ≤ ρ3σ2 , +(8) +��� 1 +|T| +� +(xi,yi)∈T +⟨v, xi⟩ +��� ≤ ρ4 +√ +v⊤Σv . +(9) +D +Proof of Theorem 5 on the private distance estimation +We first analyze the privacy. Changing a data point (xi, yi) can affect at most one partition in +{Gj}k +j=1. This would affect at most two histogram bins, increasing the count of one bin by one +and decreasing the count in another bin by one. Under such a bounded ℓ1 sensitivity, the privacy +guarantees follows from Lemma B.2. +Next, we analyze the utility. In the (private) histogram step, we claim that at most only +two consecutive bins can be occupied by any φj’s. This is also true for the private histogram, +because the private histogram of Lemma B.2 adds noise to non-empty bins only. By Lemma B.2, +if k ≥ c log(1/(δ0ζ0))/ε0, one of these two intervals (the union of which contains the true distance +∥wt − w∗∥2 +Σ + σ2) is released. This results in a multiplicative error bound of four, as the bin size +increments by a factor of two. +To show that only two bins are occupied, we show that all φj’s are close to the true distance. +We first show that each partition contains at most 2αcorrupt fraction of corrupted samples and thus +all partitions are (2¯α, 6¯α, 6ˆρ, 6ˆρ, 6ˆρ, 6ˆρ′)-corrupt good, where ˆρ(C2, K, a, ¯α) = C2K2¯α log2a(1/6¯α) +and ˆρ′(C2, K, a, ¯α) = C2K ¯α loga(1/6¯α), as defined in Definition J.6. +21 + +Let B = ⌊n/k⌋ be the sample size in each partition. Let ζ0 = ζ/2. Since the partition is +drawn uniformly at random, for each partition Gj, the number of corrupted samples α′n satisfies +α′n ∼ Hypergeometric(n, αcorruptn, n/k). The tail bound gives that with probability 1 − ζ0, +α′ ≤ αcorrupt + (k/n) log(2/ζ0) ≤ 2¯α , +where the last inequality follows from the fact that the corruption level is bounded by αcorruption ≤ ¯α +and the assumption on the sample size in Eq. (5) which implies n ≳ log(1/(δ0ζ0)) log(1/ζ0)/(¯αε0). +For a particular subset Gj, Lemma J.7 implies that if B = O((d + log(1/ζ0))/¯α2), then Gj is +(α′, 6¯α, 6ˆρ, 6ˆρ, 6ˆρ, 6ˆρ′)-corrupt good set with respect to (w∗, Σ, σ) from Assumption 1. This means +that there exists a constant C2 > 0 such that for any T1 ⊂ Sgood with |T1| ≥ (1 − 6¯α)B, we have +������ +1 +|T1| +� +i∈T1 +⟨xi, w∗ − wt⟩2 − ∥w∗ − wt∥2 +Σ +������ +≤ 6C2K2¯α log2a(1/(6¯α))∥w∗ − wt∥2 +Σ , +������ +1 +|T1| +� +i∈T1 +z2 +i − σ2 +������ +≤ 6C2K2¯α log2a(1/(6¯α))σ2 , +and +������ +1 +|T1| +� +i∈T1 +zi ⟨xi, w∗ − wt⟩ +������ +≤ 6C2K2¯α log2a(1/(6¯α))∥w∗ − wt∥Σσ . +Note that for i ∈ Sgood, bi = z2 +i + 2zi(w∗ − wt)⊤xi + (w∗ − wt)⊤xix⊤ +i (w∗ − wt). By the triangular +inequality, we know, under above conditions, +������ +1 +|T1| +� +i∈T1 +bi − ∥w∗ − wt∥2 +Σ − σ2 +������ +≤ 12C2K2¯α log2a(1/(6¯α))(∥w∗ − wt∥2 +Σ + σ2) . +(10) +Which also implies that any subset T2 ⊂ Sgood and |T2| ≤ 6¯α|Sgood|, we have +������ +1 +|T2| +� +i∈T2 +bi − ∥w∗ − wt∥2 +Σ − σ2 +������ +≤ 12C2K2 log2a(1/(6¯α))(∥w∗ − wt∥2 +Σ + σ2) . +(11) +Recall that ψj is the (1 − 3¯α)-quantile of the dataset Gj. Let T := {i ∈ Sgood : bi ≤ ψj}, where with +a slight abuse of notations, we use Sgood to denote the set of uncorrupted samples corresponding to +Gj and Sbad to denote the set of corrupted samples corresponding to Gj. Since the corruption is less +than α′, we know (1 − 3¯α − α′)B ≤ |T| ≤ (1 − 3¯α + α′)B. By our assumption that α′ ≤ 2¯α, we have +| ¯E| ≥ (3¯α − α′)B ≥ ¯αB where ¯E := Sgood \ E. Using Eq. (11) with a choice of T2 = ¯E, we get that +min +i∈ ¯E bi − ∥w∗ − wt∥2 +Σ − σ2 ≤ 12C2K2 log2a(1/(6¯α))(∥w∗ − wt∥2 +Σ + σ2) . +(12) +This implies that +ψj ≤ 12C2K2 log2a(1/(6¯α))(∥w∗ − wt∥2 +Σ + σ2). +(13) +22 + +Hence +��φj − ∥w∗ − wt∥2 +Σ − σ2�� = +������ +1 +B +� +i∈Gj +bi · 1{bi ≤ ψj} − ∥w∗ − wt∥2 +Σ − σ2 +������ += +����� +1 +B +� +i∈T +bi − ∥w∗ − wt∥2 +Σ − σ2 +����� + +������ +1 +B +� +i∈Sbad +bi · 1{bi ≤ ψj} +������ +≤ 37C2K2 · ¯α log2a(1/(6¯α))(∥w∗ − wt∥2 +Σ + σ2), +(14) +where we applied Eq. (13) and Eq. (10) in the last inequality. +On a fixed partition Gj, we showed that if B = O((d + log(1/ζ0))/¯α2) then, with probability +1 − ζ0, |φj − ∥w∗ − wt∥2 +Σ − σ2| ≤ 1 +4(∥w∗ − wt∥2 +Σ + σ2), which follows from our assumption that +37C2K2 · ¯α log2a(1/(6¯α)) ≤ 1/4. Using an union bound for all subsets, we know if B = O((d + +log(k/ζ0))/¯α2), then 1 − ζ0, |φj − ∥w∗ − wt∥2 +Σ − σ2| ≤ 1 +4(∥w∗ − wt∥2 +Σ + σ2) holds for all j ∈ [k]. Since +the upper bound lower bound ratio is 5/3 which is less than 2. All the φj must lie in two bins, +which will result in a factor of 4 multiplicative error. +E +Proof of Lemma 4.2 on the upper bound on clipped good points +Let ˆρ(C2, K, a, α) = 2C2K2α log2a(1/(2α)) and ˆρ′(C2, K, a, α) = 2C2Kα loga(1/(2α)). Lemma J.7 +implies that if n = O((d+log(1/ζ))/(α2)) with a large enough constant, then there exists a universal +constant C2 such that S3 is, with respect to (w∗, Σ, σ), (αcorrupt, 2α, ˆρ, ˆρ, ˆρ, ˆρ′)-corrupt good. The +rest of the proof is under this (deterministic) resilience condition. By the resilience property in +Eq. (7), we know for any T ⊂ Sgood with |T| ≥ (1 − 2α)n, +����� +1 +|T| +� +i∈T +(w∗ − wt)⊤xix⊤ +i (w∗ − wt) − ∥w∗ − wt∥2 +Σ +����� ≤ 2C2K2α log2a(1/(2α))∥w∗ − wt∥2 +Σ . +(15) +Let E := +� +i ∈ Sgood : (w∗ − wt)⊤xix⊤ +i (w∗ − wt) > ∥w∗ − wt∥2 +Σ(8C2K2 log2a(1/(2α)) + 1) +� +. De- +note ˜α := |E|/n. We want to show that ˜α ≤ α/2. Let T be the set of points that contain the smallest +1 − α/2 fraction in {(w∗ − wt)⊤xix⊤ +i (w∗ − wt)}i∈Sgood. We know |T| = (1 − α/2)n ≥ (1 − 2α)n. To +prove by contradiction, suppose ˜α > α/2, which means all data points in Sgood \ T are larger than +∥w∗ − wt∥2 +Σ(8C2K2 log2a(1/(2α)) + 1). From resilience property in Eq. (15), we know +1 +n +� +i∈Sgood +(w∗ − wt)⊤xix⊤ +i (w∗ − wt) += 1 +n +� +i∈T +(w∗ − wt)⊤xix⊤ +i (w∗ − wt) + 1 +n +� +i∈Sgood\T +(w∗ − wt)⊤xix⊤ +i (w∗ − wt) +≥ +� +1 − α +2 +� � +1 − 2C2K2α log2a( 1 +2α) +� +∥w∗ − wt∥2 +Σ + α +2 (8C2K2 log2a( 1 +2α) + 1)∥w∗ − wt∥2 +Σ +> (1 + 2C2K2α log2a(1/2α))∥w∗ − wt∥2 +Σ , +which contradicts Eq. (15) for Sgood. This shows ˜α ≤ α/2. +Similarly, we can show that +��� +i ∈ Sgood : z2 +t > σ2(8C2K2 log2a(1/(2α)) + 1) +��� ≤ α/2. +This +means the rest (1 − α)n points in Sgood satisfies +� +(w∗ − wt)⊤xix⊤ +i (w∗ − wt) + |zi| ≤ (∥wt − w∗∥ + +23 + +σ) +� +(8C2K2 log2a(1/(2α)) + 1). Note that for all i ∈ Sgood, we have +|x⊤ +i wt − yi| = +���x⊤ +i (wt − w∗) − zi +��� +≤ |x⊤ +i (wt − w∗)| + |zi| +≤ +�� +(w∗ − wt)⊤xix⊤ +i (w∗ − wt) + |zi| +� +. +By our assumption that C2K2 log2a(1/(2¯α)) ≥ 1 which follows from Assumption 2, we have +���� +� +i ∈ Sgood : ∥x⊤ +i wt − yi∥ ≤ (∥wt − w∗∥ + σ) +� +9C2K2 log2a(1/(2α)) +����� ≥ (1 − α)n . +(16) +F +Private norm estimation: algorithm and analysis +Algorithm 3: Private Norm Estimator +Input: S1 = {(xi, yi)}n +i=1, target privacy (ε0, δ0), failure probability ζ. +1 Let ai ← ∥xi∥2. Let ˜S = {ai}n +i=1. +2 Partition ˜S into k = ⌊C1 log(1/(δ0ζ))/ε⌋ subsets of equal size and let Gj be the j-th +partition. +3 For each j ∈ [k], denote ψj = (1/|Gj|) � +i∈Gj ai. +4 Partition [0, ∞) into bins of geometrically increasing intervals +Ω := +� +. . . , +� +2−2/4, 2−1/4� +, +� +2−1/4, 1 +� +, +� +1, 21/4� +, +� +21/4, 22/4� +, . . . +� +∪ {[0, 0]} +5 Run (ε0, δ0)-DP histogram learner of Lemma B.2 on {ψj}k +j=1 over Ω +6 if all the bins are empty then Return ⊥ +7 Let [ℓ, r] be a non-empty bin that contains the maximum number of points in the DP +histogram +8 Return ℓ +Lemma F.1. Algorithm 3 is (ε0, δ0)-DP. If {xi}n +i=1 are i.i.d. samples from (K, a)-sub-Weibull +distributions with zero mean and covariance Σ and +n = ˜O +�log2a(1/(δ0ζ)) +ε0 +� +, +with a large enough constant then Algorithm 3 returns Γ such that, with probability 1 − ζ, +1 +√ +2 Tr(Σ) ≤ Γ ≤ +√ +2 Tr(Σ) . +We provide a proof in App. F.1. +F.1 +Proof of Lemma F.1 on the private norm estimation +By Hanson-Wright inequality in Lemma J.1 and union bound, there exists constant c > 0 such that +with probability 1 − ζ, +|1 +b +b +� +i=1 +∥xi∥2 − Tr(Σ)| ≤ cK2 Tr(Σ) +�� +log(1/ζ) +b ++ log2a(1/ζ) +b +� +, +(17) +24 + +This means there exists a constant c′ > 0 such that if b ≥ c′K2 log2a(k/ζ), then for all j ∈ [k]. +|ψj − Tr(Σ)| ≤ 21/8 Tr(Σ) +(18) +With probability 1 − ζ, {ψj}k +j=1 lie in interval of size 21/4 Tr(Σ). Thus, at most two consecutive +bins are filled with {ψj}k +j=1. Denote them as I = I1∪I2. Our analysis indicates that P(ψi ∈ I) ≥ 0.99. +By private histogram in Lemma B.2, if k ≥ log(1/(δζ))/ε, |ˆpI − ˜pI| ≤ 0.01 where ˆpI is the empirical +count on I and ˜pI is the noisy count on I. Under this condition, one of these two intervals are +released. This results in multiplicative error of +√ +2. +G +Proof of the resilience in Lemma J.7 +We apply following resilience property for general distribution characterized by Orlicz function from +Zhu et al. (2019). +Lemma G.1 ((Zhu et al., 2019, Theorem 3.4)). Dataset S = {xi ∈ Rd}n +i=1 consists i.i.d. samples +from a distribution D. Suppose D is zero mean and satisfies Ex∼D +� +ψ +� +(v⊤x)2 +κ2Ex∼D[(v⊤x)2] +�� +≤ 1 for +all v ∈ Rd, where ψ(·) is Orlicz function. Let Σ = Ex∼D[xx⊤]. Suppose α ≤ ¯α, where ¯α satisfies +(1 + ¯α/2) · 2κ2¯αψ−1(2/¯α) < 1/3, ¯α ≤ 1/4. Then there exists constant c1, C2 such that if n ≥ +c1((d + log(1/ζ))/(α2)), with probability 1 − ζ, for any T ⊂ S of size |T| ≥ (1 − α)n, the following +holds: +�����Σ−1/2 +� +1 +|T| +� +i∈T +xi +������ ≤ C2κα +� +ψ−1(1/α) +(19) +and +�����Id − Σ−1/2 +� +1 +|T| +� +i∈T +xix⊤ +i +� +Σ−1/2 +����� +2 +≤ C2κ2αψ−1(1/α) . +(20) +Let ψ(t) = et1/(2a). It is easy to see that ψ(t) is a valid Orlicz function. Then if xi is (K, a)-sub- +Weibull, then we know +�����Σ−1/2 +� +1 +|T| +� +i∈T +xi +������ ≤ C2Kα +� +log2a(1/α) , +(21) +and +�����Id − Σ−1/2 +� +1 +|T| +� +i∈T +xix⊤ +i +� +Σ−1/2 +����� +2 +≤ C2K2α log2a(1/α) . +(22) +This implies +(1 − C2K2α log2a(1/α))Id ⪯ Σ−1/2 +� +1 +|T| +� +i∈T +xix⊤ +i +� +Σ−1/2 ⪯ (1 + C2K2α log2a(1/α))Id . +(23) +Using the fact that C⊤AC ⪯ C⊤BC if A ⪯ B, we know +(1 − C2K2α log2a(1/α))Σ ⪯ 1 +|T| +� +i∈T +xix⊤ +i ⪯ (1 + C2K2α log2a(1/α))Σ . +(24) +25 + +This implies resilience properties of xi and zi in Eq. (7) and Eq. (8) in Definition C.1 respectively. +Next, we show the resilience property of xizi. +By ab ≤ a2 +2 + b2 +2 , for any fixed v ∈ Rd, +E[exp +�� | ⟨xizi, v⟩ |2 +K4σ2v⊤Σv +�1/(4a)� +] ≤ E +� +exp +��| ⟨xi, v⟩ |2 +K2v⊤Σv +�1/(2a) +/2 +� +exp +�� +z2 +i +K2σ2 +�1/(2a) +/2 +�� +(25) +≤ 1 +2 +� +E +� +exp +��| ⟨xi, v⟩ |2 +K2v⊤Σv +�1/(2a)�� ++ E +� +exp +�� +z2 +i +K2σ2 +�1/(2a)��� +(26) +≤ 2 . +(27) +Since E[xizi] = 0, (Zhu et al., 2019, Lemma E.3) implies that there exists constant c1, C2 > 0 such +that if n ≥ c1(d + log(1/ζ))/(α2), with probability 1 − ζ, for any T ⊂ Sgood of size |T| ≥ (1 − α)n, +�����Σ−1 +� +1 +|T| +� +i∈T +xizi +������ ≤ C2K2σα log2a(1/α) . +(28) +H +Proof of Theorem 4 on the analysis of Algorithm 1 +The main theorem builds upon the following lemma that analyzes a (stochastic) gradient descent +method, where the randomness is from the DP noise we add and the analysis only relies on certain +deterministic conditions on the dataset including resilienece and concentration. Theorem 4 follows +in a straightforward manner by collecting Theorem 5, Lemma F.1, Lemma 4.2, and Lemma H.1. +Lemma H.1. Algorithm 1 is (ε, δ)-DP. Under Assumptions 1 and 2 for any ζ ∈ (0, 1) and +α ≥ αcorrupt satisfying K2α log2a(1/α) log(κ) ≤ c for some universal constant c > 0, if distance +threshold is small enough such that +θt +≤ +3C1/2 +2 +K loga(1/(2α)) · (∥w∗ − wt∥Σ + σ) , +(29) +and large enough such that the number of clipped clean data points is no larger than αn, at every +round, the norm threshold is large enough such that +Θ +≥ +K +� +Tr(Σ) loga(n/ζ) , +(30) +and sample size is large enough such that +n = O +� +K2d log(d/ζ) log2a(n/ζ) + d + log(1/ζ) +α2 ++ K2T 1/2d log(T/δ) loga(n/(αζ)) +εα +� +, +(31) +with a large enough constant, then the choices of a step size, η = 1/(Cλmax(Σ)) for some C ≥ 1.1, +and the number of iterations, T = ˜Θ (κ log (∥w∗∥)) , ensures that Algorithm 1 outputs wT satisfying +the following with probability 1 − ζ: +Eν1,··· ,νt∼N(0,Id)[∥wT − w∗∥2 +Σ] +≲ +K4σ2 log2(κ)α2 log4a(1/α) , +(32) +where the expectation is taken over the noise added for DP and ˜Θ(·) hides logarithmic terms in +K, σ, d, n, 1/ε, log(1/δ), 1/α. +26 + +Proof of Lemma H.1. We first prove a set of deterministic conditions on the clean dataset, which is +sufficient for the analysis of the gradient descent. +Step 1: Sufficient deterministic conditions on the clean dataset. Let Sgood be the +uncorrupted dataset for S3 and Sbad be the corrupted datapoints in S3. Let G := Sgood ∩ S3 = +S3 \ Sbad denote the clean data that remains in the input dataset. Let λmax = ∥Σ∥2. Define +ˆΣ := (1/n) � +i∈G xix⊤ +i , ˆB := Id − ηˆΣ. Lemma J.4 implies that if n = O(K2d log(d/ζ) log2a(n/ζ)), +then +0.9Σ ⪯ ˆΣ ⪯ 1.1Σ . +(33) +We pick step size η such that η ≤ 1/(1.1λmax) to ensure that η ≤ 1/∥ˆΣ∥2. Since the covariates +{xi}i∈S are not corrupted, from Lemma J.3, we know with probability 1 − ζ, for all i ∈ S3, +∥xi∥2 ≤ K2 Tr(Σ) log2a(n/ζ) . +(34) +Lemma J.7 implies that if n = O((d + log(1/ζ))/(α2)), then there exists a universal constant C2 +such that S3 is, following Definition J.6, with respect to (w∗, Σ, σ), +(αcorrupt, α, C2K2α log2a(1/α), C2K2α log2a(1/α), C2K2α log2a(1/α), C2Kα loga(1/α))-corrupt good. +Such corrupt good sets have a sufficiently large, 1 − αcorrupt, fraction of points that satisfy a good +property that we need: resilience. The rest of the proof is under Eq. (33), Eq. (34), and that Sgood +is resilient. +Step 2: Upper bounding the deterministic noise in the gradient. In this step, we +bound the deviation of the gradient from its mean. There are several sources of deviation: (i) +clipping, (ii) adversarial corruptions, and (iii) randomness of the data noise and privacy noise. +We will show that deviations from all these sources can be controlled deterministically under the +corrupt-goodness (i.e., resilience). +Let φt = ( +� +2 log(1.25/δ0)Θθt)/(ε0n), which ensures that we add enough noise to guarantee +(ε0, δ0)-DP for each step of gradient descent. This follows from the standard Gaussian mechanism +in Lemma B.1 and the fact that each gradient is clipped to the norm of Θθt, resulting in a DP +sensitivity of Θθt/n. The fact that this sensitivity scales as 1/n is one of the main reasons for +the performance gain we get over Varshney et al. (2022) that uses a minimatch of size n/κ with +sensitivity scaling as κ/n. Define g(t) +i +:= xi(x⊤ +i wt − yi). For i ∈ Sgood, we know yi = x⊤ +i w∗ + zi. Let +˜g(t) +i += clipΘ(xi)clipθt(x⊤ +i wt − yi). Note that under Eq. (34), clipΘ(xi) = xi for all i ∈ S3. +From Algorithm 1, we can write one-step update rule as follows: +wt+1 − w∗ +=wt − η +� +1 +n +� +i∈S +˜g(t) +i ++ φtνt +� +− w∗ += +� +I − η +n +� +i∈G +xix⊤ +i +� +(wt − w∗) + η +n +� +i∈G +xizi + η +n +� +i∈G +(g(t) +i +− ˜g(t) +i ) − ηφtνt − η +n +� +i∈Sbad +˜g(t) +i +(35) +Let Et := {i ∈ G : θt ≤ |x⊤ +i wt − yi|} be the set of clipped clean data points such that � +i∈G(g(t) +i +− +˜g(t) +i ) = � +i∈Et(g(t) +i +− ˜g(t) +i ). We define ˆv := (1/n) � +i∈G xizi, u(1) +t +:= (1/n) � +i∈Et xix⊤ +i (wt − w∗), +u(2) +t +:= (1/n) � +i∈Et −xizi, and u(3) +t +:= (1/n) � +i∈Sbad∪Et ˜g(t) +i . +We can further write the update rule as: +wt+1 − w∗ = ˆB(wt − w∗) + ηˆv + ηu(1) +t−1 + ηu(2) +t−1 − ηφtνt − ηu(3) +t−1 . +(36) +27 + +We bound each term one-by-one. Since G ⊂ Sgood and |G| = (1 − αcorrupt)n, using the resilience +property in Eq. (6), we know +∥Σ−1/2ˆv∥ = (1 − αcorrupt) max +∥v∥=1 Σ−1/2 +� +v, +1 +(1 − αcorrupt)n +� +i∈G +xizi +� +≤ (1 − αcorrupt)C2K2α log2a(1/α)σ +(37) +≤ C2K2α log2a(1/α)σ . +(38) +Let ˜α = |Et|/n. By assumption, we know ˜α ≤ α (which holds for the given dataset due to +Lemma 4.2), and +∥Σ−1/2u(1) +t ∥ = ∥Σ−1/2 1 +n +� +i∈Et +xix⊤ +i (wt − w∗)∥ . +From Corollary J.8, we know +�����∥Σ−1/2 1 +|Et| +� +i∈Et +xix⊤ +i (wt − w∗)∥ − ∥wt − w∗∥Σ +����� += +����� max +u:∥u∥=1 +1 +|Et| +� +i∈Et +u⊤Σ−1/2xix⊤ +i (wt − w∗)∥ − max +v:∥v∥=1 v⊤Σ1/2(wt − w∗) +����� +≤ max +u:∥u∥=1 +����� +1 +|Et| +� +i∈Et +u⊤Σ−1/2xix⊤ +i Σ−1/2Σ1/2(wt − w∗)∥ − u⊤Σ1/2(wt − w∗) +����� +≤ max +u:∥u∥=1 +����� +1 +|Et| +� +i∈Et +u⊤ � +Σ−1/2xix⊤ +i Σ−1/2 − Id +� +Σ1/2(wt − w∗)∥ +����� += +����� +1 +|Et| +� +i∈Et +� +Σ−1/2xix⊤ +i Σ−1/2 − Id +� +Σ1/2(wt − w∗) +����� +≤ +����� +1 +|Et| +� +i∈Et +� +Σ−1/2xix⊤ +i Σ−1/2 − Id +������ · +���Σ1/2(wt − w∗) +��� +≤2 − ˜α +˜α +C2K2α log2a(1/α) ∥wt − w∗∥Σ . +This implies that +∥Σ−1/2u(1) +t ∥ ≤ ∥Σ−1/2 1 +n +� +i∈E +xix⊤ +i (wt − w∗)∥ +≤ +� +˜α + 2C2K2α log2a(1/α) +� +∥wt − w∗∥Σ +≤ 3C2K2α log2a(1/α) ∥wt − w∗∥Σ , +(39) +where the last inequality follows from the fact that ˜α ≤ α and our assumption that C2K2 log2a(1/¯α) ≥ +1 from Assumption 2. Similarly, we use resilience property in Eq. (6) instead of Eq. (7), we can +show that +∥Σ−1/2u(2) +t ∥ ≤ 3C2K2α log2a(1/α)σ . +(40) +28 + +Next, we consider u(3) +t . Since |Sbad| ≤ αcorruptn and |Et| ≤ αn, using Eq. (9) and Corollary J.8, +we have +∥Σ−1/2u(3) +t ∥ = max +v:∥v∥=1 +1 +n +� +i∈Sbad∪Et +v⊤Σ−1/2xiclipθt(x⊤ +i wt − yi) +≤ 2C2Kα loga(1/α)θt +≤ 6C1.5 +2 K2α log2a(1/α)(∥wt − w∗∥Σ + σ) . +(41) +Now we use Eq. (38), Eq. (39), Eq. (40) and Eq. (41) to bound the final error from update rule +in Eq. (36). +Step 3: Analysis of the t-steps recurrence relation. We have controlled the deterministic +noise in the last step. In this step, we will upper bound the noise introduced by the Gaussian noise +for the purpose of privacy, and show the expected distance to optimum decrease every step. +We want to emphasize that most of our technical contribution is in the convergence analysis (Step +3 and Step 4). More precisely, naive linear regression analysis can only show a suboptimal error rate of +∥ ˆw−w⋆∥Σ = ˜O(κασ) with sample size n = ˜O(d/α2+κ1/2d/(εα)). Define ut = (ˆv+u(1) +t ++u(2) +t +−u(3) +t ). +This follows from Eq. (36): +wt+1 − w∗ = ˆB(wt − w∗) + ηut − ηφtνt +(42) +=(Id − ηˆΣ)(wt − w∗) + ηut − ηφtνt . +(43) +From Eq. (39), Eq. (40) and Eq. (41), it follows that +∥wt+1 − w∗∥Σ ≤ (1 − 1 +κ)∥wt − w∗∥Σ + α(σ + ∥wt − w∗∥Σ) +where we omitted constants for simplicity, which after T = ˜O(κ) iterations achieves a sub-optimal +error rate ∥wT − w∗∥Σ = ˜O(κασ). +One attempt to get around it is to take the Euclidean norm instead, which gives, after some +calculations, +E[∥wt+1 − w∗∥2] ≤ E[∥wt − w∗∥2] − η +� +∥wt − w∗∥2 +Σ − α2σ2� +. +This implies that E[∥wt+1 − w∗∥2] strictly decreases as long as ∥wt − w∗∥2 +Σ > Cα2σ2, which is +the desired statistical error level we are targeting. With this analysis, we can show that in T = ˜O(κ) +iterations, there exists at least one model wt that achieves E[∥wt − w∗∥2 +Σ] = ˜O(α2σ2) among all the +intermediate models we have seen. +However, the problem is that under differential privacy, there is no way we could select this good +model wt among T models that we have, as privacy-preserving techniques for model selection are +not accurate enough to achieve the desired level of accuracy. Hence, we came up with the following +novel analysis that does not suffer from such issues. +We can rewrite Eq. (36) or Eq. (42) as +wt+1 − w∗ = ˆB(wt − w∗) + ηut − ηφtνt +(44) += ˆBt+1(w0 − w∗) + η +t +� +i=0 +ˆBiut−i − η +t +� +i=0 +φt−i ˆBiνt−i . +(45) +29 + +Taking expectations of ˆΣ-norm square with respect to ν1, · · · , νt, we have +Eν1,...,νt∼N(0,Id)∥wt+1 − w∗∥2 +ˆΣ +(46) +≤ 2∥ ˆBt+1(w0 − w∗)∥2 +ˆΣ + 2E[∥η +t +� +i=0 +ˆBiut−i∥2 +ˆΣ] + η2 +t +� +i=0 +Tr( ˆB2i ˆΣ)E[φ2 +t−i] +(47) +≤ 2∥ ˆBt+1(w0 − w∗)∥2 +ˆΣ + 2η2E[ +t +� +i=0 +t +� +j=0 +∥ ˆBiut−i∥ˆΣ∥ ˆBjut−j∥ˆΣ] +(48) ++ η2 +t +� +i=0 +Tr( ˆB2i ˆΣ)E[φ2 +t−i] , +(49) +where at the second step we used the fact that ν1, ν2, · · · , νt are independent isotropic Gaussian. +Note that +η∥ ˆBiut−i∥ˆΣ += η∥ˆΣ1/2 ˆBi ˆΣ1/2 ˆΣ−1/2ut−i∥ +≤ η∥ˆΣ1/2 ˆBi ˆΣ1/2∥2 · ∥ˆΣ−1/2ut−i∥ +≤ η∥ˆΣ1/2 ˆBi ˆΣ1/2∥2 ˆρ(α) (∥wt−i − w∗∥ˆΣ + σ) +≤ +1 +i + 1 ˆρ(α) (∥wt−i − w∗∥ˆΣ + σ) , +where ˆρ(α) = 1.1(6C2 + 6C1.5 +2 )K2α log2a(1/α), and the second inequality follows from Eq. (39), +Eq. (40), Eq. (41) and the deterministic condition in Eq. (33). Note that the last inequality is true +because η ≤ 1/(1.1λmax) and ∥ˆΣ1/2 ˆBi ˆΣ1/2∥2 ≤ ∥Id − ηˆΣ∥i +2∥ˆΣ∥2 ≤ λmax/(i + 1) . +This implies +E[η2 +t +� +i=0 +t +� +j=0 +∥ ˆBiut−i∥ˆΣ∥ ˆBjut−j∥ˆΣ] +(50) +≤ 4 E[ +t +� +i=0 +t +� +j=0 +ˆρ(α)2 +(i + 1)(j + 1)(E[∥wt−i − w∗∥2 +ˆΣ] + E[∥wt−j − w∗∥2 +ˆΣ] + σ2) +(51) +≤ 8( +t +� +i=0 +1 +i + 1)2ˆρ(α)2(max +i +E[∥wt−i − w∗∥2 +ˆΣ] + σ2) +(52) +≤ 8(log t)2ˆρ(α)2(max +i +E[∥wt−i − w∗∥2 +ˆΣ] + σ2) , +(53) +Then, +∥ ˆBt+1(w0 − w∗)∥2 +ˆΣ = ∥ˆΣ1/2 ˆBt+1 ˆΣ−1/2 ˆΣ1/2(w0 − w∗)∥2 +≤ (1 − 1 +κ)2(t+1)∥w0 − w∗∥2 +ˆΣ ≤ e−2(t+1)/κ∥w0 − w∗∥2 +ˆΣ , +30 + +and for n ≳ (1/ε) +� +κd log(1/δ)/α, +η2 +t +� +i=0 +Tr( ˆB2i ˆΣ)E[φ2 +t−i] +(54) +≤η2 +t +� +i=0 +∥Id − ηˆΣ∥2i +2 ∥ˆΣ∥2 · 2 log(1.25/δ0)K2 Tr(Σ) log2a(n/ζ0)C2K2 log2a(1/(2α))(E[∥wt−i − w∗∥2 +Σ] + σ2) +ε2 +0n2 +(55) +≤4 +t +� +i=0 +( +1 +i + 1)2ˆρ(α)2(E[∥wt−i − w∗∥2 +ˆΣ] + σ2) . +(56) +We have +Eν1,...,νt∼N(0,Id)[∥wt+1−w∗∥2 +ˆΣ] ≤ 2e−2(t+1)/κ∥w0−w∗∥2 +ˆΣ+20(log t)2ˆρ(α)2(max +i∈[t] E[∥wt−i−w∗∥2 +ˆΣ]+σ2) . +Note that this also implies that +E[∥(wt′+t − w∗)∥2 +ˆΣ|wt′] ≤ 2e−2t/κ∥wt′ − w∗∥2 +ˆΣ + 20ˆρ(α)2 +t−1 +� +i=0 +( +1 +i + 1)2(E[∥wt′+t−i − w∗∥2 +ˆΣ|wt′] + σ2) , +(57) +which implies +E[∥(wt′+t − w∗)∥2 +ˆΣ] ≤ 2e−2t/κE[∥wt′ − w∗∥2 +ˆΣ] + 20ˆρ(α)2 +t−1 +� +i=0 +( +1 +i + 1)2(E[∥wt′+t−i − w∗∥2 +ˆΣ] + σ2) +(58) +≤ 2e−2t/κE[∥wt′ − w∗∥2 +ˆΣ] + 20(log t)2ˆρ(α)2(max +i∈[t] E[∥wt′+t−i − w∗∥2 +ˆΣ] + σ2) (59) +Step 4: End-to-end analysis of the convergence. In the last step, we shown that the amount +of estimation error decrease depends on the estimation error of the previous t steps. In order for +the estimation error to decrease by a constant factor, we will take t = κ. Roughly speaking, we will +prove that for every κ steps, the estimation error will decrease by a constant factor, if it is much +larger than O((log κ)2ˆρ(α)2σ2). This implies we will reach O((log κ)2ˆρ(α)2σ2) error with in ˜O(κ) +steps. +For any integer s ≥ 0, as long as maxi∈[(s−1)κ+1,sκ] E[∥wi − w∗∥2 +ˆΣ] ≥ 2(log κ)2ˆρ(α)2σ2, +max +i∈[sκ+1,(s+1)κ] E[∥wi − w∗∥2 +ˆΣ] ≤ ( 1 +e2 + (log κ)2ˆρ(α)2) +max +i∈[(s−1)κ+1,sκ] E[∥wi − w∗∥2 +ˆΣ] + (log 2κ)2ˆρ(α)2σ2 .(60) +Assuming ˆρ(α)2(log κ)2 ≤ 1/2 − 1/e2, the maximum expected error in a length κ sequence +decrease by a factor of 1/2 every time. +Now we bound the maximum expected error in the first length κ sequence: maxi∈[0,κ−1] E[∥wi − +w∗∥2 +ˆΣ]. Since +E[∥wi − w∗∥2 +ˆΣ] ≤ e−2i/κ∥w0 − w∗∥2 +ˆΣ + (log i)2ˆρ(α)2 +max +j∈[0,i−1] E[∥wj − w∗∥2 +ˆΣ] + (log i)2ˆρ(α)2σ2 . +31 + +As a function of i, maxj∈[0,i−1] E[∥wj − w∗∥2 +ˆΣ] only increase when it is smaller than +1 +1 − (log i)2ˆρ(α)2 (∥w0 − w∗∥2 +ˆΣ + (log i)2ˆρ(α)2σ2) . +Thus we conclude +max +i∈[0,κ−1] E[∥wi − w∗∥2 +ˆΣ] ≤ +1 +1 − (log κ)2ˆρ(α2)(∥w0 − w∗∥2 +ˆΣ + (log κ)2ˆρ(α2)σ2) +s = log(∥w∗∥/(ˆρ(α)σ)) will give us +E[∥wsκ+1 − w∗∥2 +ˆΣ] ≤ (log κ)2ˆρ(α)2σ2 . +I +Lower bounds +I.1 +Proof of Proposition 3.1 for label corruption lower bounds +We first prove the following lemma. +Lemma I.1. Consider an α label-corrupted dataset S = {(xi, yi)}n +i=1 with α < 1/2, that is generated +from either xi ∼ N(0, 1), yi ∼ N(0, 1) or xi ∼ N(0, 1), zi ∼ N(0, 1−α2), yi = αxi+zi. It is impossible +to distinguish the two hypotheses with probability larger than 1/2. +In the first case, +(xi, yi) ∼ P1 = N(0, +�1 +0 +0 +1 +� +). +In the second case, +(xi, yi) ∼ P2 = N(0, +�1 +α +α +1 +� +). +By simple calculation, it holds that DKL(P1||P2) = − 1 +2 log(1 − α2) ≤ α2/2 for all α < 1/2. Then, +Pinsker’s inequality implies that DTV (P1||P2) ≤ α/2. Since the covariate xi follows from the same +distribution in the two cases, and the total variation distance between the two cases is less than α/2. +This means there is an label corruption adversary that change α/2 fraction of yi’s in P1 to make it +identical to P2. Therefore, no algorithm can distinguish the two cases with probability better than +1/2 under α fraction of label corruption. +Since Σ = 1, σ2 ∈ [3/4, 1], the first case above has w∗ = 0, and the second case has w∗ = α, this +implies that no algorithm is able to achieve E[∥ ˆw − w∗∥Σ] < σα for all instances with ∥w∗∥ ≤ 1 +under α fraction of label corruption. +J +Technical Lemmas +Lemma J.1 (Hanson-Wright inequality for subWeibull distributions Sambale (2020)). Let S = +{xi ∈ Rd}n +i=1 be a dataset consist of i.i.d. samples from (K, a)-subWeibull distributions, then +P +������ +1 +n +n +� +i=1 +∥xi∥2 − Tr(Σ) +����� ≥ t +� +≤ 2 exp +� +− min +� +nt2 +K4(Tr(Σ))2 , +� +nt +K2 Tr(Σ) +� 1 +2a +�� +. +(61) +32 + +Lemma J.2. Let Y ∼ Lap(b). Then for all h > 0, we have P(|Y | ≥ hb) = e−h. +Lemma J.3. If x ∈ Rd is (K, a)-subWeibull for some a ∈ [1/2, ∞). Then +• for any fixed v ∈ Rd, with probability 1 − ζ, +⟨x, v⟩2 ≤ K2v⊤Σv log2a(1/ζ) . +(62) +• with probability 1 − ζ, +∥x∥2 ≤ K2 Tr(Σ) log2a(1/ζ) . +(63) +We provide a proof in Appendix J.1.1. +Lemma J.4. Dataset S = {xi ∈ Rd}n +i=1 consists i.i.d. samples from a zero mean distribution D. +Suppose D is (K, a)-subWeibull. Define Σ = Ex∼D[xx⊤]. Then there exists a constant c1 > 0 such +that with probability 1 − ζ, +����� +1 +n +n +� +i=1 +xix⊤ +i − Σ +����� ≤ c1 +� +�K2d log(d/ζ) log2a(n/ζ) +n ++ +� +K2d log(d/δ) log2a(n/ζ) +n +� +� ∥Σ∥2 . +(64) +Lemma J.5 (Lemma F.1 from Liu et al. (2022a)). Let x ∈ Rd ∼ N(0, Σ). Then there exists +universal constant C6 such that with probability 1 − ζ, +∥x∥2 ≤ C Tr(Σ) log(1/ζ) . +(65) +Definition J.6 (Corrupt good set). We say a dataset S is (αcorrupt, α, ρ1, ρ2, ρ3, ρ4)-corrupt good +with respect to (w∗, Σ, σ) if it is αcorrupt-corruption of an (α, ρ1, ρ2, ρ3, ρ4)-resilient dataset Sgood. +Lemma J.7. Under Assumptions 1 and 2, there exists positive constants c1 and C2 such that +if n ≥ c1((d + log(1/ζ))/α2, then with probability 1 − ζ, Sgood is, with respect to (w∗, Σ, σ), +(α, C2K2α log2a(1/α), C2K2α log2a(1/α), C2K2α log2a(1/α), C2Kα loga(1/α))-resilient. +We provide a proof in Appendix G. +Corollary J.8 (Lemma 10 from Steinhardt et al. (2017) and Lemma 25 from Liu et al. (2022b)). +For a (α, ρ1, ρ2, ρ3, ρ4)-resilient set S with respect to (w∗, Σ, γ) and any 0 ≤ ˜α ≤ α, the following +holds for any subset T ⊂ S of size at least ˜αn and for any unit vector v ∈ Rd: +��� 1 +|T| +� +(xi,yi)∈T +⟨v, xi⟩(yi − x⊤ +i w∗) +��� +≤ +2 − ˜α +˜α +ρ1 +√ +v⊤Σv σ , +(66) +������ +1 +|T| +� +xi∈T +⟨v, xi⟩2 − v⊤Σv +������ +≤ +2 − ˜α +˜α +ρ2v⊤Σv , +(67) +��� 1 +|T| +� +(xi,yi)∈T +(yi − x⊤ +i w∗)2 − σ2��� +≤ +2 − ˜α +˜α +ρ3 σ2 , +and +(68) +������ +1 +|T| +� +xi∈T +⟨v, xi⟩ +������ +≤ +2 − ˜α +˜α +ρ4 +√ +v⊤Σv . +(69) +33 + +J.1 +Proof of technical lemmas +J.1.1 +Proof of Lemma J.3 +Using Markov inequality, +P +� +⟨v, x⟩2 ≥ t2� += P +� +e⟨v,x⟩1/a ≥ et1/a� +(70) +≤ e−t1/aE[e⟨v,x⟩1/a] +(71) +≤ e−t1/aeK(E[⟨v,x⟩2])1/(2a) +(72) += 2 exp +� +− +� +t2 +K2E[⟨v, x⟩2] +�1/(2a)� +. +(73) +This implies for any fixed v, with probability 1 − ζ, +⟨x, v⟩2 ≤ K2v⊤E[xx⊤]v log2a(1/ζ) . +(74) +For j-th coordinate, let v = ej where j ∈ [d]. Definition 2.1 implies +E +� +�exp +� +� +� +x2 +j +K2 Tr(Σ) +�1/(2a)� +� +� +� ≤ E +� +�exp +� +� +� +x2 +j +K2Σjj +�1/(2a)� +� +� +� ≤ 2 . +(75) +Note that f(x) = xα is concave function for α ≤ 1 and x > 0. Then (a1 + · · · ak)α ≤ aα +1 + · · · aα +k +holds for any positive numbers a1, · · · , ak > 0. By our assumption that 1/(2a) ≤ 1. , we have +E[exp +�� +∥x∥2 +K2 Tr(Σ) +�1/(2a)� +] = E[exp +��x2 +1 + x2 +2 + · · · + x2 +d +K2 Tr(Σ) +�1/(2a)� +] +(76) +≤ E[ +d +� +j=1 +exp +� +� +� +x2 +j +K2 Tr(Σ) +�1/(2a)� +�] +(77) +≤ +� +� +� +� +� +� +�d +j=1 E[exp +�� +x2 +j +K2 Tr(Σ) +�1/(2a)� +] +d +� +� +� +� +� +� +d +(78) +≤ 2 . +(79) +By Markov inequality, +P (∥x∥ ≥ t) = P +� +e∥x∥1/a ≥ et1/a� +(80) +≤ e−t1/aE[e∥x∥1/a] +(81) +≤ exp +� +− +� +t2 +K2 Tr(Σ) +�1/(2a)� +. +(82) +This implies with probability 1 − ζ, +∥x∥2 ≤ K2 Tr(Σ) log2a(1/ζ) . +(83) +34 + +Figure 2: Performance of various techniques on DP linear regression. d = 10 in all the experiments. +n = 107, κ = 1 in the 2nd experiment. n = 107, σ = 1 in the 3rd experiment. +Figure 3: Non-robustness of existing techniques to adversarial corruptions. n = 107, σ = 1 in both +experiments. +K +Experiments +K.1 +DP Linear Regression +Experimental results for ϵ = 0.1 can be found in Figure 2. The observations are similar to the ϵ = 1 +case. In particular, DP-SSP has poor performance when σ is small. In other settings, DP-SSP +has better performance than DP-RobGD. +K.2 +DP Robust Linear Regression +We now illustrate the robustness of our algorithm. We consider the same experimental setup as +in Section 5 and randomly corrupt α fraction of the response variables by setting them to 1000. +Figure 3 presents the results from this experiment. It can be seen that none of the baselines are +robust to adversarial corruptions. They can be made arbitrarily bad by increasing the magnitude of +corruptions. In contrast, DP-RobGD is able to handle the corruptions well. +K.3 +Stronger adversary for DP Robust Linear Regression +In this section, we consider a stronger adversary for DP-RobGD than the one considered in +Section 5. Recall, for the adversary model considered in Section 5, DP-RobGD was able to +consistently estimate the parameter w∗ (i.e., the parameter recovery error goes down to 0 as +n → ∞). This is because the algorithm was able to easily identify the corruptions and ignore the +corresponding points while performing gradient descent. We now construct a different instance +where the corruptions are hard to identify. Consequently, DP-RobGD can no longer be consistent +against the adversary. This hard instance is inspired by the lower bound in Bakshi & Prasad (2021) +(see Theorem 6.1 of Bakshi & Prasad (2021)). This is a 2 dimensional problem where the first +35 + +Non Private OLS +DP-SSP +Non Private SGD +DP-AMBSSGD +DP-RobGD +DP-RobGD*d=10.g=1,K=1.E=0.1 +Parameter Estimation Error +100 +10 +0 +105 +106 +107 +NumberofSamplesE=0.1 +Estimation Error +10- +Parameter +10 +10-6 +10 +10-4 +10-3 +10-2 +10-1 +100 +aE=0.1 +Error +Estimation +Parameter +10- +100 +101 +KE=1 +ErTor +100 +Estimation I +Parameter +10 +10-3 +10-2 +10-1 +fraction of corruptionsα=0.1.E=1 +Parameter Estimation Error +100 +10-1 +0 +10 +101 +2 × 107 +3 × 102 +4 × 102Figure 4: Performance against the stronger adversary +covariate is sampled uniformly from [−1, 1]. The second covariate, which is uncorrelated from the +first, is sampled from a distribution with the following pdf +p(x(2)) = +� +� +� +� +� +α +2 +if x(2) ∈ {−1, 1} +1−α +2ασ +if x(2) ∈ [−σ, σ] +0 +otherwise +. +We set σ = 0.1 in our experiments. The noise zi is sampled uniformly from [−σ, σ]. We consider two +possible parameter vectors w∗ = (1, 1) and w∗ = (1, −1). It can be shown that the total variation +(TV) distance between these problem instances (each parameter vector corresponds to one problem +instance) is Θ(α) (Bakshi & Prasad, 2021). What this implies is that, one can corrupt at most α +fraction of the response variables and convert one problem instance into another. Since the distance +(in Σ norm) between the two parameter vectors is Ω(ασ), any algorithm will suffer an error of +Ω(ασ). +We generate 107 samples from this problem instance and add corruptions that convert one +problem instance to the other. Figure 4 presents the results from this experiment. It can be seen +that our algorithm works as expected. In particular, it is not consistent in this setting. Moreover, +the parameter recovery error increases with the fraction of corruptions. +L +Heavy-tailed noise +We study the heavy-tailed regression settings where the label noise zi is hypercontractive, which is +common in robust linear regression literature (Klivans et al., 2018; Liu et al., 2022b). We define +(κ2, k)-hypercontractivity as follows. This is a heavy-tailed distribution we have bound only up to +the k-th moment. +Definition L.1. For integer k ≥ 4, a distribution Pµ,Σ is (κ2, k)-hypercontractive if for all v ∈ Rd, +Ex∼PX[|⟨v, (x − µ)⟩|k] ≤ κk +2(v⊤Σv)k/2, where Σ is the covariance. +We give a formal description of our setting in Assumption 3. Note that we consider the input +vector xi to be sub-Weibull and label noise zi to be hypercontractive. +If both xi and zi are +hypercontractive, the uncorrupted set Sgood is known to be not resilient (Zhu et al., 2019; Liu et al., +2022b). However, by (Zhu et al., 2019, Lemma G.10), we can clip xi by O( +√ +d∥Σ∥2), and obtain a +(α, O(κα1−1/k), O(κα1−2/k), O(κα1−2/k), O(κα1−1/k))-resilient set (Liu et al., 2022b, Lemma 4.19). +This would result in sub-optimal error rate ˜O(κα1−2/k), which depends on condition number κ. For +convenience, in this section, we further assume that xi and zi are independent. In the dependent case, +36 + +E=1 +Error +100 +Parameter Estimation +10' +10-2 +10-4 +10-5 +106 +107 +Number of SamplesE=1 +Parameter Estimation Error +3×10 +2 × 10-1 +10-2 +10-1 +fraction of corruptionsthe only thing we need to change is the ρ1 resilience from O(α1−1/k) to O(α1−2/k) in Lemma L.2. +This would result in O(α1−3/k) error rate if we plug this new resilience in Theorem 6. +Assumption 3 ((Σ, σ2, w∗, K, a, κ2, k)-model). A multiset Sgood = {(xi ∈ Rd, yi ∈ R)}n +i=1 of n +i.i.d. samples is from a linear model yi = ⟨xi, w∗⟩ + zi, where the input vector xi is zero mean, +E[xi] = 0, with a positive definite covariance Σ := E[xix⊤ +i ] ≻ 0, and the independent label noise zi is +zero mean, E[zi] = 0, with variance σ2 := E[z2 +i ]. We assume that the marginal distribution of xi is +(K, a)-sub-Weibull and that of zi is (κ2, k)-hypercontractive, as defined above. +This is similar to the light-tailed case in Assumption 2.1. The main difference is that the noise +zi is heavy-tailed and independent of the input xi. +Assumption 4 (αcorrupt-corruption). Given a dataset Sgood = {(xi, yi)}n +i=1, an adversary inspects +all the data points, selects αcorruptn data points denoted as Sr, and replaces the labels with arbitrary +labels while keeping the covariates unchanged. We let Sbad denote this set of αcorruptn newly labelled +examples by the adversary. Let the resulting set be S := Sgood ∪ Sbad \ Sr. We further assume that +the corruption rate is bounded by αcorrupt ≤ ¯α, where ¯α is a positive constant that depends on κ2, k, +K, log(κ), a and ζ. +Compared to Assumption 2, this only difference is in the conditions on ¯α. Similar as Lemma J.7, +we have the following lemma showing that under Assumption 3, the uncorrupted dataset can Sgood +is corrupt-good, which means that it can be seen as being corrupted from a resilient set. We provide +the proof in App. L.2. +Lemma L.2. A multiset of i.i.d. labeled samples Sgood = {(xi, yi)}n +i=1 is generated from a linear +model: yi = ⟨xi, w∗⟩ + zi, where feature vector xi has zero mean and covariance E[xix⊤ +i ] = Σ ≻ 0, +independent label noise zi has zero mean and covariance E[z2 +i ] = σ2 > 0. Suppose xi is (K, a)- +sub-Weibull, zi is (κ2, k)-hypercontractive, then there exist constants c1, C2 > 0 such that, for any +0 < α ≤ ˜α ≤ c where c ∈ (0, 1/2) is some absolute constant if +n ≥ c1 +� +d +ζ2(1−1/k)α2(1−1/k) + k2α2−2/kd log d +ζ2−4/kκ2 +2 ++ κ2 +2d log d +α2/k ++ d + log(1/ζ) +˜α2 +� +, +(84) +then with probability 1 − ζ, Sgood is +(0.2α, α, C2k(ka)aKκ2α1−1/kζ−1/k, C2K2˜α log2a(1/˜α), C2k2κ2 +2α1−2/kζ−2/k, C2K ˜α loga(1/˜α))-corrupt +good with respect to (w∗, Σ, σ). +In the rest of this section, we assume we have a (O(α), α, ρ1, ρ2, ρ3, ρ4)-corrupt good set under +Assumption 3 and present following algorithm and our main theorem under this setting in Theorem 6. +37 + +We also provide the proof in App. L.1. +Algorithm 4: Robust and Private Linear Regression for heavy-tailed noise +Input: dataset S = {(xi, yi)}3n +i=1, (ε, δ), T, learning rate η, failure probability ζ, target error +rate α, distribution parameter (K, a) +1 Partition dataset S into three equal sized disjoint subsets S = S1 ∪ S2 ∪ S3. +2 δ0 ← δ/(2T), ε0 ← ε/(4 +� +T log(1/δ0)), ζ0 ← ζ/3, w0 ← 0 +3 Γ ← PrivateNormEstimator(S1, ε0, δ0, ζ0), Θ ← K +√ +2Γ loga(n/ζ0) +4 for t = 1, 2, . . . , T − 1 do +5 +γt ← RobustPrivateDistanceEstimator(S2, wt, ε0, δ0, α, ζ0) +6 +θt ← 2√2γt · +� +max{8ρ2/α, 8ρ3/α} + 1. +7 +Sample νt ∼ N (0, Id) +8 +wt+1 ← wt − η +� +1 +n +� +i∈S3 +� +clipΘ(xi)clipθt +� +w⊤ +t xi − yi +�� ++ +√ +2 log(1.25/δ0)Θθt +ε0n +· νt +� +9 Return wT +Theorem 6. Algorithm 4 is (ε, δ)-DP. Under (Σ, σ2, w∗, K, a, κ2, k)-model of Assumption 3 and +αcorrupt-corruption of Assumption 4 and for any failure probability ζ ∈ (0, 1) and target error rate +α ≥ 1.2αcorrupt, if the dataset S is (0.2α, α, ρ1, ρ2, ρ3, ρ4)-corrupt good set S with respect to (w∗, Σ, σ) +and sample size is large enough such that +n =O +� +K2d log(d/ζ) log2a(n/ζ) + K2dT 1/2 log(T/δ) loga(n/(αζ)) +� +8 max{ρ2/α, ρ3/α} + 1 +εˆρ(α) +� +, +(85) +where ˆρ(α) = max{ρ1, 3ρ2, 2ρ4 +� +8 max{ρ2/α, ρ3/α} + 1}, then the choices of a small enough step +size, η ≤ 1/(1.1λmax(Σ)), and the number of iterations, T = ˜Θ (κ log (∥w∗∥)) for a condition number +of the covariance κ := λmax(Σ)/λmin(Σ), ensures that, with probability 1 − ζ, Algorithm 1 achieves +Eν1,··· ,νt∼N(0,Id) +� +∥wT − w∗∥2 +Σ +� += +˜O +� +ˆρ2(α)σ2 � +, +(86) +where the expectation is taken over the noise added for DP, and ˜Θ(·) hides logarithmic terms in +K, κ2, σ, d, n, 1/ε, log(1/δ), 1/α, and κ. +By Lemma L.2, if we set ˜α = α1−1/k, ρ1 = C2k(ka)aKκ2α1−1/kζ−1/k, ρ2 = C2K2α1−1/k log2a(1/α1−1/k),ρ3 = +C2k2κ2 +2α1−2/kζ−2/k, and ρ4 = C2Kα1−1/k loga(1/α1−1/k), we have following corollary. +Corollary L.3. Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model +of Assumption 4, if 1.2αcorrupt ≤ α and K, a, κ2, k = O(1), then n = ˜O(d/(ζ2−2/kα2−2/k) + +κ1/2d/(εα1−1/k)) samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw−w∗∥2 +Σ = +˜O(ζ−2/kα2−4/k) with probability 1 − ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic terms +in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ. +Simiarly, if we set ˜α = α, ρ1 = C2k(ka)aKκ2α1−1/kζ−1/k, ρ2 = C2K2α log2a(1/α),ρ3 = +C2k2κ2 +2α1−2/kζ−2/k, and ρ4 = C2Kα loga(1/α), we have following corollary. +Corollary L.4. Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of +Assumption 4, if 1.2αcorrupt ≤ α and K, a, κ2, k = O(1), then n = ˜O(d/(ζ2−2/kα2−2/k)+κ1/2d/(εα)+ +(d + log(1/ζ)/α2)) samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw − +w∗∥2 +Σ = ˜O(ζ−2/kα2−2/k) with probability 1 − ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic +terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ. +38 + +As a comparison, we also apply the exponential-time robust linear regression algorithm HPTR +by Liu et al. (2022b) under our setting. +Theorem 7 ((Liu et al., 2022b, Theorem 12)). There exist positive constants c and C such that for +any ((2/11)α, α, ρ1, ρ2, ρ3, ρ4)-corrupt good set S with respect to (w∗, Σ ≻ 0, σ > 0) satisfying α < c, +ρ1 < c, ρ2 < c, ρ3 < c,and ρ2 +4 ≤ cα, HPTR achieves (1/σ)∥(ˆβ − β)∥Σ ≤ 32ρ1 with probability 1 − ζ, +if +n ≥ C d + log(1/(δζ)) +εα +. +(87) +We set ˜α = α1−1/k, ρ1 = C2k(ka)aKκ2α1−1/kζ−1/k, ρ2 = C2K2α1−1/k log2a(1/α1−1/k),ρ3 = +C2k2κ2 +2α1−2/kζ−2/k, and ρ4 = C2Kα1−1/k loga(1/α1−1/k), we have the following utility gaurentees. +Corollary L.5. Under the hypothesis of Assumption 3, there exists a constant c > 0 such that for +any α ≤ c, (ka)aKκ2α1−1/kζ−1/k ≤ c, k2κ2 +2α1−2/kζ−2/k ≤ c and K2α1−2/k log2a(1/α1−1/k) ≤ c, it +is sufficient to have a dataset of size +n = O +� +d +ζ2(1−1/k)α2(1−1/k) + k2α2−2/kd log d +ζ2−4/kκ2 +2 ++ κ2 +2d log d +α2/k +� +, +(88) +such that HPTR achieves (1/σ)∥ ˆw − w∗∥Σ = O(k(ka)aKκ2α1−1/kζ−1/k) with probability 1 − ζ. +Note that both of our result in Corollary L.3 and Corollary L.4 are suboptimal compared to +the exponential time algorithm HPTR from Corollary L.5. Suppose K, a, κ2, k, ζ = Θ(1), HPTR +achieves (1/σ)∥w∗− ˆw∥ = ˜O(α1−1/k) with sample complexities n = d/(α2(1−1/k))+(d+log(1/δ))/(εn). +However, in the analysis in Corollary L.3, Algorithm 4 achieves (1/σ)∥w∗ − ˆw∥ = ˜O(α1−2/k) with +the same sample complexities. In the analysis in Corollary L.4, Algorithm 4 achieves the same +error rate as HPTR but requires extra ˜O(d/α2) sample complexities. The suboptimality is caused +by the gradient truncation step in our algorithm. From Theorem 7, the final error rate of HPTR +only depends on the first resilience ρ1. However in Theorem 6, the final error rate of Algorithm 4 +depends on ˆρ(α) = max{ρ1, ρ2, ρ4 +� +ρ2/α}. When the noise is heavy-tailed, the bottleneck is the last +term ρ4 +� +ρ2/α ≈ α1−2/k, which is due to the truncation threshold from Eq. (98). This cannot be +tightened by using a smaller truncation threshold. Because we can construct yi, such that there are +α-fraction of points that are at the threshold level θt ≈ α−1/k(line 6 of Algorithm 4). If exponential +time complexity is allowed, we could robustly and privately estimate the average of the gradients +by directly estimating the xiyi. However, the current best efficient algorithm (Liu et al., 2021) for +estimating the mean of Gaussian with unknown covariance robustly and privately would require +O(d1.5) samples. +For a fair comparison, we also rewrite the error rates of Corollary L.3, Corollary L.4, Corollary L.5 +as the same accuracy level α and different corruption level αcorrupt respectively. +Corollary L.6. Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of +Assumption 4, if 1.2αcorrupt ≤ αk/(k−2) and K, a, κ2, k = O(1), then +n = ˜O(d/(ζ2−2/kα2(k−1)/(k−2)) + κ1/2d/(εα(k−1)/(k−2))) +samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw−w∗∥2 +Σ = ˜O(ζ−2/kα2) with +probability 1−ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) +and κ. +39 + +Corollary L.7. Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of +Assumption 4, if 1.2αcorrupt ≤ αk/(k−1) and K, a, κ2, k = O(1), then +n = ˜O(d/(ζ2−2/kα2) + κ1/2d/(εαk/(k−1)) + (d + log(1/ζ)/α2k/(k−1))) +samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw−w∗∥2 +Σ = ˜O(ζ−2/kα2) with +probability 1−ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) +and κ. +Corollary L.8 (HPTR). Under the same hypotheses of Theorem 6 and under αcorrupt-corruption +model of Assumption 4, if αcorrupt ≤ αk/(k−1) and α(k−2)/(k−1) ≤ c and K, a, κ2, k = O(1), then +n = ˜O( +d +ζ2−2/kα2 + d + log(1/(δζ)) +εαk/k−1 +) +samples are sufficient for HPTR to achieve an error rate of (1/σ2)∥ ˆw − w∗∥2 +Σ = ˜O(ζ−2/kα2) with +probability 1 − ζ, ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ. +L.1 +Proof of Theorem 6 +Proof. The proof follows similarly as the proof of Theorem 4. We only highlight the difference in +the proof. +Let Sgood be the uncorrupted dataset for S3 and Sbad be the corrupted data points in S3. Let G +denote the clean data that satisfies resilience conditions. We know |G| ≥ (1−1.2αcorrupt)n ≥ (1−α)n. +Let λmax = ∥Σ∥2. Define ˆΣ := (1/n) � +i∈G xix⊤ +i , ˆB := Id − ηˆΣ. Lemma J.4 implies that if +n = O(K2d log(d/ζ) log2a(n/ζ)), then +0.9Σ ⪯ ˆΣ ⪯ 1.1Σ . +(89) +We pick step size η such that η ≤ 1/(1.1λmax) to ensure that η ≤ 1/∥ˆΣ∥2. Since the covariates +{xi}i∈S are not corrupted, from Lemma J.3, we know with probability 1 − ζ, for all i ∈ S3, +∥xi∥2 ≤ K2 Tr(Σ) log2a(n/ζ) . +(90) +The rest of the proof is under Eq. (89), Eq. (90) and the resilience conditions. +Let φt = ( +� +2 log(1.25/δ0)Θθt)/(ε0n). Define g(t) +i +:= xi(x⊤ +i wt − yi). For i ∈ Sgood, we know +yi = x⊤ +i w∗ + zi. Let ˜g(t) +i += clipΘ(xi)clipθt(x⊤ +i wt − yi). Note that under Eq. (90), clipΘ(xi) = xi for +all i ∈ S3. +From Algorithm 4, we can write one-step update rule as follows: +wt+1 − w∗ +=wt − η +� +1 +n +� +i∈S +˜g(t) +i ++ φtνt +� +− w∗ += +� +I − η +n +� +i∈G +xix⊤ +i +� +(wt − w∗) + η +n +� +i∈G +xizi + η +n +� +i∈G +(g(t) +i +− ˜g(t) +i ) − ηφtνt − η +n +� +i∈S3\G∪Et +˜g(t) +i +(91) +Let Et := {i ∈ G : θt ≤ |x⊤ +i wt − yi|} be the set of clipped clean data points such that � +i∈G(g(t) +i +− +˜g(t) +i ) = � +i∈Et(g(t) +i +− ˜g(t) +i ). We define ˆv := (1/n) � +i∈G xizi, u(1) +t +:= (1/n) � +i∈Et xix⊤ +i (wt − w∗), +u(2) +t +:= (1/n) � +i∈Et −xizi, and u(3) +t +:= (1/n) � +i∈S3\G∪Et ˜g(t) +i . +40 + +We can further write the update rule as: +wt+1 − w∗ = ˆB(wt − w∗) + ηˆv + ηu(1) +t−1 + ηu(2) +t−1 − ηφtνt − ηu(3) +t−1 . +(92) +Since G ⊂ Sgood and |G| ≥ (1 − α)n, using the resilience property in Eq. (6), we know +∥Σ−1/2ˆv∥ = |G| max +∥v∥=1 Σ−1/2 +� +v, 1 +|G| +� +i∈G +xizi +� +≤ (1 − α)ρ1σ +(93) +≤ ρ1σ . +(94) +Let α2 = |Et|/n. Following the proof of Lemma 4.2, we can show following lemma. +Lemma L.9. Under Assumptions 3, if θt ≥ +� +max{8ρ2/α, 8ρ3/α} + 1 · (∥w∗ − wt∥Σ + σ), then +��� +� +i ∈ G : +���w⊤ +t xi − yi +��� ≥ θt +���� ≤ αn +, for all t ∈ [T]. +Similar as Theorem 5, we have following theorem. +Theorem 8. Algorithm 2 is (ε0, δ0)-DP. For an (αcorrupt, ¯α, ρ1, ρ2, ρ3, ρ4)-corrupted good dataset +S2 and an upper bound ¯α on αcorrupt that satisfy Assumption 3 and ρ1 + ρ2 + ρ3 ≤ 1/4, for any +ζ ∈ (0, 1), if +n = O +�log(1/ζ) log(1/(δ0ζ)) +¯αε0 +� +, +(95) +with a large enough constant then, with probability 1 − ζ, Algorithm 2 returns ℓ such that 1 +4(∥wt − +w∗∥2 +Σ + σ2) ≤ ℓ ≤ 4(∥wt − w∗∥2 +Σ + σ2). +This means α2 ≤ α, and we have +∥Σ−1/2u(1) +t ∥ = ∥Σ−1/2 1 +n +� +i∈Et +xix⊤ +i (wt − w∗)∥ . +41 + +From Corollary J.8, we know +�����∥Σ−1/2 1 +|Et| +� +i∈Et +xix⊤ +i (wt − w∗)∥ − ∥wt − w∗∥Σ +����� += +����� max +u:∥u∥=1 +1 +|Et| +� +i∈Et +u⊤Σ−1/2xix⊤ +i (wt − w∗)∥ − max +v:∥v∥=1 v⊤Σ1/2(wt − w∗) +����� +≤ max +u:∥u∥=1 +����� +1 +|Et| +� +i∈Et +u⊤Σ−1/2xix⊤ +i Σ−1/2Σ1/2(wt − w∗)∥ − u⊤Σ1/2(wt − w∗) +����� +≤ max +u:∥u∥=1 +����� +1 +|Et| +� +i∈Et +u⊤ � +Σ−1/2xix⊤ +i Σ−1/2 − Id +� +Σ1/2(wt − w∗)∥ +����� += +����� +1 +|Et| +� +i∈Et +� +Σ−1/2xix⊤ +i Σ−1/2 − Id +� +Σ1/2(wt − w∗) +����� +≤ +����� +1 +|Et| +� +i∈Et +� +Σ−1/2xix⊤ +i Σ−1/2 − Id +������ · +���Σ1/2(wt − w∗) +��� +≤2 − α2 +α2 +ρ2 ∥wt − w∗∥Σ . +This implies that +∥Σ−1/2u(1) +t ∥ ≤ ∥Σ−1/2 1 +n +� +i∈E +xix⊤ +i (wt − w∗)∥ +≤ (α2 + 2ρ2) ∥wt − w∗∥Σ +≤ 3ρ2 ∥wt − w∗∥Σ , +(96) +where the last inequality follows from the fact that α2 ≤ α and our assumption that α ≤ ρ2 from +Assumption 4. Similarly, we use resilience property in Eq. (6) instead of Eq. (7), we can show that +∥Σ−1/2u(2) +t ∥ ≤ 3ρ3σ . +(97) +Next, we consider u(3) +t . +Since |S3 \ G| ≤ 1.2αcorruptn and |Et| ≤ αn, using Eq. (9) and +Corollary J.8, we have +∥Σ−1/2u(3) +t ∥ = max +v:∥v∥=1 +1 +n +� +i∈Sbad∪Et +v⊤Σ−1/2xiclipθt(x⊤ +i wt − yi) +≤ 2ρ4θt +≤ 2ρ4 +� +8 max{ρ2/α, ρ3/α} + 1 · (∥wt − w∗∥Σ + σ) . +(98) +The analysis of convergence follows similarly as in Step 3 and Step 4 of the proof of Theorem 4 +except we set ˆρ(α) = max{ρ1, 3ρ2, 2ρ4 +� +8 max{ρ2/α, ρ3/α} + 1}. +The second term in Eq. (85) ensures the added Gaussian noise is small enough such that +φ2 +t ∥vt∥2 ≤ ˆρ(α)2(E[∥wt − w∗∥2 +Σ] + σ2), which is similar as in Eq. (56) +42 + +L.2 +Proof of Lemma L.2 +Proof. For any x that is (K, a)-sub-Weibull from Definition 2.1, Eq. (73) implies that for any k ≥ 1, +E[| ⟨v, x⟩ |k] = +� ∞ +0 +P(| ⟨v, x⟩ | ≥ t1/k)dt +(99) +≤ +� ∞ +0 +2 exp +� +− +t +1 +ka +(K2E[⟨v, x⟩2]) +1 +2a +� +dt +(100) += 2Kk(E[⟨v, x⟩2])k/2ka +� ∞ +0 +e−uuka−1du +(101) += 2Kk(E[⟨v, x⟩2])k/2Γ(ka + 1) +(102) +≤ 2Kk(E[⟨v, x⟩2])k/2(ka)ka +(103) +This implies that xi is also ((ka)aK, k)-hypercontractive. Since xi and zi are independent, we +have +E +���� +� +v, σ−1Σ−1/2xizi +���� +k� += E +���� +� +v, Σ−1/2xi +���� +k� +E +���σ−1zi +��k� +≤ 2(ka)kaKkκk +2 . +(104) +This means xizi is also ((ka)aKκ2, k)-hypercontractive. From Zhu et al. (2019, Lemma G.10), we +know with probability 1 − ζ, there exists S1 ⊂ Sgood with |S1| ≥ (1 − 0.1α)|Sgood|, such that for any +T ⊂ S1 with |T| ≥ (1 − α)|S1|, we have +��� 1 +|T| +� +(xi,yi)∈S +� +v, σ−1Σ−1/2xi(yi − x⊤ +i w∗) +� ��� ≤ C2k(ka)aKκ2α1−1/kζ−1/k . +(105) +Similarly, there exists S2 ⊂ Sgood with |S2| ≥ (1 − 0.1α)|Sgood|, such that for any T ⊂ S2 with +|T| ≥ (1 − α)|S2|, we have +��� 1 +|T| +� +(xi,yi)∈T +(σ−1(yi − x⊤ +i w∗))2 − 1 +��� ≤ C2k2κ2 +2α1−2/kζ−2/k . +(106) +From Lemma J.7, for any T ⊂ Sgood with |T| ≥ (1 − ˜α)|Sgood|, we have +��� 1 +|T| +� +(xi,yi)∈T +� +v, Σ−1/2xi +�2 +− 1 +��� ≤ C2K ˜α log2a(1/˜α) . +(107) +and +��� 1 +|T| +� +(xi,yi)∈T +� +v, Σ−1/2xi +� ��� ≤ C2K ˜α loga(1/˜α) . +(108) +Set S = S1 ∩ S2, we know |S| ≥ (1 − 0.2α)|Sgood| and S is +(0.2α, α, C2k(ka)aKκ2α1−1/kζ−1/k, C2K2˜α log2a(1/˜α), C2k2κ2 +2α1−2/kζ−2/k, C2K ˜α loga(1/˜α))-corrupt +good with respect to (w∗, Σ, σ). This completes the proof. +43 + diff --git a/B9FQT4oBgHgl3EQfNza6/content/tmp_files/load_file.txt b/B9FQT4oBgHgl3EQfNza6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5fd8fb1d83df977c617c3ee8b0fabd3efa711499 --- /dev/null +++ b/B9FQT4oBgHgl3EQfNza6/content/tmp_files/load_file.txt @@ -0,0 +1,2045 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf,len=2044 +page_content='Near Optimal Private and Robust Linear Regression Xiyang Liu ∗ Prateek Jain † Weihao Kong ‡ Sewoong Oh § Arun Sai Suggala ¶ Abstract We study the canonical statistical estimation problem of linear regression from n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' examples under (ε, δ)-differential privacy when some response variables are adversarially corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We propose a variant of the popular differentially private stochastic gradient descent (DP-SGD) algorithm with two innovations: a full-batch gradient descent to improve sample complexity and a novel adaptive clipping to guarantee robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When there is no adversarial corruption, this algorithm improves upon the existing state-of-the-art approach and achieves a near optimal sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under label-corruption, this is the first efficient linear regression algorithm to guarantee both (ε, δ)-DP and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Synthetic experiments confirm the superiority of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 Introduction Differential Privacy (DP) is a widely accepted notion of privacy introduced by Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2006), which is now standard in industry and government (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Erlingsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Fanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Abowd, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A query to a database is said to be (ε, δ)-differentially private if a strong adversary who knows all other entries cannot identify with high confidence whether you participated in the database or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The parameters ε and δ restrict the Type-I and Type-II errors achievable by the adversary in this hypothesis testing (Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Smaller ε > 0 and δ ∈ [0, 1] imply stronger privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Although significant advances have been made recently in understanding the utility-privacy trade-offs in canonical statistical tasks, several important questions remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide a survey in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Consider a canonical statistical task of linear regression with n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' samples, {(xi ∈ Rd, yi ∈ R)}n i=1, drawn from xi ∼ N(0, Σ), yi = x⊤ i w∗ + zi, and zi ∼ N(0, σ2) for some true parameter w ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The error is measured in ∥ ˆw − w∗∥Σ := ∥Σ1/2( ˆw − w∗)∥, which correctly accounts for the signal-to-noise ratio in each direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' in the direction of large eigenvalue of Σ, we have larger signal in xi but the noise zi remains the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' we expect smaller error in those directions, which is accounted for in ∥ ˆw − w∗∥Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When computational complexity is not concerned, the best known algorithm is introduced by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b), called High-dimensional Propose-Test-Release (HPTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For linear regression, n = ˜O(d/α2 + d/(εα)) samples are sufficient for HPTR to achieve an error of (1/σ)∥ ˆw − w∗∥Σ = α with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' After a series of work surveyed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A, Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022) achieve the ∗Paul Allen School of Computer Science & Engineering, University of Washington, xiyangl@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='edu †Google Research, prajain@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='com ‡Google Research, weihaokong@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='com §Paul Allen School of Computer Science & Engineering, University of Washington, and Google Research, sewoong@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='edu ¶Google Research, arunss@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='13273v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='LG] 30 Jan 2023 best known sample complexity for an efficient algorithm: n = ˜O(κ2d/ε + d/α2 + κd/(εα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The last term has an extra factor of κ, the condition number of the covariance Σ of the covariates, and the first term is unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In this work, we propose a novel method (Algorithm 1) that uses full-batch gradient descent with adaptive clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Furthermore, using a intuitive but intricate analysis, we improve this sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 1 (informal version of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 with no adversary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 is (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the (Σ, σ2, w∗, K, a)-model in Assumption 1, n = ˜O(d/α2 + κ1/2d/(εα)) samples are sufficient for Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 to achieve an error rate of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α), where κ := λmax(Σ)/λmin(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' That is, we can get rid of the first unnecessary term in alaysis of Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022), while also improving dependency on κ term which is quite critical for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Perhaps surprisingly, we show that the same algorithm is also robust against label-corruption, where an adversary selects arbitrary αcorrupt fraction of the data points and changes their response variables arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When computational complexity is not concerned, the best known algorithm is again HPTR that also provides optimal robustness and (ε, δ)-DP simultaneously, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', n = ˜O(d/α2 + d/(εα)) samples are sufficient for HPTR to achieve an error of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α) for any corruption bounded by αcorrupt ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that this is a stronger adversary than the label-corruption we study in this paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' this adversary can corrupt both the covariate, xi, and the response variable, yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Currently, there is no efficient algorithm that can guarantee both privacy and robustness for linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under a weaker adversary that only corrupts yi’s, we provide the first efficient algorithm that achieves both privacy and robustness with a near-optimal sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 2 (informal version of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 with adversarial label corruption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 is (ε, δ)- DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the hypotheses of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 and under αcorrupt-corruption model of Assumption 2, if αcorrupt ≤ α then n = ˜O(d/α2 + κ1/2d/(εα)) samples are sufficient for Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 to achieve an error rate of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α) , where κ := λmax(Σ)/λmin(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We focus on sub-Weibull distributions in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A similar algorithm can be applied to the case where the noise in the samples is heavy-tailed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', k-th moment bounded for k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This results in an increased sample complexity of n = ˜O(d/α2k/(k−1) + κ1/2d/(εαk/(k−1))) to achieve the same level of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We explain the heavy-tailed setting, provide detailed analysis and a proof, and discuss the results in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 3 (informal version of Coro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 is (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under (Σ, σ2, w∗, K, a, κ2, k)- model of Assumption 3 and αcorrupt-corruption of Assumption 4, if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorrupt ≤ αk/(k−1), then n = ˜O(κ1/2d/(εαk/(k−1)) + d/α2k/(k−1))) samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ)∥ ˆw − w∗∥Σ = ˜O(α), where κ := λmax(Σ)/λmin(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For a canonical problem of private linear regression under sub-Gaussian distributions, we provide a novel algorithm that achieves the state-of-the-art sample complexity and computational efficiency, improving upon the sample complexity of the prior efficient algorithms Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019) and nearly matching that of an exponential-time algorithm Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For the same problem, we show that the same algorithm is the first to achieve robustness against adversarial corruption of the response variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under a heavy-tailed distribution of the noise, we provide the first computationally efficient algorithm, to the best of our knowledge, that achieves a sample complexity close to that of an exponential-time algorithm of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This algorithm is also the first to achieve robustness against adversarial corruption of the response variables, under heavy-tailed noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 We start with the formal description of the setting in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2, where we present the prior work of Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022) and explain our main technical contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022) propose a streaming version of DP-SGD with adaptive clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Streaming algorithm ensures independence between the current iterate and the current gradient, simplifying the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Adaptive clipping finds the appropriate threshold to clip the norm of the gradients, which is an appropriate technique when there is no adversarial corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, these two algorithmic choices are sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' First, streaming DP-SGD can only use O(n/κ) samples at each round, which increases the sensitivity and leads to an extra κ1/2 factor in the sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Instead, we propose using a full-batch gradient descent and overcome the challenges in the analysis by relying on resilience (explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Together with the novel analysis technique we explain in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1, this results in the gain of κ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Next, the gradient-norm clipping is vulnerable against label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Recall that a gradient is a product of the residual, (w⊤ t xi − yi), and the covariate, xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' An adversary can target those samples with small covariates and make big changes to the residuals, while managing to evade the clipping by the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Instead, we propose clipping the residual and the covariate separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' With adaptively estimated clipping thresholds, this provides robustness against label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We present our main algorithm (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3 with theoretical analyses and justification of the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We propose a novel adaptive clipping in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 We present numerical experiments on synthetic data that demonstrates the sample efficiency of our approach in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We end with a sketch of our main proof ideas in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 Problem formulation and background In linear regression without corruption, the following assumption is standard for the uncorrupted dataset Sgood, except for the fact that we assume a more general family of (K, a)-sub-Weibull distributions that recovers the standard sub-Gaussian family as a special case when a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Assumption 1 ((Σ, σ2, w∗, K, a)-model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A multiset Sgood = {(xi ∈ Rd, yi ∈ R)}n i=1 of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' sam- ples is from a linear model yi = ⟨xi, w∗⟩ + zi, where the input vector xi is zero mean, E[xi] = 0, with a positive definite covariance Σ := E[xix⊤ i ] ≻ 0, and the (input dependent) label noise zi is zero mean, E[zi] = 0, with variance σ2 := E[z2 i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We further assume E[xizi] = 0, which is equivalent to assuming that the true parameter w∗ = Σ−1E[yixi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We assume that the marginal distribution of xi is (K, a)-sub-Weibull and that of zi is also (K, a)-sub-Weibull, as defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Sub-Weibull distributions provide Gaussian-like tail bounds determining the resilience of the dataset in Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7, which our analysis critically relies on and whose necessity is justified in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 (sub-Weibull distribution (Kuchibhotla & Chakrabortty, 2018) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For some K, a > 0, we say a random vector x ∈ Rd is from a (K, a)-sub-Weibull distribution if for all v ∈ Rd, E � exp �� ⟨v,x⟩2 K2E[⟨v,x⟩2] �1/(2a)�� ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Our goal is to estimate the unknown parameter w∗, given upper bounds on the sub-Weibull parameters, (K, a), and a corrupted dataset under the the standard definition of label corruption in (Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' There are variations in literature on the definition, which we survey in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Assumption 2 (αcorrupt-corruption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Given a dataset Sgood = {(xi, yi)}n i=1, an adversary inspects all the data points, selects αcorruptn data points denoted as Sr, and replaces the labels with arbitrary labels while keeping the covariates unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We let Sbad denote this set of αcorruptn newly labelled 3 examples by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let the resulting set be S := Sgood ∪Sbad \\Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We further assume that the corruption rate is bounded by αcorrupt ≤ ¯α, where ¯α is a known positive constant satisfying ¯α ≤ 1/10, 72C2 K2 ¯α log2a(1/(6¯α)) log(κ) ≤ 1/2, and 2C2K2 log2a(1/(2¯α)) ≥ 1 for the (K, a)-sub-Weibull distribution of interest and a positive constant C2 defined in Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7 that only depends on (K, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A vector x ∈ Rd has the Euclidean norm ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For a matrix M, we use ∥M∥2 to denote the spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The error is measured in ∥ ˆw − w∗∥Σ := ∥Σ1/2( ˆw − w∗)∥ for some PSD matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The identity matrix is denoted by Id ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ˜O(·) hides some constants terms, K, a = Θ(1), and poly-logarithmic terms in n, d, 1/ε, log(1/δ), 1/ζ, and 1/αcorrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For a vector x ∈ Rd, we define clipa(x) := x · min{1, a ∥x∥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Background on DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Differential Privacy is a standard measure of privacy leakage when data is accessed via queries, introduced by Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Two datasets S and S′ are said to be neighbors if they differ at most by one entry, which is denoted by S ∼ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A stochastic query q is said to be (ε, δ)-differentially private for some ε > 0 and δ ∈ [0, 1], if P(q(S) ∈ A) ≤ eεP(q(S) ∈ A) + δ, for all neighboring datasets S ∼ S′ and all subset A of the range of the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We build upon two widely used DP primitives, the Gaussian mechanism and the private histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A central concept in DP mechanism design is the sensitivity of a query, defined as ∆q := supS∼S′ ∥q(S) − q(S′)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We describe Gaussian mechanism and private histogram in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Comparisons with the prior work When there is no adversarial corruption, the state-of-the-art approach introduced by Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022) is based on stochastic gradient descent, where privacy is ensured by gradient norm clipping and the Gaussian mechanism to ensure privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' There are two main components in this approach: adaptive clipping and streaming SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Adaptive clipping with an appropriate threshold θt ensures that no data point is clipped while providing a bound on the sensitivity of the average mini-batch gradient, which ensures we do not add too much noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The streaming approach, where each data point is only touched once and discarded, ensures independence between the past iterate wt and the gradients at round t + 1, which the analysis critically relies on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For T = ˜Θ(κ) iterations where κ is the condition number of the covariance Σ of the covariates, the dataset S = {(xi, yi)}n i=1 is partitioned into {Bt}T−1 t=0 subsets of equal size: |Bt| = ˜Θ(n/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' At each round t < T, the gradients are clipped and averaged with additive Gaussian noise chosen to satisfy (ε, δ)-DP: wt+1 ← wt − η � 1 |Bt| � i∈Bt clipθt(xi(w⊤ t xi − yi)) + θt � 2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='25/δ) ε|Bt| νt � , (1) where νt ∼ N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022), a variation of this streaming SGD requires n = ˜O(κ2d/ε + d/α2 + κd/(εα)) to achieve an error of ∥wT − w∗∥2 Σ = O(σ2α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Our technical innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Our approach builds upon such gradient based methods but makes several important innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' First, we use full-batch gradient descent, as opposed to the streaming SGD above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Using all n samples reduces the sensitivity of the per-round gradient average, since n > |Bt| = ˜Θ(n/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This improves the sample complexity to n = ˜O(d/α2 + κ1/2d/(εα)) to achieve an error of ∥wT − w∗∥2 Σ = O(σ2α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, full-batch GD loses the independence that the streaming SGD enjoyed between wt and the samples used in the round t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This dependence makes the analysis more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We instead propose using the resilience property of sub-Weibull distributions to precisely track the bias and variance of the (dependent) full-batch gradient average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Resilience is a central concept in robust statistics which we explain in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 Next, one critical component in achieving this improved sample complexity is the new analysis technique we introduce for tracking the end-to-end gradient updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since our gradient descent algorithm is not guaranteed to make progress every step, we can not use the vanilla one-step ahead analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Taking the full end-to-end analysis by expanding the whole gradient trajectory will introduce too many correlated cross-terms which are very hard to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Therefore, we leverage an every κ-step analysis and show that the objective function at least decreases geometrically every κ steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' To be more specific, our analysis technique in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' H steps 3 and 4 opens up the iterative updates from beginning to end, and exploits the fact that λmax((ηΣ)1/2(1 − ηΣ)i(ηΣ)1/2) is upper bounded by 1/(i + 1) when ∥ηΣ∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This technique is critical in achieving the near- optimal dependence in κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This might be of independent interest to other analysis of gradient-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We refer to the beginning of step 3 in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' H for a detailed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Finally, we propose a novel clipping method that separately clips xi and (w⊤ t xi − yi) in the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is critical in achieving robustness to label-corruption, as we explain in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3 Robust and DP linear regression We introduce a gradient descent approach for linear regression with a novel adaptive clipping that ensures robustness against label-corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This achieves a near-optimal sample complexity and, for the special case of private linear regression without adversarial corruption, improves upon the state-of-the-art algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Algorithm The skeleton of our approach in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 is the general DP-SGD (Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2013) with adaptive clipping (Andrew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, the standard adaptive clipping is not robust against label-corruption under the more general (K, a)-sub-Weibull assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In particular, it is possible under sub-Weibull distribution that a positive fraction of the covariates are close to the origin, which is not possible under Gaussian data due to concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In this case, the adversary can choose to corrupt those points with small norm, ∥xi∥, making large changes in the residual, (yi −w⊤ t xi), while evading the standard clipping (by the norm of the gradient), since the norm of the gradient, ∥xi(yi − w⊤ t xi)∥ = ∥xi∥ |yi − w⊤ t xi|, can remain under the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is problematic, since the bias due to the corrupted samples in the gradient scales proportional to the magnitude of the residual (after clipping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' To this end, we propose clipping the norm and the residual separately: clipΘ(xi)clipθt � w⊤ t xi − yi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This keeps the sensitivity of gradient average bounded by Θθt, and the subsequent Gaussian mechanism in line 11 ensures (ε0, δ0)-DP at each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Applying advanced composition in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5 of T rounds, this ensures end-to-end (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Novel adaptive clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When clipping with clipΘ(xi), the only purpose of clipping the covariate by its norm, ∥xi∥, is to bound the sensitivity of the resulting clipped gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In particular, we do not need to make it robust as there is no corruption in the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Ideally, we want to select the smallest threshold Θ that does not clip any of the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the norm of a covariate is upper bounded by ∥xi∥2 ≤ K2Tr(Σ) log2a(1/ζ) with probability 1 − ζ (Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3), we estimate the unknown Tr(Σ) using Private Norm Estimator in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3 in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' F and set the norm threshold Θ = K √ 2Γ loga(n/ζ) (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The n in the logarithm ensures that the union bound holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When clipping with clipθt(w⊤ t xi − yi), the purpose of clipping the residual by its magnitude, |yi − w⊤ t xi| = |(w∗ − wt)⊤xi + zi|, is to bound the sensitivity of the gradient and also to provide robustness against label-corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We want to choose a threshold that only clips corrupt data points and at most a few clean data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In order to achieve an error (1/σ)∥wT − w∗∥Σ = O(α), 5 we know that any set of (1 − α) fraction of the clean data points is sufficient to get a good estimate of the average gradient, and we can find such a large enough set of points that satisfy |(w∗ − wt)⊤xi + zi|2 ≤ (∥wt − w∗∥2 Σ + σ2)CK2 log2a(1/(2α)) from Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' At the same time, this threshold on the residual is small enough to guarantee robustness against the label- corrupted samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We introduce the robust and private Distance Estimator in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 to estimate the unknown (squared and shifted) distance, ∥wt − w∗∥2 Σ + σ2, and set the distance threshold θt = 2√2γt � 9C2K2 log2a(1/(2α)) (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Both norm and distance estimation rely on private histogram (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2), but over a set of statistics computed on partitioned datasets, which we explain in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 1: Robust and Private Linear Regression Input: S = {(xi, yi)}3n i=1, DP parameters (ε, δ), T, learning rate η, failure probability ζ, target error α, distribution parameter (K, a) 1 Partition dataset S into three equal sized disjoint subsets S = S1 ∪ S2 ∪ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 δ0 ← δ 2T , ε0 ← ε 4√ T log(1/δ0), ζ0 ← ζ 3, w0 ← 0 3 Γ ← PrivateNormEstimator(S1, ε0, δ0, ζ0) /* using Algorithm 3, Appendix F / 4 Θ ← K √ 2Γ loga(n/ζ0) 5 for t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , T − 1 do 6 γt ← RobustPrivateDistanceEstimator(S2, wt, ε0, δ0, α, ζ0) /* using Algorithm 2 / 7 θt ← 2√2γt · � 9C2K2 log2a(1/(2α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 8 Sample νt ∼ N (0, Id) 9 ˜g(t) i ← clipΘ(xi)clipθt(x⊤ i wt − yi) 10 φt = ( � 2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='25/δ0)Θθt)/(ε0n) 11 wt+1 ← wt − η � 1 n � i∈S3 ˜g(t) i + φtνt � 12 Return wT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 Analysis We show that Algorithm 1 achieves a near-optimal sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide a proof in Appendix H and a sketch of the proof in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We address the necessity of the assumptions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3, along with some lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 1 is (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under (Σ, σ2, w∗, K, a)-model of Assumption 1 and αcorrupt- corruption of Assumption 2 and for any failure probability ζ ∈ (0, 1) and target error rate α ≥ αcorrupt, if the sample size is large enough such that n = ˜O � K2d log2a+1 �1 ζ � + d + log(1/ζ) α2 + K2dT 1/2 log( 1 δ) loga( 1 ζ ) εα � , (2) with a large enough constant where ˜O hides poly-logarithmic terms in d, n, and κ, then the choices of a step size η = 1/(Cλmax(Σ)) for any C ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 and the number of iterations, T = ˜Θ (κ log (∥w∗∥)) for a condition number of the covariance κ := λmax(Σ)/λmin(Σ), ensures that, with probability 1 − ζ, Algorithm 1 achieves Eν1,··· ,νt∼N(0,Id) � ∥wT − w∗∥2 Σ � = ˜O � K4σ2α2 log4a � 1 α � � , (3) 6 where the expectation is taken over the noise added for DP, and ˜Θ(·) hides logarithmic terms in K, σ, d, n, 1/ε, log(1/δ), 1/α, and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Omitting some constant and logarithmic terms, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 requires n = ˜O � d α2 + κ1/2d log(1/δ) εα � , (4) samples to ensure an error rate of E[∥wT − w∗∥2 Σ] = ˜O(σ2α2) for any α ≥ αcorrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The lower bound on the achievable error of σ2α2 ≥ σ2α2 corrupt is due to the label-corruption and cannot be improved, as it matches an information theoretic lower bound we provide in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the special case when the covariate follows a sub-Gaussian distribution, that is (K, 1/2)-sub-Weibull for a constant K, there is an n = Ω(d/α2 + d/(εα)) lower bound (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019), Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1), and our upper bound matches this lower bound up to a factor of κ1/2 in the second term and other logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (4) is the best known rate among all efficient private linear regression algorithms, strictly improving upon existing methods when log(1/δ) = ˜O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We discuss some exponential time algorithms that closes the κ1/2 gap in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Comparisons with the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The best existing efficient algorithm by Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022) can only handle the case where there is no adversarial corruption, and requires n = ˜O(κ2d � log(1/δ)/ε + d/α2 + κd � log(1/δ)/(εα)) to achieve an error rate of σ2α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (4), the first term dominates in its dependence in κ, which is a factor of κ larger than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The third term is larger by a factor of κ1/2 but smaller by a factor of log1/2(1/δ), compared to the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the non-private case, when ε = ∞, a recent line of work has developed algorithms for linear regression that are robust to label corruptions (Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2015, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Suggala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Dalalyan & Thompson, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Of these, Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Dalalyan & Thompson (2019) are relevant to our work as they consider the same adversary model as Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When xi’s and zi’s are sampled from N(0, Σ) and N(0, σ2), Dalalyan & Thompson (2019) proposed a Huber loss based estimator that achieves error rate of σ2α2 log2(n/δ) when n = ˜O � κ2d/α2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the same setting, Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2015) propoased a hard thresholding based estimator that achieves σ2α2 error rate with ˜O � d/α2� sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Our results in Theorem 4 match these rates, except for the sub-optimal dependence on log4a(1/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Another line of work considered both label and covariate corruptions and developed optimal algorithms for parameter recovery (Diakonikolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019c,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Pensia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Cherapanamjeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Jambulapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Klivans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Bakshi & Prasad, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Depersin, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The best existing efficient algorithm , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (Pensia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020), achieves error rate of σ2α2 log(1/α) when n = ˜O � d/α2� , and the xi and zi are sampled from N(0, I) and N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under both privacy requirements and adversarial corruption, the only algorithm with a provable guarantee is an exponential time approach, known as High-dimensional Propose-Test-Release (HPTR), of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b, Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2), which achieves a sample complexity of n = O(d/α2 + (d + log(1/δ))/(εα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Notice that there is no dependence on κ and the log(1/δ) term scales as 1/(εα) as opposed to κd1/2/(εα) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It remains an open question if computationally efficient private linear regression algorithms can achieve such a κ-independent sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Further, HPTR is robust against a stronger adversary who corrupts the covariates also and not just the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under this stronger adversary, it remains open if there is an efficient algorithm that achieves n = O(d/α2 + d/(εα)) sample complexity even for constant κ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3 Lower bounds Necessity of our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A tail assumption on the covariate xi such as Assumption 1 is 7 necessary to achieve n = O(d) sample complexity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Even when the covariance Σ is close to identity, without further assumptions on the tail of covariate x, the result in (Bassily et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2014) implies that for δ < 1/n and sufficiently large n, no (ε, δ)-DP estimator can achieve excess risk ∥ ˆw − w∗∥2 Σ better than Ω(d3/(ε2n2)) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (3) in (Wang, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that this lower bound is a factor d larger than our upper bound that benefits from the additional tail assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A tail assumption on the noise zi such as Assumption 1 is necessary to achieve n = O(d/(εα)) dependence on the sample complexity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For heavy-tailed noise, such as k-th moment bounded noise, the dependence can be significantly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b, Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5) implies that for δ = e−Θ(d) and 4-th moment bounded xi and zi, any (ε, δ)-DP estimator requires n = Ω(d/(εα2)), which is a factor of 1/α larger, to achieve excess risk E[∥ ˆw − w∗∥2 Σ] = ˜O(σ2α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The assumption that only label is corrupted is critical for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The average of the clipped gradients can be significantly more biased, if the adversary can place the covariates of the corrupted samples in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In particular, the bound on the bias of our gradient step in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (41) in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' H would no longer hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Against such strong attacks, one requires additional steps to estimate the mean of the gradients robustly and privately, similar to those used in robust private mean estimation (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Kothari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Ashtiani & Liaw, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is outside the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lower bounds under label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the αcorrupt label corruption setting (As- sumption 2), even with infinite data and without privacy constraints, no algorithm is able to learn w∗ with ℓ2 error better than αcorrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide a formal derivation for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let DΣ,σ2,w∗,K,a be a class of joint distributions on (xi, yi) from (Σ, σ2, w∗, K, a)- model in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let Sn,α be an α-corrupted dataset of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' samples from some distribution D ∈ DΣ,σ2,w∗,K,a under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let M be a class of estimators that are functions over the datasets Sn,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then there exists a positive constant c such that min n, ˆw∈M max Sn,α,D∈DΣ,σ2,w∗,K,a,w∗,K,a, E[∥ ˆw − w∗∥2 Σ] ≥ c α2 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A proof is provided in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A similar lower bound can be found in Bakshi & Prasad (2021, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 Adaptive clipping for the gradient norm In the ideal clipping thresholds for the norm and the residual, there are unknown terms which we need to estimate adaptively, (∥wt − w∗∥2 Σ + σ2) and Tr(Σ), up to constant multiplicative errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We privately estimate the (squared and shifted) distance to optimum, (∥wt − w∗∥2 Σ + σ2), with Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 and privately estimate the average input norm, E[∥xi∥2] = Tr(Σ), with Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3 in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' These are used to get the clipping thresholds in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We propose a trimmed mean approach below for distance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The norm estimator is similar and is provided in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Private distance estimation using private trimmed mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The goal is to estimate the (shifted) distance to optimum, ∥wt − w∗∥2 Σ + σ2, up to some constant multiplicative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that this is precisely the task of estimating the variance of the residual bi = yi − w⊤ t xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When there is no adversarial corruption and no privacy constraint, we can simply use the empirical variance estimator (1/n) � i∈[n](yi − w⊤ t xi)2 to obtain a good estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, the empirical variance estimator is not robust against adversarial corruptions since one outlier can make the estimate arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A classical idea is using the trimmed estimator from (Tukey & McLaughlin, 1963), 8 which throws away the 2α fraction of residuals bi with the largest magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For datasets with resilience property as assumed in this paper, this will guarantee an accurate estimate of the distance to optimum in the presence of α fraction of corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' To make the estimator private, it is tempting to simply add a Laplacian noise to the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, the sensitivity of the trimmed estimator is unknown and depends on the distance to the optimum that we aim to estimate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' this makes it challenging to determine the variance of the Laplacian noise we add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Instead, we propose to partition the dataset into k batches, compute an estimate for each batch, and form a histogram with over those k estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Using a private histogram mechanism with geometrically increasing bin sizes, we propose using the bin with the most estimates to guarantee a constant factor approximation of the distance to the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We describe the algorithm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 2: Robust and Private Distance Estimator Input: S2 = {(xi, yi)}n i=1, current wt, (ε0, δ0), ¯α, ζ 1 Let bi ← (yi − w⊤ t xi)2, ∀i ∈ [n] and ˜S ← {bi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 Partition ˜S into k = ⌈C1 log(1/(δ0ζ))/ε0⌉ subsets of equal size and let Gj be the j-th partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3 For j ∈ [k], denote ψj as the (1 − 3¯α)-quantile of Gj and φj ← 1 |Gj| � i∈Gj bi1{bi ≤ ψj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 Partition [0, ∞) into geometrically increasing intervals Ω := � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , � 2−1, 1 � , [1, 2) , � 2, 22� , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' � ∪ {[0, 0]} 5 Run (ε0, δ0)-DP histogram of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 on {φj}k j=1 over Ω 6 if all the bins are empty then Return ⊥ 7 Let [ℓ, r] be a non-empty bin that contains the maximum number of points in the DP histogram 8 return ℓ This algorithm gives an estimate of the distance up to a constant multiplicative error as we show in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide a proof in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 2 is (ε0, δ0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For an αcorrupt-corrupted dataset S2 and an upper bound ¯α on αcorrupt that satisfy Assumption 1 and 37C2K2 · ¯α log2a(1/(6¯α)) ≤ 1/4 and any ζ ∈ (0, 1), if n = O �(d + log((log(1/(δ0ζ)))/ε0ζ))(log(1/(δ0ζ))) ¯α2ε0 � , (5) with a large enough constant then, with probability 1 − ζ, Algorithm 2 returns ℓ such that 1 4(∥wt − w∗∥2 Σ + σ2) ≤ ℓ ≤ 4(∥wt − w∗∥2 Σ + σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that in Theorem 5, we only need to estimate distance up to a constant multiplicative error, as opposed to an error that depends on our final end-to-end desired level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Consequently, we require smaller sample complexity (that doesn’t depend on α) than other parts of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' While DP-STAT (Algorithm 3 in Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022)) can also be used to estimate ∥wt − w∗∥Σ + σ (and it would not change the ultimate sample complexity in its dependence on κ, d, ε, and n), there are three important improvements we make: (i) DP-STAT requires the knowledge of ∥w∗∥Σ + σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (ii) our utility guarantee has improved dependence in K and log2a(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' and (iii) Algorithm 2 is robust against label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 9 Figure 1: Performance of various techniques on DP linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' d = 10 in all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' n = 107, κ = 1 in the 2nd experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' n = 107, σ = 1 in the 3rd experiment, where κ is the condition number of Σ and σ2 is the variance of the label noise zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Upper bound on clipped good data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Using the above estimated distance to the optimum in selecting a threshold θt, we also need to ensure that we do not clip too many clean data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The tolerance in our algorithm to reach the desired level of accuracy is clipping O(α) fraction of clean data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is ensured by the following lemma, and we provide a proof in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under Assumption 1 and for all t ∈ [T], if θt ≥ � 9C2K2 log2a(1/(2α))·(∥w∗ − wt∥Σ + σ) then ��� i ∈ S3 ∩ Sgood : ��w⊤ t xi − yi �� ≥ θt ��� ≤ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 5 Experimental results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 DP Linear Regression We present experimental results comparing our proposed technique (DP-RobGD) with other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We consider non-corrupted regression in this section and defer corrupted regression to the App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We begin by describing the problem setup and the baseline algorithms first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Experiment Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We generate data for all the experiments using the following generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The parameter vector w∗ is uniformly sampled from the surface of a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The covariates {xi}n i=1 are first sampled from N(0, Σ) and then projected to unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We consider diagonal covariances Σ of the following form: Σ[0, 0] = κ, and Σ[i, i] = 1 for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Here κ ≥ 1 is the condition number of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We generate noise zi from uniform distribution over [−σ, σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Finally, the response variables are generated as follows yi = x⊤ i w∗ + zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' All the experiments presented below are repeated 5 times and the averaged results are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We set the DP parameters (ϵ, δ) as ϵ = 1, δ = min(10−6, n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Experiments for ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 in the App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Baseline Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We compare our estimator with the following baseline algorithms: Non private algorithms: ordinary least squares (OLS), one-pass stochastic gradient descent with tail-averaging (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For SGD, we use a constant step-size of 1/(2λmax) with n/T minibatch size, where T = 3κ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Private algorithms: sufficient statistics perturbation (DP-SSP) (Foulds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Wang, 2018), differentially private stochastic gradient descent (DP-AMBSSGD) (Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' DP-SSP had the best empirical performance among numerous techniques studied by Wang (2018), and DP-AMBSSGD has the best known theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The DP-SSP algorithm involves releasing XT X and XT y differentially privately and computing (� XT X)−1 � XT y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' DP-AMBSSGD is a private version of SGD where the DP noise is set adaptively according to the excess error in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For both the algorithms, we use the hyper-parameters recommended in their 10 d=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content="0=1,K=1,E=1 10- [] 10- 105 106 107 NumberofSamplesE=1 Estimation Eror [0] 10 Parameter 10-6 10] 10-4 10-3 10-2 10-1 100 aE=1 Parameter Estimation Error 10' 100 101 KNon Private OLS DP-SSP Non Private SGD DP-AMBSSGD DP-RobGD DP-RobGD*respective papers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' To improve the performance of DP-AMBSSGD, we reduce the clipping threshold recommended by the theory by a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' DP-RobGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We implement Algorithm 1 with the following key changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Instead of relying on PrivateNormEstimator to estimate Γ, we set it to its true value Tr(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is done for a fair comparison with DP-AMBSSGD which assumes the knowledge of Tr(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Next, we use 20% of the samples to compute γt in line 5 (instead of the 50% stated in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In our experiments we also present results for a variant of our algorithm called DP-RobGD* which outputs the best iterate based on γt, instead of the last iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' One could also perform tail-averaging instead of picking the best iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Both these modifications are primarily used to reduce the variance in the output of Algorithm 1 and achieved similar performance in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Figure 1 presents the performance of various algorithms as we vary n, κ, σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It can be seen that DP-RobGD outperforms DP-AMBSSGD in almost all the settings (and DP-RobGD* outperforms DP-RobGD in all cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' DP-SSP has poor performance when the noise σ is low, but performs slightly better than DP-RobGD in other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A major drawback of DP-SSP is its computational complexity which scales as O(nd2 + dω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In contrast, the computational complexity of DP-RobGD has smaller dependence on d and scales as ˜O(ndκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Thus the latter is more computationally efficient for high-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' More experimental results on both robust and private linear regression can be found in the App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 6 Sketch of the main ideas in the analysis We provide the main ideas behind the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The privacy proof is straightforward since no matter what clipping threshold we use the noise we add is always proportionally to the clipping threshold which guarantees privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the remainder, we focus on the utility analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The proof of the utility heavily relies on the resilience (Steinhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2017) (also known as stability (Diakonikolas & Kane, 2019)), which states that given a large enough sample set S, various statistics (for example, sample mean and sample variance) of any large enough subset of S will be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We define resilience in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The main effort for proving Theorem 4 lies in the analysis of the gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Without clipping and adding noise for differential privacy, convergence property of gradient descent for linear regression is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The convergence proof of noisy gradient descent is also relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, our algorithm requires both clipping and adding noise for robustness and privacy purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The key difference between our setting and the classical setting is the existence of adversarial bias and random noise in the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We give an overview of the proof of our robust and private gradient descent as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' First, we introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let g(t) i := (x⊤ i wt − yi)xi be the raw gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that when the data follows from our distributional assumption, uncorrupted samples are not clipped: clipΘ(xi) = xi for i ∈ Sgood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let G := Sgood ∩ S3 = S3 \\ Sbad denote the clean data that remains in the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We can write down one step of gradient update as follows: wt+1 − w∗ = wt − η � 1 n � i∈S ˜g(t) i + φtνt � − w∗ = � I − η n � i∈G xix⊤ i � (wt − w∗) + η n � i∈G xizi + η n � i∈G (g(t) i − ˜g(t) i ) − η n � i∈Sbad ˜g(t) i − ηφtνt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 11 In the above equation, the first term is a contraction, meaning wt is moving toward w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The second term captures the noise from the randomness in the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The third term captures the bias introduced by the clipping operation, the fourth term (η/n) � i∈Sbad ˜g(t) i captures the bias introduced by the adversarial datapoints, and the fifth term captures the added Gaussian noise for privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The second term is standard and relatively easy to control, and our main focus is on the last three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The third term (η/n) � i∈G(g(t) i − ˜g(t) i ) can be controlled using the resilience property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We prove that with our estimated threshold, the clipping will only affect a small amount of datapoints, whose contribution to the gradient is small collectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The fourth term (η/n) � i∈Sbad ˜g(t) i = (η/n) � i∈Sbad clipθt(x⊤ i wt − yi)xi can be controlled since there is only a small amount data points whose label is corrupted, the clipθt(x⊤ i wt − yi) is controlled by the clipping threshold and the xi part satisfies resilience property which implies a small, say Sbad, must have small ∥ � i∈Sbad xi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Now we have controlled the deterministic bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then, we upper bound the fifth term, which is the noise introduced by the Gaussian noise for the purpose of privacy, and show the expected prediction error decrease in every gradient step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The difficulty is that, since our clipping threshold is adaptive, the decrease of the estimation error depends on the estimation error of all the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This causes that in some iterations, the estimation error actually increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In order to get around this, we split the iterations into length κ chunks, and argue that the maximum estimation error in a chunk must be a constant factor smaller than the previous chunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This implies we will reach the desired error within ˜O(κ) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 7 Discussion We provide a novel variant of DP-SGD algorithm for differentially private linear regression under label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We show the first near-optimal rate that achieves privacy and robustness to label corruptions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When there is no label corruption, our result also improves upon the state-of-the-art method (Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022) in terms of the condition number κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Compared to (Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022), our algorithm has three innovations: 1) we introduce a novel adaptive clipping, which is critical in achieving robustness against label corruptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' and 2) we use full batch gradient descent and a novel convergence analysis to get the near-optimal sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Compared to the lower bound and upper bound from a computationally inefficient algorithm in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b), our sample complexities ˜O(d/α2 + κ1/2d/(εα)) has additional κ1/2 factor in the privacy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It remains an open question if there is an efficient algorithm to achieve the optimal rate without the κ dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Acknowledgement We thank Abhradeep Guha Thakurta for helpful discussions while working on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This work is supported in part by NSF grants CNS-2002664, DMS-2134012, CCF-2019844 as a part of NSF Institute for Foundations of Machine Learning (IFML), CNS-2112471 as a part of NSF AI Institute for Future Edge Networks and Distributed Intelligence (AI-EDGE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' References Abadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} 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Andrew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Asi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Kulkarni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Kamath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' As one of the most well-studied problem in differentially privacy, DP Empirical Risk Minimization (DP-ERM) aims to minimize the empirical risk (1/n) � i∈S ℓ(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' w) privately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The optimal excess empirical risk for approximate DP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', δ > 0) is known to be GD · √ d/(εn), where the loss ℓ is convex and G-Lipschitz with respect to the data, and D is the diameter of the convex parameter space (Bassily et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This bound can be achieved by several DP-SGD methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Bassily et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2014), with different computational complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Differentially private stochastic convex optimization considers minimizing the population risk Ex∼D[ℓ(x, w)], where data is drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' from some unknown distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Using some variations of DP-SGD, Bassily et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019) and Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2020) achieves a population risk of GD(1/√n + √ d/(εn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' DP linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Applying above results for the linear model, by observing that G = O(d) if D = O(1), the sample complexity required for achieving generalization error is n = d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Existing works for DP linear regression, for example (Vu & Slavkovic, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Kifer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Mir, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Dimitrakakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Foulds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Minami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Wang, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Sheffet, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019) typically consider deterministic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Gaussian data setting, this translates into a sample complexity of n = d3/2/(εα), where the extra d1/2 due to the fact that no statistical assumptions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' sub-Weibull data, recent work (Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022) achieved nearly optimal excess population risk d/n + d2/(ε2n2) using DP-SGD with adaptive clipping, up to extra factors on the condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is closest to our work and we provide detailed comparisons in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under Gaussian assumptions, Milionis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022) analyze linear regression algorithm with sub-optimal guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (Dwork & Lei, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Amin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Alabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b) also consider using robust statistics like Tukey median (Tukey, 1975) or Theil–Sen estimator (Theil, 1950) for differentially private regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, (Dwork & Lei, 2009) and (Amin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022) lack utility guarantees and (Alabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2020) is restricted to one-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b) achieves optimal sample complexity but takes exponential time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Robust linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Robust mean estimation and linear regression have been studied for a long time in the statistics community (Tukey & McLaughlin, 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Huber, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Tukey, 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, for high dimensional data, these estimators generalizing the notion of median to higher dimensions are typically computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Recent advances in the filter-based algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', (Diakonikolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2017, 2020, 2019a, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019), achieve nearly optimal guarantees for mean estimation in time linear in the dimension of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Motivated by the filter algorithms, Diakonikolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019c,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Pensia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Cherapanamjeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Jambulapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2020) achieved nearly optimal rate with d samples for robust linear regression, where both data xi and label yi are corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Another type of efficient methods that achieve similar rates and sample complexity in polynomial time is based on sum-of-square proofs (Klivans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Bakshi & Prasad, 2021), which can be computationally expensive in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019) and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b) achieves nearly optimal rates using d samples but require exponential time complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' An important special case of adversarial corruption is when the adversary only corrupts the response variable in supervised learning (Khetan 19 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2018) and also in unsupervised learning (Thekumparampil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For linear regression, when there is only label corruptions, (Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Dalalyan & Thompson, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022) achieve nearly optimal rates with O(d) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the oblivious label corruption model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', the adversary only corrupts a fraction of labels in complete ignorance of the data, (Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Suggala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019) provide consistent estimator ˆwn such that limn→∞ E [ �wn − w∗]2 = 0 with O(d) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Robust and private linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the settings of both DP and data corrup- tions, the only algorithm by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b) achieves nearly optimal rates α log(1/α)σ with optimal sample complexities of d/α2 + d/(εα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, their algorithm requires exponential time complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Robust and private mean estimation Based on sum-of-square proofs, recent works (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Alabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022) are able to achieve nearly optimal rates α log(1/α) with ˜O(d) samples for sub-Gaussian data with known covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' B Preliminary on differential privacy Our algorithm builds upon two DP primitive: Gaussian mechanism and private histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The Gaussian mechanism is one examples of a larger family of mechanisms known as output perturbation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In practice, it is possible to get better utility trade-off for a output perturbation mechanism by carefully designing the noise, such as the stair-case mechanism which are shown to achieve optimal utility in the variance (Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2015) and also in hypothesis testing (Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, the gain is only by constant factors, which we do not try to optimize in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide a reference for the Gaussian mechanism and private histogram below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 (Gaussian mechanism (Dwork & Roth, 2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For a query q with sensitivity ∆q, the Gaussian mechanism outputs q(S) + N(0, (∆q � 2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='25/δ)/ε)2Id) and achieves (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 (Stability-based histogram (Karwa & Vadhan, 2017, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For every K ∈ N ∪ ∞, domain Ω, for every collection of disjoint bins B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , BK defined on Ω, n ∈ N, ε ≥ 0, δ ∈ (0, 1/n), β > 0 and α ∈ (0, 1) there exists an (ε, δ)-differentially private algorithm M : Ωn → RK such that for any set of data X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , Xn ∈ Ωn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ˆpk = 1 n � Xi∈Bk 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (˜p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , ˜pK) ← M(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , Xn), and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' n ≥ min � 8 εβ log(2K/α), 8 εβ log(4/αδ) � then, P(|˜pk − ˆpk| ≤ β) ≥ 1 − α When the databse is accessed multiple times, we use the following composition theorems to account for the end-to-end privacy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3 (Parallel composition McSherry (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Consider a sequence of interactive queries {qk}K k=1 each operating on a subset Sk of the database and each satisfying (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' If Sk’s are disjoint then the composition (q1(S1), q2(S2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , qK(SK)) is (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 20 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4 (Serial composition Dwork & Roth (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' If a database is accessed with an (ε1, δ1)- DP mechanism and then with an (ε2, δ2)-DP mechanism, then the end-to-end privacy guarantee is (ε1 + ε2, δ1 + δ2)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In most modern privacy analysis of iterative processes, advanced composition theorem from Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2015) gives tight accountant for the end-to-end privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It can be improved for specific mechanisms using tighter accountants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', in Mironov (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Girgis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Gopi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5 (Advanced composition Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For ε ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='9, an end-to-end guar- antee of (ε, δ)-differential privacy is satisfied if a database is accessed k times, each with a (ε/(2 � 2k log(2/δ)), δ/(2k))-differential private mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' C Definition of resilience Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ((Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b, Definition 23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For some α ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ1 ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ2 ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' and ρ3 ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ4 ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' we say dataset Sgood = {(xi ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' yi ∈ R)}n i=1 is (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ4)-resilient with respect to (w∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' σ) for some w∗ ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' positive definite Σ ≻ 0 ∈ Rd×d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' and σ > 0 if for any T ⊂ Sgood of size |T| ≥ (1 − α)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' the following holds for all v ∈ Rd: ��� 1 |T| � (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='yi)∈T ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' xi⟩(yi − x⊤ i w∗) ��� ≤ ρ1 √ v⊤Σv σ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (6) ��� 1 |T| � xi∈T ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' xi⟩2 − v⊤Σv ��� ≤ ρ2v⊤Σv ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (7) ��� 1 |T| � (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='yi)∈T (yi − x⊤ i w∗)2 − σ2��� ≤ ρ3σ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (8) ��� 1 |T| � (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='yi)∈T ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' xi⟩ ��� ≤ ρ4 √ v⊤Σv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (9) D Proof of Theorem 5 on the private distance estimation We first analyze the privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Changing a data point (xi, yi) can affect at most one partition in {Gj}k j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This would affect at most two histogram bins, increasing the count of one bin by one and decreasing the count in another bin by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under such a bounded ℓ1 sensitivity, the privacy guarantees follows from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Next, we analyze the utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the (private) histogram step, we claim that at most only two consecutive bins can be occupied by any φj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is also true for the private histogram, because the private histogram of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 adds noise to non-empty bins only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2, if k ≥ c log(1/(δ0ζ0))/ε0, one of these two intervals (the union of which contains the true distance ∥wt − w∗∥2 Σ + σ2) is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This results in a multiplicative error bound of four, as the bin size increments by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' To show that only two bins are occupied, we show that all φj’s are close to the true distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We first show that each partition contains at most 2αcorrupt fraction of corrupted samples and thus all partitions are (2¯α, 6¯α, 6ˆρ, 6ˆρ, 6ˆρ, 6ˆρ′)-corrupt good, where ˆρ(C2, K, a, ¯α) = C2K2¯α log2a(1/6¯α) and ˆρ′(C2, K, a, ¯α) = C2K ¯α loga(1/6¯α), as defined in Definition J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 21 Let B = ⌊n/k⌋ be the sample size in each partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let ζ0 = ζ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the partition is drawn uniformly at random, for each partition Gj, the number of corrupted samples α′n satisfies α′n ∼ Hypergeometric(n, αcorruptn, n/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The tail bound gives that with probability 1 − ζ0, α′ ≤ αcorrupt + (k/n) log(2/ζ0) ≤ 2¯α , where the last inequality follows from the fact that the corruption level is bounded by αcorruption ≤ ¯α and the assumption on the sample size in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (5) which implies n ≳ log(1/(δ0ζ0)) log(1/ζ0)/(¯αε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For a particular subset Gj, Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7 implies that if B = O((d + log(1/ζ0))/¯α2), then Gj is (α′, 6¯α, 6ˆρ, 6ˆρ, 6ˆρ, 6ˆρ′)-corrupt good set with respect to (w∗, Σ, σ) from Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This means that there exists a constant C2 > 0 such that for any T1 ⊂ Sgood with |T1| ≥ (1 − 6¯α)B, we have ������ 1 |T1| � i∈T1 ⟨xi, w∗ − wt⟩2 − ∥w∗ − wt∥2 Σ ������ ≤ 6C2K2¯α log2a(1/(6¯α))∥w∗ − wt∥2 Σ , ������ 1 |T1| � i∈T1 z2 i − σ2 ������ ≤ 6C2K2¯α log2a(1/(6¯α))σ2 , and ������ 1 |T1| � i∈T1 zi ⟨xi, w∗ − wt⟩ ������ ≤ 6C2K2¯α log2a(1/(6¯α))∥w∗ − wt∥Σσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that for i ∈ Sgood, bi = z2 i + 2zi(w∗ − wt)⊤xi + (w∗ − wt)⊤xix⊤ i (w∗ − wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By the triangular inequality, we know, under above conditions, ������ 1 |T1| � i∈T1 bi − ∥w∗ − wt∥2 Σ − σ2 ������ ≤ 12C2K2¯α log2a(1/(6¯α))(∥w∗ − wt∥2 Σ + σ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (10) Which also implies that any subset T2 ⊂ Sgood and |T2| ≤ 6¯α|Sgood|, we have ������ 1 |T2| � i∈T2 bi − ∥w∗ − wt∥2 Σ − σ2 ������ ≤ 12C2K2 log2a(1/(6¯α))(∥w∗ − wt∥2 Σ + σ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (11) Recall that ψj is the (1 − 3¯α)-quantile of the dataset Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let T := {i ∈ Sgood : bi ≤ ψj}, where with a slight abuse of notations, we use Sgood to denote the set of uncorrupted samples corresponding to Gj and Sbad to denote the set of corrupted samples corresponding to Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the corruption is less than α′, we know (1 − 3¯α − α′)B ≤ |T| ≤ (1 − 3¯α + α′)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By our assumption that α′ ≤ 2¯α, we have | ¯E| ≥ (3¯α − α′)B ≥ ¯αB where ¯E := Sgood \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (11) with a choice of T2 = ¯E, we get that min i∈ ¯E bi − ∥w∗ − wt∥2 Σ − σ2 ≤ 12C2K2 log2a(1/(6¯α))(∥w∗ − wt∥2 Σ + σ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (12) This implies that ψj ≤ 12C2K2 log2a(1/(6¯α))(∥w∗ − wt∥2 Σ + σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (13) 22 Hence ��φj − ∥w∗ − wt∥2 Σ − σ2�� = ������ 1 B � i∈Gj bi · 1{bi ≤ ψj} − ∥w∗ − wt∥2 Σ − σ2 ������ = ����� 1 B � i∈T bi − ∥w∗ − wt∥2 Σ − σ2 ����� + ������ 1 B � i∈Sbad bi · 1{bi ≤ ψj} ������ ≤ 37C2K2 · ¯α log2a(1/(6¯α))(∥w∗ − wt∥2 Σ + σ2), (14) where we applied Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (13) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (10) in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' On a fixed partition Gj, we showed that if B = O((d + log(1/ζ0))/¯α2) then, with probability 1 − ζ0, |φj − ∥w∗ − wt∥2 Σ − σ2| ≤ 1 4(∥w∗ − wt∥2 Σ + σ2), which follows from our assumption that 37C2K2 · ¯α log2a(1/(6¯α)) ≤ 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Using an union bound for all subsets, we know if B = O((d + log(k/ζ0))/¯α2), then 1 − ζ0, |φj − ∥w∗ − wt∥2 Σ − σ2| ≤ 1 4(∥w∗ − wt∥2 Σ + σ2) holds for all j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the upper bound lower bound ratio is 5/3 which is less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' All the φj must lie in two bins, which will result in a factor of 4 multiplicative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' E Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 on the upper bound on clipped good points Let ˆρ(C2, K, a, α) = 2C2K2α log2a(1/(2α)) and ˆρ′(C2, K, a, α) = 2C2Kα loga(1/(2α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7 implies that if n = O((d+log(1/ζ))/(α2)) with a large enough constant, then there exists a universal constant C2 such that S3 is, with respect to (w∗, Σ, σ), (αcorrupt, 2α, ˆρ, ˆρ, ˆρ, ˆρ′)-corrupt good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The rest of the proof is under this (deterministic) resilience condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By the resilience property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (7), we know for any T ⊂ Sgood with |T| ≥ (1 − 2α)n, ����� 1 |T| � i∈T (w∗ − wt)⊤xix⊤ i (w∗ − wt) − ∥w∗ − wt∥2 Σ ����� ≤ 2C2K2α log2a(1/(2α))∥w∗ − wt∥2 Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (15) Let E := � i ∈ Sgood : (w∗ − wt)⊤xix⊤ i (w∗ − wt) > ∥w∗ − wt∥2 Σ(8C2K2 log2a(1/(2α)) + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' De- note ˜α := |E|/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We want to show that ˜α ≤ α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let T be the set of points that contain the smallest 1 − α/2 fraction in {(w∗ − wt)⊤xix⊤ i (w∗ − wt)}i∈Sgood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We know |T| = (1 − α/2)n ≥ (1 − 2α)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' To prove by contradiction, suppose ˜α > α/2, which means all data points in Sgood \\ T are larger than ∥w∗ − wt∥2 Σ(8C2K2 log2a(1/(2α)) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' From resilience property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (15), we know 1 n � i∈Sgood (w∗ − wt)⊤xix⊤ i (w∗ − wt) = 1 n � i∈T (w∗ − wt)⊤xix⊤ i (w∗ − wt) + 1 n � i∈Sgood\\T (w∗ − wt)⊤xix⊤ i (w∗ − wt) ≥ � 1 − α 2 � � 1 − 2C2K2α log2a( 1 2α) � ∥w∗ − wt∥2 Σ + α 2 (8C2K2 log2a( 1 2α) + 1)∥w∗ − wt∥2 Σ > (1 + 2C2K2α log2a(1/2α))∥w∗ − wt∥2 Σ , which contradicts Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (15) for Sgood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This shows ˜α ≤ α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Similarly, we can show that ��� i ∈ Sgood : z2 t > σ2(8C2K2 log2a(1/(2α)) + 1) ��� ≤ α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This means the rest (1 − α)n points in Sgood satisfies � (w∗ − wt)⊤xix⊤ i (w∗ − wt) + |zi| ≤ (∥wt − w∗∥ + 23 σ) � (8C2K2 log2a(1/(2α)) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that for all i ∈ Sgood, we have |x⊤ i wt − yi| = ���x⊤ i (wt − w∗) − zi ��� ≤ |x⊤ i (wt − w∗)| + |zi| ≤ �� (w∗ − wt)⊤xix⊤ i (w∗ − wt) + |zi| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By our assumption that C2K2 log2a(1/(2¯α)) ≥ 1 which follows from Assumption 2, we have ���� � i ∈ Sgood : ∥x⊤ i wt − yi∥ ≤ (∥wt − w∗∥ + σ) � 9C2K2 log2a(1/(2α)) ����� ≥ (1 − α)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (16) F Private norm estimation: algorithm and analysis Algorithm 3: Private Norm Estimator Input: S1 = {(xi, yi)}n i=1, target privacy (ε0, δ0), failure probability ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1 Let ai ← ∥xi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let ˜S = {ai}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 Partition ˜S into k = ⌊C1 log(1/(δ0ζ))/ε⌋ subsets of equal size and let Gj be the j-th partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 3 For each j ∈ [k], denote ψj = (1/|Gj|) � i∈Gj ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 4 Partition [0, ∞) into bins of geometrically increasing intervals Ω := � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , � 2−2/4, 2−1/4� , � 2−1/4, 1 � , � 1, 21/4� , � 21/4, 22/4� , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' � ∪ {[0, 0]} 5 Run (ε0, δ0)-DP histogram learner of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 on {ψj}k j=1 over Ω 6 if all the bins are empty then Return ⊥ 7 Let [ℓ, r] be a non-empty bin that contains the maximum number of points in the DP histogram 8 Return ℓ Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 3 is (ε0, δ0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' If {xi}n i=1 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' samples from (K, a)-sub-Weibull distributions with zero mean and covariance Σ and n = ˜O �log2a(1/(δ0ζ)) ε0 � , with a large enough constant then Algorithm 3 returns Γ such that, with probability 1 − ζ, 1 √ 2 Tr(Σ) ≤ Γ ≤ √ 2 Tr(Σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide a proof in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Proof of Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 on the private norm estimation By Hanson-Wright inequality in Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 and union bound, there exists constant c > 0 such that with probability 1 − ζ, |1 b b � i=1 ∥xi∥2 − Tr(Σ)| ≤ cK2 Tr(Σ) �� log(1/ζ) b + log2a(1/ζ) b � , (17) 24 This means there exists a constant c′ > 0 such that if b ≥ c′K2 log2a(k/ζ), then for all j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' |ψj − Tr(Σ)| ≤ 21/8 Tr(Σ) (18) With probability 1 − ζ, {ψj}k j=1 lie in interval of size 21/4 Tr(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Thus, at most two consecutive bins are filled with {ψj}k j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Denote them as I = I1∪I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Our analysis indicates that P(ψi ∈ I) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By private histogram in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2, if k ≥ log(1/(δζ))/ε, |ˆpI − ˜pI| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='01 where ˆpI is the empirical count on I and ˜pI is the noisy count on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under this condition, one of these two intervals are released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This results in multiplicative error of √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' G Proof of the resilience in Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7 We apply following resilience property for general distribution characterized by Orlicz function from Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ((Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Dataset S = {xi ∈ Rd}n i=1 consists i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' samples from a distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Suppose D is zero mean and satisfies Ex∼D � ψ � (v⊤x)2 κ2Ex∼D[(v⊤x)2] �� ≤ 1 for all v ∈ Rd, where ψ(·) is Orlicz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let Σ = Ex∼D[xx⊤].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Suppose α ≤ ¯α, where ¯α satisfies (1 + ¯α/2) · 2κ2¯αψ−1(2/¯α) < 1/3, ¯α ≤ 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then there exists constant c1, C2 such that if n ≥ c1((d + log(1/ζ))/(α2)), with probability 1 − ζ, for any T ⊂ S of size |T| ≥ (1 − α)n, the following holds: �����Σ−1/2 � 1 |T| � i∈T xi ������ ≤ C2κα � ψ−1(1/α) (19) and �����Id − Σ−1/2 � 1 |T| � i∈T xix⊤ i � Σ−1/2 ����� 2 ≤ C2κ2αψ−1(1/α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (20) Let ψ(t) = et1/(2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It is easy to see that ψ(t) is a valid Orlicz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then if xi is (K, a)-sub- Weibull, then we know �����Σ−1/2 � 1 |T| � i∈T xi ������ ≤ C2Kα � log2a(1/α) , (21) and �����Id − Σ−1/2 � 1 |T| � i∈T xix⊤ i � Σ−1/2 ����� 2 ≤ C2K2α log2a(1/α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (22) This implies (1 − C2K2α log2a(1/α))Id ⪯ Σ−1/2 � 1 |T| � i∈T xix⊤ i � Σ−1/2 ⪯ (1 + C2K2α log2a(1/α))Id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (23) Using the fact that C⊤AC ⪯ C⊤BC if A ⪯ B, we know (1 − C2K2α log2a(1/α))Σ ⪯ 1 |T| � i∈T xix⊤ i ⪯ (1 + C2K2α log2a(1/α))Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (24) 25 This implies resilience properties of xi and zi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (8) in Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Next, we show the resilience property of xizi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By ab ≤ a2 2 + b2 2 , for any fixed v ∈ Rd, E[exp �� | ⟨xizi, v⟩ |2 K4σ2v⊤Σv �1/(4a)� ] ≤ E � exp ��| ⟨xi, v⟩ |2 K2v⊤Σv �1/(2a) /2 � exp �� z2 i K2σ2 �1/(2a) /2 �� (25) ≤ 1 2 � E � exp ��| ⟨xi, v⟩ |2 K2v⊤Σv �1/(2a)�� + E � exp �� z2 i K2σ2 �1/(2a)��� (26) ≤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (27) Since E[xizi] = 0, (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019, Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3) implies that there exists constant c1, C2 > 0 such that if n ≥ c1(d + log(1/ζ))/(α2), with probability 1 − ζ, for any T ⊂ Sgood of size |T| ≥ (1 − α)n, �����Σ−1 � 1 |T| � i∈T xizi ������ ≤ C2K2σα log2a(1/α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (28) H Proof of Theorem 4 on the analysis of Algorithm 1 The main theorem builds upon the following lemma that analyzes a (stochastic) gradient descent method, where the randomness is from the DP noise we add and the analysis only relies on certain deterministic conditions on the dataset including resilienece and concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 4 follows in a straightforward manner by collecting Theorem 5, Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2, and Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 1 is (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under Assumptions 1 and 2 for any ζ ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 1) and α ≥ αcorrupt satisfying K2α log2a(1/α) log(κ) ≤ c for some universal constant c > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' if distance threshold is small enough such that θt ≤ 3C1/2 2 K loga(1/(2α)) · (∥w∗ − wt∥Σ + σ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (29) and large enough such that the number of clipped clean data points is no larger than αn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' at every round,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' the norm threshold is large enough such that Θ ≥ K � Tr(Σ) loga(n/ζ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (30) and sample size is large enough such that n = O � K2d log(d/ζ) log2a(n/ζ) + d + log(1/ζ) α2 + K2T 1/2d log(T/δ) loga(n/(αζ)) εα � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (31) with a large enough constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' then the choices of a step size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' η = 1/(Cλmax(Σ)) for some C ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1, and the number of iterations, T = ˜Θ (κ log (∥w∗∥)) , ensures that Algorithm 1 outputs wT satisfying the following with probability 1 − ζ: Eν1,··· ,νt∼N(0,Id)[∥wT − w∗∥2 Σ] ≲ K4σ2 log2(κ)α2 log4a(1/α) , (32) where the expectation is taken over the noise added for DP and ˜Θ(·) hides logarithmic terms in K, σ, d, n, 1/ε, log(1/δ), 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 26 Proof of Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We first prove a set of deterministic conditions on the clean dataset, which is sufficient for the analysis of the gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Step 1: Sufficient deterministic conditions on the clean dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let Sgood be the uncorrupted dataset for S3 and Sbad be the corrupted datapoints in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let G := Sgood ∩ S3 = S3 \\ Sbad denote the clean data that remains in the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let λmax = ∥Σ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Define ˆΣ := (1/n) � i∈G xix⊤ i , ˆB := Id − ηˆΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4 implies that if n = O(K2d log(d/ζ) log2a(n/ζ)), then 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='9Σ ⪯ ˆΣ ⪯ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (33) We pick step size η such that η ≤ 1/(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1λmax) to ensure that η ≤ 1/∥ˆΣ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the covariates {xi}i∈S are not corrupted, from Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3, we know with probability 1 − ζ, for all i ∈ S3, ∥xi∥2 ≤ K2 Tr(Σ) log2a(n/ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (34) Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7 implies that if n = O((d + log(1/ζ))/(α2)), then there exists a universal constant C2 such that S3 is, following Definition J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='6, with respect to (w∗, Σ, σ), (αcorrupt, α, C2K2α log2a(1/α), C2K2α log2a(1/α), C2K2α log2a(1/α), C2Kα loga(1/α))-corrupt good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Such corrupt good sets have a sufficiently large, 1 − αcorrupt, fraction of points that satisfy a good property that we need: resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The rest of the proof is under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (33), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (34), and that Sgood is resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Step 2: Upper bounding the deterministic noise in the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In this step, we bound the deviation of the gradient from its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' There are several sources of deviation: (i) clipping, (ii) adversarial corruptions, and (iii) randomness of the data noise and privacy noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We will show that deviations from all these sources can be controlled deterministically under the corrupt-goodness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', resilience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let φt = ( � 2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='25/δ0)Θθt)/(ε0n), which ensures that we add enough noise to guarantee (ε0, δ0)-DP for each step of gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This follows from the standard Gaussian mechanism in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 and the fact that each gradient is clipped to the norm of Θθt, resulting in a DP sensitivity of Θθt/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The fact that this sensitivity scales as 1/n is one of the main reasons for the performance gain we get over Varshney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022) that uses a minimatch of size n/κ with sensitivity scaling as κ/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Define g(t) i := xi(x⊤ i wt − yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For i ∈ Sgood, we know yi = x⊤ i w∗ + zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let ˜g(t) i = clipΘ(xi)clipθt(x⊤ i wt − yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (34), clipΘ(xi) = xi for all i ∈ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' From Algorithm 1, we can write one-step update rule as follows: wt+1 − w∗ =wt − η � 1 n � i∈S ˜g(t) i + φtνt � − w∗ = � I − η n � i∈G xix⊤ i � (wt − w∗) + η n � i∈G xizi + η n � i∈G (g(t) i − ˜g(t) i ) − ηφtνt − η n � i∈Sbad ˜g(t) i (35) Let Et := {i ∈ G : θt ≤ |x⊤ i wt − yi|} be the set of clipped clean data points such that � i∈G(g(t) i − ˜g(t) i ) = � i∈Et(g(t) i − ˜g(t) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We define ˆv := (1/n) � i∈G xizi, u(1) t := (1/n) � i∈Et xix⊤ i (wt − w∗), u(2) t := (1/n) � i∈Et −xizi, and u(3) t := (1/n) � i∈Sbad∪Et ˜g(t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We can further write the update rule as: wt+1 − w∗ = ˆB(wt − w∗) + ηˆv + ηu(1) t−1 + ηu(2) t−1 − ηφtνt − ηu(3) t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (36) 27 We bound each term one-by-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since G ⊂ Sgood and |G| = (1 − αcorrupt)n, using the resilience property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (6), we know ∥Σ−1/2ˆv∥ = (1 − αcorrupt) max ∥v∥=1 Σ−1/2 � v, 1 (1 − αcorrupt)n � i∈G xizi � ≤ (1 − αcorrupt)C2K2α log2a(1/α)σ (37) ≤ C2K2α log2a(1/α)σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (38) Let ˜α = |Et|/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By assumption, we know ˜α ≤ α (which holds for the given dataset due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2), and ∥Σ−1/2u(1) t ∥ = ∥Σ−1/2 1 n � i∈Et xix⊤ i (wt − w∗)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' From Corollary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' we know ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='u⊤ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i Σ−1/2 − Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ1/2(wt − w∗)∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i Σ−1/2 − Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ1/2(wt − w∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i Σ−1/2 − Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='������ · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='���Σ1/2(wt − w∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='≤2 − ˜α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='˜α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='C2K2α log2a(1/α) ∥wt − w∗∥Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This implies that ∥Σ−1/2u(1) t ∥ ≤ ∥Σ−1/2 1 n � i∈E xix⊤ i (wt − w∗)∥ ≤ � ˜α + 2C2K2α log2a(1/α) � ∥wt − w∗∥Σ ≤ 3C2K2α log2a(1/α) ∥wt − w∗∥Σ , (39) where the last inequality follows from the fact that ˜α ≤ α and our assumption that C2K2 log2a(1/¯α) ≥ 1 from Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Similarly, we use resilience property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (6) instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (7), we can show that ∥Σ−1/2u(2) t ∥ ≤ 3C2K2α log2a(1/α)σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (40) 28 Next, we consider u(3) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since |Sbad| ≤ αcorruptn and |Et| ≤ αn, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (9) and Corollary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='8, we have ∥Σ−1/2u(3) t ∥ = max v:∥v∥=1 1 n � i∈Sbad∪Et v⊤Σ−1/2xiclipθt(x⊤ i wt − yi) ≤ 2C2Kα loga(1/α)θt ≤ 6C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5 2 K2α log2a(1/α)(∥wt − w∗∥Σ + σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (41) Now we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (38), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (39), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (40) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (41) to bound the final error from update rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Step 3: Analysis of the t-steps recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We have controlled the deterministic noise in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In this step, we will upper bound the noise introduced by the Gaussian noise for the purpose of privacy, and show the expected distance to optimum decrease every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We want to emphasize that most of our technical contribution is in the convergence analysis (Step 3 and Step 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' More precisely, naive linear regression analysis can only show a suboptimal error rate of ∥ ˆw−w⋆∥Σ = ˜O(κασ) with sample size n = ˜O(d/α2+κ1/2d/(εα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Define ut = (ˆv+u(1) t +u(2) t −u(3) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (36): wt+1 − w∗ = ˆB(wt − w∗) + ηut − ηφtνt (42) =(Id − ηˆΣ)(wt − w∗) + ηut − ηφtνt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (43) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (39), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (40) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (41), it follows that ∥wt+1 − w∗∥Σ ≤ (1 − 1 κ)∥wt − w∗∥Σ + α(σ + ∥wt − w∗∥Σ) where we omitted constants for simplicity, which after T = ˜O(κ) iterations achieves a sub-optimal error rate ∥wT − w∗∥Σ = ˜O(κασ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' One attempt to get around it is to take the Euclidean norm instead, which gives, after some calculations, E[∥wt+1 − w∗∥2] ≤ E[∥wt − w∗∥2] − η � ∥wt − w∗∥2 Σ − α2σ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This implies that E[∥wt+1 − w∗∥2] strictly decreases as long as ∥wt − w∗∥2 Σ > Cα2σ2, which is the desired statistical error level we are targeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' With this analysis, we can show that in T = ˜O(κ) iterations, there exists at least one model wt that achieves E[∥wt − w∗∥2 Σ] = ˜O(α2σ2) among all the intermediate models we have seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, the problem is that under differential privacy, there is no way we could select this good model wt among T models that we have, as privacy-preserving techniques for model selection are not accurate enough to achieve the desired level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Hence, we came up with the following novel analysis that does not suffer from such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (36) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (42) as wt+1 − w∗ = ˆB(wt − w∗) + ηut − ηφtνt (44) = ˆBt+1(w0 − w∗) + η t � i=0 ˆBiut−i − η t � i=0 φt−i ˆBiνt−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (45) 29 Taking expectations of ˆΣ-norm square with respect to ν1, · · · , νt, we have Eν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=',νt∼N(0,Id)∥wt+1 − w∗∥2 ˆΣ (46) ≤ 2∥ ˆBt+1(w0 − w∗)∥2 ˆΣ + 2E[∥η t � i=0 ˆBiut−i∥2 ˆΣ] + η2 t � i=0 Tr( ˆB2i ˆΣ)E[φ2 t−i] (47) ≤ 2∥ ˆBt+1(w0 − w∗)∥2 ˆΣ + 2η2E[ t � i=0 t � j=0 ∥ ˆBiut−i∥ˆΣ∥ ˆBjut−j∥ˆΣ] (48) + η2 t � i=0 Tr( ˆB2i ˆΣ)E[φ2 t−i] , (49) where at the second step we used the fact that ν1, ν2, · · · , νt are independent isotropic Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that η∥ ˆBiut−i∥ˆΣ = η∥ˆΣ1/2 ˆBi ˆΣ1/2 ˆΣ−1/2ut−i∥ ≤ η∥ˆΣ1/2 ˆBi ˆΣ1/2∥2 · ∥ˆΣ−1/2ut−i∥ ≤ η∥ˆΣ1/2 ˆBi ˆΣ1/2∥2 ˆρ(α) (∥wt−i − w∗∥ˆΣ + σ) ≤ 1 i + 1 ˆρ(α) (∥wt−i − w∗∥ˆΣ + σ) , where ˆρ(α) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1(6C2 + 6C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5 2 )K2α log2a(1/α), and the second inequality follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (39), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (40), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (41) and the deterministic condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that the last inequality is true because η ≤ 1/(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1λmax) and ∥ˆΣ1/2 ˆBi ˆΣ1/2∥2 ≤ ∥Id − ηˆΣ∥i 2∥ˆΣ∥2 ≤ λmax/(i + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This implies E[η2 t � i=0 t � j=0 ∥ ˆBiut−i∥ˆΣ∥ ˆBjut−j∥ˆΣ] (50) ≤ 4 E[ t � i=0 t � j=0 ˆρ(α)2 (i + 1)(j + 1)(E[∥wt−i − w∗∥2 ˆΣ] + E[∥wt−j − w∗∥2 ˆΣ] + σ2) (51) ≤ 8( t � i=0 1 i + 1)2ˆρ(α)2(max i E[∥wt−i − w∗∥2 ˆΣ] + σ2) (52) ≤ 8(log t)2ˆρ(α)2(max i E[∥wt−i − w∗∥2 ˆΣ] + σ2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (53) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ∥ ˆBt+1(w0 − w∗)∥2 ˆΣ = ∥ˆΣ1/2 ˆBt+1 ˆΣ−1/2 ˆΣ1/2(w0 − w∗)∥2 ≤ (1 − 1 κ)2(t+1)∥w0 − w∗∥2 ˆΣ ≤ e−2(t+1)/κ∥w0 − w∗∥2 ˆΣ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 30 and for n ≳ (1/ε) � κd log(1/δ)/α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' η2 t � i=0 Tr( ˆB2i ˆΣ)E[φ2 t−i] (54) ≤η2 t � i=0 ∥Id − ηˆΣ∥2i 2 ∥ˆΣ∥2 · 2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='25/δ0)K2 Tr(Σ) log2a(n/ζ0)C2K2 log2a(1/(2α))(E[∥wt−i − w∗∥2 Σ] + σ2) ε2 0n2 (55) ≤4 t � i=0 ( 1 i + 1)2ˆρ(α)2(E[∥wt−i − w∗∥2 ˆΣ] + σ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (56) We have Eν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=',νt∼N(0,Id)[∥wt+1−w∗∥2 ˆΣ] ≤ 2e−2(t+1)/κ∥w0−w∗∥2 ˆΣ+20(log t)2ˆρ(α)2(max i∈[t] E[∥wt−i−w∗∥2 ˆΣ]+σ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that this also implies that E[∥(wt′+t − w∗)∥2 ˆΣ|wt′] ≤ 2e−2t/κ∥wt′ − w∗∥2 ˆΣ + 20ˆρ(α)2 t−1 � i=0 ( 1 i + 1)2(E[∥wt′+t−i − w∗∥2 ˆΣ|wt′] + σ2) , (57) which implies E[∥(wt′+t − w∗)∥2 ˆΣ] ≤ 2e−2t/κE[∥wt′ − w∗∥2 ˆΣ] + 20ˆρ(α)2 t−1 � i=0 ( 1 i + 1)2(E[∥wt′+t−i − w∗∥2 ˆΣ] + σ2) (58) ≤ 2e−2t/κE[∥wt′ − w∗∥2 ˆΣ] + 20(log t)2ˆρ(α)2(max i∈[t] E[∥wt′+t−i − w∗∥2 ˆΣ] + σ2) (59) Step 4: End-to-end analysis of the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the last step, we shown that the amount of estimation error decrease depends on the estimation error of the previous t steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In order for the estimation error to decrease by a constant factor, we will take t = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Roughly speaking, we will prove that for every κ steps, the estimation error will decrease by a constant factor, if it is much larger than O((log κ)2ˆρ(α)2σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This implies we will reach O((log κ)2ˆρ(α)2σ2) error with in ˜O(κ) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For any integer s ≥ 0, as long as maxi∈[(s−1)κ+1,sκ] E[∥wi − w∗∥2 ˆΣ] ≥ 2(log κ)2ˆρ(α)2σ2, max i∈[sκ+1,(s+1)κ] E[∥wi − w∗∥2 ˆΣ] ≤ ( 1 e2 + (log κ)2ˆρ(α)2) max i∈[(s−1)κ+1,sκ] E[∥wi − w∗∥2 ˆΣ] + (log 2κ)2ˆρ(α)2σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (60) Assuming ˆρ(α)2(log κ)2 ≤ 1/2 − 1/e2, the maximum expected error in a length κ sequence decrease by a factor of 1/2 every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Now we bound the maximum expected error in the first length κ sequence: maxi∈[0,κ−1] E[∥wi − w∗∥2 ˆΣ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since E[∥wi − w∗∥2 ˆΣ] ≤ e−2i/κ∥w0 − w∗∥2 ˆΣ + (log i)2ˆρ(α)2 max j∈[0,i−1] E[∥wj − w∗∥2 ˆΣ] + (log i)2ˆρ(α)2σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 31 As a function of i, maxj∈[0,i−1] E[∥wj − w∗∥2 ˆΣ] only increase when it is smaller than 1 1 − (log i)2ˆρ(α)2 (∥w0 − w∗∥2 ˆΣ + (log i)2ˆρ(α)2σ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Thus we conclude max i∈[0,κ−1] E[∥wi − w∗∥2 ˆΣ] ≤ 1 1 − (log κ)2ˆρ(α2)(∥w0 − w∗∥2 ˆΣ + (log κ)2ˆρ(α2)σ2) s = log(∥w∗∥/(ˆρ(α)σ)) will give us E[∥wsκ+1 − w∗∥2 ˆΣ] ≤ (log κ)2ˆρ(α)2σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' I Lower bounds I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 for label corruption lower bounds We first prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Consider an α label-corrupted dataset S = {(xi, yi)}n i=1 with α < 1/2, that is generated from either xi ∼ N(0, 1), yi ∼ N(0, 1) or xi ∼ N(0, 1), zi ∼ N(0, 1−α2), yi = αxi+zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It is impossible to distinguish the two hypotheses with probability larger than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the first case, (xi, yi) ∼ P1 = N(0, �1 0 0 1 � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the second case, (xi, yi) ∼ P2 = N(0, �1 α α 1 � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By simple calculation, it holds that DKL(P1||P2) = − 1 2 log(1 − α2) ≤ α2/2 for all α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then, Pinsker’s inequality implies that DTV (P1||P2) ≤ α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the covariate xi follows from the same distribution in the two cases, and the total variation distance between the two cases is less than α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This means there is an label corruption adversary that change α/2 fraction of yi’s in P1 to make it identical to P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Therefore, no algorithm can distinguish the two cases with probability better than 1/2 under α fraction of label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since Σ = 1, σ2 ∈ [3/4, 1], the first case above has w∗ = 0, and the second case has w∗ = α, this implies that no algorithm is able to achieve E[∥ ˆw − w∗∥Σ] < σα for all instances with ∥w∗∥ ≤ 1 under α fraction of label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' J Technical Lemmas Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 (Hanson-Wright inequality for subWeibull distributions Sambale (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let S = {xi ∈ Rd}n i=1 be a dataset consist of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' samples from (K, a)-subWeibull distributions, then P ������ 1 n n � i=1 ∥xi∥2 − Tr(Σ) ����� ≥ t � ≤ 2 exp � − min � nt2 K4(Tr(Σ))2 , � nt K2 Tr(Σ) � 1 2a �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (61) 32 Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let Y ∼ Lap(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then for all h > 0, we have P(|Y | ≥ hb) = e−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' If x ∈ Rd is (K, a)-subWeibull for some a ∈ [1/2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then for any fixed v ∈ Rd, with probability 1 − ζ, ⟨x, v⟩2 ≤ K2v⊤Σv log2a(1/ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (62) with probability 1 − ζ, ∥x∥2 ≤ K2 Tr(Σ) log2a(1/ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (63) We provide a proof in Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Dataset S = {xi ∈ Rd}n i=1 consists i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' samples from a zero mean distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Suppose D is (K, a)-subWeibull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Define Σ = Ex∼D[xx⊤].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then there exists a constant c1 > 0 such that with probability 1 − ζ, ����� 1 n n � i=1 xix⊤ i − Σ ����� ≤ c1 � �K2d log(d/ζ) log2a(n/ζ) n + � K2d log(d/δ) log2a(n/ζ) n � � ∥Σ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (64) Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5 (Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let x ∈ Rd ∼ N(0, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then there exists universal constant C6 such that with probability 1 − ζ, ∥x∥2 ≤ C Tr(Σ) log(1/ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (65) Definition J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='6 (Corrupt good set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We say a dataset S is (αcorrupt, α, ρ1, ρ2, ρ3, ρ4)-corrupt good with respect to (w∗, Σ, σ) if it is αcorrupt-corruption of an (α, ρ1, ρ2, ρ3, ρ4)-resilient dataset Sgood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under Assumptions 1 and 2, there exists positive constants c1 and C2 such that if n ≥ c1((d + log(1/ζ))/α2, then with probability 1 − ζ, Sgood is, with respect to (w∗, Σ, σ), (α, C2K2α log2a(1/α), C2K2α log2a(1/α), C2K2α log2a(1/α), C2Kα loga(1/α))-resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide a proof in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Corollary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='8 (Lemma 10 from Steinhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2017) and Lemma 25 from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For a (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' ρ4)-resilient set S with respect to (w∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' γ) and any 0 ≤ ˜α ≤ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' the following holds for any subset T ⊂ S of size at least ˜αn and for any unit vector v ∈ Rd: ��� 1 |T| � (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='yi)∈T ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' xi⟩(yi − x⊤ i w∗) ��� ≤ 2 − ˜α ˜α ρ1 √ v⊤Σv σ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (66) ������ 1 |T| � xi∈T ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' xi⟩2 − v⊤Σv ������ ≤ 2 − ˜α ˜α ρ2v⊤Σv ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (67) ��� 1 |T| � (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='yi)∈T (yi − x⊤ i w∗)2 − σ2��� ≤ 2 − ˜α ˜α ρ3 σ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' and (68) ������ 1 |T| � xi∈T ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' xi⟩ ������ ≤ 2 − ˜α ˜α ρ4 √ v⊤Σv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (69) 33 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Proof of technical lemmas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Proof of Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3 Using Markov inequality, P � ⟨v, x⟩2 ≥ t2� = P � e⟨v,x⟩1/a ≥ et1/a� (70) ≤ e−t1/aE[e⟨v,x⟩1/a] (71) ≤ e−t1/aeK(E[⟨v,x⟩2])1/(2a) (72) = 2 exp � − � t2 K2E[⟨v, x⟩2] �1/(2a)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (73) This implies for any fixed v, with probability 1 − ζ, ⟨x, v⟩2 ≤ K2v⊤E[xx⊤]v log2a(1/ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (74) For j-th coordinate, let v = ej where j ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 implies E � �exp � � � x2 j K2 Tr(Σ) �1/(2a)� � � � ≤ E � �exp � � � x2 j K2Σjj �1/(2a)� � � � ≤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (75) Note that f(x) = xα is concave function for α ≤ 1 and x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Then (a1 + · · · ak)α ≤ aα 1 + · · · aα k holds for any positive numbers a1, · · · , ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By our assumption that 1/(2a) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , we have E[exp �� ∥x∥2 K2 Tr(Σ) �1/(2a)� ] = E[exp ��x2 1 + x2 2 + · · · + x2 d K2 Tr(Σ) �1/(2a)� ] (76) ≤ E[ d � j=1 exp � � � x2 j K2 Tr(Σ) �1/(2a)� �] (77) ≤ � � � � � � �d j=1 E[exp �� x2 j K2 Tr(Σ) �1/(2a)� ] d � � � � � � d (78) ≤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (79) By Markov inequality, P (∥x∥ ≥ t) = P � e∥x∥1/a ≥ et1/a� (80) ≤ e−t1/aE[e∥x∥1/a] (81) ≤ exp � − � t2 K2 Tr(Σ) �1/(2a)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (82) This implies with probability 1 − ζ, ∥x∥2 ≤ K2 Tr(Σ) log2a(1/ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (83) 34 Figure 2: Performance of various techniques on DP linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' d = 10 in all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' n = 107, κ = 1 in the 2nd experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' n = 107, σ = 1 in the 3rd experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Figure 3: Non-robustness of existing techniques to adversarial corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' n = 107, σ = 1 in both experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' K Experiments K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 DP Linear Regression Experimental results for ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 can be found in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The observations are similar to the ϵ = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In particular, DP-SSP has poor performance when σ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In other settings, DP-SSP has better performance than DP-RobGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 DP Robust Linear Regression We now illustrate the robustness of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We consider the same experimental setup as in Section 5 and randomly corrupt α fraction of the response variables by setting them to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Figure 3 presents the results from this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It can be seen that none of the baselines are robust to adversarial corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' They can be made arbitrarily bad by increasing the magnitude of corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In contrast, DP-RobGD is able to handle the corruptions well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3 Stronger adversary for DP Robust Linear Regression In this section, we consider a stronger adversary for DP-RobGD than the one considered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Recall, for the adversary model considered in Section 5, DP-RobGD was able to consistently estimate the parameter w∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', the parameter recovery error goes down to 0 as n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is because the algorithm was able to easily identify the corruptions and ignore the corresponding points while performing gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We now construct a different instance where the corruptions are hard to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Consequently, DP-RobGD can no longer be consistent against the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This hard instance is inspired by the lower bound in Bakshi & Prasad (2021) (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 of Bakshi & Prasad (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is a 2 dimensional problem where the first 35 Non Private OLS DP-SSP Non Private SGD DP-AMBSSGD DP-RobGD DP-RobGD*d=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='g=1,K=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='E=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Parameter Estimation Error 100 10 0 105 106 107 NumberofSamplesE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Estimation Error 10- Parameter 10 10-6 10 10-4 10-3 10-2 10-1 100 aE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Error Estimation Parameter 10- 100 101 KE=1 ErTor 100 Estimation I Parameter 10 10-3 10-2 10-1 fraction of corruptionsα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='E=1 Parameter Estimation Error 100 10-1 0 10 101 2 × 107 3 × 102 4 × 102Figure 4: Performance against the stronger adversary covariate is sampled uniformly from [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The second covariate, which is uncorrelated from the first, is sampled from a distribution with the following pdf p(x(2)) = � � � � � α 2 if x(2) ∈ {−1, 1} 1−α 2ασ if x(2) ∈ [−σ, σ] 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We set σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The noise zi is sampled uniformly from [−σ, σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We consider two possible parameter vectors w∗ = (1, 1) and w∗ = (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It can be shown that the total variation (TV) distance between these problem instances (each parameter vector corresponds to one problem instance) is Θ(α) (Bakshi & Prasad, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' What this implies is that, one can corrupt at most α fraction of the response variables and convert one problem instance into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the distance (in Σ norm) between the two parameter vectors is Ω(ασ), any algorithm will suffer an error of Ω(ασ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We generate 107 samples from this problem instance and add corruptions that convert one problem instance to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Figure 4 presents the results from this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' It can be seen that our algorithm works as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In particular, it is not consistent in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Moreover, the parameter recovery error increases with the fraction of corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' L Heavy-tailed noise We study the heavy-tailed regression settings where the label noise zi is hypercontractive, which is common in robust linear regression literature (Klivans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We define (κ2, k)-hypercontractivity as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is a heavy-tailed distribution we have bound only up to the k-th moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Definition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For integer k ≥ 4, a distribution Pµ,Σ is (κ2, k)-hypercontractive if for all v ∈ Rd, Ex∼PX[|⟨v, (x − µ)⟩|k] ≤ κk 2(v⊤Σv)k/2, where Σ is the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We give a formal description of our setting in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that we consider the input vector xi to be sub-Weibull and label noise zi to be hypercontractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' If both xi and zi are hypercontractive, the uncorrupted set Sgood is known to be not resilient (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, by (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2019, Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='10), we can clip xi by O( √ d∥Σ∥2), and obtain a (α, O(κα1−1/k), O(κα1−2/k), O(κα1−2/k), O(κα1−1/k))-resilient set (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This would result in sub-optimal error rate ˜O(κα1−2/k), which depends on condition number κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For convenience, in this section, we further assume that xi and zi are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=" In the dependent case, 36 E=1 Error 100 Parameter Estimation 10' 10-2 10-4 10-5 106 107 Number of SamplesE=1 Parameter Estimation Error 3×10 2 × 10-1 10-2 10-1 fraction of corruptionsthe only thing we need to change is the ρ1 resilience from O(α1−1/k) to O(α1−2/k) in Lemma L." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This would result in O(α1−3/k) error rate if we plug this new resilience in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Assumption 3 ((Σ, σ2, w∗, K, a, κ2, k)-model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A multiset Sgood = {(xi ∈ Rd, yi ∈ R)}n i=1 of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' samples is from a linear model yi = ⟨xi, w∗⟩ + zi, where the input vector xi is zero mean, E[xi] = 0, with a positive definite covariance Σ := E[xix⊤ i ] ≻ 0, and the independent label noise zi is zero mean, E[zi] = 0, with variance σ2 := E[z2 i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We assume that the marginal distribution of xi is (K, a)-sub-Weibull and that of zi is (κ2, k)-hypercontractive, as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This is similar to the light-tailed case in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The main difference is that the noise zi is heavy-tailed and independent of the input xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Assumption 4 (αcorrupt-corruption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Given a dataset Sgood = {(xi, yi)}n i=1, an adversary inspects all the data points, selects αcorruptn data points denoted as Sr, and replaces the labels with arbitrary labels while keeping the covariates unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We let Sbad denote this set of αcorruptn newly labelled examples by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let the resulting set be S := Sgood ∪ Sbad \\ Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We further assume that the corruption rate is bounded by αcorrupt ≤ ¯α, where ¯α is a positive constant that depends on κ2, k, K, log(κ), a and ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Compared to Assumption 2, this only difference is in the conditions on ¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Similar as Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7, we have the following lemma showing that under Assumption 3, the uncorrupted dataset can Sgood is corrupt-good, which means that it can be seen as being corrupted from a resilient set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We provide the proof in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' A multiset of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' labeled samples Sgood = {(xi, yi)}n i=1 is generated from a linear model: yi = ⟨xi, w∗⟩ + zi, where feature vector xi has zero mean and covariance E[xix⊤ i ] = Σ ≻ 0, independent label noise zi has zero mean and covariance E[z2 i ] = σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Suppose xi is (K, a)- sub-Weibull, zi is (κ2, k)-hypercontractive, then there exist constants c1, C2 > 0 such that, for any 0 < α ≤ ˜α ≤ c where c ∈ (0, 1/2) is some absolute constant if n ≥ c1 � d ζ2(1−1/k)α2(1−1/k) + k2α2−2/kd log d ζ2−4/kκ2 2 + κ2 2d log d α2/k + d + log(1/ζ) ˜α2 � , (84) then with probability 1 − ζ, Sgood is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2α, α, C2k(ka)aKκ2α1−1/kζ−1/k, C2K2˜α log2a(1/˜α), C2k2κ2 2α1−2/kζ−2/k, C2K ˜α loga(1/˜α))-corrupt good with respect to (w∗, Σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the rest of this section, we assume we have a (O(α), α, ρ1, ρ2, ρ3, ρ4)-corrupt good set under Assumption 3 and present following algorithm and our main theorem under this setting in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 37 We also provide the proof in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 4: Robust and Private Linear Regression for heavy-tailed noise Input: dataset S = {(xi, yi)}3n i=1, (ε, δ), T, learning rate η, failure probability ζ, target error rate α, distribution parameter (K, a) 1 Partition dataset S into three equal sized disjoint subsets S = S1 ∪ S2 ∪ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 2 δ0 ← δ/(2T), ε0 ← ε/(4 � T log(1/δ0)), ζ0 ← ζ/3, w0 ← 0 3 Γ ← PrivateNormEstimator(S1, ε0, δ0, ζ0), Θ ← K √ 2Γ loga(n/ζ0) 4 for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' , T − 1 do 5 γt ← RobustPrivateDistanceEstimator(S2, wt, ε0, δ0, α, ζ0) 6 θt ← 2√2γt · � max{8ρ2/α, 8ρ3/α} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 7 Sample νt ∼ N (0, Id) 8 wt+1 ← wt − η � 1 n � i∈S3 � clipΘ(xi)clipθt � w⊤ t xi − yi �� + √ 2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='25/δ0)Θθt ε0n νt � 9 Return wT Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 4 is (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under (Σ, σ2, w∗, K, a, κ2, k)-model of Assumption 3 and αcorrupt-corruption of Assumption 4 and for any failure probability ζ ∈ (0, 1) and target error rate α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorrupt, if the dataset S is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2α, α, ρ1, ρ2, ρ3, ρ4)-corrupt good set S with respect to (w∗, Σ, σ) and sample size is large enough such that n =O � K2d log(d/ζ) log2a(n/ζ) + K2dT 1/2 log(T/δ) loga(n/(αζ)) � 8 max{ρ2/α, ρ3/α} + 1 εˆρ(α) � , (85) where ˆρ(α) = max{ρ1, 3ρ2, 2ρ4 � 8 max{ρ2/α, ρ3/α} + 1}, then the choices of a small enough step size, η ≤ 1/(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1λmax(Σ)), and the number of iterations, T = ˜Θ (κ log (∥w∗∥)) for a condition number of the covariance κ := λmax(Σ)/λmin(Σ), ensures that, with probability 1 − ζ, Algorithm 1 achieves Eν1,··· ,νt∼N(0,Id) � ∥wT − w∗∥2 Σ � = ˜O � ˆρ2(α)σ2 � , (86) where the expectation is taken over the noise added for DP, and ˜Θ(·) hides logarithmic terms in K, κ2, σ, d, n, 1/ε, log(1/δ), 1/α, and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' By Lemma L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2, if we set ˜α = α1−1/k, ρ1 = C2k(ka)aKκ2α1−1/kζ−1/k, ρ2 = C2K2α1−1/k log2a(1/α1−1/k),ρ3 = C2k2κ2 2α1−2/kζ−2/k, and ρ4 = C2Kα1−1/k loga(1/α1−1/k), we have following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of Assumption 4, if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorrupt ≤ α and K, a, κ2, k = O(1), then n = ˜O(d/(ζ2−2/kα2−2/k) + κ1/2d/(εα1−1/k)) samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw−w∗∥2 Σ = ˜O(ζ−2/kα2−4/k) with probability 1 − ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Simiarly, if we set ˜α = α, ρ1 = C2k(ka)aKκ2α1−1/kζ−1/k, ρ2 = C2K2α log2a(1/α),ρ3 = C2k2κ2 2α1−2/kζ−2/k, and ρ4 = C2Kα loga(1/α), we have following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of Assumption 4, if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorrupt ≤ α and K, a, κ2, k = O(1), then n = ˜O(d/(ζ2−2/kα2−2/k)+κ1/2d/(εα)+ (d + log(1/ζ)/α2)) samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw − w∗∥2 Σ = ˜O(ζ−2/kα2−2/k) with probability 1 − ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 38 As a comparison, we also apply the exponential-time robust linear regression algorithm HPTR by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2022b) under our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 7 ((Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2022b, Theorem 12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' There exist positive constants c and C such that for any ((2/11)α, α, ρ1, ρ2, ρ3, ρ4)-corrupt good set S with respect to (w∗, Σ ≻ 0, σ > 0) satisfying α < c, ρ1 < c, ρ2 < c, ρ3 < c,and ρ2 4 ≤ cα, HPTR achieves (1/σ)∥(ˆβ − β)∥Σ ≤ 32ρ1 with probability 1 − ζ, if n ≥ C d + log(1/(δζ)) εα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (87) We set ˜α = α1−1/k, ρ1 = C2k(ka)aKκ2α1−1/kζ−1/k, ρ2 = C2K2α1−1/k log2a(1/α1−1/k),ρ3 = C2k2κ2 2α1−2/kζ−2/k, and ρ4 = C2Kα1−1/k loga(1/α1−1/k), we have the following utility gaurentees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the hypothesis of Assumption 3, there exists a constant c > 0 such that for any α ≤ c, (ka)aKκ2α1−1/kζ−1/k ≤ c, k2κ2 2α1−2/kζ−2/k ≤ c and K2α1−2/k log2a(1/α1−1/k) ≤ c, it is sufficient to have a dataset of size n = O � d ζ2(1−1/k)α2(1−1/k) + k2α2−2/kd log d ζ2−4/kκ2 2 + κ2 2d log d α2/k � , (88) such that HPTR achieves (1/σ)∥ ˆw − w∗∥Σ = O(k(ka)aKκ2α1−1/kζ−1/k) with probability 1 − ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that both of our result in Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3 and Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4 are suboptimal compared to the exponential time algorithm HPTR from Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Suppose K, a, κ2, k, ζ = Θ(1), HPTR achieves (1/σ)∥w∗− ˆw∥ = ˜O(α1−1/k) with sample complexities n = d/(α2(1−1/k))+(d+log(1/δ))/(εn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, in the analysis in Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3, Algorithm 4 achieves (1/σ)∥w∗ − ˆw∥ = ˜O(α1−2/k) with the same sample complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' In the analysis in Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4, Algorithm 4 achieves the same error rate as HPTR but requires extra ˜O(d/α2) sample complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The suboptimality is caused by the gradient truncation step in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' From Theorem 7, the final error rate of HPTR only depends on the first resilience ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However in Theorem 6, the final error rate of Algorithm 4 depends on ˆρ(α) = max{ρ1, ρ2, ρ4 � ρ2/α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' When the noise is heavy-tailed, the bottleneck is the last term ρ4 � ρ2/α ≈ α1−2/k, which is due to the truncation threshold from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This cannot be tightened by using a smaller truncation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Because we can construct yi, such that there are α-fraction of points that are at the threshold level θt ≈ α−1/k(line 6 of Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' If exponential time complexity is allowed, we could robustly and privately estimate the average of the gradients by directly estimating the xiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' However, the current best efficient algorithm (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=', 2021) for estimating the mean of Gaussian with unknown covariance robustly and privately would require O(d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For a fair comparison, we also rewrite the error rates of Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3, Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4, Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='5 as the same accuracy level α and different corruption level αcorrupt respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of Assumption 4, if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorrupt ≤ αk/(k−2) and K, a, κ2, k = O(1), then n = ˜O(d/(ζ2−2/kα2(k−1)/(k−2)) + κ1/2d/(εα(k−1)/(k−2))) samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw−w∗∥2 Σ = ˜O(ζ−2/kα2) with probability 1−ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 39 Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of Assumption 4, if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorrupt ≤ αk/(k−1) and K, a, κ2, k = O(1), then n = ˜O(d/(ζ2−2/kα2) + κ1/2d/(εαk/(k−1)) + (d + log(1/ζ)/α2k/(k−1))) samples are sufficient for Algorithm 4 to achieve an error rate of (1/σ2)∥ ˆw−w∗∥2 Σ = ˜O(ζ−2/kα2) with probability 1−ζ, where κ := λmax(Σ)/λmin(Σ), ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Corollary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='8 (HPTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under the same hypotheses of Theorem 6 and under αcorrupt-corruption model of Assumption 4, if αcorrupt ≤ αk/(k−1) and α(k−2)/(k−1) ≤ c and K, a, κ2, k = O(1), then n = ˜O( d ζ2−2/kα2 + d + log(1/(δζ)) εαk/k−1 ) samples are sufficient for HPTR to achieve an error rate of (1/σ2)∥ ˆw − w∗∥2 Σ = ˜O(ζ−2/kα2) with probability 1 − ζ, ˜O(·) hides logarithmic terms in σ, d, n, 1/ε, log(1/δ), log(1/ζ) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 Proof of Theorem 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The proof follows similarly as the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We only highlight the difference in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let Sgood be the uncorrupted dataset for S3 and Sbad be the corrupted data points in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let G denote the clean data that satisfies resilience conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We know |G| ≥ (1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorrupt)n ≥ (1−α)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let λmax = ∥Σ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Define ˆΣ := (1/n) � i∈G xix⊤ i , ˆB := Id − ηˆΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='4 implies that if n = O(K2d log(d/ζ) log2a(n/ζ)), then 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='9Σ ⪯ ˆΣ ⪯ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (89) We pick step size η such that η ≤ 1/(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1λmax) to ensure that η ≤ 1/∥ˆΣ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since the covariates {xi}i∈S are not corrupted, from Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='3, we know with probability 1 − ζ, for all i ∈ S3, ∥xi∥2 ≤ K2 Tr(Σ) log2a(n/ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (90) The rest of the proof is under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (89), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (90) and the resilience conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let φt = ( � 2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='25/δ0)Θθt)/(ε0n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Define g(t) i := xi(x⊤ i wt − yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For i ∈ Sgood, we know yi = x⊤ i w∗ + zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Let ˜g(t) i = clipΘ(xi)clipθt(x⊤ i wt − yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Note that under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (90), clipΘ(xi) = xi for all i ∈ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' From Algorithm 4, we can write one-step update rule as follows: wt+1 − w∗ =wt − η � 1 n � i∈S ˜g(t) i + φtνt � − w∗ = � I − η n � i∈G xix⊤ i � (wt − w∗) + η n � i∈G xizi + η n � i∈G (g(t) i − ˜g(t) i ) − ηφtνt − η n � i∈S3\\G∪Et ˜g(t) i (91) Let Et := {i ∈ G : θt ≤ |x⊤ i wt − yi|} be the set of clipped clean data points such that � i∈G(g(t) i − ˜g(t) i ) = � i∈Et(g(t) i − ˜g(t) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' We define ˆv := (1/n) � i∈G xizi, u(1) t := (1/n) � i∈Et xix⊤ i (wt − w∗), u(2) t := (1/n) � i∈Et −xizi, and u(3) t := (1/n) � i∈S3\\G∪Et ˜g(t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 40 We can further write the update rule as: wt+1 − w∗ = ˆB(wt − w∗) + ηˆv + ηu(1) t−1 + ηu(2) t−1 − ηφtνt − ηu(3) t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (92) Since G ⊂ Sgood and |G| ≥ (1 − α)n, using the resilience property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (6), we know ∥Σ−1/2ˆv∥ = |G| max ∥v∥=1 Σ−1/2 � v, 1 |G| � i∈G xizi � ≤ (1 − α)ρ1σ (93) ≤ ρ1σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (94) Let α2 = |Et|/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Following the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2, we can show following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Lemma L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Under Assumptions 3, if θt ≥ � max{8ρ2/α, 8ρ3/α} + 1 · (∥w∗ − wt∥Σ + σ), then ��� � i ∈ G : ���w⊤ t xi − yi ��� ≥ θt ���� ≤ αn , for all t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Similar as Theorem 5, we have following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Algorithm 2 is (ε0, δ0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For an (αcorrupt, ¯α, ρ1, ρ2, ρ3, ρ4)-corrupted good dataset S2 and an upper bound ¯α on αcorrupt that satisfy Assumption 3 and ρ1 + ρ2 + ρ3 ≤ 1/4, for any ζ ∈ (0, 1), if n = O �log(1/ζ) log(1/(δ0ζ)) ¯αε0 � , (95) with a large enough constant then, with probability 1 − ζ, Algorithm 2 returns ℓ such that 1 4(∥wt − w∗∥2 Σ + σ2) ≤ ℓ ≤ 4(∥wt − w∗∥2 Σ + σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This means α2 ≤ α, and we have ∥Σ−1/2u(1) t ∥ = ∥Σ−1/2 1 n � i∈Et xix⊤ i (wt − w∗)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 41 From Corollary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' we know ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='�����∥Σ−1/2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i (wt − w∗)∥ − ∥wt − w∗∥Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='u:∥u∥=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='u⊤Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i (wt − w∗)∥ − max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='v:∥v∥=1 v⊤Σ1/2(wt − w∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='≤ max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='u:∥u∥=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='u⊤Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i Σ−1/2Σ1/2(wt − w∗)∥ − u⊤Σ1/2(wt − w∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='≤ max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='u:∥u∥=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='u⊤ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i Σ−1/2 − Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ1/2(wt − w∗)∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i Σ−1/2 − Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ1/2(wt − w∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='|Et| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i∈Et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='Σ−1/2xix⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='i Σ−1/2 − Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='������ · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='���Σ1/2(wt − w∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='≤2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='ρ2 ∥wt − w∗∥Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This implies that ∥Σ−1/2u(1) t ∥ ≤ ∥Σ−1/2 1 n � i∈E xix⊤ i (wt − w∗)∥ ≤ (α2 + 2ρ2) ∥wt − w∗∥Σ ≤ 3ρ2 ∥wt − w∗∥Σ , (96) where the last inequality follows from the fact that α2 ≤ α and our assumption that α ≤ ρ2 from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Similarly, we use resilience property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (6) instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (7), we can show that ∥Σ−1/2u(2) t ∥ ≤ 3ρ3σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (97) Next, we consider u(3) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since |S3 \\ G| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2αcorruptn and |Et| ≤ αn, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (9) and Corollary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='8, we have ∥Σ−1/2u(3) t ∥ = max v:∥v∥=1 1 n � i∈Sbad∪Et v⊤Σ−1/2xiclipθt(x⊤ i wt − yi) ≤ 2ρ4θt ≤ 2ρ4 � 8 max{ρ2/α, ρ3/α} + 1 · (∥wt − w∗∥Σ + σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (98) The analysis of convergence follows similarly as in Step 3 and Step 4 of the proof of Theorem 4 except we set ˆρ(α) = max{ρ1, 3ρ2, 2ρ4 � 8 max{ρ2/α, ρ3/α} + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (85) ensures the added Gaussian noise is small enough such that φ2 t ∥vt∥2 ≤ ˆρ(α)2(E[∥wt − w∗∥2 Σ] + σ2), which is similar as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (56) 42 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 Proof of Lemma L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' For any x that is (K, a)-sub-Weibull from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (73) implies that for any k ≥ 1, E[| ⟨v, x⟩ |k] = � ∞ 0 P(| ⟨v, x⟩ | ≥ t1/k)dt (99) ≤ � ∞ 0 2 exp � − t 1 ka (K2E[⟨v, x⟩2]) 1 2a � dt (100) = 2Kk(E[⟨v, x⟩2])k/2ka � ∞ 0 e−uuka−1du (101) = 2Kk(E[⟨v, x⟩2])k/2Γ(ka + 1) (102) ≤ 2Kk(E[⟨v, x⟩2])k/2(ka)ka (103) This implies that xi is also ((ka)aK, k)-hypercontractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' Since xi and zi are independent, we have E ���� � v, σ−1Σ−1/2xizi ���� k� = E ���� � v, Σ−1/2xi ���� k� E ���σ−1zi ��k� ≤ 2(ka)kaKkκk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (104) This means xizi is also ((ka)aKκ2, k)-hypercontractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' From Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (2019, Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='10), we know with probability 1 − ζ, there exists S1 ⊂ Sgood with |S1| ≥ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1α)|Sgood|, such that for any T ⊂ S1 with |T| ≥ (1 − α)|S1|, we have ��� 1 |T| � (xi,yi)∈S � v, σ−1Σ−1/2xi(yi − x⊤ i w∗) � ��� ≤ C2k(ka)aKκ2α1−1/kζ−1/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (105) Similarly, there exists S2 ⊂ Sgood with |S2| ≥ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='1α)|Sgood|, such that for any T ⊂ S2 with |T| ≥ (1 − α)|S2|, we have ��� 1 |T| � (xi,yi)∈T (σ−1(yi − x⊤ i w∗))2 − 1 ��� ≤ C2k2κ2 2α1−2/kζ−2/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (106) From Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='7, for any T ⊂ Sgood with |T| ≥ (1 − ˜α)|Sgood|, we have ��� 1 |T| � (xi,yi)∈T � v, Σ−1/2xi �2 − 1 ��� ≤ C2K ˜α log2a(1/˜α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (107) and ��� 1 |T| � (xi,yi)∈T � v, Σ−1/2xi � ��� ≤ C2K ˜α loga(1/˜α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' (108) Set S = S1 ∩ S2, we know |S| ≥ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2α)|Sgood| and S is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content='2α, α, C2k(ka)aKκ2α1−1/kζ−1/k, C2K2˜α log2a(1/˜α), C2k2κ2 2α1−2/kζ−2/k, C2K ˜α loga(1/˜α))-corrupt good with respect to (w∗, Σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} +page_content=' 43' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf'} diff --git a/B9FQT4oBgHgl3EQfNza6/vector_store/index.pkl b/B9FQT4oBgHgl3EQfNza6/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..865ea513ba07d44deac48242be7fa1c636983db1 --- /dev/null +++ b/B9FQT4oBgHgl3EQfNza6/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:23bf556e08c24a06de86c6004ca0fa54745b612266a65149236336be39051469 +size 309174 diff --git a/BdE2T4oBgHgl3EQfRge6/content/tmp_files/2301.03782v1.pdf.txt b/BdE2T4oBgHgl3EQfRge6/content/tmp_files/2301.03782v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f149837bd983d07313cc735a5f0803ea8ee2d47d --- /dev/null +++ b/BdE2T4oBgHgl3EQfRge6/content/tmp_files/2301.03782v1.pdf.txt @@ -0,0 +1,1366 @@ +Multiple phenotypes in HL60 leukemia cell population +Yue Wang1,2, Joseph X. Zhou3, Edoardo Pedrini3, Irit Rubin3, May Khalil3, Hong +Qian2, and Sui Huang3 +1Department of Computational Medicine, University of California, Los Angeles, +California, United States of America +2Department of Applied Mathematics, University of Washington, Seattle, +Washington, United States of America +3Institute for Systems Biology, Seattle, Washington, United States of America +Abstract +Recent studies at individual cell resolution have revealed phenotypic heterogene- +ity in nominally clonal tumor cell populations. The heterogeneity affects cell growth +behaviors, which can result in departure from the idealized exponential growth. Here +we measured the stochastic time courses of growth of an ensemble of populations of +HL60 leukemia cells in cultures, starting with distinct initial cell numbers to capture +the departure from the exponential growth model in the initial growth phase. De- +spite being derived from the same cell clone, we observed significant variations in the +early growth patterns of individual cultures with statistically significant differences in +growth kinetics and the presence of subpopulations with different growth rates that +endured for many generations. Based on the hypothesis of existence of multiple inter- +converting subpopulations, we developed a branching process model that captures the +experimental observations. +1 +Introduction +Cancer has long been considered a genetic disease caused by oncogenic mutations in so- +matic cells that confer a proliferation advantage. According to the clonal evolution theory, +accumulation of random genetic mutations produces cell clones with cancerous cell phe- +notype. Specifically, cells with the novel genotype(s) may display increased proliferative +fitness and gradually out-grow the normal cells, break down tissue homeostasis and gain +other cancer hallmarks [15]. In this view, a genetically distinct clone of cells dominates the +cancer cell population and is presumed to be uniform in terms of the phenotype of indi- +vidual cells within an isogenic clone. In this traditional paradigm, non-genetic phenotypic +variation within one clone is not taken into account. +1 +arXiv:2301.03782v1 [q-bio.PE] 10 Jan 2023 + +With the advent of systematic single-cell resolution analysis, however, non-genetic cell +heterogeneity within clonal (cancer) cell populations is found to be universal [33]. This +feature led to the consideration of the possibility of biologically (qualitatively) distinct +(meta)stable cell subpopulations due to gene expression noise, representing intra-clonal +variability of features beyond the rapid random micro-fluctuations. +Hence, transitions +between the subpopulations, as well as heterotypic interactions among them may influence +cell growth, migration, drug resistance, etc. [39, 13, 9]. Thus, an emerging view is that +cancer is more akin to an evolving ecosystem [11] in which cells form distinct subpopulations +with persistent characteristic features that determine their mode of interaction, directly +or indirectly via competition for resources [10, 36]. However, once non-genetic dynamics +is considered, cell “ecology” differs fundamentally from the classic ecological system in +macroscopic biology: the subpopulations can reversibly switch between each other whereas +species in an ecological population do not convert between each other [7]. This affords +cancer cell populations a remarkable heterogeneity, plasticity and evolvability, which may +play important roles in their growth and in the development of resistance to treatment +[30]. +Many new questions arise following the hypothesis that phenotypic heterogeneity and +transitions between phenotypes within one genetic clone are important factors in cancer. +Can tumors arise, as theoretical considerations indicate, because of a state conversion +(within one clone) to a phenotype capable of faster, more autonomous growth as opposed +to acquisition of a new genetic mutation that confers such a selectable phenotype [55, +1, 18, 34, 33, 56, 23, 41]? Is the macroscopic, apparently sudden outgrowth of a tumor +driven by a new fastest-growing clone (or subpopulation) taking off exponentially, or due +to the cell population reaching a critical mass that permits positive feedback between its +subpopulations that stimulates outgrowth, akin to a collectively autocatalytic set [17]? +Should therapy target the fastest growing subpopulations, or target the interactions and +interconversions of cancer cells? +At the core of these deliberations is the fundamental question on the mode of tumor +cell population growth that now must consider the influence of inherent phenotypic hetero- +geneity of cells and the non-genetic (hence potentially reversible) inter-conversion of cells +between the phenotypes that manifest various growth behaviors and the interplay between +these two modalities. +Traditionally tumor growth has been described as following an exponential growth law, +motivated by the notion of uniform cell division rate for each cell, i.e. a first order growth +kinetics [29]. But departure from the exponential model has long been noted. To better fit +experimental data, two major modifications have been developed, namely the Gompertz +model and the West law model [53]. While no one specific model can adequately describe +any one tumor, each model highlights certain aspects of macroscopic tumor kinetics, mainly +the maximum size and the change in growth rate at different stages. These models however +are not specifically motivated by cellular heterogeneity. Assuming non-genetic heterogene- +ity with transitions between the cell states, the population behavior is influenced by many +2 + +intrinsic and extrinsic factors that are both variable and unpredictable at the single-cell +level. Thus, unlike macroscopic population dynamics [43], tumor growth cannot be ad- +equately captured by a deterministic model, but a stochastic cell and population level +kinetic model is more realistic. +Using stochastic processes in modeling cell growth via clonal expansion has a long +history [54]. An early work is the Luria-Delbr¨uck model, which assumes cells grow deter- +ministically, with wildtype cells mutating and becoming (due to rare and quasi-irreversible +mutations) cells with a different phenotype randomly [28]. Since then, there have been +many further developments that incorporate stochastic elements into the model, such as +those proposed by Lea and Coulson [25], Koch [22], Moolgavkar and Luebeck [27], and +Dewanji et al. [8]. We can find various stochastic processes: Poisson processes [2], Markov +chains [14], and branching processes [19], or even random sums of birth-death processes [8], +all playing key roles in the mathematical theories of cellular clonal growth and evolution. +These models have been applied to clinical data on lung cancer [31], breast cancer [37], +and treatment of cancer [38]. +At single-cell resolution, another cause for departure from exponential growth is the +presence of positive (growth promoting) cell-cell interactions (Allee effect) in the early +phase of population growth, such that cell density plays a role in stimulating division, +giving rise to the critical mass dynamics [20, 24]. +To understand the intrinsic tumor growth behavior (change of tumor volume over time) +it is therefore essential to study tumor cell populations in culture which affords detailed +quantitative analysis of cell numbers over time, unaffected by the tumor microenvironment, +and to measure departure from exponential growth. +This paper focuses on stochastic +growth of clonal but phenotypically heterogeneous HL60 leukemia cells with near single-cell +sensitivities in the early phase of growth, that is, in sparse cultures. We and others have in +the past years noted that at the level of single cells, each cell behaves akin to an individual, +differently from another, which can be explained by the slow correlated transcriptome-wide +fluctuations of gene expression [4, 26]. Given the phenotypic heterogeneity and anticipated +functional consequences, grouping of cells is necessary. Such classification would require +molecular cell markers for said functional implication, but such markers are often difficult +to determine a priori. +Here, since most pertinent to cancer biology, we directly use a +functional marker that is of central relevance for cancer: cell division, which maps into cell +population growth potential — in brief “cell growth”. +Therefore, we monitored longitudinally the growth of cancer cell populations seeded at +very small numbers of cells (1, 4, or 10 cells) in statistical ensembles of microcultures (wells +on a plate of wells). We found evidence that clonal HL60 leukemia cell populations contain +subpopulations that exhibit diverse growth patterns. +Based on statistical analysis, we +propose the existence of three distinctive cell phenotypic states with respect to cell growth. +We show that a branching process model captures the population growth kinetics of a +population with distinct cell subpopulations. Our results suggest that the initial phase cell +growth (“take-off” of a cell culture) in the HL60 leukemic cells is predominantly driven by +3 + +the fast-growing cell subpopulation. Reseeding experiments revealed that the fast-growing +subpopulation could maintain its growth rate over several cell generations, even after the +placement in a new environment. Our observations underscore the need to not only target +the fast-growing cells but also the transition to them from the other cell subpopulations. +2 +Results +2.1 +Experiment of the cell population growth from distinct initial cell +numbers. +To expose the variability of growth kinetics as a function of initial cell density N0 (“initial +seed number”), HL60 cells were sorted into wells of a 384-well plate (0.084 cm2 area) +to obtain “statistical ensembles” of replicate microcultures (wells) of the same condition, +distinct only by N0. Based on prior titration experiments to determine ranges of interest +for N0 and statistical power, for this experiment we plated 80 wells with N0 = 10 cells +(N0 = 10-cell group), 80 wells with N0 = 4 cells (N0 = 4-cell group), and 80 wells with +N0 = 1 cell (N0 = 1-cell group). Cells were grown in the same conditions for 23 days (for +details of cell culture and sorting, see the Methods section). Digital images were taken +every 24 hours for each well from Day 4 on, and the area occupied by cells in each well +was determined using computational image analysis. We had previously determined that +one area unit equals approximately 500 cells. This is consistent and readily measurable +because the relatively rigid and uniformly spherical HL60 cells grow as a non-adherent +“packed” monolayer at the bottom of the well. Note that we are interested in the initial +exponential growth (and departure from it) and not in the latter phases when the culture +becomes saturated as has been the historical focus of analysis (see Introduction). +Wells that have reached at least 5 area units were considered for the characterization +of early phase (before plateau) growth kinetics by plotting the areas in logarithmic scale as +a function of time (Fig. 1). All the N0 = 10-cell wells required 3.6-4.6 days to grow from +5 area units to 50 area units (mean=4.05, standard deviation=0.23). For the N0 = 1-cell +wells, we observed a diversity of behaviors. While some of the cultures only took 3.5-5 +days to grow from 5 area units to 50 area units, others needed 6-7.2 days (mean=5.02, +standard deviation=0.75). The N0 = 4-cell wells had a mean=4.50 days and standard +deviation=0.44 to reach that same population size. +To examine the exponential growth model, in Fig. 2 (left panel), we plotted the per +capita growth rate versus cell population size, where each point represents a well (popu- +lation) at a time point. As expected, as the population became crowded, the growth rate +decreased toward zero. But in the earlier phase, many populations in the N0 = 1-cell group +had a lower per capita growth rate than those in the N0 = 10-cell group, even at the same +population size – thus departing from the expected behavior of exponential growth. The +weighted Welch’s t-test showed that the difference in these growth rates was significant +(see the Methods section). +4 + +While qualitative differences in the behaviors of cultures with different initial seeding +cell numbers N0 can be expected for biological reasons (see below), in the elementary +exponential growth model, the difference of growth rate should disappear when populations +with distinct seeding numbers are aligned for the same population size that they have +reached as in Fig. 2. A simple possibility is that the deviations of expected growth rates +emanate from difference in cell-intrinsic properties. +Some cells grew faster, with a per +capita growth rate of 0.6 ∼ 0.9 (all N0 = 10-cell wells and some N0 = 1-cell wells), while +some cells grew slower, with a per capita growth rate of 0.3 ∼ 0.5 (some of the N0 = 1- +cell wells). In other words, there is intrinsic heterogeneity in the cell population that is +not “averaged out” in the culture with low N0, and the sampling process exposes these +differences between the cells that appear to be relatively stable. +To illustrate the inherent diversity of initial growth rates, in Fig. 3 (left panel), we +display the daily cell-occupied areas plotted on a linear scale starting from Day 4. All wells +with seed of N0 = 10 or N0 = 4 cells grew exponentially. Among the N0 = 1-cell wells, +14 populations died out. Four wells in the N0 = 1-cell group had more than 10 cells on +Day 8 but never grew exponentially, and had fewer than 1000 cells after 15 days (on Day +23). For these non-growing or slow-growing N0 = 1-cell wells, the per capita growth rate +was 0 ∼ 0.2. In comparison, all the N0 = 10-cell wells needed at most 15 days to reach +the carrying capacity (around 80 area units, or 40000 cells). See Table 1 for a summary of +the N0 = 1-cell group’s growth patterns. This behavior is not idiosyncratic to the culture +system because they recapitulate a pilot experiment performed in the larger scale format +of 96-well plates (not shown). +From the above experimental observations, we asserted that there might be at least +three stable cell growth phenotypes in a population: a fast type, whose growth rate was +0.6 ∼ 0.9/day for non-crowded conditions; a moderate type, whose growth rate was 0.3 ∼ +0.5/day for non-crowded conditions; and a slow type, whose growth rate was 0 ∼ 0.2/day +for the non-crowded population. +The graphs of Fig. 3 also revealed other phenomena of growth kinetics: (1) Most +N0 = 4-cell wells plateaued by Day 14 to Day 17, but some lagged significantly behind. +(2) Similarly, four wells in the N0 = 1-cell group exhibited longer lag-times before the +exponential growth phase, and never reached half-maximal cell numbers by Day 23. These +outliers reveal intrinsic variability and were taken into account in the parameter scanning +(see the Methods section). +2.2 +Reseeding experiments revealing the enduring intrinsic growth pat- +terns. +When a well in the N0 = 1-cell group had grown to 10 cells, population behavior was +still different from those in the N0 = 10-cell group at the outset. In view of the spate of +recent results revealing phenotypic heterogeneity, we hypothesized that the difference was +cell-intrinsic as opposed to being a consequence of the environment (e.g., culture medium +5 + +Growth pattern +Well label +Day 1 +Day 8 +Day 14 +Day 19 +Day 23 +No growth, +extinction +162,167,170,176, +177,179,182,183, +186,201,234,236, +239,240 +1 +<10 +<10 +∼0 +Empty +Slow growth, +no exponential +growth +165 +1 +89 +∼300 +∼350 +∼500 +166 +1 +36 +∼110 +∼120 +∼150 +178 +1 +43 +∼140 +∼170 +∼200 +211 +1 +16 +∼90 +∼200 +∼400 +Delayed +exponential +growth +163 +1 +12 +∼130 +∼300 +∼5000 +181 +1 +44 +∼270 +∼550 +∼5500 +193 +1 +25 +∼200 +∼800 +∼9000 +204 +1 +21 +∼100 +∼600 +∼6000 +Normal +exponential +growth +200 and +many others +1 +∼130 +∼20000 +∼40000 +(full) +∼40000 +(full) +Table 1: The population of some wells in the N0 = 1-cell group in the growth experiment +with different initial cell numbers, where ∼ meant approximate cell number. These wells +illustrated different growth patterns from those wells starting with N0 = 10 or N0 = 4 +cells. Such differences implied that cells from wells with different initial cell numbers were +essentially different. +6 + +Time (days) to +reach one half area +11 +12 +13 +14 +15 +16–20 +>20 +Faster wells +26 +2 +1 +2 +1 +0 +0 +Slower wells +0 +0 +0 +1 +1 +25 +5 +Table 2: The distribution of time needed for each well to reach the “half area” population +size in the reseeding experiment. We reseeded equal numbers of cells that grew faster (from +a full well) and cells that grew slower (from a half-full well), and cultivated them under the +same new fresh medium environment to compare their intrinsic growth rates. The results +showed that faster growing cells, even reseeded, still grew faster. +in N0 = 1 vs N0 = 10 -cell wells). +To test our hypothesis and exclude differences in the culture environment as determi- +nants of growth behavior, we reseeded the cells that exhibited the different growth rates +in fresh cultures. We started with a number of N0 = 1-cell wells. After a period of almost +3 weeks, again some wells showed rapid proliferation, with cells covering the well, while +others were half full and yet others wells were almost empty. We collected cells from the full +and half-full wells and reseeded them into 32 wells each (at about N0 = 78 cells per well). +These 64 wells were monitored for another 20 days. We found that most wells reseeded +from the full well took around 11 days to reach the population size of a half-full well, while +most wells reseeded from the half-full well required around 16 ∼ 20 days to reach the same +half full well population size. Five wells reseeded from the half-full wells were far from even +reaching half full well population size by Day 20 (see Table 2). Permutation test showed +that this difference in growth rate was significant (see the Methods section). +This reseeding experiment shows that the difference in growth rate was maintained +over multiple generations, even after slowing down in the plateau phase (full well) and +was maintained when restarting a microculture at low density in fresh medium devoid of +secreted cell products. Therefore, it is plausible that there exists endogenous heterogeneity +of growth phenotypes in the clonal HL60 cell line and that these distinct growth phenotypes +are stable for at least 15 ∼ 20 cell generations. +2.3 +Quantitative analysis of experimental results. +In the experiments with different initial cell numbers N0, we observed at least three patterns +with different growth rates, and the reseeding showed that these growth patterns were +endogenous to the cells. Therefore, we propose that each growth pattern discussed above +corresponded to a cell phenotype that dominated the population: fast, moderate, and slow. +In the initial seeding of cells that varies N0, the cells were randomly chosen (by FACS); +thus, their intrinsic growth phenotypes were randomly distributed. During growth, the +population of a well would be dominated by the fastest type that existed in the seeding +cells, thus qualitatively, we have following scenarios: (1) A well in the N0 = 10-cell group +7 + +almost certainly had at least one initial cell of fast type, and the population would be +dominated by fast type cells. Different wells had almost the same growth rate, reaching +saturation at almost the same time. (2) For an N0 = 1-cell well, if the only initial cell is of +the fast type, then the population has only the fast type, and the growth pattern will be +close to that of N0 = 10-cell wells. If the only initial cell is of the moderate type, then the +population could still grow exponentially, but with a slower growth rate. This explains why +after reaching 5 area units, many but not all N0 = 1-cell wells were slower than N0 = 10- +cell wells. (3) Moreover, in such an N0 = 1-cell well with a moderate type initial cell, the +cell might not divide quite often during the first few days due to randomness of entering +the cell cycle. This would lead to a considerable delay in entering the exponential growth +phase. (4) By contrast, for an N0 = 1-cell well with a slow type initial cell, the growth rate +could be too small, and the population might die out or survive without ever entering the +exponential growth phase in duration of the experiment. (5) Most N0 = 4-cell wells had at +least one fast type initial cell, and the growth pattern was the same as N0 = 10-cell wells. +A few N0 = 4-cell wells only had moderate and slow cells, and thus had slower growth +patterns. +The above verbal argument is shown in Fig. 4 and entails mathematical modeling with +the appropriate parameters that relate the relative frequency of these cell types in the +original population, their associated growth and transition rates to examine whether it +explains the data. +2.4 +Branching process model. +To construct a quantitative dynamical model to recapitulate the growth dynamics differ- +ences from cell populations with distinct initial seed cell numbers N0, and three intrinsic +types of proliferation behaviors, we used a multi-type discrete-time branching process. +The traditional method of population dynamics based on ordinary differential equation +(ODE), which is deterministic and has continuous variables, is not suited when the cell +population is small as is the case for the earliest stage of proliferation from a few cells +being studied in our experiments. Deterministic models are also unfit because with such +small populations and measurements at single-cell resolution, stochasticity in cell activity +does not average out. The nuanced differences between individual cells cannot be captured +by a different deterministic mechanism of each individual cell, and the only information +available is the initial cell number. Thus, the unobservable nuances between cells are taken +care of by a stochastic model. +Given the small populations, our model should be purely stochastic, without determin- +istic growth. The focus is the concrete population size of a finite number (three) of types, +thus Poisson processes are not suitable. Markov chains can partially describe the propor- +tions under some conditions [47], but population sizes are known, not just their ratios, +therefore Markov chains are not necessary. Even the lifted Markov chains [48] and random +dynamical systems [52] are not applicable in this situation, since the population should be +8 + +non-negative. Branching processes can describe the population size of multiple types with +symmetric and asymmetric division, transition, and death [19]. Also, the parameters can +be temporally and spatially inhomogeneous, which is convenient. Therefore, we utilized +branching processes in our model. +In the branching process, each cell during each time interval independently and ran- +domly chooses a behavior: division, death, or stagnation in the quiescent state, whose rates +depend on the cell growth type. Denoting the growth rate and death rate of the fast type +by gF and dF respectively, and the population size of fast type cells on Day n by F(n), the +population at Day n + 1 is: +F(n + 1) = +F(n) +� +i=1 +Ai, +where Ai for different i are independent. Ai represents the descendants of a fast type cell i +after one day. It equals 2 with probability gF, 0 with probability dF, and 1 with probability +1 − gF − dF. Therefore, given F(n), the distribution of F(n + 1) is: +P[F(n + 1) = N] = +� +2a+b=N +F(n)! +a!b![F(n) − a − b]!ga +Fd[F(n)−a−b] +F +(1 − gF − dF)b, +where the summation is taken for all non-negative integer pairs (a, b) with 2a + b = N. +Moderate and slow types evolve similarly, with their corresponding growth rates gM, gS, +and death rates dM, dS. +As shown in Fig. 2, the growth rates gF, gM, and gS should be decreasing functions of +the total population. In our model, we adopted a quadratic function. +We performed a parameter scan to show that our model could reproduce experimental +phenomena for a wide range of model parameters (see details in Table 3). +The simulation results are shown on the right panels of Figs. 1–3, in comparison with +the experimental data in the left. Our model qualitatively captured the growth patterns +of groups with different initial seeding cell numbers. For example, in Fig. 2, when wells +were less than half full (cell number < 20000), most wells in the N0 = 10-cell group grew +faster than the N0 = 1-cell group even when they had the same cell number. In Fig. 3, +all wells in the N0 = 10-cell group in our model grew quickly until saturation. Similar to +the experiment, some wells in the N0 = 1-cell group in our model never grew, while some +began to take off very late. +In our model, the high extinction rate in the N0 = 1-cell group (14/80) was explained +as “bad luck” at the early stage, since birth rate and death rate were close, and a cell could +easily die without division. Another possible explanation for such a difference in growth +rates was that the population would be 10 small colonies when starting from 10 initial cells, +while starting from 1 initial cell, the population would be 1 large colony. With the same +area, 10 small colonies should have a larger total perimeter, thus larger growth space and +larger growth rate than that of 1 large colony. However, we carefully checked the photos, +9 + +Parameters +Appearance of experimental phenomena +pF +pM +pS +d +g0 +r +Feature 1 +Feature 2 +Feature 3 +Feature 4 +0.4 +0.4 +0.2 +0.01 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.4 +0.4 +0.2 +0 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.4 +0.4 +0.2 +0.05 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.4 +0.4 +0.2 +0.1 +0.5 +0.1 +No +Yes +Yes +No +0.4 +0.4 +0.2 +0.01 +0.45 +0.1 +Yes +Yes +Yes +Yes +0.4 +0.4 +0.2 +0.01 +0.6 +0.1 +Yes +Yes +Yes +Yes +0.4 +0.4 +0.2 +0.01 +0.4 +0.1 +Yes +Yes +Yes +No +0.4 +0.4 +0.2 +0.01 +0.5 +0.05 +Yes +Yes +Yes +Yes +0.4 +0.4 +0.2 +0.01 +0.5 +0 +Yes +Yes +Yes +Yes +0.4 +0.4 +0.2 +0.01 +0.5 +0.15 +Yes +Yes +Yes +No +0.4 +0.4 +0.2 +0.01 +0.5 +0.2 +No +Yes +Yes +No +0.3 +0.5 +0.2 +0.01 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.5 +0.3 +0.2 +0.01 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.4 +0.5 +0.1 +0.01 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.4 +0.3 +0.3 +0.01 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.5 +0.4 +0.1 +0.01 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.3 +0.4 +0.3 +0.01 +0.5 +0.1 +Yes +Yes +Yes +Yes +0.1 +0.1 +0.8 +0.01 +0.5 +0.1 +No +Yes +Yes +No +0.5 +0.5 +0 +0.01 +0.5 +0.1 +Yes +Yes +No +Yes +0 +0.5 +0.5 +0.01 +0.5 +0.1 +No +Yes +Yes +Yes +0.5 +0 +0.5 +0.01 +0.5 +0.1 +Yes +No +Yes +No +1 +0 +0 +0.01 +0.5 +0.1 +Yes +No +No +No +Table 3: Performance of our model with different parameters. Here we adjusted the param- +eters of our model in a wide range and observed whether the model could still reproduce +four important “features” in the experiment. This parameter scan showed that our model +is robust under perturbations on parameters. Here pF, pM, pS are the probabilities that an +initial cell is of fast, moderate, or slow type; d is the death rate; g0 is the growth factor; +r is the range of the random modifier. See the Methods section for explanations of these +parameters. +Feature 1, all wells in the N0 = 10-cell group were saturated; Feature 2, +presence of late-growing wells in the N0 = 1-cell group; Feature 3, presence of non-growing +wells in the N0 = 1-cell group; Feature 4, different growth rates at the same population +size between the N0 = 10-cell group and the N0 = 1-cell group. +10 + +and found that almost all wells produced 1 large colony with nearly the same shape, and +there was no significant relationship between colony perimeter and growth rate. +3 +Discussion +As many recent single-cell level data have shown, a tumor can contain multiple distinct +subpopulations engaging in interconversions and interactions among them that can in- +fluence cancer cell proliferation, death, migration, and other features that contribute to +malignancy [33, 55, 1, 18, 34, 56, 20, 24, 5, 32, 6]. Presence of these two intra-population +behaviors can be manifest as departure from the elementary model of exponential growth +[35] (in the early phase of population growth, far away from carrying capacity of the culture +environment which is trivially non-exponential). The exponential growth model assumes +uniformity of cell division rates across all cells (hence a population doubling rate that is +proportional to a given population size N(t)) and the absence of cell-cell interactions that +affect cell division and death rates. Investigating the “non-genetic heterogeneity” hypoth- +esis of cancer cells quantitatively is therefore paramount for understanding cancer biology +but also for elementary principles of cell population growth. +As an example, here we showed that clonal cell populations of the leukemia HL60 +cell line are heterogeneous with regard to growth behaviors of individual cells that can +be summarized in subpopulations characterized by a distinct intrinsic growth rates which +were revealed by analysis of the early population growth starting with microcultures seeded +with varying (low) cell number N0. +Since we have noted only very weak effect of cell-cell interactions on cell growth be- +haviors (Allee effect) in this cell line (as opposed to another cell tumor cell line in which +we found that departure from exponential growth could be explained by the Allee effect +[20]), we focused on the very presence among HL60 cells of subpopulations with distinct +proliferative capacity as a mechanism for the departure of the early population growth +curve from exponential growth. +The reseeding experiment demonstrated that the characteristic growth behaviors of +subpopulations could be inherited across cell generations and after moving to a new envi- +ronment (fresh culture), consistent with long-enduring endogenous properties of the cells. +This result might be explained by cells occupying distinct stable cell states (in a multi- +stable system). Thus, we introduced multiple cell types with different growth rates in our +stochastic model. Specifically, in a branching process model, we assumed the existence +of three types: fast, moderate, and slow cells. The model we built could replicate the +key features in the experimental data, such as different growth rates at the same popula- +tion size between the N0 = 10-cell group and the N0 = 1-cell group, and the presence of +late-growing and non-growing wells in the N0 = 1-cell group. +While we were able to fit the observed behaviors in which the growth rate depended not +only on N(t) but also on N0, the existence of the three or even more cell types still needs +11 + +to be verified experimentally. For instance, statistical cluster analysis of transcriptomes of +individual cells by single-cell RNA-seq [3] over the population may identify the presence +of transcriptomically distinct subpopulations that could be isolated (e.g., after association +with cell surface markers) and evaluated separately for their growth behaviors. We might +apply inference methods on such sequencing data to determine the gene regulatory relations +that lead to multiple phenotypes [50, 44], although the causal relationship might not always +be determined [49]. Besides, since the existence of transposons might affect the growth +rates, corresponding analysis should be conducted [21, 40]. +The central assumption of coexistence of multiple subpopulations in the cell line stock +must be accompanied by the second assumption that there are transitions between these +distinct cell populations. For otherwise, in the stock population the fastest growing cell +would eventually outgrow the slow growing cells. Furthermore, one has to assume a steady- +state in which the population of slow growing cells are continuously replenished from the +population of fast-growing cells. Finally, we must assume that the steady-state proportions +of the subpopulations are such that at low seeding wells with N0 = 1 cells, there is a sizable +probability that a microculture receives cells from each of the (three) presumed subtypes of +cells. The number of wells in the ensemble of replicate microcultures for each N0- condition +has been sufficiently large for us to make the observations and inform the model, but a +larger ensemble would be required to determine with satisfactory accuracy the relative +proportions of the cell types in the parental stock population. +Transitions might also have been happening during our experiment. For example, those +late growing wells in the N0 = 1-cell group could be explained by such a transition: Initially, +only slow type cells were present, but once one of these slow growing cells switched to the +moderate type, an exponential growth ensued at the same rate that is intrinsic to that of +moderate cells. +If there are transitions, what is the transition rate? Our reseeding experiments are +compatible with a relatively slow rate for interconversion of growth behaviors in that the +same growth type was maintained across 30 generations. An alternative to the principle +of transition at a constant intrinsic to each of the types of cells may be that transition +is extrinsically determined. Specifically, the seeding in the “lone” condition of N0 = 1 +may induce a dormant state, that is a transition to a slower growth mode that is then +maintained, on average over 30+ generations, with occasional return to the faster types +that account for the delayed exponential growth. The lack of experimental data might be +partially made up by inference methods [51]. +This model however would bring back the notion of “environment awareness”, or the +principle of a “critical density” for growth implemented by cell-cell interaction (Allee effect) +which we had deliberately not considered (see above) since it was not necessary. We do not +exclude this possibility which could be experimentally tested as follows: Cultivate N0 = 1- +cell wells for 20 days when the delayed exponential growth has happened in some wells, +but then use the cells of those wells with fast-growing population (which should contain of +the fast type) to restart the experiment, seeded at N0 = 10, 4, 1 cells. If wells with different +12 + +seeding numbers exhibit the same growth rates, then the growth difference in the original +experiment is solely due to preexisting (slow interconverting) cell phenotypes. If now the +N0 = 1-cell wells resumes the typical slow growth, this would indicate a density induced +transition to the slow growth type. If cell-cell interaction needs to be taken into account, +certain results in developmental biology might help, since they study the emergence of +patterns through strong cell-cell interactions [46, 45, 42]. +In the spirit of Occam’s razor, and given the technical difficulty in separate experiments +to demonstrate cell-cell interactions in HL60 cells, we were able to model the observed +behaviors with the simplest assumption of cell-autonomous properties, including existence +of multiple states (growth behaviors) and slow transitions between them but without cell +density dependence or interactions. +Taken together, we showed that one manifestation of the burgeoning awareness of ubiq- +uitous cell phenotype heterogeneity in an isogenic cell population is the presence of distinct +intrinsic types of cells that slowly interconvert among them, resulting in a stationary popu- +lation composition. The differing growth rates of the subtypes and their stable proportions +may be an elementary characteristic of a given population that by itself can account for the +departure of early population growth kinetics from the basic exponential growth model. +4 +Methods +4.1 +Setup of growth experiment with different initial cell numbers. +HL60 cells were maintained in IMDM wGln, 20% FBS(heat inactivated), 1% P/S at a +cell density between 3 × 105 and 2.5 × 106 cells/ml (GIBCO). Cells were always handled +and maintained under sterile conditions (tissue culture hood; 37◦C, 5% CO2, humidified +incubator). At the beginning of the experiment, cells were collected, washed two times in +PBS, and stained for vitality (Trypan blue GIBCO). The population of cells was first gated +for morphology and then for vitality staining. Only Trypan negative cells were sorted (BD +FACSAria II). The cells were sorted in a 384 well plate with IMDM wGln, 20% FBS(heat +inactivated), and 1% P/S (GIBCO). +Cell population growth was monitored using a Leica microscope (heated environmental +chamber and CO2 levels control) with a motorized tray. Starting from Day 4, the 384 +well plate was placed inside the environmental chamber every 24 hours. The images were +acquired in a 3 × 3 grid for each well; after acquisition, the 9 fields were stitched into a +single image. Software ImageJ was applied to identify and estimate the area occupied by +“entities” in each image. The area (proportional to cell number) was used to follow the +cell growth. +13 + +4.2 +Setup of reseeding experiment for growth pattern inheritance. +HL60 cells were cultivated for 3 weeks, and then we chose one full well and one half full +well. We supposed the full well was dominated by fast type cells, and the half-full well +was dominated by moderate type cells, which had lower growth rates. We reseeded cells +from these two wells and cultivated them in two 96-well (rows A-H, columns 1-12) plates. +In each plate, B2-B11, D2-D11, and F2-F11 wells started with 78 fast cells, while C2-C11, +E2-E11, and G2-G11 wells started with 78 moderate cells. Rows A, H, columns 1, 12 had +no cells and no media, and we found that wells in rows B, G, columns 2, 11, which were +the outmost non-empty wells, evaporated much faster than inner wells. Therefore, the +growth of cells in those wells was much slower than inner wells. Hence we only considered +inner wells, where D3-D10 and F3-F10 started with fast cells, C3-C10 and E3-E10 started +with moderate cells, namely 32 fast wells and 32 moderate wells in total. +During the +experiment, no media was added. Each day, we observed those wells to check whether +their areas exceeded one-half of the whole well. The experiment was terminated after 20 +days. +4.3 +Weighted Welch’s t-test. +The weighted Welch’s t-test is used to test the hypothesis that two populations have equal +mean, while sample values have different weights [12]. +Assume for group i (i = 1, 2), +the sample size is Ni and the jth sample is the average of cj +i independent and identically +distributed variables. Let Xj +i be the observed average for the jth sample. Set ν1 = N1 − 1, +ν2 = N2 − 1. Define +¯ +Xi +W = ( +Ni +� +j=1 +Xj +i cj)/( +Ni +� +j=1 +)cj, +s2 +i,W = +Ni[�Ni +j=1(Xj +i )2cj]/(�Ni +j=1 cj +i) − Ni( ¯ +Xi +W )2 +Ni − 1 +, +t = +¯ +X1 +W − ¯ +X2 +W +� +s2 +1,W +N1 + +s2 +2,W +N2 +, +ν = +( +s2 +1,W +N1 + +s2 +2,W +N2 )2 +s4 +1,W +N2 +1 ν1 + +s4 +2,W +N2 +2 ν2 +. +If two populations have equal mean, then t satisfies the t-distribution with degree of freedom +ν. +The weighted Welch’s t-test was applied to the growth experiment with different initial +cell numbers, in order to determine whether the growth rates during exponential phase +14 + +(5–50 area units) were different between groups. Here Xj +i corresponded to growth rate, +and cj +i corresponded to cell area. The p-value for N0 = 10-cell group vs. N0 = 4-cell group +was 2.12 × 10−8; the p-value for N0 = 10-cell group vs. N0 = 1-cell group was smaller than +10−12; the p-value for N0 = 4-cell group vs. N0 = 1-cell group was 5.35 × 10−5. Therefore, +the growth rate difference between any two groups was statistically significant. +4.4 +Permutation Test. +The permutation test is a non-parametric method to test whether two samples are signifi- +cantly different with respect to a statistic (e.g., sample mean) [16]. It is easy to calculate +and fits our situation, thus we adopt this test rather than other more complicated tests, +such as the Mann-Whitney test. +For two samples {x1, · · · , xm}, {y1, · · · , yn}, consider +the null hypothesis: the mean of x and y are the same. For these samples, calculate the +mean of the first sample: µ0 = +1 +m +� xi. Then we randomly divide these m + n samples +into two groups with size m and n: {x′ +1, · · · , x′ +m}, {y′ +1, · · · , y′ +n}, such that each permuta- +tion has equal probability. For these new samples, calculate the mean of the first sample: +µ′ +0 = 1 +m +� x′ +i. Then the two-sided p-value is defined as +p = 2 min{P(µ0 ≤ µ′ +0), 1 − P(µ0 ≤ µ′ +0)}. +If µ0 is an extreme value in the distribution of µ′ +0, then the two sample means are different. +In the reseeding experiment, the mean time of exceeding half well for the fast group +was 11.4375 days. For all +�64 +32 +� +possible result combinations, only 7 combinations had equal +or less mean time. Thus the p-value was 2 × 7/ +�64 +32 +� += 7.6 × 10−18. This indicated that the +growth rate difference between fast group and moderate group was significant. +4.5 +Model Details. +The simulation time interval was half day, but we only utilized the results in full days. For +each initial cell, the probabilities of being fast, moderate or slow type, pF, pM, pS, were 0.4, +0.4, 0.2. +Each half day, a fast type cell had probability d to die, and probability gF to divide. +The division produced two fast cells, capturing the intrinsic growth behavior that is to +some extent inheritable. Denote the total cell number of previous day as N, then +gF = g0(1 − N2/C2) + δ, +where δ is a random variable that satisfies the uniform distribution on [−r, r], and it is a +constant for all cells in the same well. If gF < 0, set gF = 0. If gF > 1 − d, set gF = 1 − d. +In the simulation displayed, death rate d = 0.01, carrying capacity C = 40000, growth +factor g0 = 0.5, and the range of random modifier r = 0.1. +Each half day, a moderate type cell had probability d to die, and probability gM to +divide. The division produced two moderate cells. gM = gF/1.5. +15 + +Similarly, each half day, a slow type cell had probability d to die, and probability gS to +divide. The division produced two slow-growing cells. gS = gF/3. +4.6 +Parameter scan. +Since growth is measured by the area covered by cells, we could not experimentally verify +most assumptions of our model, or determine the values of parameters. +Therefore, we +performed a parameter scan by evaluating the performance of our model for different sets +of parameters. +We adjusted 6 parameters: initial type probabilities pF, pM, pS, death +rate d, growth factor g0, and random modifier r. We checked whether these 4 features +observable in the experiment could be reproduced: growth of all wells in the N0 = 10-cell +group to saturation; existence of late-growing wells in the N0 = 1-cell group; existence of +non-growing wells in the N0 = 1-cell group; difference in growth rates in the N0 = 10-cell +group and the N0 = 1-cell group at the same population size. Table 3 shows the results +of the performance of simulations with the various parameter sets. 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Model- +ing the development of metastases from primary and locally recurrent tumors: Com- +parison with a clinical data base for prostatic cancer. +Cancer Res. 53, 13 (1993), +2987–2993. +[54] Zheng, Q. Progress of a half century in the study of the Luria–Delbr¨uck distribution. +Math. Biosci. 162, 1 (1999), 1–32. +[55] Zhou, D., Wang, Y., and Wu, B. +A multi-phenotypic cancer model with cell +plasticity. J. Theor. Biol. 357 (2014), 35–45. +20 + +[56] Zhou, J. X., Pisco, A. O., Qian, H., and Huang, S. Nonequilibrium population +dynamics of phenotype conversion of cancer cells. PLOS ONE 9, 12 (2014), e110714. +21 + +Figure 1: Growth curves of the experiment (left) and simulation (right), starting from +the time of reaching 5 area units (experiment) or having 2500 cells (simulation), with a +logarithm scale for the y-axis. The time required for reaching 5 area units was determined +by exponential extrapolation, as reliable imaging started at > 5 area units. The x-axis is +the time from reaching 5 area units (experiment) or 2500 cells (simulation). Red, green, +or blue curves correspond to 10, 4, or 1 initial cell(s). Although starting from the same +population level, patterns are different for distinct initial cell numbers. The N0 = 1-cell +group has higher diversity. +22 + +experimental +80 +cell area +40 +20 +10-cell group +4-cell group +10 +1-cell group +5 +0 +5 +10 +15 +time (day) +80 +cell area +40 +20simulation +40000 +cell number +20000 +10000 +5000 +2500 +0 +5 +10 +15 +time (day) +40000 +ell number +20000 +100005 +0 +5 +10 +15 +time (day) +80 +cell area +40 +20 +10 +5 +0 +5 +10 +15 +time (day)8 +QQQ +2500 +0 +5 +10 +15 +time (day) +40000 +cell number +20000 +10000 +5000 +2500 +0 +5 +10 +15 +time (day)20 +40 +60 +80 +cell area +0 +0.5 +1 +1.5 +growth rate +experimental +10-cell group +4-cell group +1-cell group +0 +20 +40 +60 +80 +cell area +0 +0.5 +1 +1.5 +growth rate +1 +2 +3 +4 +cell number +104 +0 +0.5 +1 +1.5 +growth rate +simulation +0 +1 +2 +3 +4 +cell number +104 +0 +0.5 +1 +1.5 +growth rate +Figure 2: Per capita growth rate (averaged within one day) vs. cell population for the +experiment (left) and simulation (right). Each point represents one well in one day. Red, +green, or blue points correspond to 10, 4, or 1 initial cell(s). +23 + +0 +5 +10 +15 +20 +time (day) +0 +20 +40 +60 +80 +cell area +experimental +10-cell group +4-cell group +1-cell group +0 +5 +10 +15 +20 +time (day) +5 +10 +20 +40 +80 +cell area +0 +5 +10 +15 +20 +time (day) +0 +1 +2 +3 +4 +cell number +104 +simulation +0 +5 +10 +15 +20 +time (day) +2500 +5000 +10000 +20000 +40000 +cel number +Figure 3: Growth curves of the experiments with different initial cell numbers N0 (left) +and growth curves of corresponding simulation (right). Each curve describes the change in +the cell population (measured by area or number) over a well along time. Red, green, or +blue curves correspond to N0 = 10, 4, or 1 initial cell(s). +24 + +Figure 4: Schematic illustration of the qualitative argument: Three cell types and growth +patterns (three colors) with different seeding numbers. One N0 = 10-cell well will have +at least one fast type cell with high probability, which will dominate the population. One +N0 = 1-cell well can only have one cell type, thus in the microculture ensemble of replicate +wells, three possible growth patterns for wells can be observed. +25 + +fast +moderate +slow +fast +fast +moderate +slow \ No newline at end of file diff --git a/BdE2T4oBgHgl3EQfRge6/content/tmp_files/load_file.txt b/BdE2T4oBgHgl3EQfRge6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0503eaa0d1b6ed3091f1dd5a3cbdbe69d1761fb4 --- /dev/null +++ b/BdE2T4oBgHgl3EQfRge6/content/tmp_files/load_file.txt @@ -0,0 +1,1040 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf,len=1039 +page_content='Multiple phenotypes in HL60 leukemia cell population Yue Wang1,2, Joseph X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Zhou3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Edoardo Pedrini3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Irit Rubin3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' May Khalil3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Hong Qian2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' and Sui Huang3 1Department of Computational Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' United States of America 2Department of Applied Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' United States of America 3Institute for Systems Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' United States of America Abstract Recent studies at individual cell resolution have revealed phenotypic heterogene- ity in nominally clonal tumor cell populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The heterogeneity affects cell growth behaviors, which can result in departure from the idealized exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Here we measured the stochastic time courses of growth of an ensemble of populations of HL60 leukemia cells in cultures, starting with distinct initial cell numbers to capture the departure from the exponential growth model in the initial growth phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' De- spite being derived from the same cell clone, we observed significant variations in the early growth patterns of individual cultures with statistically significant differences in growth kinetics and the presence of subpopulations with different growth rates that endured for many generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Based on the hypothesis of existence of multiple inter- converting subpopulations, we developed a branching process model that captures the experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 1 Introduction Cancer has long been considered a genetic disease caused by oncogenic mutations in so- matic cells that confer a proliferation advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' According to the clonal evolution theory, accumulation of random genetic mutations produces cell clones with cancerous cell phe- notype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Specifically, cells with the novel genotype(s) may display increased proliferative fitness and gradually out-grow the normal cells, break down tissue homeostasis and gain other cancer hallmarks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In this view, a genetically distinct clone of cells dominates the cancer cell population and is presumed to be uniform in terms of the phenotype of indi- vidual cells within an isogenic clone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In this traditional paradigm, non-genetic phenotypic variation within one clone is not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='03782v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='PE] 10 Jan 2023 With the advent of systematic single-cell resolution analysis, however, non-genetic cell heterogeneity within clonal (cancer) cell populations is found to be universal [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This feature led to the consideration of the possibility of biologically (qualitatively) distinct (meta)stable cell subpopulations due to gene expression noise, representing intra-clonal variability of features beyond the rapid random micro-fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Hence, transitions between the subpopulations, as well as heterotypic interactions among them may influence cell growth, migration, drug resistance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' [39, 13, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Thus, an emerging view is that cancer is more akin to an evolving ecosystem [11] in which cells form distinct subpopulations with persistent characteristic features that determine their mode of interaction, directly or indirectly via competition for resources [10, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' However, once non-genetic dynamics is considered, cell “ecology” differs fundamentally from the classic ecological system in macroscopic biology: the subpopulations can reversibly switch between each other whereas species in an ecological population do not convert between each other [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This affords cancer cell populations a remarkable heterogeneity, plasticity and evolvability, which may play important roles in their growth and in the development of resistance to treatment [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Many new questions arise following the hypothesis that phenotypic heterogeneity and transitions between phenotypes within one genetic clone are important factors in cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Can tumors arise, as theoretical considerations indicate, because of a state conversion (within one clone) to a phenotype capable of faster, more autonomous growth as opposed to acquisition of a new genetic mutation that confers such a selectable phenotype [55, 1, 18, 34, 33, 56, 23, 41]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Is the macroscopic, apparently sudden outgrowth of a tumor driven by a new fastest-growing clone (or subpopulation) taking off exponentially, or due to the cell population reaching a critical mass that permits positive feedback between its subpopulations that stimulates outgrowth, akin to a collectively autocatalytic set [17]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Should therapy target the fastest growing subpopulations, or target the interactions and interconversions of cancer cells?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' At the core of these deliberations is the fundamental question on the mode of tumor cell population growth that now must consider the influence of inherent phenotypic hetero- geneity of cells and the non-genetic (hence potentially reversible) inter-conversion of cells between the phenotypes that manifest various growth behaviors and the interplay between these two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Traditionally tumor growth has been described as following an exponential growth law, motivated by the notion of uniform cell division rate for each cell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' a first order growth kinetics [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' But departure from the exponential model has long been noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' To better fit experimental data, two major modifications have been developed, namely the Gompertz model and the West law model [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' While no one specific model can adequately describe any one tumor, each model highlights certain aspects of macroscopic tumor kinetics, mainly the maximum size and the change in growth rate at different stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' These models however are not specifically motivated by cellular heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Assuming non-genetic heterogene- ity with transitions between the cell states, the population behavior is influenced by many 2 intrinsic and extrinsic factors that are both variable and unpredictable at the single-cell level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Thus, unlike macroscopic population dynamics [43], tumor growth cannot be ad- equately captured by a deterministic model, but a stochastic cell and population level kinetic model is more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Using stochastic processes in modeling cell growth via clonal expansion has a long history [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' An early work is the Luria-Delbr¨uck model, which assumes cells grow deter- ministically, with wildtype cells mutating and becoming (due to rare and quasi-irreversible mutations) cells with a different phenotype randomly [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Since then, there have been many further developments that incorporate stochastic elements into the model, such as those proposed by Lea and Coulson [25], Koch [22], Moolgavkar and Luebeck [27], and Dewanji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We can find various stochastic processes: Poisson processes [2], Markov chains [14], and branching processes [19], or even random sums of birth-death processes [8], all playing key roles in the mathematical theories of cellular clonal growth and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' These models have been applied to clinical data on lung cancer [31], breast cancer [37], and treatment of cancer [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' At single-cell resolution, another cause for departure from exponential growth is the presence of positive (growth promoting) cell-cell interactions (Allee effect) in the early phase of population growth, such that cell density plays a role in stimulating division, giving rise to the critical mass dynamics [20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' To understand the intrinsic tumor growth behavior (change of tumor volume over time) it is therefore essential to study tumor cell populations in culture which affords detailed quantitative analysis of cell numbers over time, unaffected by the tumor microenvironment, and to measure departure from exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This paper focuses on stochastic growth of clonal but phenotypically heterogeneous HL60 leukemia cells with near single-cell sensitivities in the early phase of growth, that is, in sparse cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We and others have in the past years noted that at the level of single cells, each cell behaves akin to an individual, differently from another, which can be explained by the slow correlated transcriptome-wide fluctuations of gene expression [4, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Given the phenotypic heterogeneity and anticipated functional consequences, grouping of cells is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Such classification would require molecular cell markers for said functional implication, but such markers are often difficult to determine a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Here, since most pertinent to cancer biology, we directly use a functional marker that is of central relevance for cancer: cell division, which maps into cell population growth potential — in brief “cell growth”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, we monitored longitudinally the growth of cancer cell populations seeded at very small numbers of cells (1, 4, or 10 cells) in statistical ensembles of microcultures (wells on a plate of wells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We found evidence that clonal HL60 leukemia cell populations contain subpopulations that exhibit diverse growth patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Based on statistical analysis, we propose the existence of three distinctive cell phenotypic states with respect to cell growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We show that a branching process model captures the population growth kinetics of a population with distinct cell subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Our results suggest that the initial phase cell growth (“take-off” of a cell culture) in the HL60 leukemic cells is predominantly driven by 3 the fast-growing cell subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Reseeding experiments revealed that the fast-growing subpopulation could maintain its growth rate over several cell generations, even after the placement in a new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Our observations underscore the need to not only target the fast-growing cells but also the transition to them from the other cell subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2 Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='1 Experiment of the cell population growth from distinct initial cell numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' To expose the variability of growth kinetics as a function of initial cell density N0 (“initial seed number”), HL60 cells were sorted into wells of a 384-well plate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='084 cm2 area) to obtain “statistical ensembles” of replicate microcultures (wells) of the same condition, distinct only by N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Based on prior titration experiments to determine ranges of interest for N0 and statistical power, for this experiment we plated 80 wells with N0 = 10 cells (N0 = 10-cell group), 80 wells with N0 = 4 cells (N0 = 4-cell group), and 80 wells with N0 = 1 cell (N0 = 1-cell group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Cells were grown in the same conditions for 23 days (for details of cell culture and sorting, see the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Digital images were taken every 24 hours for each well from Day 4 on, and the area occupied by cells in each well was determined using computational image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We had previously determined that one area unit equals approximately 500 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This is consistent and readily measurable because the relatively rigid and uniformly spherical HL60 cells grow as a non-adherent “packed” monolayer at the bottom of the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Note that we are interested in the initial exponential growth (and departure from it) and not in the latter phases when the culture becomes saturated as has been the historical focus of analysis (see Introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Wells that have reached at least 5 area units were considered for the characterization of early phase (before plateau) growth kinetics by plotting the areas in logarithmic scale as a function of time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' All the N0 = 10-cell wells required 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='6-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='6 days to grow from 5 area units to 50 area units (mean=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='05, standard deviation=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For the N0 = 1-cell wells, we observed a diversity of behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' While some of the cultures only took 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5-5 days to grow from 5 area units to 50 area units, others needed 6-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='2 days (mean=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='02, standard deviation=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The N0 = 4-cell wells had a mean=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='50 days and standard deviation=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='44 to reach that same population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' To examine the exponential growth model, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2 (left panel), we plotted the per capita growth rate versus cell population size, where each point represents a well (popu- lation) at a time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' As expected, as the population became crowded, the growth rate decreased toward zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' But in the earlier phase, many populations in the N0 = 1-cell group had a lower per capita growth rate than those in the N0 = 10-cell group, even at the same population size – thus departing from the expected behavior of exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The weighted Welch’s t-test showed that the difference in these growth rates was significant (see the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 4 While qualitative differences in the behaviors of cultures with different initial seeding cell numbers N0 can be expected for biological reasons (see below), in the elementary exponential growth model, the difference of growth rate should disappear when populations with distinct seeding numbers are aligned for the same population size that they have reached as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' A simple possibility is that the deviations of expected growth rates emanate from difference in cell-intrinsic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Some cells grew faster, with a per capita growth rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='6 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='9 (all N0 = 10-cell wells and some N0 = 1-cell wells), while some cells grew slower, with a per capita growth rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='3 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 (some of the N0 = 1- cell wells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In other words, there is intrinsic heterogeneity in the cell population that is not “averaged out” in the culture with low N0, and the sampling process exposes these differences between the cells that appear to be relatively stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' To illustrate the inherent diversity of initial growth rates, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 3 (left panel), we display the daily cell-occupied areas plotted on a linear scale starting from Day 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' All wells with seed of N0 = 10 or N0 = 4 cells grew exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Among the N0 = 1-cell wells, 14 populations died out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Four wells in the N0 = 1-cell group had more than 10 cells on Day 8 but never grew exponentially, and had fewer than 1000 cells after 15 days (on Day 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For these non-growing or slow-growing N0 = 1-cell wells, the per capita growth rate was 0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In comparison, all the N0 = 10-cell wells needed at most 15 days to reach the carrying capacity (around 80 area units, or 40000 cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' See Table 1 for a summary of the N0 = 1-cell group’s growth patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This behavior is not idiosyncratic to the culture system because they recapitulate a pilot experiment performed in the larger scale format of 96-well plates (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' From the above experimental observations, we asserted that there might be at least three stable cell growth phenotypes in a population: a fast type, whose growth rate was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='6 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='9/day for non-crowded conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' a moderate type, whose growth rate was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='3 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5/day for non-crowded conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' and a slow type, whose growth rate was 0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='2/day for the non-crowded population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The graphs of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 3 also revealed other phenomena of growth kinetics: (1) Most N0 = 4-cell wells plateaued by Day 14 to Day 17, but some lagged significantly behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' (2) Similarly, four wells in the N0 = 1-cell group exhibited longer lag-times before the exponential growth phase, and never reached half-maximal cell numbers by Day 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' These outliers reveal intrinsic variability and were taken into account in the parameter scanning (see the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='2 Reseeding experiments revealing the enduring intrinsic growth pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' When a well in the N0 = 1-cell group had grown to 10 cells, population behavior was still different from those in the N0 = 10-cell group at the outset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In view of the spate of recent results revealing phenotypic heterogeneity, we hypothesized that the difference was cell-intrinsic as opposed to being a consequence of the environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' culture medium 5 Growth pattern Well label Day 1 Day 8 Day 14 Day 19 Day 23 No growth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' extinction 162,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='167,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='170,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='176,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 177,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='179,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='182,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='183,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 186,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='234,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='236,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 239,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='240 1 <10 <10 ∼0 Empty Slow growth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='no exponential ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='(full) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='Table 1: The population of some wells in the N0 = 1-cell group in the growth experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='with different initial cell numbers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' where ∼ meant approximate cell number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' These wells illustrated different growth patterns from those wells starting with N0 = 10 or N0 = 4 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Such differences implied that cells from wells with different initial cell numbers were essentially different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 6 Time (days) to reach one half area 11 12 13 14 15 16–20 >20 Faster wells 26 2 1 2 1 0 0 Slower wells 0 0 0 1 1 25 5 Table 2: The distribution of time needed for each well to reach the “half area” population size in the reseeding experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We reseeded equal numbers of cells that grew faster (from a full well) and cells that grew slower (from a half-full well), and cultivated them under the same new fresh medium environment to compare their intrinsic growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The results showed that faster growing cells, even reseeded, still grew faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' in N0 = 1 vs N0 = 10 -cell wells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' To test our hypothesis and exclude differences in the culture environment as determi- nants of growth behavior, we reseeded the cells that exhibited the different growth rates in fresh cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We started with a number of N0 = 1-cell wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' After a period of almost 3 weeks, again some wells showed rapid proliferation, with cells covering the well, while others were half full and yet others wells were almost empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We collected cells from the full and half-full wells and reseeded them into 32 wells each (at about N0 = 78 cells per well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' These 64 wells were monitored for another 20 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We found that most wells reseeded from the full well took around 11 days to reach the population size of a half-full well, while most wells reseeded from the half-full well required around 16 ∼ 20 days to reach the same half full well population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Five wells reseeded from the half-full wells were far from even reaching half full well population size by Day 20 (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Permutation test showed that this difference in growth rate was significant (see the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This reseeding experiment shows that the difference in growth rate was maintained over multiple generations, even after slowing down in the plateau phase (full well) and was maintained when restarting a microculture at low density in fresh medium devoid of secreted cell products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, it is plausible that there exists endogenous heterogeneity of growth phenotypes in the clonal HL60 cell line and that these distinct growth phenotypes are stable for at least 15 ∼ 20 cell generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='3 Quantitative analysis of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In the experiments with different initial cell numbers N0, we observed at least three patterns with different growth rates, and the reseeding showed that these growth patterns were endogenous to the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, we propose that each growth pattern discussed above corresponded to a cell phenotype that dominated the population: fast, moderate, and slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In the initial seeding of cells that varies N0, the cells were randomly chosen (by FACS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' thus, their intrinsic growth phenotypes were randomly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' During growth, the population of a well would be dominated by the fastest type that existed in the seeding cells, thus qualitatively, we have following scenarios: (1) A well in the N0 = 10-cell group 7 almost certainly had at least one initial cell of fast type, and the population would be dominated by fast type cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Different wells had almost the same growth rate, reaching saturation at almost the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' (2) For an N0 = 1-cell well, if the only initial cell is of the fast type, then the population has only the fast type, and the growth pattern will be close to that of N0 = 10-cell wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If the only initial cell is of the moderate type, then the population could still grow exponentially, but with a slower growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This explains why after reaching 5 area units, many but not all N0 = 1-cell wells were slower than N0 = 10- cell wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' (3) Moreover, in such an N0 = 1-cell well with a moderate type initial cell, the cell might not divide quite often during the first few days due to randomness of entering the cell cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This would lead to a considerable delay in entering the exponential growth phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' (4) By contrast, for an N0 = 1-cell well with a slow type initial cell, the growth rate could be too small, and the population might die out or survive without ever entering the exponential growth phase in duration of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' (5) Most N0 = 4-cell wells had at least one fast type initial cell, and the growth pattern was the same as N0 = 10-cell wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' A few N0 = 4-cell wells only had moderate and slow cells, and thus had slower growth patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The above verbal argument is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 4 and entails mathematical modeling with the appropriate parameters that relate the relative frequency of these cell types in the original population, their associated growth and transition rates to examine whether it explains the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='4 Branching process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' To construct a quantitative dynamical model to recapitulate the growth dynamics differ- ences from cell populations with distinct initial seed cell numbers N0, and three intrinsic types of proliferation behaviors, we used a multi-type discrete-time branching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The traditional method of population dynamics based on ordinary differential equation (ODE), which is deterministic and has continuous variables, is not suited when the cell population is small as is the case for the earliest stage of proliferation from a few cells being studied in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Deterministic models are also unfit because with such small populations and measurements at single-cell resolution, stochasticity in cell activity does not average out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The nuanced differences between individual cells cannot be captured by a different deterministic mechanism of each individual cell, and the only information available is the initial cell number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Thus, the unobservable nuances between cells are taken care of by a stochastic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Given the small populations, our model should be purely stochastic, without determin- istic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The focus is the concrete population size of a finite number (three) of types, thus Poisson processes are not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Markov chains can partially describe the propor- tions under some conditions [47], but population sizes are known, not just their ratios, therefore Markov chains are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Even the lifted Markov chains [48] and random dynamical systems [52] are not applicable in this situation, since the population should be 8 non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Branching processes can describe the population size of multiple types with symmetric and asymmetric division, transition, and death [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Also, the parameters can be temporally and spatially inhomogeneous, which is convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, we utilized branching processes in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In the branching process, each cell during each time interval independently and ran- domly chooses a behavior: division, death, or stagnation in the quiescent state, whose rates depend on the cell growth type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Denoting the growth rate and death rate of the fast type by gF and dF respectively, and the population size of fast type cells on Day n by F(n), the population at Day n + 1 is: F(n + 1) = F(n) � i=1 Ai, where Ai for different i are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Ai represents the descendants of a fast type cell i after one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' It equals 2 with probability gF, 0 with probability dF, and 1 with probability 1 − gF − dF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, given F(n), the distribution of F(n + 1) is: P[F(n + 1) = N] = � 2a+b=N F(n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' [F(n) − a − b]!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='ga Fd[F(n)−a−b] F (1 − gF − dF)b, where the summation is taken for all non-negative integer pairs (a, b) with 2a + b = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Moderate and slow types evolve similarly, with their corresponding growth rates gM, gS, and death rates dM, dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2, the growth rates gF, gM, and gS should be decreasing functions of the total population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In our model, we adopted a quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We performed a parameter scan to show that our model could reproduce experimental phenomena for a wide range of model parameters (see details in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The simulation results are shown on the right panels of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 1–3, in comparison with the experimental data in the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Our model qualitatively captured the growth patterns of groups with different initial seeding cell numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 2, when wells were less than half full (cell number < 20000), most wells in the N0 = 10-cell group grew faster than the N0 = 1-cell group even when they had the same cell number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 3, all wells in the N0 = 10-cell group in our model grew quickly until saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Similar to the experiment, some wells in the N0 = 1-cell group in our model never grew, while some began to take off very late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In our model, the high extinction rate in the N0 = 1-cell group (14/80) was explained as “bad luck” at the early stage, since birth rate and death rate were close, and a cell could easily die without division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Another possible explanation for such a difference in growth rates was that the population would be 10 small colonies when starting from 10 initial cells, while starting from 1 initial cell, the population would be 1 large colony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' With the same area, 10 small colonies should have a larger total perimeter, thus larger growth space and larger growth rate than that of 1 large colony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' However, we carefully checked the photos, 9 Parameters Appearance of experimental phenomena pF pM pS d g0 r Feature 1 Feature 2 Feature 3 Feature 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='01 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='1 Yes No No No Table 3: Performance of our model with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Here we adjusted the param- eters of our model in a wide range and observed whether the model could still reproduce four important “features” in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This parameter scan showed that our model is robust under perturbations on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Here pF, pM, pS are the probabilities that an initial cell is of fast, moderate, or slow type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' d is the death rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' g0 is the growth factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' r is the range of the random modifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' See the Methods section for explanations of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Feature 1, all wells in the N0 = 10-cell group were saturated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Feature 2, presence of late-growing wells in the N0 = 1-cell group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Feature 3, presence of non-growing wells in the N0 = 1-cell group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Feature 4, different growth rates at the same population size between the N0 = 10-cell group and the N0 = 1-cell group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 10 and found that almost all wells produced 1 large colony with nearly the same shape, and there was no significant relationship between colony perimeter and growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 3 Discussion As many recent single-cell level data have shown, a tumor can contain multiple distinct subpopulations engaging in interconversions and interactions among them that can in- fluence cancer cell proliferation, death, migration, and other features that contribute to malignancy [33, 55, 1, 18, 34, 56, 20, 24, 5, 32, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Presence of these two intra-population behaviors can be manifest as departure from the elementary model of exponential growth [35] (in the early phase of population growth, far away from carrying capacity of the culture environment which is trivially non-exponential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The exponential growth model assumes uniformity of cell division rates across all cells (hence a population doubling rate that is proportional to a given population size N(t)) and the absence of cell-cell interactions that affect cell division and death rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Investigating the “non-genetic heterogeneity” hypoth- esis of cancer cells quantitatively is therefore paramount for understanding cancer biology but also for elementary principles of cell population growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' As an example, here we showed that clonal cell populations of the leukemia HL60 cell line are heterogeneous with regard to growth behaviors of individual cells that can be summarized in subpopulations characterized by a distinct intrinsic growth rates which were revealed by analysis of the early population growth starting with microcultures seeded with varying (low) cell number N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Since we have noted only very weak effect of cell-cell interactions on cell growth be- haviors (Allee effect) in this cell line (as opposed to another cell tumor cell line in which we found that departure from exponential growth could be explained by the Allee effect [20]), we focused on the very presence among HL60 cells of subpopulations with distinct proliferative capacity as a mechanism for the departure of the early population growth curve from exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The reseeding experiment demonstrated that the characteristic growth behaviors of subpopulations could be inherited across cell generations and after moving to a new envi- ronment (fresh culture), consistent with long-enduring endogenous properties of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This result might be explained by cells occupying distinct stable cell states (in a multi- stable system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Thus, we introduced multiple cell types with different growth rates in our stochastic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Specifically, in a branching process model, we assumed the existence of three types: fast, moderate, and slow cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The model we built could replicate the key features in the experimental data, such as different growth rates at the same popula- tion size between the N0 = 10-cell group and the N0 = 1-cell group, and the presence of late-growing and non-growing wells in the N0 = 1-cell group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' While we were able to fit the observed behaviors in which the growth rate depended not only on N(t) but also on N0, the existence of the three or even more cell types still needs 11 to be verified experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For instance, statistical cluster analysis of transcriptomes of individual cells by single-cell RNA-seq [3] over the population may identify the presence of transcriptomically distinct subpopulations that could be isolated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=', after association with cell surface markers) and evaluated separately for their growth behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We might apply inference methods on such sequencing data to determine the gene regulatory relations that lead to multiple phenotypes [50, 44], although the causal relationship might not always be determined [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Besides, since the existence of transposons might affect the growth rates, corresponding analysis should be conducted [21, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The central assumption of coexistence of multiple subpopulations in the cell line stock must be accompanied by the second assumption that there are transitions between these distinct cell populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For otherwise, in the stock population the fastest growing cell would eventually outgrow the slow growing cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Furthermore, one has to assume a steady- state in which the population of slow growing cells are continuously replenished from the population of fast-growing cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Finally, we must assume that the steady-state proportions of the subpopulations are such that at low seeding wells with N0 = 1 cells, there is a sizable probability that a microculture receives cells from each of the (three) presumed subtypes of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The number of wells in the ensemble of replicate microcultures for each N0- condition has been sufficiently large for us to make the observations and inform the model, but a larger ensemble would be required to determine with satisfactory accuracy the relative proportions of the cell types in the parental stock population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Transitions might also have been happening during our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For example, those late growing wells in the N0 = 1-cell group could be explained by such a transition: Initially, only slow type cells were present, but once one of these slow growing cells switched to the moderate type, an exponential growth ensued at the same rate that is intrinsic to that of moderate cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If there are transitions, what is the transition rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Our reseeding experiments are compatible with a relatively slow rate for interconversion of growth behaviors in that the same growth type was maintained across 30 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' An alternative to the principle of transition at a constant intrinsic to each of the types of cells may be that transition is extrinsically determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Specifically, the seeding in the “lone” condition of N0 = 1 may induce a dormant state, that is a transition to a slower growth mode that is then maintained, on average over 30+ generations, with occasional return to the faster types that account for the delayed exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The lack of experimental data might be partially made up by inference methods [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This model however would bring back the notion of “environment awareness”, or the principle of a “critical density” for growth implemented by cell-cell interaction (Allee effect) which we had deliberately not considered (see above) since it was not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We do not exclude this possibility which could be experimentally tested as follows: Cultivate N0 = 1- cell wells for 20 days when the delayed exponential growth has happened in some wells, but then use the cells of those wells with fast-growing population (which should contain of the fast type) to restart the experiment, seeded at N0 = 10, 4, 1 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If wells with different 12 seeding numbers exhibit the same growth rates, then the growth difference in the original experiment is solely due to preexisting (slow interconverting) cell phenotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If now the N0 = 1-cell wells resumes the typical slow growth, this would indicate a density induced transition to the slow growth type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If cell-cell interaction needs to be taken into account, certain results in developmental biology might help, since they study the emergence of patterns through strong cell-cell interactions [46, 45, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In the spirit of Occam’s razor, and given the technical difficulty in separate experiments to demonstrate cell-cell interactions in HL60 cells, we were able to model the observed behaviors with the simplest assumption of cell-autonomous properties, including existence of multiple states (growth behaviors) and slow transitions between them but without cell density dependence or interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Taken together, we showed that one manifestation of the burgeoning awareness of ubiq- uitous cell phenotype heterogeneity in an isogenic cell population is the presence of distinct intrinsic types of cells that slowly interconvert among them, resulting in a stationary popu- lation composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The differing growth rates of the subtypes and their stable proportions may be an elementary characteristic of a given population that by itself can account for the departure of early population growth kinetics from the basic exponential growth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 4 Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='1 Setup of growth experiment with different initial cell numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' HL60 cells were maintained in IMDM wGln, 20% FBS(heat inactivated), 1% P/S at a cell density between 3 × 105 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 × 106 cells/ml (GIBCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Cells were always handled and maintained under sterile conditions (tissue culture hood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 37◦C, 5% CO2, humidified incubator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' At the beginning of the experiment, cells were collected, washed two times in PBS, and stained for vitality (Trypan blue GIBCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The population of cells was first gated for morphology and then for vitality staining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Only Trypan negative cells were sorted (BD FACSAria II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The cells were sorted in a 384 well plate with IMDM wGln, 20% FBS(heat inactivated), and 1% P/S (GIBCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Cell population growth was monitored using a Leica microscope (heated environmental chamber and CO2 levels control) with a motorized tray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Starting from Day 4, the 384 well plate was placed inside the environmental chamber every 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The images were acquired in a 3 × 3 grid for each well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' after acquisition, the 9 fields were stitched into a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Software ImageJ was applied to identify and estimate the area occupied by “entities” in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The area (proportional to cell number) was used to follow the cell growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='2 Setup of reseeding experiment for growth pattern inheritance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' HL60 cells were cultivated for 3 weeks, and then we chose one full well and one half full well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We supposed the full well was dominated by fast type cells, and the half-full well was dominated by moderate type cells, which had lower growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We reseeded cells from these two wells and cultivated them in two 96-well (rows A-H, columns 1-12) plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In each plate, B2-B11, D2-D11, and F2-F11 wells started with 78 fast cells, while C2-C11, E2-E11, and G2-G11 wells started with 78 moderate cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Rows A, H, columns 1, 12 had no cells and no media, and we found that wells in rows B, G, columns 2, 11, which were the outmost non-empty wells, evaporated much faster than inner wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, the growth of cells in those wells was much slower than inner wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Hence we only considered inner wells, where D3-D10 and F3-F10 started with fast cells, C3-C10 and E3-E10 started with moderate cells, namely 32 fast wells and 32 moderate wells in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' During the experiment, no media was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Each day, we observed those wells to check whether their areas exceeded one-half of the whole well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The experiment was terminated after 20 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='3 Weighted Welch’s t-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The weighted Welch’s t-test is used to test the hypothesis that two populations have equal mean, while sample values have different weights [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Assume for group i (i = 1, 2), the sample size is Ni and the jth sample is the average of cj i independent and identically distributed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Let Xj i be the observed average for the jth sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Set ν1 = N1 − 1, ν2 = N2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Define ¯ Xi W = ( Ni � j=1 Xj i cj)/( Ni � j=1 )cj, s2 i,W = Ni[�Ni j=1(Xj i )2cj]/(�Ni j=1 cj i) − Ni( ¯ Xi W )2 Ni − 1 , t = ¯ X1 W − ¯ X2 W � s2 1,W N1 + s2 2,W N2 , ν = ( s2 1,W N1 + s2 2,W N2 )2 s4 1,W N2 1 ν1 + s4 2,W N2 2 ν2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If two populations have equal mean, then t satisfies the t-distribution with degree of freedom ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The weighted Welch’s t-test was applied to the growth experiment with different initial cell numbers, in order to determine whether the growth rates during exponential phase 14 (5–50 area units) were different between groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Here Xj i corresponded to growth rate, and cj i corresponded to cell area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The p-value for N0 = 10-cell group vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' N0 = 4-cell group was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='12 × 10−8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' the p-value for N0 = 10-cell group vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' N0 = 1-cell group was smaller than 10−12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' the p-value for N0 = 4-cell group vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' N0 = 1-cell group was 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='35 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, the growth rate difference between any two groups was statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='4 Permutation Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The permutation test is a non-parametric method to test whether two samples are signifi- cantly different with respect to a statistic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=', sample mean) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' It is easy to calculate and fits our situation, thus we adopt this test rather than other more complicated tests, such as the Mann-Whitney test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For two samples {x1, · · · , xm}, {y1, · · · , yn}, consider the null hypothesis: the mean of x and y are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For these samples, calculate the mean of the first sample: µ0 = 1 m � xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Then we randomly divide these m + n samples into two groups with size m and n: {x′ 1, · · · , x′ m}, {y′ 1, · · · , y′ n}, such that each permuta- tion has equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For these new samples, calculate the mean of the first sample: µ′ 0 = 1 m � x′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Then the two-sided p-value is defined as p = 2 min{P(µ0 ≤ µ′ 0), 1 − P(µ0 ≤ µ′ 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If µ0 is an extreme value in the distribution of µ′ 0, then the two sample means are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In the reseeding experiment, the mean time of exceeding half well for the fast group was 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='4375 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For all �64 32 � possible result combinations, only 7 combinations had equal or less mean time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Thus the p-value was 2 × 7/ �64 32 � = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='6 × 10−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' This indicated that the growth rate difference between fast group and moderate group was significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 Model Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The simulation time interval was half day, but we only utilized the results in full days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' For each initial cell, the probabilities of being fast, moderate or slow type, pF, pM, pS, were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Each half day, a fast type cell had probability d to die, and probability gF to divide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The division produced two fast cells, capturing the intrinsic growth behavior that is to some extent inheritable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Denote the total cell number of previous day as N, then gF = g0(1 − N2/C2) + δ, where δ is a random variable that satisfies the uniform distribution on [−r, r], and it is a constant for all cells in the same well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If gF < 0, set gF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' If gF > 1 − d, set gF = 1 − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' In the simulation displayed, death rate d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='01, carrying capacity C = 40000, growth factor g0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5, and the range of random modifier r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Each half day, a moderate type cell had probability d to die, and probability gM to divide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The division produced two moderate cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' gM = gF/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 15 Similarly, each half day, a slow type cell had probability d to die, and probability gS to divide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The division produced two slow-growing cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' gS = gF/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='6 Parameter scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Since growth is measured by the area covered by cells, we could not experimentally verify most assumptions of our model, or determine the values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Therefore, we performed a parameter scan by evaluating the performance of our model for different sets of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We adjusted 6 parameters: initial type probabilities pF, pM, pS, death rate d, growth factor g0, and random modifier r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' We checked whether these 4 features observable in the experiment could be reproduced: growth of all wells in the N0 = 10-cell group to saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' existence of late-growing wells in the N0 = 1-cell group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' existence of non-growing wells in the N0 = 1-cell group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' difference in growth rates in the N0 = 10-cell group and the N0 = 1-cell group at the same population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Table 3 shows the results of the performance of simulations with the various parameter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Within a wide range of parameters, our model is able to replicate the experimental results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 1–3, indicating that our model is robust under perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Acknowledgements We would like to thank Ivana Bozic, Yifei Liu, Georg Luebeck, Weili Wang, Yuting Wei and Lingxue Zhu for helpful advice and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' References [1] Angelini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=', Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' X.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=', Qian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=', and Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Nonequilibrium population dynamics of phenotype conversion of cancer cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' PLOS ONE 9, 12 (2014), e110714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 21 Figure 1: Growth curves of the experiment (left) and simulation (right), starting from the time of reaching 5 area units (experiment) or having 2500 cells (simulation), with a logarithm scale for the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The time required for reaching 5 area units was determined by exponential extrapolation, as reliable imaging started at > 5 area units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The x-axis is the time from reaching 5 area units (experiment) or 2500 cells (simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Red, green, or blue curves correspond to 10, 4, or 1 initial cell(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Although starting from the same population level, patterns are different for distinct initial cell numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' The N0 = 1-cell group has higher diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 22 experimental 80 cell area 40 20 10-cell group 4-cell group 10 1-cell group 5 0 5 10 15 time (day) 80 cell area 40 20simulation 40000 cell number 20000 10000 5000 2500 0 5 10 15 time (day) 40000 ell number 20000 100005 0 5 10 15 time (day) 80 cell area 40 20 10 5 0 5 10 15 time (day)8 QQQ 2500 0 5 10 15 time (day) 40000 cell number 20000 10000 5000 2500 0 5 10 15 time (day)20 40 60 80 cell area 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 growth rate experimental 10-cell group 4-cell group 1-cell group 0 20 40 60 80 cell area 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 growth rate 1 2 3 4 cell number 104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 growth rate simulation 0 1 2 3 4 cell number 104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content='5 growth rate Figure 2: Per capita growth rate (averaged within one day) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' cell population for the experiment (left) and simulation (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Each point represents one well in one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Red, green, or blue points correspond to 10, 4, or 1 initial cell(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 23 0 5 10 15 20 time (day) 0 20 40 60 80 cell area experimental 10-cell group 4-cell group 1-cell group 0 5 10 15 20 time (day) 5 10 20 40 80 cell area 0 5 10 15 20 time (day) 0 1 2 3 4 cell number 104 simulation 0 5 10 15 20 time (day) 2500 5000 10000 20000 40000 cel number Figure 3: Growth curves of the experiments with different initial cell numbers N0 (left) and growth curves of corresponding simulation (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Each curve describes the change in the cell population (measured by area or number) over a well along time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' Red, green, or blue curves correspond to N0 = 10, 4, or 1 initial cell(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 24 Figure 4: Schematic illustration of the qualitative argument: Three cell types and growth patterns (three colors) with different seeding numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' One N0 = 10-cell well will have at least one fast type cell with high probability, which will dominate the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' One N0 = 1-cell well can only have one cell type, thus in the microculture ensemble of replicate wells, three possible growth patterns for wells can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} +page_content=' 25 fast moderate slow fast fast moderate slow' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfRge6/content/2301.03782v1.pdf'} diff --git a/BdE2T4oBgHgl3EQfRge6/vector_store/index.pkl b/BdE2T4oBgHgl3EQfRge6/vector_store/index.pkl new file mode 100644 index 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b/BdE4T4oBgHgl3EQf5Q5A/content/tmp_files/2301.05321v1.pdf.txt @@ -0,0 +1,2198 @@ +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell +populations via crowding control: stability, +robustness and quasi-dedifferentiation +Cristina Parigini1,2,3 and Philip Greulich1,2* +1*School of Mathematical Sciences, University of Southampton, +Southampton, United Kingdom. +2Institute for Life Sciences, University of Southampton, +Southampton, United Kingdom. +3Te P¯unaha ¯Atea - Space Institute, University of Auckland, +Auckland, New Zealand. +*Corresponding author(s). E-mail(s): p.s.greulich@soton.ac.uk; +Contributing authors: cristina.parigini@auckland.ac.nz; +Abstract +To maintain renewing epithelial tissues in a healthy, homeostatic state, +(stem) cell divisions and differentiation need to be tightly regulated. +Mechanisms of homeostatic control often rely on crowding control: cells +are able to sense the cell density in their environment (via various +molecular and mechanosensing pathways) and respond by adjusting +division, differentiation, and cell state transitions appropriately. Here +we determine, via a mathematically rigorous framework, which general +conditions for the crowding feedback regulation (i) must be minimally +met, and (ii) are sufficient, to allow the maintenance of homeosta- +sis in renewing tissues. We show that those conditions naturally allow +for a degree of robustness toward disruption of regulation. Further- +more, intrinsic to this feedback regulation is that stem cell identity is +established collectively by the cell population, not by individual cells, +which implies the possibility of ‘quasi-dedifferentiation’, in which cells +committed to differentiation may reacquire stem cell properties upon +depletion of the stem cell pool. These findings can guide future exper- +imental campaigns to identify specific crowding feedback mechanisms. +Keywords: keyword1, Keyword2, Keyword3, Keyword4 +1 +arXiv:2301.05321v1 [q-bio.TO] 12 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Homeostatic regulation of renewing tissue cell populations via crowding control +1 Introduction +Many adult tissues are renewing, that is, terminally differentiated cells are +steadily removed and replaced by new cells produced by the division of cycling +cells (stem cells and progenitor cells), which then differentiate. In order to +maintain those tissues in a healthy, homeostatic state, (stem) cell divisions +and differentiation must be tightly balanced. Adult stem cells are the key +players in maintaining and renewing such tissues due to their ability to produce +cells through cell division and differentiation persistently [1]. However, the +underlying cell-intrinsic and extrinsic factors that regulate a homeostatic state +are complex and not always well understood. +Several experimental studies have identified mechanisms and pathways that +regulate homeostasis. For example, cell crowding can trigger delamination and +thus loss of cells in Drosophila back [2], and differentiation in cultured human +colon, various zebrafish epiderimises, and canine kidney cells [3, 4]. On the +other hand, cell crowding can affect cell proliferation: overcrowding can inhibit +proliferation [5], whereas a reduction in the cell density, obtained, for example, +by stretching a tissue [6] causes an increase in proliferative activity (both +shown in cultured canine kidney cells). Although the mechanisms to mediate +this regulation are not always clear, experimental studies on mechanosensing +showed that cell overcrowding reduces cell motility and consequently produces +a compression on cells that inhibits cell proliferation [5, 7]. Another mechanism +utilising crowding feedback is the competition for limited growth signalling +factors [8]. More specifically, in the mouse germ line, cells in the niche respond +to a growth factor (FGF5) that promotes proliferation over differentiation, +which they deplete upon being exposed to it. Therefore, the more cells are +in the niche, the less FGF5 is available per cell, and the less proliferation (or +more differentiation) occurs. +Despite differing in the involved molecular pathways and many other +details, all these regulatory mechanisms are, in essence, sensing the cell den- +sity in their environment and responding by adjusting their propensities to +divide, differentiate, die, or emigrate from the tissue. This class of mechanisms, +for which cell fate propensities depend on the cell density, can be classified as +crowding feedback regulation: the cell density determines the cells’ prolifera- +tion and differentiation, which affects their population dynamics and thus the +cell density. However, the crowding response to changes in cell density cannot +be arbitrary in order to maintain homeostasis. It must provide a (negative) +feedback, in the sense that cells sense the cell density and adjust proliferation, +differentiation, and cell loss, such that the cell density is decreased if it is too +high and increased if it is too low. For simple tissues consisting of a single +cell type with a unique cell state, it is relatively straightforward to give the +conditions for crowding feedback to maintain homeostasis successfully. In this +case, when the cell division rate decreases with cell density and differentiation +and or death rate increase with cell density, a homeostatic state is maintained. +However, such conclusions are not as simple to make when a tissue consists of +a complex lineage hierarchy and a multitude of underlying cellular states. In + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +3 +the latter, more realistic case, conditions for successful homeostatic regulation +– in which case we speak of crowding control – may take more complex forms. +Previous studies based on mathematical modelling have shed some light +on quantitative mechanisms for homeostatic control [9–13]. In particular, in +[13], a mathematical assessment of crowding feedback modelling shows that +a (dynamic) homeostatic state exists under reasonable biological conditions. +Nevertheless, the case of dynamic homeostasis considered there may not nec- +essarily be a steady state but could also exhibit oscillations in cell numbers +(as does realistically happen in the uterus during the menstrual cycle). While +the criterion presented in [13] provides a valid sufficient condition for dynamic +homeostasis, it relies on a rather abstract mathematical quantity – the domi- +nant eigenvalue of the dynamical matrix – that is difficult, if not impossible, +to measure in reality. +Here, we wish to generalise previous findings and seek to identify general +conditions for successful homeostatic control if propensities for cell division, +differentiation, and loss are responsive to variations in cell density. More +precisely, we derive conditions that must be minimally fulfilled (necessary +conditions) and conditions which are sufficient, to ensure that homeostasis pre- +vails. To identify and formulate those conditions, we note that homeostasis is +a property of the tissue cell population dynamics, which can be mathemati- +cally expressed as a dynamical system. Even if a numerically exact formulation +of the dynamics may not be possible, one can formulate generic yet mathe- +matically rigorous conditions by referring to the criteria for the existence of +stable steady states in the cell population dynamics of renewing tissues. We +will derive those conditions by mathematical, analytical means, augmented by +a numerical analysis testing the limits of those conditions. +We will also show that homeostatic control by crowding feedback possesses +inherent robustness to failures and perturbations of the regulatory pathways, +which may occur through external influences (e.g. wide-spread biochemical fac- +tors) and genetic mutations. Finally, we will assess the response of cells when +the pool of stem cells is depleted. Crucially, we find that inherent to crowd- +ing feedback control is that formerly committed progenitor cells reacquire +self-renewal capacity without substantial changes in their internal states. Ded- +ifferentiation has been widely reported under conditions of tissue regeneration +[14, 15] or when stem cells are depleted [16–19], which is usually thought to +involve a substantial reprogramming of the cell-intrinsic states towards a stem +cell type. On the other hand, our analysis suggests the possibility of “quasi”- +dedifferentiation, the reversion from a committed cell to a stem cell by a +mere quantitative adjustment of the pacing of proliferation and differentiation, +without a substantial qualitative change in its expression profiles. + +Springer Nature 2021 LATEX template +4 +Homeostatic regulation of renewing tissue cell populations via crowding control +2 Modelling of tissue cell dynamics under +crowding feedback +We seek to assess the conditions for homeostasis in renewing tissue cell pop- +ulations, that is, either a steady state of the tissue cell population (strict +homeostasis) or long-term, bounded oscillations or fluctuations (dynamic +homeostasis), which represent well-defined constraints on the dynamics of the +tissue cell population. To this end, we will here derive a formal, mathematical +representation of the tissue cell dynamics under crowding feedback. +The cell population is fully defined by (i) the number of cells, (ii) the +internal (biochemical and mechanical) states of each cell, and (iii) the spatial +position of cells. We assume that a cell’s behaviour can depend on the cell +density and the states of cells in its close cellular environment. As we examine +a situation close to a homeostatic state, we assume that the cell density is +homogeneous over the range of interaction between cells, which expands over +a volume V . Hence, the cell density ρ is proportional to the average number +of cells, ¯n, in that volume, ρ = +¯n +V . Similarly, we define the number of cells +in internal state i as ni, and the cell density of cells in internal state i as +ρi = ¯ni +V , where ¯ni is the expected value of ni. As we consider only the crowding +feedback response of cells, which only accounts for the cell densities ρi but +not the explicit position of cells, the spatial configuration (iii) is not relevant +to our considerations. Thus, the configuration of the cell population and its +time evolution is entirely determined by the average number of cells in each +state i, as a function of time t, ¯ni(t). The configuration of cell numbers ni +can change only through three processes: (1) cell division, whereby it must be +distinguished between the cell state of daughter cells, (2) the transition from +one cell state to another, (3) loss of a cell, through cell death or emigration +out of the tissue. Following the lines of Refs. [13, 20] and denoting as Xi,j,k a +cell in internal states i, j, k, respectively, we can formalise these events as: +cell division: Xi +λirjk +i +−−−→ Xj + Xk +(1) +cell state transition: Xi +ωij +−−→ Xj +(2) +cell loss: Xi +γi +−→ ∅ , +(3) +where the symbols above the arrows denote the dynamical rates of the transi- +tions, i.e. the average frequency at which such events occur. In particular, γi +is the rate at which a cell in state i is lost, ωij the rate at which a cell changes +its state from i to j and λirjk +i +denotes the rate at which a cell i divides to pro- +duce two daughter cells, one in state j and one in state k (i = j, j = k, k = i +are possible). For later convenience, we distinguish here the overall rate of cell +division in state i, λi and the probability rjk +i +that such a division produces +daughter cells in states j and k. +Since we consider a situation where cells can respond to the cell densities +ρi via crowding feedback, all the rates and probabilities (λi, γi, ωij, rjk +i ) may + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +5 +depend on the cell densities of either state j, ρj. For convenience, we discretise +the number of states in case the state space is a continuum and only distinguish +states which have substantially different propensities (λi, γi, ωij, rjk +i ). Without +loss of generality, we assume that there are m states, that is, i, j, k = 1, ..., m +(for a rigorous argument for the discretisation of the state space, see [13]). +The rates given above denote the average number of events happening per +time unit. Thus, we can express the total rate of change of the average (i.e. +expected) number of cells ¯ni(t), that is, the derivative ˙¯ni = d¯ni +dt , in terms of +the rates of those events. This defines a set of ordinary differential equations. +Following the lines of Refs. [13, 20], we can write ˙ni as, +˙¯ni = +�� +j +ωji¯nj + λj +�� +k +rik +j + rki +j +� +¯nj +� +− ¯ni +� +λi + γi + +� +j +ωij +� +, +(4) +where for convenience, we did not write the time dependence explicitly, i.e. +ni = ni(t), and all parameters may depend on the cell densities ρj. Since V is +constant, we can divide by V to equivalently express this in terms of the cell +state densities, ρi = ¯ni +V , and then write Eq. (4) compactly as, +d +dtρ(t) = A(ρ(t)) ρ(t) +(5) +where ρ = (ρ1, ρ2, ...) is the vector of cell state densities and A(ρ) is the matrix, +A = +� +� +λ1 − � +j̸=1 κ1j − γ1 +κ21 +κ31 +· · · +κ12 +λ2 − � +j̸=2 κ2j − γ2 κ32 +· · · +κ1m +κ2m +· · · λm − � +j̸=m κmj − γm +� +� , +(6) +in which κij = λi2rj +i + ωij, with rj +i = � +k(rjk +i ++ rkj +i )/2, is the total transition +rate, that combines all transitions from Xi to Xj by cell divisions and direct +state transitions (again, all parameters may depend on ρ, as therefore also +does A). We can thus generally write the elements of the matrix A, aij with +i, j = 1, ..., m as +aij = +� λi − γi − � +k̸=i κik +for i = j +κji +for i ̸= j +(7) +We now make the mild assumption that divisions of the form Xi → Xj+Xk +are effectively three events, namely, cell duplication, Xi → Xi + Xi coupled to +cell state changes, Xi → Xj and Xi → Xk, if j ̸= i or k ̸= i. In this view, the +parameters relevant for crowding feedback are the total cell state transition +propensities κij and the cell division rate λi, as in (6), instead of ωij and rjk +i . +These equations describe a dynamical system which, for given initial con- +ditions, determines the time evolution of the cell densities, ρi(t). Crucially, + +Springer Nature 2021 LATEX template +6 +Homeostatic regulation of renewing tissue cell populations via crowding control +this description allows for a rigorous mathematical definition of what a home- +ostatic state is, and to apply tools of dynamical systems analysis to determine +the circumstances under which a homeostatic state prevails. In particular, we +define a strict homeostatic state as a steady state of the system, (5), when the +cell numbers – and thus cell densities, given that V is fixed – in each state +do not change, mathematically expressed as dρ +dt = 0 (a fixed point of the sys- +tem). A dynamic homeostatic state is when cell densities may also oscillate +or fluctuate but remain bounded and thus possess a finite long-term average +cell population (in which case the system either approaches a steady state or +limit cycles – that is, oscillations – or chaotic but bounded behaviour). Based +on these definitions, we can now analyse under which circumstances crowd- +ing feedback can maintain those states, which in the case of strict homeostasis +requires, in addition, that the corresponding steady state is stable. +2.1 Cell types and lineage hierarchies +According to [13], cell population dynamics of the type (5) can be associated +with a cell state network, in which each state is a node, and the nodes are +connected through cell state transition (direct transitions and cell divisions). +Furthermore, by decomposing this network in strongly connected components +(SCCs), the cell fate model can be viewed as a directed acyclic network [21], +generally called the condensed network. Here, we follow the definitions of [13] +and define a cell type as an SCC of the cell state network, so that any cell states +connected via cyclic cell state trajectories (sequences of cell state transitions) +are of the same type, and the condensed network of cell types represents the +cell lineage hierarchy. This definition ensures that cells of the same type have +the same lineage potential (outgoing cell state trajectories) and that the stages +of the cell cycle are associated with the same cell type. In this context, we +will in the following also speak of differentiation when a cell state transition +between different cell types occurs. +Each cell type can be classified as self-renewing, declining or hyper- +proliferating, depending on the dominant eigenvalue µ (called growth parame- +ter) of the dynamical matrix A (from Eq. (5) ff.) reduced to that SCC. This +is µ = 0 for self-renewing cell types, when cell numbers of that type remain +constant over time, µ < 0 (µ > 0) for the declining (hyperproliferating) types +when cell numbers decline (increase) in the long term [13]. Importantly, for the +population dynamics to be strictly homeostatic, which means that a steady +state of model (5) exists, the cell type network must fulfil strict rules. These +are: (i) at least one self-renewing cell type (with µ = 0) must exist; (ii) self- +renewing cell types must stay at an apex of the condensed network; (iii) all +the other cells must be of declining types. This means that the critical task of +homeostatic control is to ensure that the kinetic parameters of the cell type at +the apex of the cell lineage hierarchy are fine-tuned to maintain exactly µ = 0. +Therefore, we can restrict our analysis to find conditions for the cell type +at the lineage hierarchy’s apex to be self-renewing, which we will do in the +following. Other cell types simply need that differentiation (transition towards + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +7 +another cell type) or loss is faster than proliferation, so that they become +declining cell types, µ < 0, but those rates do not require fine-tuning and thus +trivially regulated. We note that when we consider only cell states of the type +at the apex of the cell lineage hierarchy, any differentiation event is – according +to this restricted model – a cell loss event and included as event occurring with +rates γi. Given that cell loss from a cell type at the lineage apex is rare, we +will therefore in the following also denote the rates γi simply as differentiation +rates. +3 Results +We will now determine necessary and sufficient conditions for the establish- +ment of strict and dynamical homeostasis when subject to crowding feedback, +which we here define through the derivatives of the dynamical parameters +λi, rjk +i , ωij, γi as a function of the cell densities. As argued before, we only need +to consider cell types at an apex of the cell type network, which, for home- +ostasis to prevail, must have a growth parameter (i.e. dominant eigenvalue of +matrix A in Eq. (6)) µ = 0. Furthermore, we assume that the apex cell type +resides in a separate stem cell niche. Therefore, the parameters only depend +on cell densities ρi of states associated with that cell type, i.e. we can write +A = A(ρ), where ρ = � +i∈S ρi comprises only cell states of the apex cell type +S. Provided that, the matrix elements are functions of ρ, and therefore also µ +is a function of ρ. Thus, self-renewal corresponds to a non-trivial fixed point, +ρ∗, of Eq. (5), restricted to cell type S, for which the dominant eigenvalue of +A is zero, that is µ(ρ∗) = 0 (ρ∗ = � +i∈S ρ∗ +i ). +For convenience, we will often generally refer to parameters as αi, i = +1, ..., 2m + m2, where αi stands for any of the parameters, {λi, γi, κij|i, j = +1, ..., m}, respectively1. Hence, we study which conditions the functions αi(ρ) +must meet to maintain homeostasis. In particular, we study how those param- +eters qualitatively change with the cell density – increase or decrease – that +is, we study how the sign and magnitude of derivatives α′ +i := +dαi +dρ +affects +homeostasis. +A crucial property of the matrix A(ρ) is that it is always a Metzler matrix, +since all its off-diagonal elements, κij ≥ 0. Since the cell state network of a +cell type is strongly connected, we can further state that A(ρ) is irreducible. +Notably, for irreducible Metzler matrices holds the Perron-Frobenius theorem +[22], and thus A(ρ) possesses a simple, real dominant eigenvalue µ. Besides, +it as left and right eigenvectors associated with µ, respectively indicated as v +and w, which are strictly positive, that is, all their entries are vi > 0, wi > 0. +From this follows that the partial derivative of the dominant eigenvalue µ by +1More +precisely, +αi|i=1,..,m +:= +λi, αi|i=m+1,..,2m +:= +γi−m, αi|i=2m+1,..,2m+m2 +:= +κ⌊(i−2m)/m⌋,i−⌊(i−2m)/m⌋m + +Springer Nature 2021 LATEX template +8 +Homeostatic regulation of renewing tissue cell populations via crowding control +the i, j-th element of A, aij = [A]ij is always positive: +∂µ +∂aij += viwj +vw > 0 +(8) +where the left equality is according to [23] and is generally valid for simple +eigenvalues. Here, v is assumed to be in row form, and vw thus corresponds +to a scalar product. +3.1 Sufficient condition for dynamic homeostasis +In [13], it was shown that a dynamic homeostatic state, where cell numbers +may change over time but stay bounded, is assured if, 2 +µ′(ρ) < 0 for all ρ > 0. +(9) +This sufficient condition requires that the dominant eigenvalue of A as a func- +tion of the cell density, µ(ρ), is a strictly decreasing function of cell density. +Also, the range of this function must be sufficiently large so that it has a root, +i.e. a value ρ∗ with µ(ρ∗) = 0 must exist for the function µ(ρ). +Assuming that a non-trivial steady state, ρ∗ > 0, exists, we now translate +the sufficient condition for a dynamic homeostatic state, Eq. (9), into condi- +tions on the parameters as a function of the cell density, αi(ρ). In particular, +we can write, +µ′(ρ) = +� +ij +∂µ +∂aij +∂aij +∂ρ = +� +ij +viwj +vw a′ +ij = +� +i +viwi +vw a′ +ii + +� +i,j̸=i +viwj +vw a′ +ij += +� +i +viwi +vw +� +�λ′ +i − γ′ +i − +� +j̸=i +κ′ +ij +� +� + +� +i,j̸=i +vjwi +vw κ′ +ij , +(10) +where we used Eq. (8) and the explicit forms of aij, the elements of the matrix +A according to Eq. (7). Provided that all the parameters depend on ρ, condition +(9) results in: +0 > µ′ =⇒ 0 > +� +i +viwi (λ′ +i − γ′ +i) + wi +� +j̸=i +(vj − vi)κ′ +ij +for all ρ > 0 , +(11) +While we cannot give an explicit general expression for the dominant eigen- +vectors v, w, this condition is sufficiently fulfilled if each term of the sum on +the right-hand side of Eq. (11) is negative. More restrictively, we have Eq. (11) +2In [13], this condition, defined through dependency on cell number, can be directly translated +into a condition on the cell density derivative if the volume is assumed as a constant. + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +9 +sufficiently fulfilled if +� +� +� +� +� +λ′ +i ≤ 0, γ′ +i ≥ 0 for all i +λ′ +i < 0 or γ′ +i > 0 at for least one i +κ′ +ij = 0 for all i, j +for ρ > 0 +(12) +This means that, excluding rates that are zero, which are biologically mean- +ingless, if no direct state transitions within a cell type are subject to crowding +feedback (κ′ +ij = 0), while all (non-zero) cell division rates depend negatively +on ρ (λ′ +i < 0), and differentiation rates depend positively (γ′ +i > 0), for all +attainable levels of ρ, then dynamical homeostasis is ensured. +Alternatively, we can rewrite Eq. (11) as +0 > +� +i +viwi +vw +� +�λ′ +i − γ′ +i − +� +j̸=i +κ′ +ij + +� +j̸=i +vj +vi +κ′ +ij +� +� +for all ρ > 0 , +(13) +which, due to +vj +vi +> 0, implies another sufficient condition for dynamic +homeostasis: +� +� +� +� +� +λ′ +i ≤ 0, γ′ +i ≥ 0 for all i +λ′ +i < 0 or γ′ +i > 0 at for least one i +κ′ +ij ≤ 0 with |� +j κ′ +ij| ≤ γ′ +i − λ′ +i for all i, j +(14) +The above condition is less restrictive than Eq. (12), allowing for some non- +zero crowding feedback dependency of state transition rates κij, as long as the +crowding feedback strength of the total outgoing transition rate of each state +does not outweigh the feedback on proliferation and differentiation rate of that +state (if there is). +3.2 Necessary condition for strict homeostasis +We now consider the circumstances under which a strict homeostatic is main- +tained, that is, when a steady state of the cell population exists and is +asymptotically stable. +A necessary condition for the existence of a steady state ρ∗ (irrespective +of stability) has been given in [13], namely, that the cell type at the apex of +the lineage hierarchy is self-renewing, i.e. its dynamical matrix A has µ = 0. +µ depends on the cell density ρ of the apex cell type, since the dynamical +parameters αi and thus A depend on ρ. As before, it is required that µ(ρ∗) +has sufficient range so that a value ρ∗ with µ(ρ∗) = 0 exists. This condition is +fulfilled if the range of the feedback parameters αi(ρ) is sufficiently large. In +that case there exists an eigenvector ρ∗ with A(ρ∗)ρ∗ = 0, which can be chosen +by normalisation to fulfil � +i∈S ρ∗ +i = ρ∗. Thus, ρ∗ is a fixed point (steady state) + +Springer Nature 2021 LATEX template +10 +Homeostatic regulation of renewing tissue cell populations via crowding control +of the cell population system (5). Hence, we need to establish what is required +for this state to be asymptotically stable. +To start with, we give the Jacobian matrix of the system (5) at the fixed +point ρ∗ : +[J]ij = ∂[A(ρ)ρ]i +∂ρj +���� +ρ=ρ∗ += a∗ +ij + ηi , +(15) +where +ηi = +� +k +a′ +ikρ∗ +k . +(16) +Here and in the following, we assume the derivatives to be taken at the steady +state, i.e. a′ +ij := +daij +dρ |ρ=ρ∗. The eigenvalues of the Jacobian matrix J at ρ∗ +determine the stability of the steady state ρ∗: it is asymptotically stable if and +only if the real part of all eigenvalues of J(ρ∗) is negative. +The Routh-Hurwitz theorem [24] states that for a polynomial to have only +roots with negative real part, all its coefficients must necessarily be positive. +Given that the eigenvalues of the Jacobian matrix J are the roots of its char- +acteristic polynomial, a necessary condition for ρ∗ to be asymptotically stable +is that the coefficients of the characteristic polynomial of J are all positive. +Let us start by considering a self-renewing cell type with exactly two cell +states being at the apex of a lineage hierarchy. This system has a 2 × 2 dynam- +ical matrix A and Jacobian J, whereby A is irreducible and has dominant +eigenvalue µA = 0. The characteristic polynomial of a generic 2×2 matrix, M, +is +P M(s) = s2 + pM +1 s + pM +0 . +(17) +with pM +1 = −tr(M) and pM +0 = det(M). In particular, since A has an eigenvalue +zero, +pA +0 = det(A) = a11a22 − a12a21 = 0 . +(18) +From this follows that the right and left eigenvectors to the matrix A +associated with the dominant eigenvalue µA = 0, w and v, are: +w = +� +−a22 +a21 +� +and v = +� +−a22 a12 +� +. +(19) +From the Jacobian matrix J, we get equivalently, +pJ +0 = det(J) = (a21 − a22)(−a22η1 + a12η2) +a22 += vη |w| +a22 +, +(20) +with the L1-norm |w| = w1 + w2 = −a22 + a213. Here we used the form of J +in Eq. (15) with η = (η1, η2) from (16), as well as the relations (18) and (19), +and we factorised the determinant. +3Note that aii is always negative or zero + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +11 +From Eq. (10), we can further establish: +µ′ = +� +ij +viwj +vw a′ +ij = +� +ij +|w| +ρ∗ +viρ∗ +j +vw a′ +ij = |w| +ρ∗ +vη +vw +(21) += − a22pJ +0 +ρ∗pJ +1 a22 +. +(22) +Here, we used that ρ∗ is a dominant right eigenvector, and thus ρ∗ = +ρ∗ +|w|w, +and furthermore we used the definition of ηi = � +j a′ +ijρ∗ +j, we substituted Eq. +(20), and used that vw = a2 +22 + a12a21 = −pA +1 a22. Finally, we get: +pJ +0 = −µ′ρ∗pA +1 . +(23) +Notably, we can show that this relation also holds for higher dimensions by +explicitly computing the coefficients of characteristic polynomials pA,J +i +, the +eigenvalues and eigenvectors, and then evaluating both sides of the equation. +For systems with three states, this can be done analytically. For systems with +4,5, and 6 states we tested relation (23) numerically by generating N =1000 +random matrices with entries chosen from a uniform distribution4. In each +case, this relation was fulfilled. Hence we are confident that this relation holds +up to 6 states, and it is reasonable to expect this to hold also for larger systems. +Since A has a simple dominant eigenvalue µA = 0, we can factorise one term +from the characteristic polynomial, P(s) = sQ(s) knowing that all roots of +Q(s) are negative. Applying the Routh-Hurwitz necessary condition to Q(s), it +follows that the coefficients of the polynomial Q, pQ +i > 0, where i = 1, 2, ..., n− +1. Thus, pA +1 > 0 and considering that ρ∗ > 0 by definition, then for having +pJ +0 > 0 we must require µ′ < 0. Therefore, a necessary condition for a stable, +strict homeostatic state is +0 > µ′ =⇒ 0 > +� +i +viwi (λ′ +i − γ′ +i) + wi +� +j̸=i +(vj − vi)κ′ +ij +������ +ρ=ρ∗ +, +(24) +where on the right-hand side, we used Eq. (11). This condition is bound to the +validity of Eq. (23), that is, we can show it analytically for up to three states +and numerically up to 6 states. Nonetheless, we also expect this to be true for +larger systems. +One way to satisfy this necessary condition is if at ρ = ρ∗ +� +� +� +� +� +λ′ +i ≤ 0, γ′ +i ≥ 0 for all i +λ′ +i < 0 or γ′ +i > 0 at for least one i +κ′ +ij = 0 +. +(25) +4The diagonal elements of the random matrix are tuned using a local optimiser (fmincon +function of Matab) so that the matrix has a zero dominant eigenvalue. + +Springer Nature 2021 LATEX template +12 +Homeostatic regulation of renewing tissue cell populations via crowding control +Notably, the necessary conditions (24) and (25) only differ from the suffi- +cient conditions for dynamic heterogeneity, Eqs. (11) and (12), by needing to +be fulfilled only at the steady-state cell density ρ∗, whereas to ensure dynamic +homeostasis, those should be valid for a sufficiently large range of ρ. +3.3 Sufficient condition for strict homeostasis +Now we assess under which circumstances a strict homeostatic state is assured +to prevail. +First of all, the necessary conditions from above need to be fulfilled. In +particular, the parameter functions αi(ρ) must have a sufficient range so that +µ(ρ) has a root, i.e. ρ∗ with µ(ρ∗) = 0 exists, from which the existence of +a steady state follows. The question now is whether we can find sufficient +conditions assuring that the fixed point ρ∗ with � +i ρ∗ +i = ρ∗ is (asymptotically) +stable. +Let us define a matrix B(x), x = (x1, ..., xm) with bij(x) = [B]ij(x) = +a∗ +ij +xi. Hence, B(xi = 0) = A(ρ∗) and B(xi = ηi) = J, where J, the Jacobian +matrix, and ηi are defined as in (15) and (16), respectively. We consider now the +dominant eigenvalue as function of the entries of B, µ[B] := µ({bij}|i,j=1,...,m) +(the square brackets are chosen to denote the difference from the function +µ(ρ)). For sufficiently small ηi, we can then express the dominant eigenvalue +of the Jacobian matrix J, µ[J], relative to the dominant eigenvalues of A∗ := +A(ρ∗) as, +µ[J] = µ[A∗] + +� +i +∂µ +∂xi +|xi=0 ηi + O(η2) , +(26) +with, +∂µ +∂xi +|xi=0 = +� +ij +∂µ +∂bij +∂bij +∂xi +|xi=0 = +� +ij +∂µ +∂aij +|B=A∗ , +(27) +since for x = 0, bij = aij for all i, j. It follows that for sufficiently small5 ηi, +and if all ηi < 0, we have +µJ = µ[A∗](ρ∗) + +� +i +∂µB +∂xi +|xi=0ηi + O(η2 +i ) ≈ +� +i +∂µA +∂aij +ηi < 0 +(28) +since all ∂µA +∂aij > 0 (according to Eq. (8)) and µA(ρ∗) = 0. Hence, since µJ < 0, +the steady state ρ∗ is asymptotically stable if all ηi < 0. Thus, we get a +sufficient condition for asymptotic stability of the steady state ρ∗: +0 > ηi = ρ∗ +i (λ′ +i − γ′ +i) + +� +k̸=i +(κ′ +kiρ∗ +k − κ′ +ikρ∗ +i ) > −ϵi for all i +(29) +5That is, there exist ϵi > 0 so that this is valid for any |ηi| < ϵi + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +13 +where ϵi > 0 is sufficiently small. As this is an asymptotically stable steady +state, it corresponds to a controlled strict homeostatic state. In this case, +even if the cell numbers are disturbed (to some degree), the cell population is +regulated to return to the strict homeostatic state. +Notably, condition (29) is fulfilled if, +� +� +� +� +� +� +� +� +� +λ′ +i ≤ 0, γ′ +i ≥ 0 for all i +λ′ +i < 0 or γ′ +i > 0 at for least one i +κ′ +ij = 0 +and |λ′ +i|, |γ′ +i|, < ϵi +(30) +Furthermore, we may soften the condition on κij to +κ′ +ij +κ′ +ji < +ρ∗ +j +ρ∗ +i to allow also +some degree of feedback for the κij. +The conditions (30) are very similar to the ones for dynamic homeostasis, +(12), but here these conditions only need to be fulfilled at ρ = ρ∗, whereas +for dynamic homeostasis they need to be fulfilled for a sufficient range of ρ. +Moreover, in addition to the qualitative nature of the feedback (related to +the signs of λ′ +i, γ′ +i), the ‘strength’ of the crowding feedback, i.e. the absolute +values of λ′ +i, γ′ +i must not be ‘too large’, that is, smaller than ϵi. We cannot, in +general and for all system sizes, give a definite value for the feedback strength +bound ϵi below which strict homeostasis is assured. Nevertheless, by using the +sufficient stability criterion based on the Routh-Hurwitz criterion [24] we can +identify those bounds for systems with up to three cell states, which guides +expectations for larger systems. The details of this criterion and the necessary +derivations are shown in Appendix A. There, we show that for systems with one +or two cell states, ϵi = ∞, which means that asymptotic stability is ensured for +arbitrary feedback strengths. For systems with three cell states, we can assure +that ϵi = ∞ if certain further conditions are met (see Eq. (A13)). Otherwise, +ϵi can be determined implicitly from the roots of a quadratic form (Eq. (A14)), +and thus stability may depend on the magnitude of the feedback. In principle, +such bounds can be found for larger systems too, but the algebraic complexity +of this process renders it unfeasible to do this in practical terms. +3.4 Robustness to perturbations and failures +Now, we wish to assess the robustness of the above crowding control mecha- +nism, i.e. what occurs if it is disrupted, for example, by the action of toxins, +other environmental cues, or by cell mutations. More precisely, we will study +what happens if one or more feedback pathways, here characterised as a param- +eter αi with α′ +i ̸= 0 fulfilling the conditions for (dynamic or strict) homeostatic +control, is failing, that is, it becomes α′ +i = 0. We will first address the case +of tissue-extrinsic factors, i.e. those affecting all the cells in the tissue, and +then the case of single-cell mutations. In the latter case, only a single cell +would initially show a dysregulated behaviour, yet, if this confers a proliferative +advantage, it can lead to hyperplasia and possibly cancer [25–27]. + +Springer Nature 2021 LATEX template +14 +Homeostatic regulation of renewing tissue cell populations via crowding control +First, we note that the sufficient condition for strict homeostasis, given +by Eq. (30), is overly restrictive. In a tissue cell type under crowding feed- +back control with λ′ +i < 0 and γ′ +i > 0 for more than one i, there is a degree +of redundancy. That is, if the feedback is removed for one or more of these +parameters (changing to λ′ +i = 0 and, or γ′ +i = 0), then the sufficient condition +for a strict homeostatic state remains fulfilled as long as at least one λ′ +i or +γ′ +i remains non-zero. This possible redundancy confers a degree of robustness, +meaning that feedback pathways can be removed – setting α′ +i = 0 – without +losing homeostatic control. Since the necessary conditions, Eqs. (24), are even +less restrictive, tissue homeostasis may even tolerate more severe disruptions +that reverse some feedback pathways, e.g. switching from λ′ +i < 0 to λ′ +i > 0, +as long as other terms in the sum on the right-hand side of (24) compensate +for this changed sign, ensuring that the sum as a whole is negative. In any +case, it is important to remind the underlying assumption for which a non- +trivial steady state exists. In case the variability of the kinetic parameters is +not enough to assure the condition µ(ρ∗ = 0), then the tissue will degenerate, +either shrinking and eventually disappearing or indefinitely growing. +From the above considerations, we conclude that if crowding control applies +to more than one parameter αi, that is, α′ +i ̸= 0 with appropriate sign and +magnitude, homeostasis is potentially robust to feedback disruption. This may +include a simple variation of the feedback function α′ +i but also perturbation in +the feedback functions shape and complete feedback failure, α′ +i = 0. +An illustrative example of this situation is shown in Figure 1. Here, the time +evolution of the cell density is shown for a three-state cell fate model, which has +been computed by integration of the dynamical system (5) (the details of this +model are given in Appendix B as Eq. (B15) and illustrated in Figure B1). Four +kinetic parameters are regulated via crowding control satisfying the sufficient +condition for strict homeostasis, (30). Then, starting from this homeostatic +configuration, feedback disruption is introduced at a time equal to zero. In one +case (“Single failure”), a single kinetic parameter suffers a complete failure of +the type α′ +i = 0. In this case, the remaining feedback functions compensate +for this failure, and a new homeostatic condition is achieved. Instead, in the +second case (“Multiple failures”), failures are applied so that three of the four +kinetic parameters initially regulated do not adjust with cell density6. Notably, +the only feedback function left satisfies the condition for asymptotic stability, +(30). Nevertheless, the variability of this kinetic parameter is not enough to +assure the existence of a steady state, since in this case, the function µ(ρ) does +not possess any root. Hence µ > 0 for all ρ, leading to an indefinite growth of +the cell population. Additional test cases are presented in Appendix B.2. +So far, we modelled the feedback dysregulation as acting on a global scale, +thus changing the whole tissue’s dynamics behaviour. This situation represents +a feedback mechanism affected by cell-extrinsic signals, in which any dysregu- +lation applies to all the cells in the same way. However, dysregulation can also +6Only in this example, feedback control fails upon multiple failures, while in general, multiple +failures may still be compensated to maintain homeostatic control. + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +15 +-10 +0 +10 +20 +30 +40 +0 +2 +4 +6 +8 +10 +Homeostasis +Single failure +Multiple failures +-10 +0 +10 +20 +30 +40 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +Homeostasis +Single failure +Multiple failures +Fig. 1 +Cell dynamics in terms of cell density, scaled by the steady state in the homeostatic +case, as a function of time (left) and the corresponding variation of the dominant eigenvalue µ +(right). Time is scaled by the inverse of ¯α = mini α∗ +i . The homeostatic model is perturbed at +a time equal to zero to include feedback failure of the type α′ +i = 0. In the case where only one +feedback function fails (“Single failure”), the system is able to achieve and maintain a new +homeostatic state, characterised by a constant cell density and a zero dominant eigenvalue. +In case more than one feedback fails (“Multiple failures”), the cell dynamics are unstable +since a steady state does not exist and µ > 0 for all ρ. The cell fate model corresponds to +model (B15) with parameters given in Table B1 and Table B2. +act at the single-cell level, for example, when DNA mutations occur. In this +case, the impact of the dysregulation is slightly different, as explained in the +following. +Suppose, upon disruption of crowding control in a single cell, for example, +by DNA mutations, a sufficient number of crowding feedback pathways remain +so that there is a steady state and the sufficient condition (30) is still fulfilled. +In that case, homeostasis is retained, just as when this occurs in a tissue-wide +disruption. However, if the homeostatic control of that single cell fails such that +the cell becomes hyperproliferative, µ > 0, or declining, µ < 0, the tissue may +still remain homeostatic. If µ < 0, the single mutated cell will be lost, upon +which only a population of crowding controlled cells remain, which remain in +homeostasis. If µ > 0 in a single cell, hyper-proliferation is not ensured either: +while the probability for mutated cells to grow in numbers is larger than to +decline, due to the low numbers, mere randomness can lead to the loss of the +mutated cell with a non-zero probability, which results in the extinction of +the dysregulated mutant7. In that case, the mutant cells go extinct and the +tissue remains homeostatic despite the disruption of homeostatic control in +the mutated cells; a stark contrast to disruption on the tissue level. Otherwise, +if the mutant clone (randomly) survives, it will continue to hyper-proliferate +and eventually dominate the tissue, which is thus rendered non-homeostatic. +However, the tissue divergence time scale may be much longer than the case +where the same dysregulation occurs in all cells. +The deterministic cell population model (5) is suitable for describing the +average cell numbers. Nevertheless, it fails to describe the stochastic nature of +7For example, in the case of a single state with cell division rate λ and loss rate γ – a simple +branching process – the probability for a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ, +which is less than certainty. + +Springer Nature 2021 LATEX template +16 +Homeostatic regulation of renewing tissue cell populations via crowding control +single-cell fate choice. Thus, assessing a single cell’s impact on tissue dynamics +requires stochastic modelling. To that end, we implemented this situation as +a Markov process with the same rates as the tissue cell population dynamics +model8 (see Appendix B.3 for more details). +In Figure 2, we show numerical simulation results of a stochastic version +of the model used for previous results in Figure 1, depicted in terms of tissue +cell density as a function of time. Here, two possible realisations of the same +stochastic process are presented. We note that the initially homeostatic tissue +results in stochastic fluctuations of the cell density, which remain, on average, +constant. At a time equal to zero, a single cell in this tissue switches behaviour, +presenting multiple failures which, if applied to all the cells, would determine +the growth of the tissue (corresponding to Multiple failures curve in Figure 1). +In one instance of the stochastic simulation, however, the mutated clone goes +extinct after some time, leaving a tissue globally unaffected by the mutation. +On the other hand, in another instance, the mutated clone prevails, leading to +the growth of the tissue cell population. The fact that vastly different outcomes +can occur with the same parameters and starting conditions demonstrates the +impact of stochasticity in the case of a single-cell mutation. +-10 +0 +10 +20 +30 +40 +50 +1 +1.2 +1.4 +Homeostasis +Multiple Failure (instance #1) +Multiple Failure (instance #2) +Fig. 2 +Numerical simulation results of a stochastic version of the model used in Figure 1 +upon disruption of crowding control in a single cell, mimicking a DNA mutation. At a +time equal to 0, the initially homeostatic model is disrupted with a single cell presenting +multiple failures in the feedback control, as in Figure 1. Two instances of simulations run +with identical parameters are presented. The rescaled cell density ρ/ρ∗ is shown as a function +of the time, scaled by the inverse of ¯α = mini α∗ +i . Whilst the mutated cell and its progeny go +extinct in one instance (#1), in the other (#2), mutated cells prevail and hyper-proliferate +so that tissue homeostasis is lost. The simulation stops when the clone goes extinct or when +instability is detected. Full details of the simulation are provided in Appendix B.3. +3.5 Quasi-dedifferentiation +In the previous section, we addressed the case where external or cell-intrinsic +factors disrupt homeostatic control in self-renewing cells of a tissue. However, +8While a Markov process is an approximation which not necessarily reflects the probability +distribution of subsequent event times realistically, it is often sufficient to assess the qualitative +behaviour of a system with low numbers, subject to random influences from the environment. + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +17 +situations such as injury, poisoning, or cell radiation might also affect home- +ostasis in other ways. An example is when stem cells are completely depleted +from the tissue. In this context, many studies about tissue regeneration after +injury report evidence of cell plasticity [17, 18], when committed cells regain +the potential of the previously depleted stem cells. Cell dedifferentiation is just +an example where differentiated cells return to an undifferentiated state as a +response to tissue damage. Lineage tracing experiments confirmed this feature +in vivo in several cases [16, 28–30]. +In the following, we assess how committed progenitor cells respond to the +depletion of the stem cell pool if they are under crowding feedback control. +Without loss of generality, let us consider an initially homeostatic scenario +where there is a self-renewing (i.e. stem) cell type (S) – with growth param- +eter µ = 0 – at the apex of a lineage hierarchy, and a committed progenitor +cell type (C) – with µ < 0, but with at least one state that has a non-zero +cell division rate – below type S in the hierarchy, as depicted in Figure 3. +Based on this cell fate model, S-cells proliferate and differentiate into C-cells +while maintaining the S-cell population. The C-cells also proliferate and dif- +ferentiate into other downstream cell types which we do not explicitly consider +here. C-cells do not maintain their own population; only the steady influx of +new cells of that type via differentiation of S-cells into C-cells maintains the +latter population (see [13]). We further assume that both S- and C-cells are +under appropriate crowding control, fulfilling both the sufficient conditions for +dynamic homeostasis, (12), and for stable, strict homeostasis, (30). +Based on the above modelling, we can write the dynamics of the cell +densities belonging to the committed progenitor type as, +d +dtρc = Ac(ρc)ρc + u , +(31) +where ρc = (ρms+1, ρms+2, .., ρms+mc) are the cell densities in the committed +C-type, with ms being the number of states of the self-renewing S-type. Ac is +the dynamical matrix restricted to states in the C-type and ui = �ms +j=1 κjiρj +is a constant vector quantifying the influx of cells into the C-type. +First, we note that the Jacobian matrix of a committed cell type, described +by (31), J = +� +∂A(ρc)ρc +∂ρj +� +j=ms+1,...,ms+mc +, has the same form as a cell type at the +apex of the hierarchy, since u does not depend on the densities ρms+1,...,ms+mc. +From this follows that if C-cells are regulated by crowding control, fulfilling the +conditions (30), then also the population of C-cells is stable around a steady +state ρ∗ +c, albeit with a growth parameter µc(ρ∗ +c) < 09. +We now consider the scenario where all stem cells are depleted at some +point, as was experimentally done in [16, 18]. This would stop any replen- +ishment of C-cells through differentiation of S-cells, corresponding to setting +9This can be seen when multiplying the steady state condition for (31), Ac(ρs, ρc)ρc + u = 0 +with a positive left dominant eigenvector v, giving, µcvρ∗ +c + vu = 0. Since ρ∗ and v have all +positive entries and u is non-negative, this equation can only be fulfilled for µc < 0. + +Springer Nature 2021 LATEX template +18 +Homeostatic regulation of renewing tissue cell populations via crowding control +Fig. 3 Sketch representative of the quasi-dedifferentiation scenario. A homeostatic system +enclosed in the black box comprises two cell types: a stem cell type, S, (blue) and a com- +mitted cell type, C, (green). In the unperturbed homeostatic scenario, S is self-renewing, +characterised by a growth parameter at the steady state µ∗ = 0, and C is transient, with +a growth parameter at the steady state µ∗ < 0. Both cell types are subject to crowding +control, fulfilling both conditions (12), and (30). By removing the stem cell type XS, the +committed cell type acquires self-renewing property through crowding control, effectively +becoming a stem cell type (see Figure 4). +u = 0 in (31). Hence we end up with the dynamics ˙ρc = A(ρc)ρc. Now, assum- +ing that the function µ(ρ) has sufficient range, so that µ(ρ∗∗ +c ) = 0 for some +ρ∗∗ +c , and provided that A(ρc) is under crowding control fulfilling the sufficient +conditions for asymptotic stability of a steady state, then, following our argu- +ments from section 3.3, the population of C-cells will attain a stable steady +state. In other words, those previously committed cells become self-renewing +cells. Also, since they now reside at the apex of the lineage hierarchy (given +that S-cells are absent), they effectively become stem cells. +Hence, under crowding control, previously committed progenitor cells +(committed cells that can divide) will automatically become stem cells if the +original stem cells are depleted. Commonly, such a reversion of a committed cell +type to a stem cell type would be called ‘dedifferentiation’ or ‘reprogramming’. +However, in this case, no genuine reversion of cell states occurs; previously +committed cells do not transition back to states associated with the stem cell +type. Instead, they respond by crowding feedback and adjust their dynamical +rates so that µ becomes zero, hence attaining a self-renewing cell type. Cru- +cially, this new stem cell type is fundamentally different to the original one +and still most similar to the original committed type. We call this process +quasi-dedifferentiation. The quasi-dedifferentiation follows the same reversion +of proliferative potential as in ‘genuine’ dedifferentiation but without explicit +reversion in the cell state trajectories. +The following numerical example illustrates this situation. We focus on the +cell dynamics of a single C-type regulated via crowding feedback (detail of +the model are provided in Appendix B.4). The cell density as a function of +the time, shown in Figure 4, is obtained by integrating the corresponding cell +population model according to Eq. (5). The system is initially in a homeostatic +condition, meaning that there is a constant influx of cells from some upstream +self-renewing types. Such upstream types are assumed to be properly regulated +such that this cell influx is constant over time. At a time equal to zero, the cell +influx becomes suddenly zero, representing an instantaneous removal of all the + +Xo- +HomeostasisSpringer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +19 +-10 +0 +10 +20 +30 +40 +0 +0.2 +0.4 +0.6 +0.8 +1 +Homeostasis +Quasi-dedifferentiation +-10 +0 +10 +20 +30 +40 +-0.2 +-0.1 +0 +Homeostasis +Quasi-dedifferentiation +Fig. 4 Cell dynamics of an initially committed cell type C (µ < 0) upon removal of all stem +cells. (Left) Cell density scaled by the steady-state density as a function of time. (Right) +Corresponding variation of the dominant eigenvalue µc. Time is scaled by the inverse of +¯α = mini α∗ +i . It is assumed that a stem cell type, S, initially resides in the lineage hierarchy +above the committed cell type (as in Figure 3). S cells differentiate into C cells, which is +modelled as a constant cell influx of C-cells (S is not explicitly simulated). At a time equal +to zero, a sudden depletion of S cells is modelled by stopping the cell influx. After some +transitory phase, the cell population stabilises around a new steady state and becomes self- +renewing with µc = 0. The full description of the dynamical model, which corresponds to +model (B15) with parameters given in Table B1, is reported in Appendix B.4. +self-renewing cells from the tissue. A new homeostatic condition is achieved +after a transitory phase thanks to the crowding feedback acting on the C- +type. This example demonstrates how an initially committed cell type, i.e. +with µc < 0, regulated via crowding feedback, might be able to switch, upon +disruption, to a self-renewing behaviour µc = 0. +4 Discussion +For maintaining healthy adult tissue, the tissue cell population must be +maintained in a homeostatic state. Here, we assessed one of the most com- +mon generalised regulation mechanisms of homeostasis, which we refer to as +crowding feedback. Based on this, progenitor cells (stem cells and committed +progenitors) adjust their propensities to divide, differentiate, and die, accord- +ing to the surrounding density of cells, which they sense via biochemical or +mechanical signals. For this purpose, we used a generic mathematical model +introduced before in Refs. [13, 20], which describes tissue cell population +dynamics in the most generic way, including cell divisions, cell state transi- +tions, and cell loss / differentiation. Based on this model, we rigorously define +what is meant when speaking of a ‘homeostatic state’, introducing two notions: +a strict homeostasis is a steady state of the tissue cell population dynamics, +while dynamical homeostasis allows, in addition to strict homeostasis, for oscil- +lations and fluctuations, as long as a finite long-term average cell population +is maintained (such as the endometrium during the menstrual cycle). +By analysing this dynamical system, we find several sufficient and necessary +conditions for homeostasis. These conditions are formulated in terms of how +the propensities of cell division, differentiation, and cell state changes, of cells + +Springer Nature 2021 LATEX template +20 +Homeostatic regulation of renewing tissue cell populations via crowding control +whose type is at the apex of an adult cell lineage hierarchy, may depend on +their cell density. We find that when, for a wide range of cell density values, +the cell division propensity of at least one state decreases with cell density or +the differentiation propensity increases with it, while other propensities (e.g. +of cell state transitions) are not affected by the cell density, then dynamic +homeostasis prevails (12). For strict homeostasis to prevail, this only needs +to be fulfilled at the steady state itself, but in addition, the magnitude of +the feedback strength may not be too large (30). We can derive explicit and +implicit expressions for the bound on feedback strength for systems of two +and three-cell states but cannot do so for arbitrary systems. Furthermore, we +find that a necessary condition for strict homeostasis is that the conditions for +dynamic homeostasis are met at least at the steady state cell density. +A direct consequence of the conditions we found is that they allow for a +considerable degree of redundancy when more than one propensity depends +appropriately on the cell density. Hence feedback pathways, that is, cell dynam- +ics parameters depending on the cell density, may serve as ‘back-ups’ to each +other if one fails. We demonstrate that this confers robustness to the home- +ostatic system in that one or more crowding feedback pathways may fail, yet +the tissue remains in homeostasis. +Finally, we assess how crowding feedback regulation affects the response of +committed progenitor cells to a complete depletion of all stem cells. We showed +that committed cells which can divide and are under appropriate crowding +feedback control (that is, meeting the sufficient conditions (12) and (30)), will +necessarily, without additional mechanisms or assumptions, reacquire stem cell +identity, that is, become self-renewing and are at the apex of the lineage hierar- +chy. Notably, while this process resembles that of dedifferentiation, it does not +involve explicit reprogramming, in that the cell state transitions are reversed. +Instead, only the cell fate propensities adjust to the changing environment by +balancing proliferation and differentiation as is required for self-renewal. While +these are purely theoretical considerations, and such a process has not yet +been experimentally found, we predict that it must necessarily occur under the +appropriate conditions. This can be measured by assessing the gene expression +profiles (e.g. via single-cell RNA sequencing) of cells that ‘dedifferentiate’, i.e. +reacquire stemness after depletion of stem cells. Moreover, those considerations +yield further, more general insights: +• Stem cell identity is neither the property of individual cells nor is it strictly +associated with particular cell types or states. Any cell that can divide and +differentiate, committed or not, may become a stem cell under appropriate +environmental control. +• From the latter follows that stemness is a property determined by the +environment, not the cell itself. +• ‘Cell plasticity’ might need to be seen in a wider context. Usually, cell +plasticity is associated with a change of a cell’s type when subjected to +environmental cues, which involves a substantial remodelling of the cell’s +morphology and biochemical state. However, we see that a committed cell + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +21 +may turn into a stem cell simply by adjusting the pace of the cell cycle +and differentiation processes to the environment. This may not require +substantial changes in the cell’s state. +This exemplifies that homeostatic control through crowding feedback is not +only a way to render homeostasis stable and robust, but also to create stem +cell identities as a collective property of the tissue cell population. +Acknowledgments. +We thank Ben MacArthur and Ruben Sanchez-Garcia +for valuable discussions. +Declarations +PG is supported by an MRC New Investigator Award, Grant number +MR/R026610/1. The code generated for numerical computations in the cur- +rent study is available on Github, https://github.com/cp4u17/Feedback. No +other data was generated for this work. +Contributions are as follows: C.P. and P.G. conceptualised the paper, C.P. +and P.G. did the mathematical analysis, C.P. did the numerical analysis, P.G. +supervised the work. +The authors have no competing interests to declare that are relevant to the +content of this article. +Appendix A +Asymptotic stability assessment +based on Routh-Hurwitz +A.1 +Background +In control system theory, a commonly used method for assessing the stability +of a linear system is the Routh-Hurtwiz (RH) criterion [24]. It is an algebraic +criterion providing a necessary and sufficient condition on the parameters of a +dynamic system of arbitrary order to ensure the dynamics are asymptotically +stable. In particular, the criterion defines a set of conditions on the coefficients, +pi, of the characteristic polynomial, P(s), written as +P(s) = sn + +n +� +i=1 +pisn−i , +(A1) +in which n corresponds to the dimension of the system. Note that the notation +used in this section, based on that from [24], is different from that of the main +text, where pi is the polynomial coefficient of ith order. +A first result of the RH criterion is that a necessary condition for the +dynamical system to be asymptotically stable is that all the coefficients must +be positive, that is, +pi > 0, for all i . +(A2) + +Springer Nature 2021 LATEX template +22 +Homeostatic regulation of renewing tissue cell populations via crowding control +Additional conditions on the polynomial coefficients are added for a necessary +and sufficient condition. These conditions are based on Routh’s array, written +as +� +����� +1 p2 p4 ... 0 +p1 p3 ... +b1 b2 ... +c1 +... +� +����� +, +(A3) +in which the first two rows contain all the coefficients of the characteristic +polynomial, and the following ones are recursively computed as +bi = − +det +� +1 +p2i +p1 p2i+1 +� +p1 +, +(A4) +ci = − +det +� +p1 p2i+1 +b1 +bi +� +b1 +, +(A5) +and so on until a zero is encountered. The RH criterion states that the system is +asymptotically stable if and only if the elements in the first column of Routh’s +array are positive. +Based on that, it can be easily shown that for a second-order polynomial, +the necessary condition (A2) is also sufficient for asymptotic stability (a.s.) +since b1 = p1p2, which means that +The system is a. s. +⇐⇒ pi > 0, for i = 1, 2 . +(A6) +Instead, the necessary and sufficient condition for a polynomial of order three +results in +The system is a. s. +⇐⇒ pi > 0, for i = 1, 2, 3 and p1p2 − p3 > 0 . +(A7) +The same reasoning can be applied to higher-order dynamics to derive +additional conditions on the coefficients pi. +A.2 +Verification of the necessary condition for +asymptotic stability +The Matlab code for verifying (23) is provided in https://github.com/ +cp4u17/Feedback.git. +The strategy used is to evaluate each term in Eq. (23) and simply compare +the left and right-hand sides of the equation. We followed a symbolic approach +(based on the Matlab symbolic toolbox) for an arbitrary three-state model. A +numerical approach was used instead for higher-order dynamics, specifically +4, 5 and 6 state cell fate models. To do so, we randomly defined the cell +dynamical matrix at the steady state, A(ρ∗), and its derivative with respect + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +23 +to ρ. Entries were chosen from a uniform distribution and, for assuring a zero +dominant eigenvalue for A(ρ∗), a local optimiser (fmincon function of Matlab) +was used to find appropriate diagonal elements. For each dimension of the cell +fate model, we tested up to 1000 random cases. +A.3 +Sufficient condition for asymptotic stability +In this section, we will indicate with the superscripts A and J the coefficients of +the characteristic polynomial expressed as Eq. (A1) respectively of the matrix +of the dynamical system, Eq. (6), and those of the Jacobian matrix, Eq. (15). +For a two and three-state system, the following relations can be alge- +braically derived +pJ +1 = pA +1 − +� +i +ηi . +(A8) +where ηi is according to Eq. (16). Again, considering that pA +1 > 0, if all ηi ≤ 0 +then pJ +1 > 0. +Hence, the above relation implies that in a two-state system, the RH cri- +terion given by Eq. (A6) is fulfilled when η ≤ 0, with at least one negative +component (otherwise J = A) and therefore the system is asymptotically sta- +ble. We recall that asking ηi ≤ 0 without further constraints is equivalent to +the previously derived condition (30) with ϵi = ∞. +For applying the RH criterion to a three-state cell dynamic system, given +by Eq. (A7), we need to evaluate the sign of pJ +2 and then that of pJ +1 pJ +2 − pJ +3 . +To do so, we first write +pJ +2 = pA +2 − +� +i +fiηi , +(A9) +in which fi = � +j aji − Tr(A) for i = 1, 2, 3. Since the off-diagonal elements +are non-negative, and the trace of A is negative, then fi > 0 for i = 1, 2, 3. +That means that if all ηi ≤ 0 then pJ +2 > 0. Concerning the term pJ +1 pJ +2 − pJ +3 , +this can be written as a quadratic form in η = +� +η1, η2, η3 +� +as +pJ +1 pJ +2 − pJ +3 = Q(η) = ηT AQη + bT +Qη + cQ , +(A10) +in which +AQ = +� +� +f1 f1 f1 +f2 f2 f2 +f3 f3 f3 +� +� , +(A11) +bQ = −pA +1 +� +� +f1 +f2 +f3 +� +� − pA +2 +vw +� +� +v3(w3 − w1) + v2(w2 − w1) +v3(w3 − w2) + v1(w1 − w2) +v2(w2 − w3) + v1(w1 − w3) +� +� , +(A12) +and cQ = pA +1 pA +2 . Here, v = (v1, v2, v3) is a left dominant eigenvector and +w = (w1, w2, w3) a right dominant eigenvector. +We now note that the matrix AQ is semidefinite positive since two eigen- +values are zero (the rows are two-fold degenerate) and one is positive, equal +to Tr(AQ) = � +i fi, and cQ > 0. We now distinguish two cases, depending on + +Springer Nature 2021 LATEX template +24 +Homeostatic regulation of renewing tissue cell populations via crowding control +the sign of bQ elements. First, if bQ ≤ 0, then Q(η) > 0 for any η ≤ 0. Since +fi, pA +1 , pA +2 , vw > 0, we get a sufficient condition for bQ ≤ 0, namely, +0 ≤ v3(w3 − w1) + v2(w2 − w1) +(A13) +0 ≤ v3(w3 − w2) + v1(w1 − w2) +0 ≤ v2(w2 − w3) + v1(w1 − w3) +In that case, asymptotic stability and thus crowding feedback control is assured +for any η < 0, and thus the bound for feedback strength is ϵi = ∞ for i = +1, 2, 3. +Otherwise, if there is at least one positive element in bQ, then Q(η) > 0 +only if |ηi| < ϵi, where ϵ = (ϵ1, ϵ2, ϵ3) are the absolute values of the solutions +to the equation Q(η) = 0, that is – given that ηi are negative – the solution to, +0 = ϵT AQϵ − bT +Qϵ + cQ . +(A14) +Importantly, we note that the elements of bQ depend uniquely on the proper- +ties of the dynamical system and therefore, they can be determined without +requiring the knowledge of the parameter derivatives, i.e. the specific crowding +feedback dependencies. +The Matlab code for verifying (A8), (A9) and (A10) is provided in +https://github.com/cp4u17/Feedback.git. +Appendix B +Test case +B.1 +Asymptotic stability +This section reports the details of the model used for numerical examples +presented in the main text. The cell dynamics correspond to the following +three-state cell fate model +X1 +λ1 +−→ X1 + X1, +X1 +ω13 +−−→ X3, +X1 +γ1 +−→ ∅ +X2 +ω21 +−−→ X1, +X2 +ω23 +−−→ X3, +X2 +γ2 +−→ ∅ +X3 +λ3 +−→ X3 + X3, +X3 +ω31 +−−→ X1, +X3 +ω32 +−−→ X2, +(B15) +whose network is shown in Figure B1. In such a model, for simplicity, we only +consider symmetric self-renewing divisions so that κij = ωij. Also, we apply +the crowding feedback to division rates, λi, and differentiation rates γi. In this +way, it is straightforward to apply the sufficient condition (30) for asymptotic +stability since κ′ +ij = 0 for all i, j. +Hence, each kinetic parameter of the type αi ∈ {λj, γj}j=1,...,3 is expressed +as a function of ρ, whilst those of the type αi ∈ {κjk}j,k=1,...,3 are constant. In +particular, we chose a Hill function [31] where αi(ρ) = ci + kiρni/(Kni +i ++ ρni) +in case αi is a differentiation rate, so that α′ +i = ∂αi/∂ρ > 0, and αi(ρ) = + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +25 +ci +ki/(Kni +i +ρ/ni) in case it is a proliferation rate, so that α′ +i < 0. According +to (30) this choice assures that, if there is a value ρ = ρ∗ for which µ(ρ∗) = 0, +this corresponds to an asymptotically stable steady state. +The parameter values used in our example are reported in Table B1, and +the profiles of the proliferation and differentiation rates as a function of ρ are +shown in Figure B2. Based on these values, the steady state corresponds to +ρ∗ = 1 (arbitrary unit). As expected, the dominant eigenvalue of the Jacobian +at the steady state is negative (µJ = −1.21). +To test the dynamical behaviour of the tissue cell population, we numer- +ically solved the system of ODEs (5) for different initial conditions based on +the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function). +The results are shown in Figure B3 as the time evolution of ρ, normalised +by the steady-state ρ∗, (left panels), and of the dominant eigenvalue, µ (right +panels). The label H indicates an initial condition corresponding to the self- +renewing state ρ∗, that is, the system is initially in homeostasis. In the +simulations labelled as P− and P+, we applied perturbation in the initial +state ρ∗ = (ρ∗ +1, ρ∗ +2, ρ∗ +3), which are, respectively, +� +0.8ρ∗ +1, 0.75ρ∗ +2, 0.85ρ∗ +3 +� +and +� +1.5ρ∗ +1 1.1ρ∗ +2 1.2ρ∗ +3 +� +. As expected, in all these cases, the feedback’s effect is sta- +bilising the system so that it returns to the steady state upon perturbation, +ρ → ρ∗, (asymptotic stability) and thus regains self-renewal property, µ → 0, +over time. +Fig. B1 +Cell state network representing a cell type composed of three states. The links +represent direct transitions, ωij; symmetric divisions occur with rates λi and differentiation +with rate γi, where subscripts i, j = 1, 2, 3 indicate the corresponding cell state, as per model +(B15). +B.2 +Failure of feedback function +Based on the cell fate model regulated via crowding feedback described in +the previous section, we assess the impact of failure in one or more feedback +functions. In particular, the failure of the crowding regulation is modelled, +assuming one or more kinetic parameters as a constant. Five different failure +test cases are assessed. For doing so, we chose αi = (1 + C)α∗ +i being constant +instead of depending on ρ, in which α∗ is the value at the steady state when + +M +Y1 +XI +23 +1 +013 +021 +1 +031 +- +X +X2 +023 +032Springer Nature 2021 LATEX template +26 +Homeostatic regulation of renewing tissue cell populations via crowding control +k +K +n +α∗ +α′ +λ1 +0.74 +0.57 +2.00 +0.61 +-0.84 +λ3 +7.79 +2.07 +2.00 +1.53 +-0.56 +γ1 +3.07 +1.22 +2.00 +1.28 +1.48 +γ2 +2.28 +0.43 +2.00 +1.97 +0.61 +κ13 +– +0.95 +0.00 +κ21 +– +1.44 +0.00 +κ23 +– +1.71 +0.00 +κ31 +– +2.03 +0.00 +κ32 +– +1.35 +0.00 +Table B1 +Values of the Hill function parameters describing the kinetic parameters in +case of homeostasis regulation via crowding feedback for the cell fate model (B15). The +generic kinetic parameters (represented as αi in the right columns of the table) are a +function of the total cell density, ρ, and are given by γi(ρ) = c + kρn/(Kn + ρn) and +λi(ρ) = c + k/(Kn + ρn) with i = 1, 2, 3. A common value c = 0.05 is assumed. State +transition rates ωij, are constant and equal to κij. For such a cell fate dynamics, the steady +state is ρ∗ = 1. The unit of the kinetic parameter is arbitrary and therefore omitted. Unless +specified otherwise, these values apply to all the numerical examples presented in this work. +0 +0.5 +1 +1.5 +2 +0 +0.5 +1 +1.5 +2 +2.5 +0 +0.5 +1 +1.5 +2 +-4 +-2 +0 +2 +4 +Fig. B2 +Proliferation and differentiation rates (left panels, with α as a generic placeholder +for parameters), and their derivative with respect to ρ (right panels) as functions of cell +density normalised by the steady-state ρ∗ for the cell fate model (B15) schematised in +Figure B1. The profiles in the left panel correspond to Hill functions defined in Table B1. +there are no failures (reported in Table B1) and C is a constant (reported in +Table B2). Five test cases, indicated as F1−5, are assessed. +In test case F1, only one feedback fails. Three of the four kinetic parameters +fail in cases F2−4. Finally, F5 represents a case where all the feedback functions +fail. The corresponding variability of the dominant eigenvalue, µ, as a function +of the cell density is shown in Figure B4. It is clear that whilst F1−4 cases +satisfy the sufficient condition for strict homeostasis, (30), in test cases F5, +the dominant eigenvalue being constant means that there is no homeostatic +regulation. Importantly, there is no steady state in test cases F2,4 since the +dominant eigenvalue is always positive in one case or negative in the other. +Based on these assumptions, we numerically solved the system of ODEs +(5) using the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 +function). The failure test cases start at time 0 from an initially homeostatic +condition, H. The results are shown in Figure B5 as the time evolution of +ρ, normalised by the homeostatic steady-state, ρ∗, (left panels), and of the + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +27 +0 +5 +10 +15 +0.8 +1 +1.2 +1.4 +H +P- +P+ +0 +5 +10 +15 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +H +P- +P+ +Fig. B3 +Effect of perturbation of homeostasis under crowding control, when feedback +parameters are according to Table B1. (Left) Cell density ρ, scaled by the steady-state ρ∗, +as a function of time. (Right) Corresponding variation of the dominant eigenvalue µ. Time +is scaled by the inverse of ¯α = mini α∗ +i . Three different initial condition are tested: H, +corresponds to the steady state ρ∗ = (ρ∗ +1, ρ∗ +2, ρ∗ +3), P− to +�0.8ρ∗ +1, 0.75ρ∗ +2, 0.85ρ∗ +3 +� +and P+ +to +�1.5ρ∗ +1, 1.1ρ∗ +1, 1.2ρ∗ +1 +� +. Since the steady state is asymptotically stable, thanks to crowding +control, the cell population remain in, or return to, a homeostatic state characterised by +µ = 0. +Parameter +F1 +F2 +F3 +F4 +F5 +λ1 ++5% ++5% ++5% +-20% +-5% +λ3 +- ++5% ++5% +-20% +-5% +γ1 +- +-5% +- ++20% +-5% +γ2 +- +- +-5% +- +-5% +Table B2 +Value of the constant C in the feedback failure test cases. Whenever a failure +in the feedback of one kinetic parameter α occurs, that parameter is modelled as a +constant, α = (1 + C)α∗, in which the steady-state value, α∗, is reported in Table B1. Test +cases F1 and F2 correspond to those presented in the main text (Figure 1). +dominant eigenvalue, µ, (right panels). Note that the cases F1,2 correspond +respectively to the Single failure and Multiple failures reported in the +main text (Figure 1). +In two cases, F1,3, despite a single or multiple feedback functions failing, a +new homeostatic condition is reached after some time, where µ = 0. However, +suppose a different set of feedback fails, like in F2,4, such that the dominant +eigenvalue is respectively positive or negative for any ρ. In that case, no steady +state can be attained, and the tissue cell population will hyper-proliferate or +decline in the long term. Hence, even if the condition for asymptotic stability +is met, there is no steady state. Finally, if homeostasis is not regulated at +all, as in F5, then the population dynamics only depend on the value of the +dominant eigenvalue (the cell dynamical model (5) turns linear). In the case +shown, µ > 0 and therefore, the cell population diverges. +B.3 +Single cell mutation scenario +To assess the tissue dynamics with a single-cell mutation, as presented in the +main text, we modelled the clonal dynamics, namely, the dynamics of single +cells and their progeny. For doing so, we considered the model (B15) as a + +Springer Nature 2021 LATEX template +28 +Homeostatic regulation of renewing tissue cell populations via crowding control +0 +0.5 +1 +1.5 +2 +-2 +0 +2 +4 +H +F1 +F2 +F3 +F4 +F5 +Fig. B4 +Variation of the dominant eigenvalue µ as a function of the cell density, ρ, +normalised by the reference homeostatic state value, ρ∗. The curve H corresponds to the +reference homeostatic model presented in Appendix B.1. The other curves, F1−5, represent +different sets of feedback failure, as reported in Table B2. +-10 +0 +10 +20 +30 +40 +0 +0.5 +1 +1.5 +2 +2.5 +H +F1 +F2 +F3 +F4 +F5 +-10 +0 +10 +20 +30 +40 +-0.6 +-0.4 +-0.2 +0 +0.2 +H +F1 +F2 +F3 +F4 +F5 +Fig. B5 +Failure of feedback control. (Left) Cell density, scaled by the steady state in the +homeostatic case, as a function of time. (Right) Corresponding variation of the dominant +eigenvalue µ. Time is scaled by the inverse of ¯α = mini α∗ +i . The homeostatic model, H, is +perturbed at a time equal to zero to include the feedback failure reported in Table B2. Whilst +in F1,3, the regulation is able to achieve and maintain a new homeostatic state (µ = 0), +the remaining case fails to regulate the cell population, leading to an indefinite growth or +shrinking of the tissue. +Markov process with the same numerical rates as before, but now events are +treated as stochastic. Then, we run numerical simulations using the Gillespie +algorithm [32] to evaluate this model. In particular, the results presented in +this work are based on 100 independent instances, where each instance is a +possible realisation of the stochastic process. We chose a total cell number +N0 = 5000 as the initial condition (cell density is based on unitary volume). +In real tissues, the number of cells could be a few orders of magnitude larger. +However, this number is sufficiently large to avoid the extinction of the process +in the time scale analysed, so once rescaled, these dynamics are representative +of those in the tissue. All the simulations are stopped when the mutated clone +goes extinct or divergence of the dynamics is detected, defined as reaching +N = 5N0. + +Springer Nature 2021 LATEX template +Homeostatic regulation of renewing tissue cell populations via crowding control +29 +From an implementation point of view, we consider a cell fate model +represented by two disconnected cell state networks to model the tissue dynam- +ics, including the mutated cell. One network corresponds to the unperturbed +test case H, and the other to the dysregulated one, F2 (both described in +Appendix B.2). The simulation starts with N0 cells in the H network, dis- +tributed in each state proportionally to the expected steady-state distribution +in the tissue, and no cells in the F2 network. Thus, since the two networks +are disconnected, F2 remains empty, and the simulation represents the tissue +dynamics before the dysregulation. At a time equal to zero, we moved one +cell from a random state in the H network to the corresponding state in the +F2 one. This simulation represents the tissue dynamics, including the single +mutated cell. +In Figure B6 (left), all the trajectories where the mutated clones go extinct +are shown. In these cases, the tissue dynamics remain globally unaffected by +the mutation. Due to the stochastic nature of the process, mutant clones can +go extinct even if the growth parameter is positive. That is, even in cases where +divergence would be observed for the tissue-wide disruption. However, this does +not occur in all the instances. The right panel of the same figure shows those +instances where the mutated clone does not go extinct and eventually prevails, +resulting in diverging cell population dynamics. For the chosen parameters, +this divergence of the mutated clone is detected in 6% of all cases. Surprisingly, +only a few clones survive despite a proliferative advantage, but this is plausible +for a small fitness advantage (For example, in the case of a single state with +cell division rate λ and loss rate γ – a simple branching process [33] – the +probability for the a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ, +which can be very low for λ ≈ γ). +In the main text (Figure 2), only one profile for each scenario is shown, +respectively. They correspond to instance #24 for the homeostatic case and +instance #43 for the diverging case. +B.4 +Quasi-dedifferentiation +The numerical example presented in the main text is based on the same cell +fate model described in Appendix B.1. To model the dynamics of a committed +cell type, we choose a constant non-negative u = +� +0.02 0.07 0.06 +�T to model +for the cell influx. For such a model, the steady state, ρ∗ +c, is asymptotically +stable. +The figures presented in the main text are based on the numerical integra- +tion of the system of ordinary differential equation (31). In particular, we used +the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function). +References +[1] National Institute of Health: Stem Cell Basics (2016). https://stemcells. +nih.gov/info/basics + +Springer Nature 2021 LATEX template +30 +Homeostatic regulation of renewing tissue cell populations via crowding control +-5 +0 +5 +10 +15 +20 +0.8 +0.9 +1 +1.1 +1.2 +H +F2 +0 +20 +40 +60 +80 +0.8 +0.9 +1 +1.1 +1.2 +H +F2 +Fig. B6 +Results of numerical simulations of the stochastic process representing the cell +dynamics, according to section B.3. 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Cambridge University +Press, Cambridge (2005). https://doi.org/10.2277/0521832209. http:// +pure.iiasa.ac.at/7598/ + diff --git a/BdE4T4oBgHgl3EQf5Q5A/content/tmp_files/load_file.txt b/BdE4T4oBgHgl3EQf5Q5A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a597e0d49351ada93199a16e6891b9341716cd02 --- /dev/null +++ b/BdE4T4oBgHgl3EQf5Q5A/content/tmp_files/load_file.txt @@ -0,0 +1,1134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf,len=1133 +page_content='Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control: stability, robustness and quasi-dedifferentiation Cristina Parigini1,2,3 and Philip Greulich1,2* 1*School of Mathematical Sciences, University of Southampton, Southampton, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 2Institute for Life Sciences, University of Southampton, Southampton, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3Te P¯unaha ¯Atea - Space Institute, University of Auckland, Auckland, New Zealand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' E-mail(s): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='greulich@soton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Contributing authors: cristina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='parigini@auckland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='nz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Abstract To maintain renewing epithelial tissues in a healthy, homeostatic state, (stem) cell divisions and differentiation need to be tightly regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Mechanisms of homeostatic control often rely on crowding control: cells are able to sense the cell density in their environment (via various molecular and mechanosensing pathways) and respond by adjusting division, differentiation, and cell state transitions appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here we determine, via a mathematically rigorous framework, which general conditions for the crowding feedback regulation (i) must be minimally met, and (ii) are sufficient, to allow the maintenance of homeosta- sis in renewing tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We show that those conditions naturally allow for a degree of robustness toward disruption of regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Further- more, intrinsic to this feedback regulation is that stem cell identity is established collectively by the cell population, not by individual cells, which implies the possibility of ‘quasi-dedifferentiation’, in which cells committed to differentiation may reacquire stem cell properties upon depletion of the stem cell pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' These findings can guide future exper- imental campaigns to identify specific crowding feedback mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Keywords: keyword1, Keyword2, Keyword3, Keyword4 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='05321v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='TO] 12 Jan 2023 Springer Nature 2021 LATEX template 2 Homeostatic regulation of renewing tissue cell populations via crowding control 1 Introduction Many adult tissues are renewing, that is, terminally differentiated cells are steadily removed and replaced by new cells produced by the division of cycling cells (stem cells and progenitor cells), which then differentiate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In order to maintain those tissues in a healthy, homeostatic state, (stem) cell divisions and differentiation must be tightly balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Adult stem cells are the key players in maintaining and renewing such tissues due to their ability to produce cells through cell division and differentiation persistently [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, the underlying cell-intrinsic and extrinsic factors that regulate a homeostatic state are complex and not always well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Several experimental studies have identified mechanisms and pathways that regulate homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For example, cell crowding can trigger delamination and thus loss of cells in Drosophila back [2], and differentiation in cultured human colon, various zebrafish epiderimises, and canine kidney cells [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' On the other hand, cell crowding can affect cell proliferation: overcrowding can inhibit proliferation [5], whereas a reduction in the cell density, obtained, for example, by stretching a tissue [6] causes an increase in proliferative activity (both shown in cultured canine kidney cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Although the mechanisms to mediate this regulation are not always clear, experimental studies on mechanosensing showed that cell overcrowding reduces cell motility and consequently produces a compression on cells that inhibits cell proliferation [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Another mechanism utilising crowding feedback is the competition for limited growth signalling factors [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' More specifically, in the mouse germ line, cells in the niche respond to a growth factor (FGF5) that promotes proliferation over differentiation, which they deplete upon being exposed to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Therefore, the more cells are in the niche, the less FGF5 is available per cell, and the less proliferation (or more differentiation) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Despite differing in the involved molecular pathways and many other details, all these regulatory mechanisms are, in essence, sensing the cell den- sity in their environment and responding by adjusting their propensities to divide, differentiate, die, or emigrate from the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This class of mechanisms, for which cell fate propensities depend on the cell density, can be classified as crowding feedback regulation: the cell density determines the cells’ prolifera- tion and differentiation, which affects their population dynamics and thus the cell density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, the crowding response to changes in cell density cannot be arbitrary in order to maintain homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' It must provide a (negative) feedback, in the sense that cells sense the cell density and adjust proliferation, differentiation, and cell loss, such that the cell density is decreased if it is too high and increased if it is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For simple tissues consisting of a single cell type with a unique cell state, it is relatively straightforward to give the conditions for crowding feedback to maintain homeostasis successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this case, when the cell division rate decreases with cell density and differentiation and or death rate increase with cell density, a homeostatic state is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, such conclusions are not as simple to make when a tissue consists of a complex lineage hierarchy and a multitude of underlying cellular states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 3 the latter, more realistic case, conditions for successful homeostatic regulation – in which case we speak of crowding control – may take more complex forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Previous studies based on mathematical modelling have shed some light on quantitative mechanisms for homeostatic control [9–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, in [13], a mathematical assessment of crowding feedback modelling shows that a (dynamic) homeostatic state exists under reasonable biological conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Nevertheless, the case of dynamic homeostasis considered there may not nec- essarily be a steady state but could also exhibit oscillations in cell numbers (as does realistically happen in the uterus during the menstrual cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' While the criterion presented in [13] provides a valid sufficient condition for dynamic homeostasis, it relies on a rather abstract mathematical quantity – the domi- nant eigenvalue of the dynamical matrix – that is difficult, if not impossible, to measure in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here, we wish to generalise previous findings and seek to identify general conditions for successful homeostatic control if propensities for cell division, differentiation, and loss are responsive to variations in cell density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' More precisely, we derive conditions that must be minimally fulfilled (necessary conditions) and conditions which are sufficient, to ensure that homeostasis pre- vails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To identify and formulate those conditions, we note that homeostasis is a property of the tissue cell population dynamics, which can be mathemati- cally expressed as a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Even if a numerically exact formulation of the dynamics may not be possible, one can formulate generic yet mathe- matically rigorous conditions by referring to the criteria for the existence of stable steady states in the cell population dynamics of renewing tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We will derive those conditions by mathematical, analytical means, augmented by a numerical analysis testing the limits of those conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We will also show that homeostatic control by crowding feedback possesses inherent robustness to failures and perturbations of the regulatory pathways, which may occur through external influences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' wide-spread biochemical fac- tors) and genetic mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Finally, we will assess the response of cells when the pool of stem cells is depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Crucially, we find that inherent to crowd- ing feedback control is that formerly committed progenitor cells reacquire self-renewal capacity without substantial changes in their internal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Ded- ifferentiation has been widely reported under conditions of tissue regeneration [14, 15] or when stem cells are depleted [16–19], which is usually thought to involve a substantial reprogramming of the cell-intrinsic states towards a stem cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' On the other hand, our analysis suggests the possibility of “quasi”- dedifferentiation, the reversion from a committed cell to a stem cell by a mere quantitative adjustment of the pacing of proliferation and differentiation, without a substantial qualitative change in its expression profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 Homeostatic regulation of renewing tissue cell populations via crowding control 2 Modelling of tissue cell dynamics under crowding feedback We seek to assess the conditions for homeostasis in renewing tissue cell pop- ulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' either a steady state of the tissue cell population (strict homeostasis) or long-term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' bounded oscillations or fluctuations (dynamic homeostasis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' which represent well-defined constraints on the dynamics of the tissue cell population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To this end, we will here derive a formal, mathematical representation of the tissue cell dynamics under crowding feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The cell population is fully defined by (i) the number of cells, (ii) the internal (biochemical and mechanical) states of each cell, and (iii) the spatial position of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We assume that a cell’s behaviour can depend on the cell density and the states of cells in its close cellular environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' As we examine a situation close to a homeostatic state, we assume that the cell density is homogeneous over the range of interaction between cells, which expands over a volume V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, the cell density ρ is proportional to the average number of cells, ¯n, in that volume, ρ = ¯n V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Similarly, we define the number of cells in internal state i as ni, and the cell density of cells in internal state i as ρi = ¯ni V , where ¯ni is the expected value of ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' As we consider only the crowding feedback response of cells, which only accounts for the cell densities ρi but not the explicit position of cells, the spatial configuration (iii) is not relevant to our considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, the configuration of the cell population and its time evolution is entirely determined by the average number of cells in each state i, as a function of time t, ¯ni(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The configuration of cell numbers ni can change only through three processes: (1) cell division, whereby it must be distinguished between the cell state of daughter cells, (2) the transition from one cell state to another, (3) loss of a cell, through cell death or emigration out of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Following the lines of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' [13, 20] and denoting as Xi,j,k a cell in internal states i, j, k, respectively, we can formalise these events as: cell division: Xi λirjk i −−−→ Xj + Xk (1) cell state transition: Xi ωij −−→ Xj (2) cell loss: Xi γi −→ ∅ , (3) where the symbols above the arrows denote the dynamical rates of the transi- tions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' the average frequency at which such events occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, γi is the rate at which a cell in state i is lost, ωij the rate at which a cell changes its state from i to j and λirjk i denotes the rate at which a cell i divides to pro- duce two daughter cells, one in state j and one in state k (i = j, j = k, k = i are possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For later convenience, we distinguish here the overall rate of cell division in state i, λi and the probability rjk i that such a division produces daughter cells in states j and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since we consider a situation where cells can respond to the cell densities ρi via crowding feedback, all the rates and probabilities (λi, γi, ωij, rjk i ) may Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 5 depend on the cell densities of either state j, ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For convenience, we discretise the number of states in case the state space is a continuum and only distinguish states which have substantially different propensities (λi, γi, ωij, rjk i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Without loss of generality, we assume that there are m states, that is, i, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=', m (for a rigorous argument for the discretisation of the state space, see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The rates given above denote the average number of events happening per time unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, we can express the total rate of change of the average (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' expected) number of cells ¯ni(t), that is, the derivative ˙¯ni = d¯ni dt , in terms of the rates of those events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This defines a set of ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Following the lines of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' [13, 20], we can write ˙ni as, ˙¯ni = �� j ωji¯nj + λj �� k rik j + rki j � ¯nj � − ¯ni � λi + γi + � j ωij � , (4) where for convenience, we did not write the time dependence explicitly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' ni = ni(t), and all parameters may depend on the cell densities ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since V is constant, we can divide by V to equivalently express this in terms of the cell state densities, ρi = ¯ni V , and then write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (4) compactly as, d dtρ(t) = A(ρ(t)) ρ(t) (5) where ρ = (ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=') is the vector of cell state densities and A(ρ) is the matrix, A = � � λ1 − � j̸=1 κ1j − γ1 κ21 κ31 · · κ12 λ2 − � j̸=2 κ2j − γ2 κ32 · · κ1m κ2m · · λm − � j̸=m κmj − γm � � , (6) in which κij = λi2rj i + ωij, with rj i = � k(rjk i + rkj i )/2, is the total transition rate, that combines all transitions from Xi to Xj by cell divisions and direct state transitions (again, all parameters may depend on ρ, as therefore also does A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We can thus generally write the elements of the matrix A, aij with i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=', m as aij = � λi − γi − � k̸=i κik for i = j κji for i ̸= j (7) We now make the mild assumption that divisions of the form Xi → Xj+Xk are effectively three events, namely, cell duplication, Xi → Xi + Xi coupled to cell state changes, Xi → Xj and Xi → Xk, if j ̸= i or k ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this view, the parameters relevant for crowding feedback are the total cell state transition propensities κij and the cell division rate λi, as in (6), instead of ωij and rjk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' These equations describe a dynamical system which, for given initial con- ditions, determines the time evolution of the cell densities, ρi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Crucially, Springer Nature 2021 LATEX template 6 Homeostatic regulation of renewing tissue cell populations via crowding control this description allows for a rigorous mathematical definition of what a home- ostatic state is, and to apply tools of dynamical systems analysis to determine the circumstances under which a homeostatic state prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, we define a strict homeostatic state as a steady state of the system, (5), when the cell numbers – and thus cell densities, given that V is fixed – in each state do not change, mathematically expressed as dρ dt = 0 (a fixed point of the sys- tem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A dynamic homeostatic state is when cell densities may also oscillate or fluctuate but remain bounded and thus possess a finite long-term average cell population (in which case the system either approaches a steady state or limit cycles – that is, oscillations – or chaotic but bounded behaviour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on these definitions, we can now analyse under which circumstances crowd- ing feedback can maintain those states, which in the case of strict homeostasis requires, in addition, that the corresponding steady state is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 Cell types and lineage hierarchies According to [13], cell population dynamics of the type (5) can be associated with a cell state network, in which each state is a node, and the nodes are connected through cell state transition (direct transitions and cell divisions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Furthermore, by decomposing this network in strongly connected components (SCCs), the cell fate model can be viewed as a directed acyclic network [21], generally called the condensed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here, we follow the definitions of [13] and define a cell type as an SCC of the cell state network, so that any cell states connected via cyclic cell state trajectories (sequences of cell state transitions) are of the same type, and the condensed network of cell types represents the cell lineage hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This definition ensures that cells of the same type have the same lineage potential (outgoing cell state trajectories) and that the stages of the cell cycle are associated with the same cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this context, we will in the following also speak of differentiation when a cell state transition between different cell types occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Each cell type can be classified as self-renewing, declining or hyper- proliferating, depending on the dominant eigenvalue µ (called growth parame- ter) of the dynamical matrix A (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (5) ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=') reduced to that SCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This is µ = 0 for self-renewing cell types, when cell numbers of that type remain constant over time, µ < 0 (µ > 0) for the declining (hyperproliferating) types when cell numbers decline (increase) in the long term [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Importantly, for the population dynamics to be strictly homeostatic, which means that a steady state of model (5) exists, the cell type network must fulfil strict rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' These are: (i) at least one self-renewing cell type (with µ = 0) must exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (ii) self- renewing cell types must stay at an apex of the condensed network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (iii) all the other cells must be of declining types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This means that the critical task of homeostatic control is to ensure that the kinetic parameters of the cell type at the apex of the cell lineage hierarchy are fine-tuned to maintain exactly µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Therefore, we can restrict our analysis to find conditions for the cell type at the lineage hierarchy’s apex to be self-renewing, which we will do in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Other cell types simply need that differentiation (transition towards Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 7 another cell type) or loss is faster than proliferation, so that they become declining cell types, µ < 0, but those rates do not require fine-tuning and thus trivially regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We note that when we consider only cell states of the type at the apex of the cell lineage hierarchy, any differentiation event is – according to this restricted model – a cell loss event and included as event occurring with rates γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Given that cell loss from a cell type at the lineage apex is rare, we will therefore in the following also denote the rates γi simply as differentiation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3 Results We will now determine necessary and sufficient conditions for the establish- ment of strict and dynamical homeostasis when subject to crowding feedback, which we here define through the derivatives of the dynamical parameters λi, rjk i , ωij, γi as a function of the cell densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' As argued before, we only need to consider cell types at an apex of the cell type network, which, for home- ostasis to prevail, must have a growth parameter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' dominant eigenvalue of matrix A in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (6)) µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Furthermore, we assume that the apex cell type resides in a separate stem cell niche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Therefore, the parameters only depend on cell densities ρi of states associated with that cell type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' we can write A = A(ρ), where ρ = � i∈S ρi comprises only cell states of the apex cell type S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Provided that, the matrix elements are functions of ρ, and therefore also µ is a function of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, self-renewal corresponds to a non-trivial fixed point, ρ∗, of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (5), restricted to cell type S, for which the dominant eigenvalue of A is zero, that is µ(ρ∗) = 0 (ρ∗ = � i∈S ρ∗ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For convenience, we will often generally refer to parameters as αi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=', 2m + m2, where αi stands for any of the parameters, {λi, γi, κij|i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=', m}, respectively1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, we study which conditions the functions αi(ρ) must meet to maintain homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, we study how those param- eters qualitatively change with the cell density – increase or decrease – that is, we study how the sign and magnitude of derivatives α′ i := dαi dρ affects homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A crucial property of the matrix A(ρ) is that it is always a Metzler matrix, since all its off-diagonal elements, κij ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since the cell state network of a cell type is strongly connected, we can further state that A(ρ) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Notably, for irreducible Metzler matrices holds the Perron-Frobenius theorem [22], and thus A(ρ) possesses a simple, real dominant eigenvalue µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Besides, it as left and right eigenvectors associated with µ, respectively indicated as v and w, which are strictly positive, that is, all their entries are vi > 0, wi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' From this follows that the partial derivative of the dominant eigenvalue µ by 1More precisely, αi|i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='.,m := λi, αi|i=m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='.,2m := γi−m, αi|i=2m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='.,2m+m2 := κ⌊(i−2m)/m⌋,i−⌊(i−2m)/m⌋m Springer Nature 2021 LATEX template 8 Homeostatic regulation of renewing tissue cell populations via crowding control the i, j-th element of A, aij = [A]ij is always positive: ∂µ ∂aij = viwj vw > 0 (8) where the left equality is according to [23] and is generally valid for simple eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here, v is assumed to be in row form, and vw thus corresponds to a scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 Sufficient condition for dynamic homeostasis In [13], it was shown that a dynamic homeostatic state, where cell numbers may change over time but stay bounded, is assured if, 2 µ′(ρ) < 0 for all ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (9) This sufficient condition requires that the dominant eigenvalue of A as a func- tion of the cell density, µ(ρ), is a strictly decreasing function of cell density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Also, the range of this function must be sufficiently large so that it has a root, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' a value ρ∗ with µ(ρ∗) = 0 must exist for the function µ(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Assuming that a non-trivial steady state, ρ∗ > 0, exists, we now translate the sufficient condition for a dynamic homeostatic state, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (9), into condi- tions on the parameters as a function of the cell density, αi(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, we can write, µ′(ρ) = � ij ∂µ ∂aij ∂aij ∂ρ = � ij viwj vw a′ ij = � i viwi vw a′ ii + � i,j̸=i viwj vw a′ ij = � i viwi vw � �λ′ i − γ′ i − � j̸=i κ′ ij � � + � i,j̸=i vjwi vw κ′ ij , (10) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (8) and the explicit forms of aij, the elements of the matrix A according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Provided that all the parameters depend on ρ, condition (9) results in: 0 > µ′ =⇒ 0 > � i viwi (λ′ i − γ′ i) + wi � j̸=i (vj − vi)κ′ ij for all ρ > 0 , (11) While we cannot give an explicit general expression for the dominant eigen- vectors v, w, this condition is sufficiently fulfilled if each term of the sum on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (11) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' More restrictively, we have Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (11) 2In [13], this condition, defined through dependency on cell number, can be directly translated into a condition on the cell density derivative if the volume is assumed as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 9 sufficiently fulfilled if � � � � � λ′ i ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' γ′ i ≥ 0 for all i λ′ i < 0 or γ′ i > 0 at for least one i κ′ ij = 0 for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' j for ρ > 0 (12) This means that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' excluding rates that are zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' which are biologically mean- ingless,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' if no direct state transitions within a cell type are subject to crowding feedback (κ′ ij = 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' while all (non-zero) cell division rates depend negatively on ρ (λ′ i < 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' and differentiation rates depend positively (γ′ i > 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' for all attainable levels of ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' then dynamical homeostasis is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Alternatively, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (11) as 0 > � i viwi vw � �λ′ i − γ′ i − � j̸=i κ′ ij + � j̸=i vj vi κ′ ij � � for all ρ > 0 , (13) which, due to vj vi > 0, implies another sufficient condition for dynamic homeostasis: � � � � � λ′ i ≤ 0, γ′ i ≥ 0 for all i λ′ i < 0 or γ′ i > 0 at for least one i κ′ ij ≤ 0 with |� j κ′ ij| ≤ γ′ i − λ′ i for all i, j (14) The above condition is less restrictive than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (12), allowing for some non- zero crowding feedback dependency of state transition rates κij, as long as the crowding feedback strength of the total outgoing transition rate of each state does not outweigh the feedback on proliferation and differentiation rate of that state (if there is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 Necessary condition for strict homeostasis We now consider the circumstances under which a strict homeostatic is main- tained, that is, when a steady state of the cell population exists and is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A necessary condition for the existence of a steady state ρ∗ (irrespective of stability) has been given in [13], namely, that the cell type at the apex of the lineage hierarchy is self-renewing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' its dynamical matrix A has µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' µ depends on the cell density ρ of the apex cell type, since the dynamical parameters αi and thus A depend on ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' As before, it is required that µ(ρ∗) has sufficient range so that a value ρ∗ with µ(ρ∗) = 0 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This condition is fulfilled if the range of the feedback parameters αi(ρ) is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In that case there exists an eigenvector ρ∗ with A(ρ∗)ρ∗ = 0, which can be chosen by normalisation to fulfil � i∈S ρ∗ i = ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, ρ∗ is a fixed point (steady state) Springer Nature 2021 LATEX template 10 Homeostatic regulation of renewing tissue cell populations via crowding control of the cell population system (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, we need to establish what is required for this state to be asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To start with, we give the Jacobian matrix of the system (5) at the fixed point ρ∗ : [J]ij = ∂[A(ρ)ρ]i ∂ρj ���� ρ=ρ∗ = a∗ ij + ηi , (15) where ηi = � k a′ ikρ∗ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (16) Here and in the following, we assume the derivatives to be taken at the steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' a′ ij := daij dρ |ρ=ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The eigenvalues of the Jacobian matrix J at ρ∗ determine the stability of the steady state ρ∗: it is asymptotically stable if and only if the real part of all eigenvalues of J(ρ∗) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The Routh-Hurwitz theorem [24] states that for a polynomial to have only roots with negative real part, all its coefficients must necessarily be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Given that the eigenvalues of the Jacobian matrix J are the roots of its char- acteristic polynomial, a necessary condition for ρ∗ to be asymptotically stable is that the coefficients of the characteristic polynomial of J are all positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Let us start by considering a self-renewing cell type with exactly two cell states being at the apex of a lineage hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This system has a 2 × 2 dynam- ical matrix A and Jacobian J, whereby A is irreducible and has dominant eigenvalue µA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The characteristic polynomial of a generic 2×2 matrix, M, is P M(s) = s2 + pM 1 s + pM 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (17) with pM 1 = −tr(M) and pM 0 = det(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, since A has an eigenvalue zero, pA 0 = det(A) = a11a22 − a12a21 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (18) From this follows that the right and left eigenvectors to the matrix A associated with the dominant eigenvalue µA = 0, w and v, are: w = � −a22 a21 � and v = � −a22 a12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (19) From the Jacobian matrix J, we get equivalently, pJ 0 = det(J) = (a21 − a22)(−a22η1 + a12η2) a22 = vη |w| a22 , (20) with the L1-norm |w| = w1 + w2 = −a22 + a213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here we used the form of J in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (15) with η = (η1, η2) from (16), as well as the relations (18) and (19), and we factorised the determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3Note that aii is always negative or zero Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 11 From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (10), we can further establish: µ′ = � ij viwj vw a′ ij = � ij |w| ρ∗ viρ∗ j vw a′ ij = |w| ρ∗ vη vw (21) = − a22pJ 0 ρ∗pJ 1 a22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (22) Here, we used that ρ∗ is a dominant right eigenvector, and thus ρ∗ = ρ∗ |w|w, and furthermore we used the definition of ηi = � j a′ ijρ∗ j, we substituted Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (20), and used that vw = a2 22 + a12a21 = −pA 1 a22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Finally, we get: pJ 0 = −µ′ρ∗pA 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (23) Notably, we can show that this relation also holds for higher dimensions by explicitly computing the coefficients of characteristic polynomials pA,J i , the eigenvalues and eigenvectors, and then evaluating both sides of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For systems with three states, this can be done analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For systems with 4,5, and 6 states we tested relation (23) numerically by generating N =1000 random matrices with entries chosen from a uniform distribution4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In each case, this relation was fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence we are confident that this relation holds up to 6 states, and it is reasonable to expect this to hold also for larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since A has a simple dominant eigenvalue µA = 0, we can factorise one term from the characteristic polynomial, P(s) = sQ(s) knowing that all roots of Q(s) are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Applying the Routh-Hurwitz necessary condition to Q(s), it follows that the coefficients of the polynomial Q, pQ i > 0, where i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=', n− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, pA 1 > 0 and considering that ρ∗ > 0 by definition, then for having pJ 0 > 0 we must require µ′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Therefore, a necessary condition for a stable, strict homeostatic state is 0 > µ′ =⇒ 0 > � i viwi (λ′ i − γ′ i) + wi � j̸=i (vj − vi)κ′ ij ������ ρ=ρ∗ , (24) where on the right-hand side, we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This condition is bound to the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (23), that is, we can show it analytically for up to three states and numerically up to 6 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Nonetheless, we also expect this to be true for larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' One way to satisfy this necessary condition is if at ρ = ρ∗ � � � � � λ′ i ≤ 0, γ′ i ≥ 0 for all i λ′ i < 0 or γ′ i > 0 at for least one i κ′ ij = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (25) 4The diagonal elements of the random matrix are tuned using a local optimiser (fmincon function of Matab) so that the matrix has a zero dominant eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 12 Homeostatic regulation of renewing tissue cell populations via crowding control Notably, the necessary conditions (24) and (25) only differ from the suffi- cient conditions for dynamic heterogeneity, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (11) and (12), by needing to be fulfilled only at the steady-state cell density ρ∗, whereas to ensure dynamic homeostasis, those should be valid for a sufficiently large range of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='3 Sufficient condition for strict homeostasis Now we assess under which circumstances a strict homeostatic state is assured to prevail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' First of all, the necessary conditions from above need to be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, the parameter functions αi(ρ) must have a sufficient range so that µ(ρ) has a root, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' ρ∗ with µ(ρ∗) = 0 exists, from which the existence of a steady state follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The question now is whether we can find sufficient conditions assuring that the fixed point ρ∗ with � i ρ∗ i = ρ∗ is (asymptotically) stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Let us define a matrix B(x), x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=', xm) with bij(x) = [B]ij(x) = a∗ ij +xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, B(xi = 0) = A(ρ∗) and B(xi = ηi) = J, where J, the Jacobian matrix, and ηi are defined as in (15) and (16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We consider now the dominant eigenvalue as function of the entries of B, µ[B] := µ({bij}|i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=',m) (the square brackets are chosen to denote the difference from the function µ(ρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For sufficiently small ηi, we can then express the dominant eigenvalue of the Jacobian matrix J, µ[J], relative to the dominant eigenvalues of A∗ := A(ρ∗) as, µ[J] = µ[A∗] + � i ∂µ ∂xi |xi=0 ηi + O(η2) , (26) with, ∂µ ∂xi |xi=0 = � ij ∂µ ∂bij ∂bij ∂xi |xi=0 = � ij ∂µ ∂aij |B=A∗ , (27) since for x = 0, bij = aij for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' It follows that for sufficiently small5 ηi, and if all ηi < 0, we have µJ = µ[A∗](ρ∗) + � i ∂µB ∂xi |xi=0ηi + O(η2 i ) ≈ � i ∂µA ∂aij ηi < 0 (28) since all ∂µA ∂aij > 0 (according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (8)) and µA(ρ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, since µJ < 0, the steady state ρ∗ is asymptotically stable if all ηi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, we get a sufficient condition for asymptotic stability of the steady state ρ∗: 0 > ηi = ρ∗ i (λ′ i − γ′ i) + � k̸=i (κ′ kiρ∗ k − κ′ ikρ∗ i ) > −ϵi for all i (29) 5That is, there exist ϵi > 0 so that this is valid for any |ηi| < ϵi Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 13 where ϵi > 0 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' As this is an asymptotically stable steady state, it corresponds to a controlled strict homeostatic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this case, even if the cell numbers are disturbed (to some degree), the cell population is regulated to return to the strict homeostatic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Notably, condition (29) is fulfilled if, � � � � � � � � � λ′ i ≤ 0, γ′ i ≥ 0 for all i λ′ i < 0 or γ′ i > 0 at for least one i κ′ ij = 0 and |λ′ i|, |γ′ i|, < ϵi (30) Furthermore, we may soften the condition on κij to κ′ ij κ′ ji < ρ∗ j ρ∗ i to allow also some degree of feedback for the κij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The conditions (30) are very similar to the ones for dynamic homeostasis, (12), but here these conditions only need to be fulfilled at ρ = ρ∗, whereas for dynamic homeostasis they need to be fulfilled for a sufficient range of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Moreover, in addition to the qualitative nature of the feedback (related to the signs of λ′ i, γ′ i), the ‘strength’ of the crowding feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' the absolute values of λ′ i, γ′ i must not be ‘too large’, that is, smaller than ϵi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We cannot, in general and for all system sizes, give a definite value for the feedback strength bound ϵi below which strict homeostasis is assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Nevertheless, by using the sufficient stability criterion based on the Routh-Hurwitz criterion [24] we can identify those bounds for systems with up to three cell states, which guides expectations for larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The details of this criterion and the necessary derivations are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' There, we show that for systems with one or two cell states, ϵi = ∞, which means that asymptotic stability is ensured for arbitrary feedback strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For systems with three cell states, we can assure that ϵi = ∞ if certain further conditions are met (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Otherwise, ϵi can be determined implicitly from the roots of a quadratic form (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A14)), and thus stability may depend on the magnitude of the feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In principle, such bounds can be found for larger systems too, but the algebraic complexity of this process renders it unfeasible to do this in practical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 Robustness to perturbations and failures Now, we wish to assess the robustness of the above crowding control mecha- nism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' what occurs if it is disrupted, for example, by the action of toxins, other environmental cues, or by cell mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' More precisely, we will study what happens if one or more feedback pathways, here characterised as a param- eter αi with α′ i ̸= 0 fulfilling the conditions for (dynamic or strict) homeostatic control, is failing, that is, it becomes α′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We will first address the case of tissue-extrinsic factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' those affecting all the cells in the tissue, and then the case of single-cell mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In the latter case, only a single cell would initially show a dysregulated behaviour, yet, if this confers a proliferative advantage, it can lead to hyperplasia and possibly cancer [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 Homeostatic regulation of renewing tissue cell populations via crowding control First, we note that the sufficient condition for strict homeostasis, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (30), is overly restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In a tissue cell type under crowding feed- back control with λ′ i < 0 and γ′ i > 0 for more than one i, there is a degree of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' That is, if the feedback is removed for one or more of these parameters (changing to λ′ i = 0 and, or γ′ i = 0), then the sufficient condition for a strict homeostatic state remains fulfilled as long as at least one λ′ i or γ′ i remains non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This possible redundancy confers a degree of robustness, meaning that feedback pathways can be removed – setting α′ i = 0 – without losing homeostatic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since the necessary conditions, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (24), are even less restrictive, tissue homeostasis may even tolerate more severe disruptions that reverse some feedback pathways, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' switching from λ′ i < 0 to λ′ i > 0, as long as other terms in the sum on the right-hand side of (24) compensate for this changed sign, ensuring that the sum as a whole is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In any case, it is important to remind the underlying assumption for which a non- trivial steady state exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In case the variability of the kinetic parameters is not enough to assure the condition µ(ρ∗ = 0), then the tissue will degenerate, either shrinking and eventually disappearing or indefinitely growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' From the above considerations, we conclude that if crowding control applies to more than one parameter αi, that is, α′ i ̸= 0 with appropriate sign and magnitude, homeostasis is potentially robust to feedback disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This may include a simple variation of the feedback function α′ i but also perturbation in the feedback functions shape and complete feedback failure, α′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' An illustrative example of this situation is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here, the time evolution of the cell density is shown for a three-state cell fate model, which has been computed by integration of the dynamical system (5) (the details of this model are given in Appendix B as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (B15) and illustrated in Figure B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Four kinetic parameters are regulated via crowding control satisfying the sufficient condition for strict homeostasis, (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Then, starting from this homeostatic configuration, feedback disruption is introduced at a time equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In one case (“Single failure”), a single kinetic parameter suffers a complete failure of the type α′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this case, the remaining feedback functions compensate for this failure, and a new homeostatic condition is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Instead, in the second case (“Multiple failures”), failures are applied so that three of the four kinetic parameters initially regulated do not adjust with cell density6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Notably, the only feedback function left satisfies the condition for asymptotic stability, (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Nevertheless, the variability of this kinetic parameter is not enough to assure the existence of a steady state, since in this case, the function µ(ρ) does not possess any root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence µ > 0 for all ρ, leading to an indefinite growth of the cell population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Additional test cases are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' So far, we modelled the feedback dysregulation as acting on a global scale, thus changing the whole tissue’s dynamics behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This situation represents a feedback mechanism affected by cell-extrinsic signals, in which any dysregu- lation applies to all the cells in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, dysregulation can also 6Only in this example, feedback control fails upon multiple failures, while in general, multiple failures may still be compensated to maintain homeostatic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 15 10 0 10 20 30 40 0 2 4 6 8 10 Homeostasis Single failure Multiple failures 10 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='15 Homeostasis Single failure Multiple failures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 1 Cell dynamics in terms of cell density, scaled by the steady state in the homeostatic case, as a function of time (left) and the corresponding variation of the dominant eigenvalue µ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Time is scaled by the inverse of ¯α = mini α∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The homeostatic model is perturbed at a time equal to zero to include feedback failure of the type α′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In the case where only one feedback function fails (“Single failure”), the system is able to achieve and maintain a new homeostatic state, characterised by a constant cell density and a zero dominant eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In case more than one feedback fails (“Multiple failures”), the cell dynamics are unstable since a steady state does not exist and µ > 0 for all ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The cell fate model corresponds to model (B15) with parameters given in Table B1 and Table B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' act at the single-cell level, for example, when DNA mutations occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this case, the impact of the dysregulation is slightly different, as explained in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Suppose, upon disruption of crowding control in a single cell, for example, by DNA mutations, a sufficient number of crowding feedback pathways remain so that there is a steady state and the sufficient condition (30) is still fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In that case, homeostasis is retained, just as when this occurs in a tissue-wide disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, if the homeostatic control of that single cell fails such that the cell becomes hyperproliferative, µ > 0, or declining, µ < 0, the tissue may still remain homeostatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' If µ < 0, the single mutated cell will be lost, upon which only a population of crowding controlled cells remain, which remain in homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' If µ > 0 in a single cell, hyper-proliferation is not ensured either: while the probability for mutated cells to grow in numbers is larger than to decline, due to the low numbers, mere randomness can lead to the loss of the mutated cell with a non-zero probability, which results in the extinction of the dysregulated mutant7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In that case, the mutant cells go extinct and the tissue remains homeostatic despite the disruption of homeostatic control in the mutated cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' a stark contrast to disruption on the tissue level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Otherwise, if the mutant clone (randomly) survives, it will continue to hyper-proliferate and eventually dominate the tissue, which is thus rendered non-homeostatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, the tissue divergence time scale may be much longer than the case where the same dysregulation occurs in all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The deterministic cell population model (5) is suitable for describing the average cell numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Nevertheless, it fails to describe the stochastic nature of 7For example, in the case of a single state with cell division rate λ and loss rate γ – a simple branching process – the probability for a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ, which is less than certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 16 Homeostatic regulation of renewing tissue cell populations via crowding control single-cell fate choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, assessing a single cell’s impact on tissue dynamics requires stochastic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To that end, we implemented this situation as a Markov process with the same rates as the tissue cell population dynamics model8 (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='3 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In Figure 2, we show numerical simulation results of a stochastic version of the model used for previous results in Figure 1, depicted in terms of tissue cell density as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here, two possible realisations of the same stochastic process are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We note that the initially homeostatic tissue results in stochastic fluctuations of the cell density, which remain, on average, constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' At a time equal to zero, a single cell in this tissue switches behaviour, presenting multiple failures which, if applied to all the cells, would determine the growth of the tissue (corresponding to Multiple failures curve in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In one instance of the stochastic simulation, however, the mutated clone goes extinct after some time, leaving a tissue globally unaffected by the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' On the other hand, in another instance, the mutated clone prevails, leading to the growth of the tissue cell population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The fact that vastly different outcomes can occur with the same parameters and starting conditions demonstrates the impact of stochasticity in the case of a single-cell mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 10 0 10 20 30 40 50 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 Homeostasis Multiple Failure (instance #1) Multiple Failure (instance #2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 2 Numerical simulation results of a stochastic version of the model used in Figure 1 upon disruption of crowding control in a single cell, mimicking a DNA mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' At a time equal to 0, the initially homeostatic model is disrupted with a single cell presenting multiple failures in the feedback control, as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Two instances of simulations run with identical parameters are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The rescaled cell density ρ/ρ∗ is shown as a function of the time, scaled by the inverse of ¯α = mini α∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Whilst the mutated cell and its progeny go extinct in one instance (#1), in the other (#2), mutated cells prevail and hyper-proliferate so that tissue homeostasis is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The simulation stops when the clone goes extinct or when instability is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Full details of the simulation are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 Quasi-dedifferentiation In the previous section, we addressed the case where external or cell-intrinsic factors disrupt homeostatic control in self-renewing cells of a tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, 8While a Markov process is an approximation which not necessarily reflects the probability distribution of subsequent event times realistically, it is often sufficient to assess the qualitative behaviour of a system with low numbers, subject to random influences from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 17 situations such as injury, poisoning, or cell radiation might also affect home- ostasis in other ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' An example is when stem cells are completely depleted from the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this context, many studies about tissue regeneration after injury report evidence of cell plasticity [17, 18], when committed cells regain the potential of the previously depleted stem cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Cell dedifferentiation is just an example where differentiated cells return to an undifferentiated state as a response to tissue damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Lineage tracing experiments confirmed this feature in vivo in several cases [16, 28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In the following, we assess how committed progenitor cells respond to the depletion of the stem cell pool if they are under crowding feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Without loss of generality, let us consider an initially homeostatic scenario where there is a self-renewing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' stem) cell type (S) – with growth param- eter µ = 0 – at the apex of a lineage hierarchy, and a committed progenitor cell type (C) – with µ < 0, but with at least one state that has a non-zero cell division rate – below type S in the hierarchy, as depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on this cell fate model, S-cells proliferate and differentiate into C-cells while maintaining the S-cell population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The C-cells also proliferate and dif- ferentiate into other downstream cell types which we do not explicitly consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' C-cells do not maintain their own population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' only the steady influx of new cells of that type via differentiation of S-cells into C-cells maintains the latter population (see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We further assume that both S- and C-cells are under appropriate crowding control, fulfilling both the sufficient conditions for dynamic homeostasis, (12), and for stable, strict homeostasis, (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on the above modelling, we can write the dynamics of the cell densities belonging to the committed progenitor type as, d dtρc = Ac(ρc)ρc + u , (31) where ρc = (ρms+1, ρms+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='., ρms+mc) are the cell densities in the committed C-type, with ms being the number of states of the self-renewing S-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Ac is the dynamical matrix restricted to states in the C-type and ui = �ms j=1 κjiρj is a constant vector quantifying the influx of cells into the C-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' First, we note that the Jacobian matrix of a committed cell type, described by (31), J = � ∂A(ρc)ρc ∂ρj � j=ms+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=',ms+mc , has the same form as a cell type at the apex of the hierarchy, since u does not depend on the densities ρms+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=',ms+mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' From this follows that if C-cells are regulated by crowding control, fulfilling the conditions (30), then also the population of C-cells is stable around a steady state ρ∗ c, albeit with a growth parameter µc(ρ∗ c) < 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We now consider the scenario where all stem cells are depleted at some point, as was experimentally done in [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This would stop any replen- ishment of C-cells through differentiation of S-cells, corresponding to setting 9This can be seen when multiplying the steady state condition for (31), Ac(ρs, ρc)ρc + u = 0 with a positive left dominant eigenvector v, giving, µcvρ∗ c + vu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since ρ∗ and v have all positive entries and u is non-negative, this equation can only be fulfilled for µc < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 18 Homeostatic regulation of renewing tissue cell populations via crowding control Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 3 Sketch representative of the quasi-dedifferentiation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A homeostatic system enclosed in the black box comprises two cell types: a stem cell type, S, (blue) and a com- mitted cell type, C, (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In the unperturbed homeostatic scenario, S is self-renewing, characterised by a growth parameter at the steady state µ∗ = 0, and C is transient, with a growth parameter at the steady state µ∗ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Both cell types are subject to crowding control, fulfilling both conditions (12), and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' By removing the stem cell type XS, the committed cell type acquires self-renewing property through crowding control, effectively becoming a stem cell type (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' u = 0 in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence we end up with the dynamics ˙ρc = A(ρc)ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Now, assum- ing that the function µ(ρ) has sufficient range, so that µ(ρ∗∗ c ) = 0 for some ρ∗∗ c , and provided that A(ρc) is under crowding control fulfilling the sufficient conditions for asymptotic stability of a steady state, then, following our argu- ments from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='3, the population of C-cells will attain a stable steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In other words, those previously committed cells become self-renewing cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Also, since they now reside at the apex of the lineage hierarchy (given that S-cells are absent), they effectively become stem cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, under crowding control, previously committed progenitor cells (committed cells that can divide) will automatically become stem cells if the original stem cells are depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Commonly, such a reversion of a committed cell type to a stem cell type would be called ‘dedifferentiation’ or ‘reprogramming’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, in this case, no genuine reversion of cell states occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' previously committed cells do not transition back to states associated with the stem cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Instead, they respond by crowding feedback and adjust their dynamical rates so that µ becomes zero, hence attaining a self-renewing cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Cru- cially, this new stem cell type is fundamentally different to the original one and still most similar to the original committed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We call this process quasi-dedifferentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The quasi-dedifferentiation follows the same reversion of proliferative potential as in ‘genuine’ dedifferentiation but without explicit reversion in the cell state trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The following numerical example illustrates this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We focus on the cell dynamics of a single C-type regulated via crowding feedback (detail of the model are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The cell density as a function of the time, shown in Figure 4, is obtained by integrating the corresponding cell population model according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The system is initially in a homeostatic condition, meaning that there is a constant influx of cells from some upstream self-renewing types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Such upstream types are assumed to be properly regulated such that this cell influx is constant over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' At a time equal to zero, the cell influx becomes suddenly zero, representing an instantaneous removal of all the Xo- HomeostasisSpringer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 19 10 0 10 20 30 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='8 1 Homeostasis Quasi-dedifferentiation 10 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 0 Homeostasis Quasi-dedifferentiation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 4 Cell dynamics of an initially committed cell type C (µ < 0) upon removal of all stem cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (Left) Cell density scaled by the steady-state density as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (Right) Corresponding variation of the dominant eigenvalue µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Time is scaled by the inverse of ¯α = mini α∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' It is assumed that a stem cell type, S, initially resides in the lineage hierarchy above the committed cell type (as in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' S cells differentiate into C cells, which is modelled as a constant cell influx of C-cells (S is not explicitly simulated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' At a time equal to zero, a sudden depletion of S cells is modelled by stopping the cell influx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' After some transitory phase, the cell population stabilises around a new steady state and becomes self- renewing with µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The full description of the dynamical model, which corresponds to model (B15) with parameters given in Table B1, is reported in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' self-renewing cells from the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A new homeostatic condition is achieved after a transitory phase thanks to the crowding feedback acting on the C- type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This example demonstrates how an initially committed cell type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' with µc < 0, regulated via crowding feedback, might be able to switch, upon disruption, to a self-renewing behaviour µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 4 Discussion For maintaining healthy adult tissue, the tissue cell population must be maintained in a homeostatic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here, we assessed one of the most com- mon generalised regulation mechanisms of homeostasis, which we refer to as crowding feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on this, progenitor cells (stem cells and committed progenitors) adjust their propensities to divide, differentiate, and die, accord- ing to the surrounding density of cells, which they sense via biochemical or mechanical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For this purpose, we used a generic mathematical model introduced before in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' [13, 20], which describes tissue cell population dynamics in the most generic way, including cell divisions, cell state transi- tions, and cell loss / differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on this model, we rigorously define what is meant when speaking of a ‘homeostatic state’, introducing two notions: a strict homeostasis is a steady state of the tissue cell population dynamics, while dynamical homeostasis allows, in addition to strict homeostasis, for oscil- lations and fluctuations, as long as a finite long-term average cell population is maintained (such as the endometrium during the menstrual cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' By analysing this dynamical system, we find several sufficient and necessary conditions for homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' These conditions are formulated in terms of how the propensities of cell division, differentiation, and cell state changes, of cells Springer Nature 2021 LATEX template 20 Homeostatic regulation of renewing tissue cell populations via crowding control whose type is at the apex of an adult cell lineage hierarchy, may depend on their cell density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We find that when, for a wide range of cell density values, the cell division propensity of at least one state decreases with cell density or the differentiation propensity increases with it, while other propensities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' of cell state transitions) are not affected by the cell density, then dynamic homeostasis prevails (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For strict homeostasis to prevail, this only needs to be fulfilled at the steady state itself, but in addition, the magnitude of the feedback strength may not be too large (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We can derive explicit and implicit expressions for the bound on feedback strength for systems of two and three-cell states but cannot do so for arbitrary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Furthermore, we find that a necessary condition for strict homeostasis is that the conditions for dynamic homeostasis are met at least at the steady state cell density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A direct consequence of the conditions we found is that they allow for a considerable degree of redundancy when more than one propensity depends appropriately on the cell density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence feedback pathways, that is, cell dynam- ics parameters depending on the cell density, may serve as ‘back-ups’ to each other if one fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We demonstrate that this confers robustness to the home- ostatic system in that one or more crowding feedback pathways may fail, yet the tissue remains in homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Finally, we assess how crowding feedback regulation affects the response of committed progenitor cells to a complete depletion of all stem cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We showed that committed cells which can divide and are under appropriate crowding feedback control (that is, meeting the sufficient conditions (12) and (30)), will necessarily, without additional mechanisms or assumptions, reacquire stem cell identity, that is, become self-renewing and are at the apex of the lineage hierar- chy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Notably, while this process resembles that of dedifferentiation, it does not involve explicit reprogramming, in that the cell state transitions are reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Instead, only the cell fate propensities adjust to the changing environment by balancing proliferation and differentiation as is required for self-renewal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' While these are purely theoretical considerations, and such a process has not yet been experimentally found, we predict that it must necessarily occur under the appropriate conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This can be measured by assessing the gene expression profiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' via single-cell RNA sequencing) of cells that ‘dedifferentiate’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' reacquire stemness after depletion of stem cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Moreover, those considerations yield further, more general insights: Stem cell identity is neither the property of individual cells nor is it strictly associated with particular cell types or states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Any cell that can divide and differentiate, committed or not, may become a stem cell under appropriate environmental control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' From the latter follows that stemness is a property determined by the environment, not the cell itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' ‘Cell plasticity’ might need to be seen in a wider context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Usually, cell plasticity is associated with a change of a cell’s type when subjected to environmental cues, which involves a substantial remodelling of the cell’s morphology and biochemical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, we see that a committed cell Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 21 may turn into a stem cell simply by adjusting the pace of the cell cycle and differentiation processes to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This may not require substantial changes in the cell’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This exemplifies that homeostatic control through crowding feedback is not only a way to render homeostasis stable and robust, but also to create stem cell identities as a collective property of the tissue cell population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We thank Ben MacArthur and Ruben Sanchez-Garcia for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Declarations PG is supported by an MRC New Investigator Award, Grant number MR/R026610/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The code generated for numerical computations in the cur- rent study is available on Github, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='com/cp4u17/Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' No other data was generated for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Contributions are as follows: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' conceptualised the paper, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' did the mathematical analysis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' did the numerical analysis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' supervised the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The authors have no competing interests to declare that are relevant to the content of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Appendix A Asymptotic stability assessment based on Routh-Hurwitz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 Background In control system theory, a commonly used method for assessing the stability of a linear system is the Routh-Hurtwiz (RH) criterion [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' It is an algebraic criterion providing a necessary and sufficient condition on the parameters of a dynamic system of arbitrary order to ensure the dynamics are asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, the criterion defines a set of conditions on the coefficients, pi, of the characteristic polynomial, P(s), written as P(s) = sn + n � i=1 pisn−i , (A1) in which n corresponds to the dimension of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Note that the notation used in this section, based on that from [24], is different from that of the main text, where pi is the polynomial coefficient of ith order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A first result of the RH criterion is that a necessary condition for the dynamical system to be asymptotically stable is that all the coefficients must be positive, that is, pi > 0, for all i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A2) Springer Nature 2021 LATEX template 22 Homeostatic regulation of renewing tissue cell populations via crowding control Additional conditions on the polynomial coefficients are added for a necessary and sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' These conditions are based on Routh’s array, written as � ����� 1 p2 p4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 0 p1 p3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' b1 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' c1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' � ����� , (A3) in which the first two rows contain all the coefficients of the characteristic polynomial, and the following ones are recursively computed as bi = − det � 1 p2i p1 p2i+1 � p1 , (A4) ci = − det � p1 p2i+1 b1 bi � b1 , (A5) and so on until a zero is encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The RH criterion states that the system is asymptotically stable if and only if the elements in the first column of Routh’s array are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on that, it can be easily shown that for a second-order polynomial, the necessary condition (A2) is also sufficient for asymptotic stability (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=') since b1 = p1p2, which means that The system is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' ⇐⇒ pi > 0, for i = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A6) Instead, the necessary and sufficient condition for a polynomial of order three results in The system is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' ⇐⇒ pi > 0, for i = 1, 2, 3 and p1p2 − p3 > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A7) The same reasoning can be applied to higher-order dynamics to derive additional conditions on the coefficients pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 Verification of the necessary condition for asymptotic stability The Matlab code for verifying (23) is provided in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='com/ cp4u17/Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The strategy used is to evaluate each term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (23) and simply compare the left and right-hand sides of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We followed a symbolic approach (based on the Matlab symbolic toolbox) for an arbitrary three-state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A numerical approach was used instead for higher-order dynamics, specifically 4, 5 and 6 state cell fate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To do so, we randomly defined the cell dynamical matrix at the steady state, A(ρ∗), and its derivative with respect Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 23 to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Entries were chosen from a uniform distribution and, for assuring a zero dominant eigenvalue for A(ρ∗), a local optimiser (fmincon function of Matlab) was used to find appropriate diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For each dimension of the cell fate model, we tested up to 1000 random cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='3 Sufficient condition for asymptotic stability In this section, we will indicate with the superscripts A and J the coefficients of the characteristic polynomial expressed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A1) respectively of the matrix of the dynamical system, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (6), and those of the Jacobian matrix, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For a two and three-state system, the following relations can be alge- braically derived pJ 1 = pA 1 − � i ηi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A8) where ηi is according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Again, considering that pA 1 > 0, if all ηi ≤ 0 then pJ 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, the above relation implies that in a two-state system, the RH cri- terion given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A6) is fulfilled when η ≤ 0, with at least one negative component (otherwise J = A) and therefore the system is asymptotically sta- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We recall that asking ηi ≤ 0 without further constraints is equivalent to the previously derived condition (30) with ϵi = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For applying the RH criterion to a three-state cell dynamic system, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A7), we need to evaluate the sign of pJ 2 and then that of pJ 1 pJ 2 − pJ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To do so, we first write pJ 2 = pA 2 − � i fiηi , (A9) in which fi = � j aji − Tr(A) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since the off-diagonal elements are non-negative, and the trace of A is negative, then fi > 0 for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' That means that if all ηi ≤ 0 then pJ 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Concerning the term pJ 1 pJ 2 − pJ 3 , this can be written as a quadratic form in η = � η1, η2, η3 � as pJ 1 pJ 2 − pJ 3 = Q(η) = ηT AQη + bT Qη + cQ , (A10) in which AQ = � � f1 f1 f1 f2 f2 f2 f3 f3 f3 � � , (A11) bQ = −pA 1 � � f1 f2 f3 � � − pA 2 vw � � v3(w3 − w1) + v2(w2 − w1) v3(w3 − w2) + v1(w1 − w2) v2(w2 − w3) + v1(w1 − w3) � � , (A12) and cQ = pA 1 pA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Here, v = (v1, v2, v3) is a left dominant eigenvector and w = (w1, w2, w3) a right dominant eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We now note that the matrix AQ is semidefinite positive since two eigen- values are zero (the rows are two-fold degenerate) and one is positive, equal to Tr(AQ) = � i fi, and cQ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We now distinguish two cases, depending on Springer Nature 2021 LATEX template 24 Homeostatic regulation of renewing tissue cell populations via crowding control the sign of bQ elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' First, if bQ ≤ 0, then Q(η) > 0 for any η ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since fi, pA 1 , pA 2 , vw > 0, we get a sufficient condition for bQ ≤ 0, namely, 0 ≤ v3(w3 − w1) + v2(w2 − w1) (A13) 0 ≤ v3(w3 − w2) + v1(w1 − w2) 0 ≤ v2(w2 − w3) + v1(w1 − w3) In that case, asymptotic stability and thus crowding feedback control is assured for any η < 0, and thus the bound for feedback strength is ϵi = ∞ for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Otherwise, if there is at least one positive element in bQ, then Q(η) > 0 only if |ηi| < ϵi, where ϵ = (ϵ1, ϵ2, ϵ3) are the absolute values of the solutions to the equation Q(η) = 0, that is – given that ηi are negative – the solution to, 0 = ϵT AQϵ − bT Qϵ + cQ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (A14) Importantly, we note that the elements of bQ depend uniquely on the proper- ties of the dynamical system and therefore, they can be determined without requiring the knowledge of the parameter derivatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' the specific crowding feedback dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The Matlab code for verifying (A8), (A9) and (A10) is provided in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='com/cp4u17/Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Appendix B Test case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 Asymptotic stability This section reports the details of the model used for numerical examples presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The cell dynamics correspond to the following three-state cell fate model X1 λ1 −→ X1 + X1, X1 ω13 −−→ X3, X1 γ1 −→ ∅ X2 ω21 −−→ X1, X2 ω23 −−→ X3, X2 γ2 −→ ∅ X3 λ3 −→ X3 + X3, X3 ω31 −−→ X1, X3 ω32 −−→ X2, (B15) whose network is shown in Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In such a model, for simplicity, we only consider symmetric self-renewing divisions so that κij = ωij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Also, we apply the crowding feedback to division rates, λi, and differentiation rates γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In this way, it is straightforward to apply the sufficient condition (30) for asymptotic stability since κ′ ij = 0 for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, each kinetic parameter of the type αi ∈ {λj, γj}j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=',3 is expressed as a function of ρ, whilst those of the type αi ∈ {κjk}j,k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=',3 are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, we chose a Hill function [31] where αi(ρ) = ci + kiρni/(Kni i + ρni) in case αi is a differentiation rate, so that α′ i = ∂αi/∂ρ > 0, and αi(ρ) = Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 25 ci +ki/(Kni i +ρ/ni) in case it is a proliferation rate, so that α′ i < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' According to (30) this choice assures that, if there is a value ρ = ρ∗ for which µ(ρ∗) = 0, this corresponds to an asymptotically stable steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The parameter values used in our example are reported in Table B1, and the profiles of the proliferation and differentiation rates as a function of ρ are shown in Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on these values, the steady state corresponds to ρ∗ = 1 (arbitrary unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' As expected, the dominant eigenvalue of the Jacobian at the steady state is negative (µJ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To test the dynamical behaviour of the tissue cell population, we numer- ically solved the system of ODEs (5) for different initial conditions based on the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The results are shown in Figure B3 as the time evolution of ρ, normalised by the steady-state ρ∗, (left panels), and of the dominant eigenvalue, µ (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The label H indicates an initial condition corresponding to the self- renewing state ρ∗, that is, the system is initially in homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In the simulations labelled as P− and P+, we applied perturbation in the initial state ρ∗ = (ρ∗ 1, ρ∗ 2, ρ∗ 3), which are, respectively, � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='8ρ∗ 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='75ρ∗ 2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='85ρ∗ 3 � and � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5ρ∗ 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1ρ∗ 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2ρ∗ 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' As expected, in all these cases, the feedback’s effect is sta- bilising the system so that it returns to the steady state upon perturbation, ρ → ρ∗, (asymptotic stability) and thus regains self-renewal property, µ → 0, over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B1 Cell state network representing a cell type composed of three states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The links represent direct transitions, ωij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' symmetric divisions occur with rates λi and differentiation with rate γi, where subscripts i, j = 1, 2, 3 indicate the corresponding cell state, as per model (B15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 Failure of feedback function Based on the cell fate model regulated via crowding feedback described in the previous section, we assess the impact of failure in one or more feedback functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, the failure of the crowding regulation is modelled, assuming one or more kinetic parameters as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Five different failure test cases are assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For doing so, we chose αi = (1 + C)α∗ i being constant instead of depending on ρ, in which α∗ is the value at the steady state when M Y1 XI 23 1 013 021 1 031 X X2 023 032Springer Nature 2021 LATEX template 26 Homeostatic regulation of renewing tissue cell populations via crowding control k K n α∗ α′ λ1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='84 λ3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='56 γ1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='48 γ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='61 κ13 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 κ21 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 κ23 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 κ31 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 κ32 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='00 Table B1 Values of the Hill function parameters describing the kinetic parameters in case of homeostasis regulation via crowding feedback for the cell fate model (B15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The generic kinetic parameters (represented as αi in the right columns of the table) are a function of the total cell density, ρ, and are given by γi(ρ) = c + kρn/(Kn + ρn) and λi(ρ) = c + k/(Kn + ρn) with i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' A common value c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='05 is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' State transition rates ωij, are constant and equal to κij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For such a cell fate dynamics, the steady state is ρ∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The unit of the kinetic parameter is arbitrary and therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Unless specified otherwise, these values apply to all the numerical examples presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 2 4 2 0 2 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B2 Proliferation and differentiation rates (left panels, with α as a generic placeholder for parameters), and their derivative with respect to ρ (right panels) as functions of cell density normalised by the steady-state ρ∗ for the cell fate model (B15) schematised in Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The profiles in the left panel correspond to Hill functions defined in Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' there are no failures (reported in Table B1) and C is a constant (reported in Table B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Five test cases, indicated as F1−5, are assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In test case F1, only one feedback fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Three of the four kinetic parameters fail in cases F2−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Finally, F5 represents a case where all the feedback functions fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The corresponding variability of the dominant eigenvalue, µ, as a function of the cell density is shown in Figure B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' It is clear that whilst F1−4 cases satisfy the sufficient condition for strict homeostasis, (30), in test cases F5, the dominant eigenvalue being constant means that there is no homeostatic regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Importantly, there is no steady state in test cases F2,4 since the dominant eigenvalue is always positive in one case or negative in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Based on these assumptions, we numerically solved the system of ODEs (5) using the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The failure test cases start at time 0 from an initially homeostatic condition, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The results are shown in Figure B5 as the time evolution of ρ, normalised by the homeostatic steady-state, ρ∗, (left panels), and of the Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 27 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 H P- P+ 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 H P- P+ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B3 Effect of perturbation of homeostasis under crowding control, when feedback parameters are according to Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (Left) Cell density ρ, scaled by the steady-state ρ∗, as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (Right) Corresponding variation of the dominant eigenvalue µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Time is scaled by the inverse of ¯α = mini α∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Three different initial condition are tested: H, corresponds to the steady state ρ∗ = (ρ∗ 1, ρ∗ 2, ρ∗ 3), P− to �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='8ρ∗ 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='75ρ∗ 2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='85ρ∗ 3 � and P+ to �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5ρ∗ 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1ρ∗ 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2ρ∗ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Since the steady state is asymptotically stable, thanks to crowding control, the cell population remain in, or return to, a homeostatic state characterised by µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Parameter F1 F2 F3 F4 F5 λ1 +5% +5% +5% 20% 5% λ3 +5% +5% 20% 5% γ1 5% +20% 5% γ2 5% 5% Table B2 Value of the constant C in the feedback failure test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Whenever a failure in the feedback of one kinetic parameter α occurs, that parameter is modelled as a constant, α = (1 + C)α∗, in which the steady-state value, α∗, is reported in Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Test cases F1 and F2 correspond to those presented in the main text (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' dominant eigenvalue, µ, (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Note that the cases F1,2 correspond respectively to the Single failure and Multiple failures reported in the main text (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In two cases, F1,3, despite a single or multiple feedback functions failing, a new homeostatic condition is reached after some time, where µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, suppose a different set of feedback fails, like in F2,4, such that the dominant eigenvalue is respectively positive or negative for any ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In that case, no steady state can be attained, and the tissue cell population will hyper-proliferate or decline in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Hence, even if the condition for asymptotic stability is met, there is no steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Finally, if homeostasis is not regulated at all, as in F5, then the population dynamics only depend on the value of the dominant eigenvalue (the cell dynamical model (5) turns linear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In the case shown, µ > 0 and therefore, the cell population diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='3 Single cell mutation scenario To assess the tissue dynamics with a single-cell mutation, as presented in the main text, we modelled the clonal dynamics, namely, the dynamics of single cells and their progeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For doing so, we considered the model (B15) as a Springer Nature 2021 LATEX template 28 Homeostatic regulation of renewing tissue cell populations via crowding control 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 2 2 0 2 4 H F1 F2 F3 F4 F5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B4 Variation of the dominant eigenvalue µ as a function of the cell density, ρ, normalised by the reference homeostatic state value, ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The curve H corresponds to the reference homeostatic model presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The other curves, F1−5, represent different sets of feedback failure, as reported in Table B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' 10 0 10 20 30 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='5 H F1 F2 F3 F4 F5 10 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 H F1 F2 F3 F4 F5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B5 Failure of feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (Left) Cell density, scaled by the steady state in the homeostatic case, as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' (Right) Corresponding variation of the dominant eigenvalue µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Time is scaled by the inverse of ¯α = mini α∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The homeostatic model, H, is perturbed at a time equal to zero to include the feedback failure reported in Table B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Whilst in F1,3, the regulation is able to achieve and maintain a new homeostatic state (µ = 0), the remaining case fails to regulate the cell population, leading to an indefinite growth or shrinking of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Markov process with the same numerical rates as before, but now events are treated as stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Then, we run numerical simulations using the Gillespie algorithm [32] to evaluate this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, the results presented in this work are based on 100 independent instances, where each instance is a possible realisation of the stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' We chose a total cell number N0 = 5000 as the initial condition (cell density is based on unitary volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In real tissues, the number of cells could be a few orders of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, this number is sufficiently large to avoid the extinction of the process in the time scale analysed, so once rescaled, these dynamics are representative of those in the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' All the simulations are stopped when the mutated clone goes extinct or divergence of the dynamics is detected, defined as reaching N = 5N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Homeostatic regulation of renewing tissue cell populations via crowding control 29 From an implementation point of view, we consider a cell fate model represented by two disconnected cell state networks to model the tissue dynam- ics, including the mutated cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' One network corresponds to the unperturbed test case H, and the other to the dysregulated one, F2 (both described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The simulation starts with N0 cells in the H network, dis- tributed in each state proportionally to the expected steady-state distribution in the tissue, and no cells in the F2 network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Thus, since the two networks are disconnected, F2 remains empty, and the simulation represents the tissue dynamics before the dysregulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' At a time equal to zero, we moved one cell from a random state in the H network to the corresponding state in the F2 one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' This simulation represents the tissue dynamics, including the single mutated cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In Figure B6 (left), all the trajectories where the mutated clones go extinct are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In these cases, the tissue dynamics remain globally unaffected by the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Due to the stochastic nature of the process, mutant clones can go extinct even if the growth parameter is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' That is, even in cases where divergence would be observed for the tissue-wide disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' However, this does not occur in all the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The right panel of the same figure shows those instances where the mutated clone does not go extinct and eventually prevails, resulting in diverging cell population dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For the chosen parameters, this divergence of the mutated clone is detected in 6% of all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Surprisingly, only a few clones survive despite a proliferative advantage, but this is plausible for a small fitness advantage (For example, in the case of a single state with cell division rate λ and loss rate γ – a simple branching process [33] – the probability for the a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ, which can be very low for λ ≈ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In the main text (Figure 2), only one profile for each scenario is shown, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' They correspond to instance #24 for the homeostatic case and instance #43 for the diverging case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='4 Quasi-dedifferentiation The numerical example presented in the main text is based on the same cell fate model described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' To model the dynamics of a committed cell type, we choose a constant non-negative u = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='06 �T to model for the cell influx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' For such a model, the steady state, ρ∗ c, is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The figures presented in the main text are based on the numerical integra- tion of the system of ordinary differential equation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' In particular, we used the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' References [1] National Institute of Health: Stem Cell Basics (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' https://stemcells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='gov/info/basics Springer Nature 2021 LATEX template 30 Homeostatic regulation of renewing tissue cell populations via crowding control 5 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 H F2 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2 H F2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' B6 Results of numerical simulations of the stochastic process representing the cell dynamics, according to section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The cell density, scaled by the steady state in the homeostatic case, as a function of the time is shown for 100 random instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Each shown trajectory is the result of a different instance of the stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' At a time equal to zero, the cell mutation is modelled as a switch of a single random cell from the homeostatic H cell dynamics to the F2 model assessed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' On the left panel, only the trajectories for which the mutated clone goes extinct are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' The right panel shows the trajectories in which the mutated clone prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' Dynamics are scaled by ¯α = mini{α∗ i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf'} +page_content=' [2] Marinari, E.' metadata={'source': 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Postgraduate School, Slovenia, +Jamova cesta 39 +{abdul.sittar, dunja.mladenic}@ijs.si + +Abstract. Detection of news propagation barriers, being economical, +cultural, political, time zonal, or geographical, is still an open research +issue. We present an approach to barrier detection in news spreading +by utilizing Wikipedia-concepts and metadata associated with each bar- +rier. Solving this problem can not only convey the information about the +coverage of an event but it can also show whether an event has been +able to cross a specific barrier or not. Experimental results on IPoNews +dataset (dataset for information spreading over the news) reveals that +simple classification models are able to detect barriers with high accu- +racy. We believe that our approach can serve to provide useful insights +which pave the way for the future development of a system for predicting +information spreading barriers over the news. + +Keywords: news propagation · news spreading barriers · cultural bar- +rier · economical barriers · geographical barrier · political barrier · time +zone barrier · classification methods + +1 Introduction +The phenomenon of event-centric news spreading due to globalization has been +exposed internationally [8]. International events capture attention from all cor- +ners of the world. News agencies play their part to bring our attentions on some +events and not on others. Varying nature of living styles, cultures, economic con- +ditions, time zone, and geographical juxtaposition of countries present a signifi- +cant role in process of publishing news related to different events [3, 6, 13, 19–21]. +For example, publishing about sports events could be dependent on culture, epi- +demic events can reach firstly to neighboring countries due to geographic prox- +imity and, news on a luxury product may be relevant for economically strong +countries due to demand of wealthy people. We represent this differentiation +along with different barriers. These barriers include but are not limited to 1) +Economic Barrier, 2) Cultural Barrier, 3) Political Barrier, 4) Geographical Bar- +rier, and 5) Time Zone Barrier. Detection of the overpass of these barriers does +Copyright © 2021 for this paper by its authors. Use permitted under Creative +Commons License Attribution 4.0 International (CC BY 4.0). + +2 +A. Sittar et al. + + +not only tell us the area where the broadcasting of an event reached, but it also +shows us events-location relation as countries have different culture, economic +conditions, geographical placement on the globe, political point of view, and +time zone. Following are the definitions of news crossing these barriers: +Cultural Barrier. If we identify the coverage of specific event-centric news by +publishers that are surrounded by different cultures, then we can say that the +news related to the event crossed cultural barriers. +Political Barrier. If news about a specific event is disseminated from publishers +having different political alignment, we can say that the news related to that +event crossed the political barrier. +Geographical Barrier. We say that some news related to a specific event +overpasses geographical barriers if that event gets attention by publishers of +countries located in different geographical regions. +Time Zone Barrier. We can claim that event-centric news has crossed the +time zone barrier if it has been published by publishers located in different time +zones. +Economic Barrier. It can be asserted that a piece of event-centric news has +crossed economic barriers if it is published in countries having different economic +conditions. +In this paper, we propose a methodology for detection of different barriers +during information propagation in form of news that utilize data (IPoNews) [18] +related to three contrasting events (earthquake, Global warming, and FIFA world +cup) in different domains (natural disasters, climate changes, and sports) in 5 +different languages: English, Slovene, Portuguese, German, and Spanish. + +1.1 Contributions +Following are the main scientific contributions of this paper: + +– A novel methodology for barrier detection in news spreading. +– Experimental comparison of several simple classification models that can +serve as a baseline. + +1.2 Problem Statement +Observing the spreading of news on a particular event over time, we want to +predict whether a barrier (cultural, political, geographical, time zone, economi- +cal) is likely to hamper information while information propagates over the news +(binary classification). + +2 RELATED WORK + +Multiple barriers come across event-centric news specifically when the news is +concerned about international or national events. According to news flow theo- +ries, multiple determinants impact international news spreading. The economic + +Using the profile of publishers to predict barriers across news articles +3 + + +power of a country is one of the factors that influence news spreading. Moreover, +economic variations has different influence for different events (e.g. protests, con- +flicts, disasters) [15]. The magnitude of economic interactivity between countries +can also impact the news flow [21]. Economic growth/income level shows the eco- +nomic condition of a country. Multiple organizations are working on generating +prosperity and welfare index on yearly basis. Among them, “The Legatum Pros- +perity Index” and “Human Development Index” are popular 1, 2. Geographical +representation of entities and events has been utilized extensively in the past +to detect local, global, and critical events [3, 13, 19, 20]. It has been said that +countries with close distance share culture and language up to a certain extent +which can further unfold interesting facts about shared tendencies in informa- +tion spreading [15, 16]. + +News agencies tend to follow the national context in which journalists op- +erate. One of the related examples is the SARS epidemic study which found +that cross-national contextual values such as political and economic situations +impact the news selection [5]. It will be true to say that fake news is produced +based on many factors and it is surrounded by a paramount factor that is polit- +ical effect [11]. A great amount of work regarding fake news dwells on different +strategies and few studies considered political alignment to have a compelling +effect on news spreading [4, 12]. [12] strongly proved it to be a major strategy +in news agencies to control the news and change accordingly due to the involve- +ment of journalists and political actors. Countries that share common culture +are expected to have heavier news flow about between them reporting on similar +events [21]. Many quantitative studies found demographic, psychological, socio- +cultural, source, system, and content-related aspects [1]. Many models have tried +to explain cultural differences between societies. Hofstede’s national culture di- +mensions (HNCD) has been widely used and cited in different disciplines [7, 9]. + +News classification for different kinds of problems is a well-known topic since +the past and features used to classify varies depending upon the problem. [17] +used news content and user profile to classify the news whether it is fake or +not. [2] calculated TF-IDF score and Word2Vec score of most frequent words +and used them as features to classify into one of the five categories (state, econ- +omy, entertainment, international, and sports). Similarly, [14] performed part- +of-speech (POS) tagging at sentences level and used them as features, and built +supervised learning classifiers to classify news articles based on their location. +Mostly classifier trained to utilize popular supervised learning methods such as +Random Forest, Support Vector Machine (SVM), Naive Bayes, k-Nearest Neigh- +bour (kNN), and Decision Tree. In this work, we used the profile of each barrier +for each news publisher (see section 3.5) and most frequent 300 Wikipedia con- +cepts from the dataset that appeared in the list of news articles related to three +contrasting events (earthquake, Global Warming, and FIFA world cup). We also + +1 http://hdr.undp.org/en/content/human-development-index-hdi +2 https://www.prosperity.com/ + +4 +A. Sittar et al. + +≥ + +compared the results of popular classifiers such as SVM, Random Forest, Deci- +sion Tree, Naive Bayes, and kNN (see Section 5.4). + +3 DATA DESCRIPTION + +3.1 Dataset + +We utilized dataset ”A dataset for information spreading over the news (IPoNews)” +that consists of pairs of news articles that were labeled based on the level of their +similarity, as described in [18]. This dataset was collected from Event Registry, +a platform that identifies events by collecting related articles written in differ- +ent languages from tens of thousands of news sources [10]. The similarity score +among cross-lingual news articles was calculated using concept-based similar- +ity employing Wikifier service3. [18] describes the criteria when information is +considered to be propagated. Statistics of the data set are shown in table 3. + +Table 1. Statistics about dataset + +Dataset Domain +Event type +Articles per Language Total Articles + +1 + +Sports + +FIFA World Cup +Eng Spa Ger Slv Por + +2682 +983 762 711 10 216 +2 +Natural Disaster Earthquake +941 999 937 19 251 +3147 +3 +Climate Changes Global Warming 996 298 545 8 +97 +1944 + + +The dataset contains a list of pairs of news articles annotated with one of +the labels such as ”information-Propagated”, ”Unsure”, or ”Information-Not- +Propagated” (see Table 2). The information is considered to be propagated if the +cosine similarity score of the two articles in the pair is above a predefined thresh- +old ( 0.7 for Information-Propagated, < 0.4 for Information-not-Propagated, +otherwise Unsure). We restructured the original dataset to include only exam- +ples labeled as spreading information. In this way, we have pair of news articles +where we observe information spreading from one to the other. Furthermore, for +each example, instead of having a pair of articles, we kept only the article that +was published earlier. In this way, each example contains an article that spreads +information. + +Table 2. Articles with metadata + +from +to +weight Class +from-publisher to-publisher from-pub-uri +to-pub-uri +Por44 +Por43 +0.627 +Unsure +ClicRBS +SAPO 24 +jornald.clicrbs.com.br 24.sapo.pt +English881 English880 1 +Information-Propagated +Sky News +247 Wall St. +news.sky.com +247wallst.com +English258 English329 0.313 +Information-Not-Propagated Sify +4-traders +sify.com +4-traders.com +English793 English787 0.238 +Information-Not-Propagated Bioengineer.org 7NEWS Sydney scienmag.com +7news.com.au +German237 German236 0.979 +Information-Propagated +watson +watson +aargauerzeitung.ch +aargauerzeitung.ch + + +3 http://wikifier.org/info.html, https://github.com/abdulsittar/IPoNews + +Using the profile of publishers to predict barriers across news articles +5 + + +3.2 Statistics after restructuring the data + +The original dataset describes in Section 3 contains pairs of articles along with +the information on whether there was the propagation of information related to a +specific event or not. We used only examples labeled as propagating information +4. Based on the available metadata for articles, we ignored articles that do not +have metadata information in our database (see Section 3.4). Table 3 shows the +statistics for each barrier after filtering the original dataset. + +Table 3. Statistics about barrier + +Dataset Domain +Event type +Articles for each barrier + +1 + +Sports + +FIFA World Cup +Time-Zone Cultural Political Geographical Economical +724 +699 +143 +726 +634 +2 +Natural Disaster Earthquake +1102 +1113 +227 +1113 +1010 +3 +Climate Changes Global Warming 586 +445 +108 +487 +463 + + + + +3.3 Wikipedia Concepts as Features + +As our dataset already mention (see Section 3) if information in news is spread- +ing from an article to another based on Wikipedia-concepts, we utilized the +most frequent (top 300) Wikipedia-concepts as features. Figure 1 portrays these +Wikipedia-concepts for all three events in form of word clouds. + + + + + + + + +Fig. 1. Word clouds of most frequent words related to earthquake, FIFA +World Cup and Global Warming events respectively. + + + + +3.4 Barriers Knowledge + +Barriers knowledge refers to a database that contains metadata about each bar- +rier. Figure 3 shows schema of database and Table 4 presents barriers along with +their characteristics. Each barrier depends on one main information that is the +country name of the headquarter of the news publishers. Since the utilized data +4 https://doi.org/10.5281/zenodo.3950064 + +DEBpresident +United +Yor +Wart +overnmen +States +nameGermanname +football +SWar +ummerWorld +assoclation +Unitednationa +CUDFIFAASSO +ation +FIFAWorldYorKname +War +States +IInchEarthFranceUnited +United +New +Globa +disambiguation6 +A. Sittar et al. + + +set already contains headquarter of publishers therefore we fetched the coun- +try associated with headquarters. For economical barrier, we fetched economical +profile for each country using “”The Legatum Prosperity Index”” 5. Cultural +differences among different regions were collected using Hofstede’s national cul- +ture dimensions (HNCD). For time zone and geographical barrier, we stored +general UTC-offset, latitude, and longitude. For political barrier we are using +the political alignment of the newspaper/magazine that we determined based on +Wikipedia infobox at their Wikipedia page. For instance, for Austrian newspa- +per ”Der Standard” we find social liberalism as political alignment (See Figure +2), for British newspaper ”Daily Mail” we find right-wing as political alignment, +for German ”Stern” magazine there is no information in its Wikipedia infobox +on the political alignment thus we label political alignment as unknown. + + + +Fig. 2. Three Wikipedia infobox for three different newspapers/magazines +with political alignment + + + +5 https://www.prosperity.com/ + +Der Standard +DERSTANDARD +Type +Daily newspaper +Owner(s) +Oscar Bronner +Publisher +Oscar Bronner +Martin Kotynek +Founded +19 October 1988: 32 years +ago +Political +Social liberalism +alignment +Headguarters +Vienna +Circulation +86,000 (2013) +Website +www.derstandard.de +www.derstandard.at DailyMail +DailumlailFREE +MICHELIN +SO MUCH +FOR THE +BONFIRE +OF THE +QUANGOS! +aplasticheart +DailyMail frontpageon 4August 2010 +Type +Dailynewspaper +Format +Tabloid +Owner(s) +DailyMail and General Trust +Founder(s) +AlfredHarmsworthandHarold +Harmsworth +Publisher +DMGMedia +Editor +GeordieGreig +Founded +4 May1896:124 years ago +Political +Right-wing[1]2][3] +alignment +Language +English +Headquarters Northcliffe House +2 Derry Street +LondonW85TT +Circulation +1.134.184(asofFebruary +2020)[4] +ISSN +0307-7578 +OCLC +16310567 +number +Website +www.dailymail.co.ukStern +? +stern +? +stern +KRERSMID +Alein in turopa +IXABE +HRSECIEENE +Sternmagazinecoveron18February2016 +Editor +FlorianGless,Anna-Beeke +Gretemeier +Categories Newsmagazine +FrequencyWeekly +Circulation390,000(2020) +Year +1948 +founded +Firstissue +1August1948,72yearsago +Company +Gruner+Jahr +Country +Gemany +Basedin +Hamburg +Language +Geman +Website +www.stern.de +ISSN +0039-1239Using the profile of publishers to predict barriers across news articles +7 + + +3.5 Features for Individual Barrier +We represented each barrier with a specific profile containing a list of features. +Table 4 depicts the list of features for each barrier. Economic and cultural bar- +riers consist of a vector of length 11 and 6 features whereas geographical, time +zone, and political only contain 1 or 2 features such as latitude-longitude, UTC- +offset, and political alignment. + + + +Fig. 3. Database Schema for Barriers + + + + +3.6 Dataset Annotation +We queried the metadata information for each article and generated a CSV file +for each barrier. We annotated each article based on that meta information to be +used for model training and classification. For economic and cultural barriers, we +calculated cosine similarity between vectors of economical values and vectors of +cultural values. Score greater than the threshold value of 0.9 labeled as FALSE +otherwise TRUE. We set the lowest value as a threshold based on the fact that +if two countries have a little gap concerning culture or economical values then +there exists a barrier. For geographical barriers, we compared the latitude and +longitude of the country of each publisher. If a country name or lat/lat appeared +to be the same then we annotated it with FALSE otherwise TRUE. Lastly, for + +Enterprise Conditions +Social Capital +Education +EconomicQuality +Marketaccessand +Individualistic cuiture +Living Conditions +Economic +infrastructure +Power distance +Profile +Governance +Natural Environment +afety +Fam +Health +nty +Cultural Value's +Has +Has +induigence vs +restraint +Headquarter +Has +Country +Has +Geographical +Values +(Lat/Lon) +Has +name +Has +Longitude +Latitude +Political barrier +Time-Zone +Political +UTC-offset +Alignment8 +A. Sittar et al. + + +Table 4. Features of each barrier + +Barrier +Features + +Economic +Rank, Safety-Security, +Personal-Freedom, Governance, Social-Capital, Investment-Environment, +Enterprise-Conditions, Market-Infrastructure, Economic-Quality, +Living-Conditions, Health, Education, Natural-Environment + +Cultural +Power-Distance, +Uncertainty-Avoidance-By-Individuals, Individualistic-Cultures, +Masculinity-Femininity, Long-Term-Orientation, Indulgence-Restraint +Geographical Latitude, Longitude +Time Zone +UTC-offset +Political +Political-Alignment + + + + +time-zone and political barriers, we followed the same process that was for the +geographical barrier. if political alignment or UTC-offset appeared to be the +same for a pair then it is annotated with FALSE otherwise TRUE. Figure 4 +depicts the class distribution for each barrier. We can notice unbalanced class +distribution with majority of the examples being False. This is especially true +for Cultural and Political barrier with 91 percent of example being False. Thus +in our evaluation we rely more on F1 measure than classification accuracy. + + + + + + + +Fig. 4. Class Distribution for Each Barrier + +2000 +True +2014 +False +1500 +1588 +1599 +1324 +1000 +948 +670 +500 +478 +408 +203 +42 +0 +I Barrier +nical +olitical Barrier +Econom +TimeUsing the profile of publishers to predict barriers across news articles +9 + + +4 MATERIALS AND METHODS +4.1 Problem Modeling +For each barrier, we have a list of news articles where each article is associated +with 300 Wikipedia-concepts and features related to that barrier. The task is to +predict the status S of each barrier B. +S = f (C, B) +f is the learning function for barrier detection, C is donating here Wikipedia- +concepts related to an article and B is the list of features related to a specific +barrier (see Table 4). + +4.2 Methodology +We utilized dataset IPoNews [18] and built a database on top of this dataset +that includes barrier knowledge. Figure 5 explains the overall process of model +construction from news articles to results generation. We created a list of in- +stances using the most frequent Wikipedia-concepts based on news articles and +joined them along with barrier knowledge. After performing the annotation (see +Section 3.6), we trained popular classification models and generated the results +on test data (see Section 5.4). + + + +Fig. 5. Steps for Model Construction + + + +5 EXPERIMENTAL EVALUATION +5.1 Baselines +We used the following methods as baselines for all our models. +– Uniform: Generates predictions uniformly at random. +– Stratified: Generates predictions by respecting the training set’s class dis- +tribution. +– Most Frequent: Always predicts the most frequent label in the training +set. + +Barrier's +Results +knowledge +Testset +Newsarticles (IPoNews) +Metadata +Barriers'Annotation +Wikipediaconcepts +Model Construction +Trainset10 +A. Sittar et al. + +sum +sum +sum +sum + +5.2 Classification Methods + +We trained popular classification models for each barrier such as SVM, kNN, +Decision Tree, Random Forest, and Naive Bayes using Scikit-Learn. We applied +a stratified 10-fold cross-validator to split the dataset for training and testing. +For Random Forest, kNN, and Decision Tree, we varied the size of n-estimator, +value of k, and max-leafs and chosen the one with the best score on test data +respectively. Implementation of this methodology to barrier detection can be +found on GitHub 6. + +5.3 Evaluation Metric + +Due to imbalance in the class distribution for all barriers, we used micro averaged +precision and recall to evaluate our models. 7 +– Micro-Precision: The precision of average contributions from each class is +calculated in micro-precision whereas the following question is answered by +precision: What proportion of positive predictions was correct? It is defined +as: + +TruePositivesum + +Micro − Precision = TruePositive ++ FalsePositive +– Micro-Recall: Recall of average contributions from each class is calculated +in micro-recall whereas the following question is answered by recall: What +proportion of actual positives was predicted correctly? It is defined as: + +TruePositivesum + +Micro − Recall = TruePositive ++ FalseNegative + +5.4 Results and Analysis + +Table 5 shows the results of all the classifiers for each barrier along with baselines. +Analysis of the experimental results show that overall all the machine learning +models outperform the three baselines. For all the barriers, we can notice Micro- +Recall is equal to Micro-Precision. The best performing baseline is the ”Most- +frequent” with Micro-F1 for economic, cultural, geographical, time zone, and +political barrier equal to 0.70, 0.90, 0.58, 0.70, and 0.90 respectively. The best +performing models on all the barriers are Decision Tree, Random Forest, and +kNN. Looking at Micro-F1, we can see that on the Economic and Cultural +barrier kNN achieved the best performance of 0.75 and 0.95 respectively. On +Geographical barriers, kNN and Decision Tree performed the best achieving 0.81. +On Time-Zone, the best performing classifier is Random Forest with Micro-F1 +6 https://github.com/cleopatra-itn/BarrierDetection-Classification +7 https://peltarion.com/knowledge-center/documentation/evaluation- +view/classification-loss-metrics/micro-recall + +Using the profile of publishers to predict barriers across news articles +11 + + +0.83. On Political barriers, SVM, kNN, and Random Forest achieve the best +Micro-F1 score of 0.97. +In terms of classification accuracy, we can see that Random Forest outper- +forms the baselines as well as the other four classifiers for the first four barriers. +Notice that Random forest performs better than decision tree but takes more +time. Naive-Bayes achieves a little bit lower classification accuracy than the Deci- +sion Tree for the first four barriers. On the political barrier Naive-Bayes achieves +the best classification accuracy (0.98) but lower Micro-F1 (0.66). + +6 CONCLUSIONS AND FUTURE WORK + +It is highly important to detect the barriers while information propagates specif- +ically through the news. For journalists, marketers, and social scientists, the phe- +nomenon of knowing which barrier appeared most frequently for what type of +events, is significantly helpful to solve business and marketing problems. In this +regard, we proposed a simple methodology. Though its results are good enough +for three types of events, we would like to enhance features as well as events. We +used only Wikipedia-concepts and meta information to detect barriers. In the +future, we would like to use DMoz categories provided by Event Registry [10], +and transformation of the text of news articles as a feature for barrier detection. +Currently geographical and time zone barriers are calculated in a binary way ei- +ther the same or different. In the future, we would like to introduce the distance +between countries and between time zones as labels instead of the currently used +binary labeling. + +7 ACKNOWLEDGMENTS + +The research described in this paper was supported by the Slovenian research +agency under the project J2-1736 Causalify and co-financed by the Republic +of Slovenia and the European Union’s Horizon 2020 research and innovation +program under the Marie Sk-lodowska-Curie grant agreement No 812997. + +12 +A. Sittar et al. + + + +Table 5. Classifiers’ comparison with baselines + +Barrier +Model +CA Mic-Pre Mic-Rec Mic-F1 +Economic +Uniform +0.50 0.50 +0.49 +0.49 + +Stratified +0.58 0.59 +0.57 +0.59 + +Most Frequent 0.70 0.70 +0.70 +0.70 + +SVM +0.66 0.69 +0.69 +0.69 + +kNN +0.70 0.75 +0.75 +0.75 + +Decision Tree +0.69 0.73 +0.73 +0.73 + +Random Forest 0.74 0.74 +0.74 +0.74 + +Naive Bayes +0.61 0.63 +0.63 +0.63 + + +Cultural +Uniform +0.50 0.50 +0.49 +0.50 + +Stratified +0.83 0.83 +0.83 +0.83 + +Most Frequent 0.90 0.90 +0.90 +0.90 + +SVM +0.84 0.93 +0.93 +0.93 + +kNN +0.55 0.95 +0.95 +0.95 + +Decision Tree +0.90 0.94 +0.94 +0.94 + +Random Forest 0.93 0.93 +0.93 +0.93 + +Naive Bayes +0.83 0.51 +0.51 +0.51 + + +Geographical Uniform +0.49 0.50 +0.50 +0.50 + +Stratified +0.50 0.51 +0.51 +0.51 + +Most Frequent 0.58 0.58 +0.58 +0.58 + +SVM +0.81 0.76 +0.76 +0.76 + +kNN +0.79 0.81 +0.81 +0.81 + +Decision Tree +0.78 0.81 +0.81 +0.81 + +Random Forest 0.79 0.79 +0.79 +0.79 + +Naive Bayes +0.76 0.79 +0.79 +0.79 + + +Time Zone +Uniform +0.49 0.49 +0.49 +0.49 + +Stratified +0.59 0.58 +0.58 +0.58 + +Most Frequent 0.70 0.70 +0.70 +0.70 + +SVM +0.78 0.77 +0.77 +0.77 + +kNN +0.70 0.78 +0.78 +0.78 + +Decision Tree +0.80 0.81 +0.81 +0.81 + +Random Forest 0.83 0.83 +0.83 +0.83 + +Naive Bayes +0.72 0.64 +0.64 +0.64 + + +Political +Uniform +0.51 0.52 +0.50 +0.50 + +Stratified +0.84 0.83 +0.81 +0.82 + +Most Frequent 0.90 0.90 +0.90 +0.90 + +SVM +0.79 0.97 +0.97 +0.97 + +kNN +0.62 0.97 +0.97 +0.97 + +Decision Tree +0.79 0.91 +0.91 +0.91 + +Random Forest 0.97 0.97 +0.97 +0.97 + +Naive Bayes +0.98 0.66 +0.66 +0.66 + +Using the profile of publishers to predict barriers across news articles +13 + + +References +1. 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GeoInformatica pp. 1–31 (2020) +21. Wu, H.D.: A brave new world for international news? exploring the determinants of +the coverage of foreign news on us websites. International Communication Gazette +69(6), 539–551 (2007) + diff --git a/BtFKT4oBgHgl3EQfXS78/content/tmp_files/2301.11794v1.pdf.txt b/BtFKT4oBgHgl3EQfXS78/content/tmp_files/2301.11794v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2819d7db4d91bfe04e304e323aafe5d834e445dd --- /dev/null +++ b/BtFKT4oBgHgl3EQfXS78/content/tmp_files/2301.11794v1.pdf.txt @@ -0,0 +1,1249 @@ +Thermodynamic features of the 1D dilute Ising model +in the external magnetic field +A.V. Shadrina,∗, Yu.D. Panova +aInstitute of Natural Sciences and Mathematics, Ural Federal University, 620002, 19 Mira +street, Ekaterinburg, Russia +Abstract +We consider the effects of the magnetic field on the frustrated phase states of +the dilute Ising chain, especially, the behavior of the magnetic entropy change +and the isentropic dependence of the temperature on the magnetic field, which +are the key parameters of the magnetocaloric effect. The found temperature +dependences of entropy demonstrate the nonequivalence of frustrated phases in +the antiferromagnetic and ferromagnetic cases. In the antiferromagnetic case, +the nonzero magnetic field at certain parameters causes a charge ordering for +nonmagnetic impurities at a half-filling, while in the ferromagnetic case, the +magnetic field reduces the frustration of the ground state only partially. It is +also shown, that impurities radically change the magnetic Gr¨uneisen parameter +in comparison with the case of a pure Ising chain. +Keywords: +dilute Ising chain, frustrated magnets, magnetic entropy change +1. Introduction +One of the remarkable features of low-dimensional systems, such as deco- +rated Ising models [1–7], the anisotropic Potts chain [8], the diamond Hubbard +chain [9], is the presence, under certain parameters, of a frustrated ground state +for which the residual entropy is nonzero. Despite the absence of a real phase +transition at finite temperatures according to the Perron–Frobenius theorem for +square real matrices [10] the thermodynamic behavior near the boundaries be- +tween different phases of the ground state for these systems can exhibit striking +features. As shown in [11], if one of the phases has a nonzero residual entropy +that preserves continuity at the boundary with the other phase, then the ther- +modynamic characteristics of the system will demonstrate pseudo-transitions at +a finite temperature. Entropy, heat capacity, magnetization, and susceptibility +have similar features to the behavior of these properties at conventional phase +transitions, including the presence of quasicritical exponents [12]. +∗Corresponding author +Email address: shadrin.anton@urfu.ru (A.V. Shadrin) +Preprint submitted to Journal of Magnetism and Magnetic Materials +January 30, 2023 +arXiv:2301.11794v1 [cond-mat.stat-mech] 27 Jan 2023 + +Recently, frustrated magnetic systems have also attracted the attention of +researchers due to the enhanced magnetocaloric effect in the vicinity of finite- +field transitions [13, 14]. Besides to geometric factors, the impurities are also +the reason for the existence of frustrations in the magnetic system. The simplest +example of a magnetic system that is frustrated by impurities is a diluted Ising +chain. The Hamiltonian of 1D diluted Ising model can be written in the following +form +H = −J +� +i +Sz,iSz,i+1 + V +� +i +P0,iP0,i+1 − h +� +i +Sz,i − µ +� +i +P0,i. +(1) +Here we use the S = 1 pseudospin operators. The states for a given lattice +site with the pseudospin projections Sz = ±1 correspond to the two magnetic +states with the conventional spin projections sz = ±1/2, while the state with +Sz = 0 corresponds to the charged nonmagnetic state. Sz,i is a z-projection of +the on-site pseudospin operator, P0,i = 1 − S2 +z,i is the projection operator onto +the Sz = 0 state, J is the exchange constant, V > 0 is the inter-site correlation +parameter for impurities, h is an external magnetic field, and µ is a chemical +potential for impurities. Further we will assume that nonmagnetic impurities +are mobile, which corresponds to the annealed system. +As well known [15], +V = V0 + V1 − 2V01 describes the interaction for a more general case: +V0 +� +i +P0,iP0,i+1 + V1 +� +i +P1,iP1,i+1 + V01 +� +i +� +P0,iP1,i+1 + P1,iP0,i+1 +� +, +(2) +where P1 = S2 +z, is the projection operator onto magnetic states. The solutions +and various thermodynamic properties of the 1D dilute Ising model at zero +external magnetic field was found in [16–20]. If h ̸= 0, then there are no explicit +analytical expressions for various thermodynamic functions of the model (1). It +is known the account of magnetic field for the S = 1 Ising chain significantly +expands the list of possible phase states of the system and leads to various +features of thermodynamic behavior [21, 22]. +In the present paper, we consider the effects of the magnetic field on the +frustrated phase states of the model (1). We focused on the behavior of entropy +and, in particular, on the magnetic entropy change, which is the key parameter +of the magnetocaloric effect. +Also, we explore the isentropic dependence of +the temperature on the magnetic field. The paper is organized as follows. We +briefly describe the methods in section 2, and section 3 the results, including +the ground state phase diagram, and their discussions are given. Conclusions +are presented in section 4. +2. Methods +We define the transfer matrix for the model (1) as +τ = +� +� +xz +z1/2 t1/2 +x−1 +z1/2 t1/2 +y−1t +z−1/2 t1/2 +x−1 +z−1/2 t1/2 +xz−1 +� +� , +(3) +2 + +where x = eβJ, y = eβV , z = eβh, t = eβµ and β = 1/T, and we assume kB = 1. +From (3), we found the characteristic equation for the eigenvalues λi: +λ3 − λ2 � +ty−1 + x(z + z−1) +� +− λ +� +x2 − x−2 + t +� +xy−1 − 1 +� +(z + z−1) +� +− 2t(x − x−1) − tx−2y−1 = 0. +(4) +The eigenvalues in a general case are cumbersome functions, but at h = 0 they +could be reduced to the known expressions [20]: +λ1,2 += +1 +2 +� +x + x−1 + y−1t +� +± +� +2t + 1 +4 +� +x + x−1 − y−1t +�2�1/2 +, +λ3 += +x − x−1. +(5) +According to the Perron–Frobenius theorem [10], there is only one maximum +eigenvalue, λ1, and in the thermodynamic limit we obtain the grand potential +and the entropy in the following form: +Ω = Nω = −NT ln λ1, +S = − +� ∂ω +∂T +� +h,µ += ln λ1 + T +λ1 +�∂λ1 +∂T +� +h,µ +. +(6) +The grand potential and entropy found depend on parameters J, V , h, µ, and +T. But in the present problem, it is more convenient to use the concentration n +of impurities as an external parameter. The dependence n(µ) can be obtained +from the equation +n = − +�∂ω +∂µ +� +T,h += T +λ1 +�∂λ1 +∂µ +� +T,h +. +(7) +In a general case h ̸= 0, we used numerical methods to get the inverse +dependence µ(n) and fix the concentration of impurities n at all temperatures. +If h = 0, we obtain the explicit expressions [20]: +µ = ln +� +y +� +x + x−1� g + m +g − m +� +, +(8) +S = 1 +2 ln 2 (1 + 2g)2 +1 − 4m2 ++ g + 2m2 +1 + 2g +ln y − m ln +�� +x + x−1� g + m +g − m +� +− (1 − 2m) (g − m) +� +x − x−1� +(1 + 2g) (x + x−1) ln x, +(9) +where +g = +� +m2 + 1 +2 +�1 +4 − m2 +� +y−1 � +x + x−1��1/2 +, +(10) +and we introduced the deviation of the concentration of impurities from half- +filling, m = n − 1/2. +3 + +The knowledge of the entropy from Eqs. (6,7) gives an opportunity to explore +magnetocaloric properties of the dilute Ising chain for a given n. We explore the +magnetic entropy change, the isentropic dependencies of the temperature on the +magnetic field and the magnetic Gr¨uneisen parameter, which can be calculated +from the relation +Γmag = 1 +T +�∂T +∂h +� +S,n += − 1 +T +(∂S/∂h)T,n +(∂S/∂T)h,n +. +(11) +The explicit expression that we use to calculate Γmag for a given n in variables +(T, h, µ) has the following form: +Γmag = − 1 +T +� +(∂S/∂h)T,µ (∂n/∂µ)T,h − (∂S/∂µ)T,h (∂n/∂h)T,µ +(∂S/∂T)h,µ (∂n/∂µ)T,h − (∂S/∂µ)T,h (∂n/∂T)h,µ +� +. +(12) +3. Results +3.1. Phase diagram at zero temperature. +The phase diagram of the dilute Ising chain in longitudinal magnetic field at +zero temperature is shown in Fig. 1 for the J − h plane. The limiting case for +the Ising chain without impurities, m = −1/2, is given in Fig. 1(a). Two ground +states, the ferromagnetic (FM) state with magnetic moment oriented towards +the field, and the antiferromagnetic (AFM) state with zero magnetic moment, +are separated by the critical value of magnetic field |hc| = −2J at which the spin- +flip transition occurs. Figures 1(b) and 1(c) show the cases of a weakly diluted +chain, −1/2 < m < 0, and a strongly diluted chain, 0 ≤ m < 1/2, respectively. +Dilution with impurities leads to the appearance of two new boundary lines on +the J − h plane, J = V and |h| = −J − V . If J > V , the ground state is +represented by macroscopic FM domains (or drops) separated by macroscopic +impurity domains. +Similarly, the AFM domains arise when J < −V − |h|. +Schematically, this is shown in Fig. 1(b,c), where the arrows correspond to the +spins, and the circles correspond to the impurities. For both FM and AFM +phases, the entropy is zero. +Analysis of the ground state of the model (1) in zero magnetic field shows [19, +20] that phases with the nonzero residual entropy exist at |J| < V and at +|J| = V . For the weak exchange, when |J| < V , the spin correlation length is +always finite, but the impurity correlation length with the temperature lowering +tends to infinity at the half-filling concentration, m = 0, due to the formation of +charge ordering [20]. If |J| = V and h = 0, the spin correlation length and the +impurity correlation length are finite for all values of the impurity concentration +and temperature. +In magnetic field, if −V − |h| < J < V , the residual entropy is also not zero, +so the ground state is frustrated. If −1/2 < m < 0, the ground state of the +chain is a set of finite AFM or FM spin clusters separated by impurities. This +state we call frustrated ferromagnetic (FR-FM) or frustrated antiferromagnetic +(FR-AFM) respectively. +The FR-FM and FR-AFM states separated by the +4 + +J +J +J +h +h +h +V +V +-V +-V +|h|=-V-J +|h|=-V-J +|h|=-2J +|h|=-2J +0 Ј m < 0.5 +-0.5 < m < 0 +m = -0.5 +(b) +(a) +(c) +AFM +AFM +AFM +FR-PM +FR-AFM +FR-FM +FM +FM +FM +0 +0 +0 +Figure 1: +Phase diagram at zero temperature for a dilute Ising chain in a longitudinal +magnetic field: (a) the Ising chain without impurities, n = 0, (b) the case of a weakly diluted +chain, (c) the case of a strongly diluted chain. +5 + +spin-flip line, |h| = −2J, as it is shown in Fig. 1(b). In the strongly diluted +case, 0 ≤ m < 1/2, there are single spins separated by impurity clusters, and +the system exhibits a paramagnetic response, which is uniform over the entire +range −V −|h| < J < V . This frustrated paramagnetic (FR-PM) state is shown +in Fig. 1(c). +3.2. The magnetic entropy change. +Temperature dependences of the entropy S and the magnetic entropy change, +∆S = S(h = 0)−S(h ̸= 0), are shown in Fig. 2 for the antiferromagnetic (AFM) +sign of the exchange constant, J < 0, and in Fig. 3 for the ferromagnetic (FM) +sign, J > 0. The correlation parameter for impurities V accepted and used as +a positive scaling factor. +Fig. 2 shows the temperature dependences of the entropy for J/V = −1, +h = 0 in panel (a), and for J/V = −1, h/V = 0.5 in panel (b). If at h = 0 the +entropy monotonically depends on |m| and has a maximum at m = 0, at h ̸= 0 +the dependence on |m| has a local minimum at m = 0. The magnetic entropy +change for J/V = −1 is shown in panel (c). The maximum ∆S for all m is +achieved at T = 0 and also has a minimum with ∆S < 0 at finite temperature +for small values of impurity concentrations. +It is worth noting that in the AFM chain, for any value of the applied +magnetic field, we get zero entropy at m = 0, because there is only one way to +minimize the energy: alternating spins and charges, and all spins are oriented +along the magnetic field. In a certain sense, in this case we get a kind of magneto- +electric effect: an external magnetic field causes a charge ordering. This also +gives us the maximum change in entropy at half-filling. +The temperature dependences of the entropy for J/V = −0, 5, h = 0 are +shown in Fig. 2(d), and for J/V = −0.5, h/V = 0.5 in Fig. 2(e). +The de- +pendences of S on |m| have a local minimum at m = 0 both at h = 0 and at +h/V = 0.5. The magnetic entropy change for J/V = −0.5 is shown in Fig. 2(f). +In a contrast to the previous case, the ∆S dependences show a maximum at +finite temperature for some m ≥ 0, and also show a minimum with ∆S < 0 at +finite temperature in some range for m < 0. +Fig. 3 shows the temperature dependences of the entropy for J/V = 1, h = 0 +in panel (a), and for J/V = 1, h/V = 0.5 in panel (b). The magnetic entropy +change for J/V = 1 is shown in panel (c). Both at h = 0 and h ̸= 0 the entropy +monotonically depends on |m| and has a maximum at m = 0. The magnetic +entropy change has a maximum at finite temperature for some m < 0, and near +the m = 0 it also has a local minimum at a finite temperature. In the case +of FM, we will not get the same effect at h > 0 as for J/V = −1, because it +makes no sense to split the spin clusters into more than one spin in order to +minimize the energy. But the entropy is still slightly reduced, because there is +no ordering chaos for different the spin clusters: they will all be oriented by a +magnetic field. +The temperature dependences of the entropy for J/V = 0.5, h = 0 are +shown Fig. 3 in panel (d), and for J/V = 0.5, h/V = 0.5 in panel (e). The +6 + +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +S +S +S +S +T/V +T/V +T/V +T/V +T/V +T/V +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +DS +DS +(a) +(b) +(c) +(d) +(e) +(f) +0.5 +0.4 +0.3 +0.2 +0.1 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0. +0. +-0.4 +-0.5 +-0.3 +0.5 +0.4 +0.3 +0.2 +0.1 +-0.1 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +-0.5 +-0.4 +0. +0. +-0.3 +-0.2 +-0.1 +-0.2 +-0.5 +-0.1 +-0.2 +-0.3 +-0.4 +0. +-0.5 +0. +-0.4 +-0.3 +-0.2 +-0.1 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.1 +-0.2 +-0.3 +-0.4 +-0.5 +0.1 +0.2 +0.3 +0.4 0.5 +Figure 2: (color online) Temperature dependences of the entropy S and the magnetic entropy +change ∆S in the AFM case (J < 0). Panels (a), (b), and (c) correspond to J/V = −1; +(d), (e), (f) – to J/V = −0.5. Panels (a) and (d) show the entropy S at h = 0, (b) and (e) +– at h/V = 0.5, (c) and (f) – the magnetic entropy change ∆S = S(h = 0) − S(h = 0.5). +The numbers near lines correspond to the deviation of the impurity concentration n from +half-filling, m = n − 1/2. Solid (dashed) lines correspond to m ≤ 0 (m > 0). +7 + +S +S +S +S +T/V +T/V +T/V +T/V +T/V +T/V +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.1 +0.2 +0.3 +DS +DS +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 (d) +(a) +(b) +(c) +(e) +(f) +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +-0.1 +0. +0. +-0.5 +-0.5 +-0.4 +-0.4 +-0.3 +-0.3 +-0.2 +-0.2 +-0.1 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +-0.5 +-0.5 +-0.4 +-0.4 +-0.3 +-0.3 +0. +0. +0.2 +0.2 +0.1 +0.1 +-0.1 +-0.1 +-0.2 +0.5 +-0.5 +0.4 +-0.4 +-0.3 +-0.2 +-0.1 +0. +0.3 +0.2 +0.1 +-0.5 +-0.4 +-0.3 +0. +-0.2 +-0.1 +0.5 +0.4 +0.3 +0.2 +0.1 +-0.2 +Figure 3: (color online) Temperature dependences of the entropy S and the magnetic entropy +change ∆S in the FM case (J > 0). Panels (a), (b), and (c) correspond to J/V = 1; (d), +(e), (f) – to J/V = 0.5. +Panels (a) and (d) show the entropy S at h = 0, (b) and (e) – +at h/V = 0.5, (c) and (f) – the magnetic entropy change ∆S = S(h = 0) − S(h = 0.5). +The numbers near lines correspond to the deviation of the impurity concentration n from +half-filling, m = n − 1/2. Solid (dashed) lines correspond to m ≤ 0 (m > 0). +magnetic entropy change for J/V = 0.5 is shown in panel (f). Qualitatively, +the behavior of entropy differs from J/V = 1 case at some region near |m| = 0, +where the tendency to the charge ordering causes the decreasing of S. The ∆S +dependences also show local maxima at finite temperature in some range for +m < 0, and a monotonic behavior with maximal value at T = 0 for m ≥ 0. The +magnetic entropy change for FM case is always positive. +The concentration dependences of entropy at T/V = 0.05 shown in Fig. 4 +allow estimating approximately the features of the residual entropy S0. The +dependences of S0 on m have the following form [20]: +S0 = ln +�1 +2 + g0 +� ++ 1 +2 ln +2 +1 +4 − m2 − m ln g0 + m +g0 − m, +|J|/V = 1, +(13) +S0 = 1 +2 ln +1 +2 + |m| +1 +2 − |m| + |m| ln +1 +4 − m2 +8m2 ++ 1 +2 ln 2, +|J|/V < 1, +(14) +where +g0 = +1 +√ +2 +�1 +4 + m2 +�1/2 +. +(15) +These expressions depend only on |m| and are identical for the AFM and FM +cases. The curves of S(h = 0) in Fig. 4(a) and (c), and in Fig. 4(b) and (d) +8 + +(d) +S +0.8 +0.0 +0.2 +0.4 +0.6 +S +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +m +m +m +m +S +0.5 +0.5 +0.5 +0.5 +0.8 +0.0 +0.2 +0.4 +0.6 +0.25 +0.25 +0.25 +0.25 +-0.25 +-0.25 +-0.25 +-0.25 +-0.5 +-0.5 +-0.5 +-0.5 +S +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +(a) +(c) +(b) +DS +DS +DS +DS +S(h=0) +S(h=0) +S(h=0) +S(h=0) +S(h№0) +S(h№0) +S(h№0) +S(h№0) +0.0 +0.0 +0.0 +0.0 +Figure 4: (color online) The dependence of the entropy S (solid lines) and the magnetic entropy +change ∆S (dashed lines) on the deviation of the impurity concentration n from half-filling, +m = n − 1/2, at T/V = 0.05. Panel (a) corresponds to J/V = −1, (b) – to J/V = −0.5, (c) +– to J/V = 1, and (d) – to J/V = 0.5. +confirm this property. For h ̸= 0 the concentration dependences of the residual +entropy become asymmetric with respect to m = 0 for AFM case, but save +the symmetry in FM case. The same dependence for the AFM and FM cases +holds only at m > 0 for |J| < V , when the ground state consists of single spins +separated by nonmagnetic impurities, and the sign of the exchange constant has +no effect. +3.3. The isentropic dependence of the temperature on the magnetic field. +Fig. 5 shows the isentropic lines in the h − T parameter plane for the fer- +romagnetic sign of exchange constant, J > 0. +For the Ising chain without +impurities, the isentropes slope near the critical field hc = 0 is almost vertical +that leads to extremely high and narrow peak of the Gr¨uneisen parameter [23], +which is proportional to e2J/T at h ∝ T e−2J/T . The impurities change this +9 + +h/V +T/ V +T/ V +T/ V +0.01 +0.1 +0.3 +0.55 +0.65 +0.75 +0.85 +0.95 +0.5 +0.5 +0.55 +0.55 +0.6 +0.6 +0.65 +0.7 +-3 +10 +-6 +10 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.1 +0.3 +0.5 +0.7 +0.1 +0.3 +0.5 +0.7 +0.1 +0.3 +0.5 +0.7 +n=0 +n=0.25 +n=0.75 +0.1 +0.135 +0.15 +0.2 +0.25 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.2 +0.5 +0.65 +0.75 +0.85 +h/V +0.1 +0.3 +0.5 +0.7 +T/ V +0.1 +0.3 +0.5 +0.7 +T/ V +0.1 +0.3 +0.5 +0.7 +T/ V +-5 +10 +0.01 +0.05 +-8 +10 +-3 +10 +n=0 +n=0.03 +n=0.5 +(a) +(b) +Figure 5: +The isentropic lines in the h − T parameter plane (a) for J/V = 1.3 and (b) for +J/V = 0.5. The value of the impurity concentration n is given in the frame. The numbers +next to the lines show the entropy values. +picture drastically: the entropy value increases by several orders of magnitude, +and the isentropes slope near hc = 0 remains finite. +The magnetic Gr¨uneisen parameter can be rewritten [24] in the scaling form +as +Γmag = −Gr +1 +h − hc +, +(16) +where −Gr is a prefactor, and hc is a critical magnetic field. Fig. 6 shows the +value −Gr for hc = 0 as a function of T and h for J/V = 1.3, n = 0.03 in +panel (a), and for J/V = 0.5, n = 0.25 in panel (b). As can be seen, in both +cases, impurities lead to suppression of the singular behavior of the magnetic +Gr¨uneisen parameter which is observed for the Ising chain without impurities. +At low temperatures, in the FR-FM state, the system behaves like an ideal +paramagnet near h = 0 with a prefactor value −Gr = 1 [24]. +Fig. 7 shows the isentropic lines in the h − T parameter plane for the anti- +ferromagnetic sign of exchange constant, J < 0. The case of a moderate value +of the exchange constant, J/V = −1.3, is given in panel (a). In the absence of +impurities, there is practically no dependence of entropy on the magnetic field. +Impurities lead to an increase in the entropy of the system by several orders +of magnitude and the appearance of two critical values of the magnetic field, +10 + +(a) +(b) +Figure 6: (color online) The prefactor of the Gr¨uneisen parameter −Gr (a) for J/V = 1.3, +n = 0.03, (b) for J/V = 0.5, n = 0.25. +|hc| = −J − V , which correspond to the transition lines from AFM to FR-AFM +or FR-PM states. It is worth to note, that for comparable values of |J| and +V , the critical field |hc| = −J − V can be much smaller than the spin-flip field +|hc| = −2J. The case of a small exchange constant, J/V = −0.15, is given +in panel (b). Without impurities, the system has two critical spin-flip fields, +|hc| = −2J. Impurities lead to the appearance of a critical field hc = 0. As +a result, there are three critical fields for a weakly diluted case, and only one +critical field hc = 0 for a strongly diluted case. +Fig. 8 shows the prefactor of the magnetic Gr¨uneisen parameter as a function +of T and h near the corresponding critical fields: for J/V = −1.3, n = 0.25, +hc/V = −1 − J/V = 0.3 in panel (a), for J/V = −0.15, n = 0.25, hc = 0 in +panel (b), and for J/V = −0.15, n = 0.25, hc/V = −2J/V = 0.3 in panel (c). +In all cases, the prefactor tends to −Gr = 1 at sufficiently low temperatures. +4. Conclusion +We examined the effects of the magnetic field on the frustrated phase states +of the dilute Ising chain. +The temperature dependences of entropy and the +magnetic entropy change show the nonequivalence of frustrated phases in AFM +and FM cases. The largest effect is achieved when |J|/V = 1. In the AFM case, +J/V = −1, the nonzero magnetic field causes a charge ordering for nonmagnetic +impurities and leads to the maximal value of the magnetic entropy change at a +half-filling. In the FM case, J/V = 1, the magnetic field reduces the frustration +of the ground state only partially. Impurities radically change the magnetic +Gr¨uneisen parameter in comparison with the case of a pure Ising chain. They +suppress the singular behavior of Γmag near h = 0 in the FM case and produce +the paramagnetic behavior in the FR-FM case. In the AFM case, additional +values of the critical magnetic field for Γmag appear, which are associated with +the transition line from AFM to frustrated ground state. In the FR-AFM state, +Γmag exhibits paramagnetic behavior at h = 0. +11 + +0.6 +0.4 +0.8 +0.2 +0.6 +0.0 +-0.5 +0.4 T/ V +0.0 +0.2 +h/V +0.51.0 +0.8 +0.5 +0.6 +0.0 +-0.5 +0.4 +T/V +0.0 +0.2 +h/ V +0.5h/V +T/ V +T/ V +T/ V +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.1 +0.1 +0.1 +0.2 +0.2 +0.2 +0.3 +0.3 +0.3 +0.4 +0.4 +0.4 +0.5 +0.1 +0.2 +0.3 +0.1 +0.2 +0.3 +0.1 +0.2 +0.3 +0.4 +0.4 +0.5 +0.5 +0.5 +0.5 +0.6 +0.6 +0.6 +0.6 +0.7 +0.8 +0.48 +0.48 +0.5 +0.5 +0.55 +0.55 +0.6 +0.6 +0.65 +n=0 +n=0.25 +n=0.75 +h/V +T/ V +T/ V +T/ V +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.05 +0.05 +0.05 +0.1 +0.1 +0.1 +0.15 +0.15 +0.15 +0.2 +0.2 +0.2 +0.1 +0.3 +0.4 +0.4 +0.4 +0.5 +0.6 +0.1 +0.25 +0.4 +0.5 +0.5 +0.5 +0.6 +-13 +10 +-10 +10 +-8 +10 +-6 +10 +-5 +10 +n=0 +n=0.25 +n=0.75 +(a) +(b) +Figure 7: +The isentropic lines in the h − T parameter plane (a) for J/V = −1.3 and (b) for +J/V = −0.15. The value of the impurity concentration n is given in the frame. The numbers +next to the lines show the entropy values. +(a) +(b) +(c) +Figure 8: (color online) The prefactor of the Gr¨uneisen parameter −Gr near the critical field +(a) for J/V = −1.3, n = 0.25, hc/V = 0.3, (b) for J/V = −0.15, n = 0.25, hc = 0, (c) for +J/V = −0.15, n = 0.25, hc = 0.3. +12 + +1.0 +0.5 +0.20 +0.0 +0.15 +0.0 +T/V +0.10 +0.2 +h/ V +0.4 +0.051.0 +0.3 +0.5 +0.2 +0.0 +T/V +-0.1 +0.1 +0.0 +h/ V +0.11.0 +0.5 +0.3 +0.0 +0.2 +T/V +0.2 +0.1 +0.3 +h/ V +0.4 +0.5Acknowledgments +This work was supported by the Ministry of Education and Science of the +Russian Federation, project FEUZ-2020-0054. +References +[1] L. ˇCanov´a, J. Streˇcka, M. Jaˇsˇcur, Geometric frustration in the class of ex- +actly solvable Ising–Heisenberg diamond chains, Journal of Physics: Con- +densed Matter 18 (20) (2006) 4967–4984. doi:10.1088/0953-8984/18/ +20/020. +[2] J. Torrico, M. Rojas, S. M. de Souza, O. Rojas, N. S. 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Lang, Cooling through +quantum criticality and many-body effects in condensed matter and cold +gases, International Journal of Modern Physics B 28 (26) (2014) 1–35. +doi:10.1142/S0217979214300175. +15 + diff --git a/BtFKT4oBgHgl3EQfXS78/content/tmp_files/load_file.txt b/BtFKT4oBgHgl3EQfXS78/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c6d38c6db0d4695ba9c7a55cd45678e4cdce3ca --- /dev/null +++ b/BtFKT4oBgHgl3EQfXS78/content/tmp_files/load_file.txt @@ -0,0 +1,928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf,len=927 +page_content='Thermodynamic features of the 1D dilute Ising model in the external magnetic field A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Shadrina,∗, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Panova aInstitute of Natural Sciences and Mathematics, Ural Federal University, 620002, 19 Mira street, Ekaterinburg, Russia Abstract We consider the effects of the magnetic field on the frustrated phase states of the dilute Ising chain, especially, the behavior of the magnetic entropy change and the isentropic dependence of the temperature on the magnetic field, which are the key parameters of the magnetocaloric effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The found temperature dependences of entropy demonstrate the nonequivalence of frustrated phases in the antiferromagnetic and ferromagnetic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the antiferromagnetic case, the nonzero magnetic field at certain parameters causes a charge ordering for nonmagnetic impurities at a half-filling, while in the ferromagnetic case, the magnetic field reduces the frustration of the ground state only partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' It is also shown, that impurities radically change the magnetic Gr¨uneisen parameter in comparison with the case of a pure Ising chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Keywords: dilute Ising chain, frustrated magnets, magnetic entropy change 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Introduction One of the remarkable features of low-dimensional systems, such as deco- rated Ising models [1–7], the anisotropic Potts chain [8], the diamond Hubbard chain [9], is the presence, under certain parameters, of a frustrated ground state for which the residual entropy is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Despite the absence of a real phase transition at finite temperatures according to the Perron–Frobenius theorem for square real matrices [10] the thermodynamic behavior near the boundaries be- tween different phases of the ground state for these systems can exhibit striking features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' As shown in [11], if one of the phases has a nonzero residual entropy that preserves continuity at the boundary with the other phase, then the ther- modynamic characteristics of the system will demonstrate pseudo-transitions at a finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Entropy, heat capacity, magnetization, and susceptibility have similar features to the behavior of these properties at conventional phase transitions, including the presence of quasicritical exponents [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' ∗Corresponding author Email address: shadrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='anton@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='ru (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Shadrin) Preprint submitted to Journal of Magnetism and Magnetic Materials January 30, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='11794v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='stat-mech] 27 Jan 2023 Recently, frustrated magnetic systems have also attracted the attention of researchers due to the enhanced magnetocaloric effect in the vicinity of finite- field transitions [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Besides to geometric factors, the impurities are also the reason for the existence of frustrations in the magnetic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The simplest example of a magnetic system that is frustrated by impurities is a diluted Ising chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The Hamiltonian of 1D diluted Ising model can be written in the following form H = −J � i Sz,iSz,i+1 + V � i P0,iP0,i+1 − h � i Sz,i − µ � i P0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (1) Here we use the S = 1 pseudospin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The states for a given lattice site with the pseudospin projections Sz = ±1 correspond to the two magnetic states with the conventional spin projections sz = ±1/2, while the state with Sz = 0 corresponds to the charged nonmagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Sz,i is a z-projection of the on-site pseudospin operator, P0,i = 1 − S2 z,i is the projection operator onto the Sz = 0 state, J is the exchange constant, V > 0 is the inter-site correlation parameter for impurities, h is an external magnetic field, and µ is a chemical potential for impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Further we will assume that nonmagnetic impurities are mobile, which corresponds to the annealed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' As well known [15], V = V0 + V1 − 2V01 describes the interaction for a more general case: V0 � i P0,iP0,i+1 + V1 � i P1,iP1,i+1 + V01 � i � P0,iP1,i+1 + P1,iP0,i+1 � , (2) where P1 = S2 z, is the projection operator onto magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The solutions and various thermodynamic properties of the 1D dilute Ising model at zero external magnetic field was found in [16–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' If h ̸= 0, then there are no explicit analytical expressions for various thermodynamic functions of the model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' It is known the account of magnetic field for the S = 1 Ising chain significantly expands the list of possible phase states of the system and leads to various features of thermodynamic behavior [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the present paper, we consider the effects of the magnetic field on the frustrated phase states of the model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' We focused on the behavior of entropy and, in particular, on the magnetic entropy change, which is the key parameter of the magnetocaloric effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Also, we explore the isentropic dependence of the temperature on the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' We briefly describe the methods in section 2, and section 3 the results, including the ground state phase diagram, and their discussions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Conclusions are presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Methods We define the transfer matrix for the model (1) as τ = � � xz z1/2 t1/2 x−1 z1/2 t1/2 y−1t z−1/2 t1/2 x−1 z−1/2 t1/2 xz−1 � � , (3) 2 where x = eβJ, y = eβV , z = eβh, t = eβµ and β = 1/T, and we assume kB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' From (3), we found the characteristic equation for the eigenvalues λi: λ3 − λ2 � ty−1 + x(z + z−1) � − λ � x2 − x−2 + t � xy−1 − 1 � (z + z−1) � − 2t(x − x−1) − tx−2y−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (4) The eigenvalues in a general case are cumbersome functions, but at h = 0 they could be reduced to the known expressions [20]: λ1,2 = 1 2 � x + x−1 + y−1t � ± � 2t + 1 4 � x + x−1 − y−1t �2�1/2 , λ3 = x − x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (5) According to the Perron–Frobenius theorem [10], there is only one maximum eigenvalue, λ1, and in the thermodynamic limit we obtain the grand potential and the entropy in the following form: Ω = Nω = −NT ln λ1, S = − � ∂ω ∂T � h,µ = ln λ1 + T λ1 �∂λ1 ∂T � h,µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (6) The grand potential and entropy found depend on parameters J, V , h, µ, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' But in the present problem, it is more convenient to use the concentration n of impurities as an external parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The dependence n(µ) can be obtained from the equation n = − �∂ω ∂µ � T,h = T λ1 �∂λ1 ∂µ � T,h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (7) In a general case h ̸= 0, we used numerical methods to get the inverse dependence µ(n) and fix the concentration of impurities n at all temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' If h = 0, we obtain the explicit expressions [20]: µ = ln � y � x + x−1� g + m g − m � , (8) S = 1 2 ln 2 (1 + 2g)2 1 − 4m2 + g + 2m2 1 + 2g ln y − m ln �� x + x−1� g + m g − m � − (1 − 2m) (g − m) � x − x−1� (1 + 2g) (x + x−1) ln x, (9) where g = � m2 + 1 2 �1 4 − m2 � y−1 � x + x−1��1/2 , (10) and we introduced the deviation of the concentration of impurities from half- filling, m = n − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 3 The knowledge of the entropy from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (6,7) gives an opportunity to explore magnetocaloric properties of the dilute Ising chain for a given n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' We explore the magnetic entropy change, the isentropic dependencies of the temperature on the magnetic field and the magnetic Gr¨uneisen parameter, which can be calculated from the relation Γmag = 1 T �∂T ∂h � S,n = − 1 T (∂S/∂h)T,n (∂S/∂T)h,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (11) The explicit expression that we use to calculate Γmag for a given n in variables (T, h, µ) has the following form: Γmag = − 1 T � (∂S/∂h)T,µ (∂n/∂µ)T,h − (∂S/∂µ)T,h (∂n/∂h)T,µ (∂S/∂T)h,µ (∂n/∂µ)T,h − (∂S/∂µ)T,h (∂n/∂T)h,µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Phase diagram at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The phase diagram of the dilute Ising chain in longitudinal magnetic field at zero temperature is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 1 for the J − h plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The limiting case for the Ising chain without impurities, m = −1/2, is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Two ground states, the ferromagnetic (FM) state with magnetic moment oriented towards the field, and the antiferromagnetic (AFM) state with zero magnetic moment, are separated by the critical value of magnetic field |hc| = −2J at which the spin- flip transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Figures 1(b) and 1(c) show the cases of a weakly diluted chain, −1/2 < m < 0, and a strongly diluted chain, 0 ≤ m < 1/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Dilution with impurities leads to the appearance of two new boundary lines on the J − h plane, J = V and |h| = −J − V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' If J > V , the ground state is represented by macroscopic FM domains (or drops) separated by macroscopic impurity domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Similarly, the AFM domains arise when J < −V − |h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Schematically, this is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 1(b,c), where the arrows correspond to the spins, and the circles correspond to the impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' For both FM and AFM phases, the entropy is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Analysis of the ground state of the model (1) in zero magnetic field shows [19, 20] that phases with the nonzero residual entropy exist at |J| < V and at |J| = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' For the weak exchange, when |J| < V , the spin correlation length is always finite, but the impurity correlation length with the temperature lowering tends to infinity at the half-filling concentration, m = 0, due to the formation of charge ordering [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' If |J| = V and h = 0, the spin correlation length and the impurity correlation length are finite for all values of the impurity concentration and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In magnetic field, if −V − |h| < J < V , the residual entropy is also not zero, so the ground state is frustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' If −1/2 < m < 0, the ground state of the chain is a set of finite AFM or FM spin clusters separated by impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' This state we call frustrated ferromagnetic (FR-FM) or frustrated antiferromagnetic (FR-AFM) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The FR-FM and FR-AFM states separated by the 4 J J J h h h V V V V |h|=-V-J |h|=-V-J |h|=-2J |h|=-2J 0 Ј m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 < m < 0 m = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 (b) (a) (c) AFM AFM AFM FR-PM FR-AFM FR-FM FM FM FM 0 0 0 Figure 1: Phase diagram at zero temperature for a dilute Ising chain in a longitudinal magnetic field: (a) the Ising chain without impurities, n = 0, (b) the case of a weakly diluted chain, (c) the case of a strongly diluted chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 5 spin-flip line, |h| = −2J, as it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the strongly diluted case, 0 ≤ m < 1/2, there are single spins separated by impurity clusters, and the system exhibits a paramagnetic response, which is uniform over the entire range −V −|h| < J < V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' This frustrated paramagnetic (FR-PM) state is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The magnetic entropy change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Temperature dependences of the entropy S and the magnetic entropy change, ∆S = S(h = 0)−S(h ̸= 0), are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 2 for the antiferromagnetic (AFM) sign of the exchange constant, J < 0, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 3 for the ferromagnetic (FM) sign, J > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The correlation parameter for impurities V accepted and used as a positive scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 2 shows the temperature dependences of the entropy for J/V = −1, h = 0 in panel (a), and for J/V = −1, h/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' If at h = 0 the entropy monotonically depends on |m| and has a maximum at m = 0, at h ̸= 0 the dependence on |m| has a local minimum at m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The magnetic entropy change for J/V = −1 is shown in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The maximum ∆S for all m is achieved at T = 0 and also has a minimum with ∆S < 0 at finite temperature for small values of impurity concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' It is worth noting that in the AFM chain, for any value of the applied magnetic field, we get zero entropy at m = 0, because there is only one way to minimize the energy: alternating spins and charges, and all spins are oriented along the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In a certain sense, in this case we get a kind of magneto- electric effect: an external magnetic field causes a charge ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' This also gives us the maximum change in entropy at half-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The temperature dependences of the entropy for J/V = −0, 5, h = 0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 2(d), and for J/V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, h/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The de- pendences of S on |m| have a local minimum at m = 0 both at h = 0 and at h/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The magnetic entropy change for J/V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In a contrast to the previous case, the ∆S dependences show a maximum at finite temperature for some m ≥ 0, and also show a minimum with ∆S < 0 at finite temperature in some range for m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 3 shows the temperature dependences of the entropy for J/V = 1, h = 0 in panel (a), and for J/V = 1, h/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The magnetic entropy change for J/V = 1 is shown in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Both at h = 0 and h ̸= 0 the entropy monotonically depends on |m| and has a maximum at m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The magnetic entropy change has a maximum at finite temperature for some m < 0, and near the m = 0 it also has a local minimum at a finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the case of FM, we will not get the same effect at h > 0 as for J/V = −1, because it makes no sense to split the spin clusters into more than one spin in order to minimize the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' But the entropy is still slightly reduced, because there is no ordering chaos for different the spin clusters: they will all be oriented by a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The temperature dependences of the entropy for J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, h = 0 are shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 3 in panel (d), and for J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, h/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 S S S S T/V T/V T/V T/V T/V T/V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 Figure 2: (color online) Temperature dependences of the entropy S and the magnetic entropy change ∆S in the AFM case (J < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Panels (a), (b), and (c) correspond to J/V = −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (d), (e), (f) – to J/V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Panels (a) and (d) show the entropy S at h = 0, (b) and (e) – at h/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, (c) and (f) – the magnetic entropy change ∆S = S(h = 0) − S(h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The numbers near lines correspond to the deviation of the impurity concentration n from half-filling, m = n − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Solid (dashed) lines correspond to m ≤ 0 (m > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 7 S S S S T/V T/V T/V T/V 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 Figure 3: (color online) Temperature dependences of the entropy S and the magnetic entropy change ∆S in the FM case (J > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Panels (a), (b), and (c) correspond to J/V = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (d), (e), (f) – to J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Panels (a) and (d) show the entropy S at h = 0, (b) and (e) – at h/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, (c) and (f) – the magnetic entropy change ∆S = S(h = 0) − S(h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The numbers near lines correspond to the deviation of the impurity concentration n from half-filling, m = n − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Solid (dashed) lines correspond to m ≤ 0 (m > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' magnetic entropy change for J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 is shown in panel (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Qualitatively, the behavior of entropy differs from J/V = 1 case at some region near |m| = 0, where the tendency to the charge ordering causes the decreasing of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The ∆S dependences also show local maxima at finite temperature in some range for m < 0, and a monotonic behavior with maximal value at T = 0 for m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The magnetic entropy change for FM case is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The concentration dependences of entropy at T/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='05 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 4 allow estimating approximately the features of the residual entropy S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The dependences of S0 on m have the following form [20]: S0 = ln �1 2 + g0 � + 1 2 ln 2 1 4 − m2 − m ln g0 + m g0 − m, |J|/V = 1, (13) S0 = 1 2 ln 1 2 + |m| 1 2 − |m| + |m| ln 1 4 − m2 8m2 + 1 2 ln 2, |J|/V < 1, (14) where g0 = 1 √ 2 �1 4 + m2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' (15) These expressions depend only on |m| and are identical for the AFM and FM cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The curves of S(h = 0) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 4(a) and (c), and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 4(b) and (d) 8 (d) S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='6 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='7 m m m m S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='7 (a) (c) (b) DS DS DS DS S(h=0) S(h=0) S(h=0) S(h=0) S(h№0) S(h№0) S(h№0) S(h№0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 Figure 4: (color online) The dependence of the entropy S (solid lines) and the magnetic entropy change ∆S (dashed lines) on the deviation of the impurity concentration n from half-filling, m = n − 1/2, at T/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Panel (a) corresponds to J/V = −1, (b) – to J/V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, (c) – to J/V = 1, and (d) – to J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' confirm this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' For h ̸= 0 the concentration dependences of the residual entropy become asymmetric with respect to m = 0 for AFM case, but save the symmetry in FM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The same dependence for the AFM and FM cases holds only at m > 0 for |J| < V , when the ground state consists of single spins separated by nonmagnetic impurities, and the sign of the exchange constant has no effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The isentropic dependence of the temperature on the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 5 shows the isentropic lines in the h − T parameter plane for the fer- romagnetic sign of exchange constant, J > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' For the Ising chain without impurities, the isentropes slope near the critical field hc = 0 is almost vertical that leads to extremely high and narrow peak of the Gr¨uneisen parameter [23], which is proportional to e2J/T at h ∝ T e−2J/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The impurities change this 9 h/V T/ V T/ V T/ V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='85 h/V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='7 T/ V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='7 T/ V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='7 T/ V 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='05 8 10 3 10 n=0 n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='03 n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 (a) (b) Figure 5: The isentropic lines in the h − T parameter plane (a) for J/V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 and (b) for J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The value of the impurity concentration n is given in the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The numbers next to the lines show the entropy values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' picture drastically: the entropy value increases by several orders of magnitude, and the isentropes slope near hc = 0 remains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The magnetic Gr¨uneisen parameter can be rewritten [24] in the scaling form as Γmag = −Gr 1 h − hc , (16) where −Gr is a prefactor, and hc is a critical magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 6 shows the value −Gr for hc = 0 as a function of T and h for J/V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='03 in panel (a), and for J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25 in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' As can be seen, in both cases, impurities lead to suppression of the singular behavior of the magnetic Gr¨uneisen parameter which is observed for the Ising chain without impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' At low temperatures, in the FR-FM state, the system behaves like an ideal paramagnet near h = 0 with a prefactor value −Gr = 1 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 7 shows the isentropic lines in the h − T parameter plane for the anti- ferromagnetic sign of exchange constant, J < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The case of a moderate value of the exchange constant, J/V = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3, is given in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the absence of impurities, there is practically no dependence of entropy on the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Impurities lead to an increase in the entropy of the system by several orders of magnitude and the appearance of two critical values of the magnetic field, 10 (a) (b) Figure 6: (color online) The prefactor of the Gr¨uneisen parameter −Gr (a) for J/V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='03, (b) for J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' |hc| = −J − V , which correspond to the transition lines from AFM to FR-AFM or FR-PM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' It is worth to note, that for comparable values of |J| and V , the critical field |hc| = −J − V can be much smaller than the spin-flip field |hc| = −2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The case of a small exchange constant, J/V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='15, is given in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Without impurities, the system has two critical spin-flip fields, |hc| = −2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Impurities lead to the appearance of a critical field hc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' As a result, there are three critical fields for a weakly diluted case, and only one critical field hc = 0 for a strongly diluted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 8 shows the prefactor of the magnetic Gr¨uneisen parameter as a function of T and h near the corresponding critical fields: for J/V = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25, hc/V = −1 − J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 in panel (a), for J/V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='15, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25, hc = 0 in panel (b), and for J/V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='15, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='25, hc/V = −2J/V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='3 in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In all cases, the prefactor tends to −Gr = 1 at sufficiently low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Conclusion We examined the effects of the magnetic field on the frustrated phase states of the dilute Ising chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The temperature dependences of entropy and the magnetic entropy change show the nonequivalence of frustrated phases in AFM and FM cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' The largest effect is achieved when |J|/V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the AFM case, J/V = −1, the nonzero magnetic field causes a charge ordering for nonmagnetic impurities and leads to the maximal value of the magnetic entropy change at a half-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the FM case, J/V = 1, the magnetic field reduces the frustration of the ground state only partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' Impurities radically change the magnetic Gr¨uneisen parameter in comparison with the case of a pure Ising chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' They suppress the singular behavior of Γmag near h = 0 in the FM case and produce the paramagnetic behavior in the FR-FM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the AFM case, additional values of the critical magnetic field for Γmag appear, which are associated with the transition line from AFM to frustrated ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' In the FR-AFM state, Γmag exhibits paramagnetic behavior at h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='4 T/ V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='2 h/V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf'} +page_content='5 0.' metadata={'source': 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--git a/CNFAT4oBgHgl3EQfsx5b/content/tmp_files/2301.08660v1.pdf.txt b/CNFAT4oBgHgl3EQfsx5b/content/tmp_files/2301.08660v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a6e545a8905c804f3036b17ced67a082f8b945b --- /dev/null +++ b/CNFAT4oBgHgl3EQfsx5b/content/tmp_files/2301.08660v1.pdf.txt @@ -0,0 +1,1189 @@ +1 + +A BIG-DATA DRIVEN FRAMEWORK TO ESTIMATING VEHICLE VOLUME BASED +ON MOBILE DEVICE LOCATION DATA + +Mofeng Yang1, Weiyu Luo2, Mohammad Ashoori3, Jina Mahmoudi4, Chenfeng Xiong5*, Jiawei +Lu6, Guangchen Zhao7, Saeed Saleh Namadi8, Songhua Hu9 and Aliakbar Kabiri10 + + +1. Ph.D. (mofeng@umd.edu) +2. Graduate Research Assistant (wyl@umd.edu) +3. Graduate Research Assistant (mashoori@umd.edu) +4. Ph.D., P.E., Research Scientist (zhina@umd.edu) +5. Assistant Professor (Chenfeng.Xiong@Villanova.edu), *Corresponding Author +6. Graduate Research Assistant (jiaweil9@asu.edu) +7. Graduate Research Assistant (gczhao@umd.edu) +8. Graduate Research Assistant (saeed@umd.edu) +9. Graduate Research Assistant (hsonghua@umd.edu) +10. Graduate Research Assistant (kabiri@umd.edu) + + +1-4, 7-10: Maryland Transportation Institute (MTI), Department of Civil and Environmental +Engineering, 1173 Glenn Martin Hall, University of Maryland, College Park MD 20742, USA. +5: Department of Civil and Environmental Engineering, College of Engineering, Villanova +University, Villanova, PA 19085, USA +6. School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, +AZ 85281, USA + + +Words Count: 4,623 + 2 Tables (250*2) = 5,123 + +Submission Date: 07/31/2022 + + + + +2 + +ABSTRACT + +Vehicle volume serves as a critical metric and the fundamental basis for traffic signal control, +transportation project prioritization, road maintenance plans and more. Traditional methods of +quantifying vehicle volume rely on manual counting, video cameras, and loop detectors at a limited +number of locations. These efforts require significant labor and cost for expansions. Researchers +and private sector companies have also explored alternative solutions such as probe vehicle data, +while still suffering from a low penetration rate. In recent years, along with the technological +advancement in mobile sensors and mobile networks, Mobile Device Location Data (MDLD) have +been growing dramatically in terms of the spatiotemporal coverage of the population and its +mobility. This paper presents a big-data driven framework that can ingest terabytes of MDLD and +estimate vehicle volume at a larger geographical area with a larger sample size. The proposed +framework first employs a series of cloud-based computational algorithms to extract multimodal +trajectories and trip rosters. A scalable map matching and routing algorithm is then applied to snap +and route vehicle trajectories to the roadway network. The observed vehicle counts on each +roadway segment are weighted and calibrated against ground truth control totals, i.e., Annual +Vehicle-Miles of Travel (AVMT), and Annual Average Daily Traffic (AADT). The proposed +framework is implemented on the all-street network in the state of Maryland using MDLD for the +entire year of 2019. Results indicate that our proposed framework produces reliable vehicle +volume estimates and also demonstrate its transferability and the generalization ability. + +Keywords: mobile device location data; big data analytics; vehicle volume; cloud computing; map +matching and routing. + + + + + +3 + +1. INTRODUCTION + +Vehicle volume measures the amount of traffic traveling through a roadway segment given a +specific period of time. It serves as a critical metric and the fundamental basis for various +transportation applications including traffic signal control, transportation project prioritization and +road maintenance plan. Traditional methods to quantify vehicle volume rely on manual counting, +video cameras, and loop detectors at a limited number of locations, a practice that requires +significant human labor and a high cost for expansions (1-5). Researchers and private sector +companies have also explored alternative solutions such as probe vehicle data, while still suffering +from the low penetration rate issue (6-10). + +In the past two decades, along with the technological advancement in mobile sensors and mobile +networks, mobile device location data (MDLD) have been growing dramatically in terms of +coverage and size, with broader spatiotemporal coverage of the population and its mobility. A +series of research studies have demonstrated the usefulness of MDLD for enhancing the traditional +travel survey and have revealed its potential to substitute surveys (11, 12). At the same time, +obtaining travel statistics solely based on MDLD is also worth investigating to reduce human labor +and cost. However, MDLD do not include any ground truth information such as trip origins and +destinations, travel modes, and trip purposes, which requires computational algorithms to be +developed and validated against the existing travel surveys. More importantly, unlike travel +surveys which collect information from representative samples to obtain population-representative +statistics, MDLD contain all available mobile devices with uneven data quality. + +This study was conducted as part of the Vulnerable Road User Density Exposure Dashboard +project (https://mti.umd.edu/sdi) - an interactive dashboard that utilizes MDLD to provide data +and insights on multimodal volume and safety risk exposure of vulnerable road users (e.g., +pedestrians, bicycles) at intersections and roadway segments within Maryland. In this study, we +present a big-data driven framework that ingests terabytes of MDLD and estimates vehicle volume +for all roadway segments. First, a series of cloud-based computational algorithms are applied— +including but not limited to—a trip and tour identification algorithm to mine travel behavior +information and a travel mode imputation model that impute multimodal trajectories from MDLD. +A map matching and routing algorithm is then applied to snap and route vehicle trajectories to the +roadway network. The observed vehicle counts on each roadway segment are weighted to match +the Annual Vehicle Miles of Travel (AVMT) by county, urban/rural status, and functional classes. +Further, a random forest regression model is used to calibrate the weighted vehicle volume against +the Annual Average Daily Traffic (AADT) acquired from loop detectors. The proposed framework +is implemented on the all-street network in the state of Maryland using MDLD data for the entire +year of 2019. + +2. LITERATURE REVIEW + +2.1. Application of Mobile Device Location Data in Transportation Research + +The appearance of MDLD in the transportation industry started in the 1990s. Since the mid-1990s, +researchers began installing Global Positioning System (GPS) data loggers in vehicles to +supplement travel surveys (13-15). With high-frequency in-vehicle GPS data, this approach can + +4 + +significantly improve the accuracy of travel surveys by recording the exact origin and destination +as well as the departure and arrival times. However, only a small number of vehicles can be +sampled with this technique, a drawback limiting its capability. Similarly, the wearable GPS, +which was introduced in the early 2000s, allowed respondents to report non-vehicle travel modes +while still suffering from small sample size issues (16, 17). In the past decade, private sector +entities such as INRIX and RITIS also started to incorporate the probe vehicle data into their +commercial products (18-21). Nonetheless, the low penetration rate (i.e., 2%-10%) of the +commercial probe vehicle data remains the core challenge with respect to drawing the whole +picture of travel patterns. + +As mentioned above, despite having high precision, traditional MDLD usually suffer from small- +sample-size issues, which significantly limits the usefulness of the data. Since mobile devices, +such as smartphones and tablets, have become more popular, MDLD generated from these devices +have a greater potential for being used in transportation applications. These new types of MDLD, +namely cellular data and Location-Based Service (LBS) data, offer a more extensive +spatiotemporal coverage and a larger sample size. The cellular data are generated through +communication between cellphones and cell towers (22) and can be further categorized into Call +Detail Record (CDR) and sightings (11). The CDR data can only capture the cell tower location, +whereas the sightings provide the exact latitude and longitude values. Both types of cellular data +have been widely applied to research topics such as travel behavior, human mobility, and social +networks in the past two decades (23-31). Despite the large volume of data, cellular data are limited +by their spatial and temporal resolution, which is determined by the density of cell towers and +users’ cellphone usage levels (32). On a positive note, however, cellular data require less advanced +phones and can raise fewer user privacy concerns. The LBS data provide the exact locations +generated when a mobile application updates the device’s location with the most accurate sources, +based on the existing location sensors such as Wi-Fi, Bluetooth, cellular tower, and GPS (11, 23- +25, 33, 34). Many applications have been developed using the LBS data. For instance, a recent +smartphone-enhanced travel survey conducted in the U.S. used a mobile application, rMove +developed by Resource Systems Group (RSG), to collect high-frequency location data and allow +the respondents to recall their trips by showing the trajectories in rMove (35-38). Additionally, +Airsage leveraged LBS data to develop a traffic platform that can estimate traffic flow, speed, +congestion and road user sociodemographic for every road and time of day (39). Further, the +Maryland Transportation Institute (MTI) at the University of Maryland (UMD) developed the +COVID-19 Impact Analysis Platform (https://data.covid.umd.edu) to provide insight on COVID- +19’s impact on mobility, health, economy, and society across the U.S. (40-43). + +2.2. Vehicle Volume Estimation Methods + +2.2.1. Estimating Vehicle Volume with Loop Detectors + +Loop detectors are widely used to record traffic volumes and occupancy levels. These sensors are +usually buried under the pavements to detect the induction change from the presence of a vehicle. +Kwon et al. 2003 developed an algorithm using data from single loop detectors to estimate truck +traffic volumes (1). The results showed a 5.7% error compared with the ground truth highway data. +Loop detector data were also applied together with probe vehicle data to estimate queue length (44) +and vehicle volume at a city-wide scale (45). Although proven to be efficient in estimating vehicle + +5 + +volume, the high installation and maintenance cost of loop detectors limit their capability of being +scaled up to cover the entire transportation network. Therefore, loop detector datasets are often +incomplete and mostly unavailable at minor arterials and local streets. + +2.2.2. Estimating Vehicle Volume with Probe Vehicle Data + +In the past two decades, MDLD have gained significant attention and have been utilized for +estimating various traffic characteristics including vehicle volume. With the development of +MDLD, estimating vehicle volume at the city scale became a reality. Probe vehicles can record +their trajectory data with high granularity (i.e., 1Hz). Based on the trajectory data obtained from +probe vehicles, a wide range of methods can be used by researchers to solve transportation +problems. Zhao et al. proposed novel methods to estimate queue length and vehicle volume based +on the probability theory without prior information about the penetration rate or queue length +distribution (6). Guo et al. estimated vehicle volume and queue length at signalized intersections +and proposed a new framework to optimize traffic signal control operations (7). Sekuła et al. +applied several machine learning and neural networks to estimate historical hourly vehicle volume +between sparsely located sensors based on the probe vehicle data (8). Shockwave theories were +also applied to probe vehicle data by a few studies (9, 10). + +2.2.3. Estimating Vehicle Volume with Mobile Device Location Data +Many studies have been conducted focusing on estimating traffic flow and detecting congestion +using cellular data (46, 47). Xing et al. 2019 utilized CDR with Time Difference of Arrival (TDOA) +positioning technique in order to estimate multimodal traffic volumes on different types of urban +roadways by identifying three modes of travel – namely, drive alone, carpooling, and bus (48). +The results showed that compared with the ground truth vehicle volume obtained from License +Plate Recognition (LPR) cameras, the mean relative error was in the range of 17.1% to 25.7%, +depending on the roadway type. Despite significant advances in positioning techniques, cellular +data still suffers from low accuracy issues, whereas LBS data have a noticeable advantage due to +utilizing different sources to accurately locate the user – a feature that has resulted in an increased +usage of this type of data by researchers and the private sector for estimating vehicle volume. Fan +et al. 2019 developed a computing framework alongside a heuristic map matching algorithm to +estimate Vehicle Miles of Travel (VMT) and AADT for the state of Maryland using INRIX data. +The results showed an R2 of 0.878 when fitting the estimated AADT with the ground truth AADT +(49). Moreover, a number of state agencies conducted rigorous evaluations of vehicle volume +obtained through traditional methods as well as from MDLD obtained by private sector companies. +They found the latter to be a promising source for supplementing current surveys and traditional +methods (50). +3. THE BIG-DATA DRIVEN VEHICLE VOLUME ESTIMATION FRAMEWORK + +3.1. Overview of the Framework + +In this study, we propose a big-data driven vehicle volume estimation framework, which offers the +capability of efficiently estimating vehicle volume ingested from terabytes of MDLD. Figure 1 +shows the proposed framework. The proposed framework is built on Amazon Web Services +(AWS). MDLD and all supporting data are stored in Simple Cloud Storage (S3). All algorithms + +6 + +are developed based on Apache Spark, which uses Resilient Distributed Datasets (RDD), and are +coded in PySpark using the Elastic MapReduce (EMR) services. In the cloud environment, MDLD +are spliced into RDDs given the number of executors (43, 49). At the same time, all external data +sources (i.e., K-D Tree, network, routing engine) are broadcasted into all executors for master and +core nodes. The same algorithms are applied to each RDD along with the broadcasted variables, +and the results are aggregated and outputted into S3. + + +Figure 1. The Big-Data Driven Vehicle Volume Estimation Framework + +3.2. Trip End Identification and Travel Mode Imputation + +Trip is the basic unit of analysis for almost all transportation applications. However, MDLD +usually do not contain any trip-related information. Therefore, in this study, a trip end +identification algorithm is used to extract trip-level information from the MDLD, including trip +start location, trip end location, departure time, and arrival time. Then, a travel mode imputation +model is further applied to infer four travel modes–namely, the air, drive, rail, and nonmotorized +modes based on heuristic rules and a random forest model. Detailed descriptions of the trip end +identification algorithm and the travel mode imputation model can be found in the following +references (12, 51). + +3.3. Map Matching and Routing + +To ensure flexibility and scalability of our map matching and routing method across the entire +U.S., we extract the drivable network from OpenStreetMap (OSM) using the latest open-source +Python package osm2gmns. The osm2gmns package can parse roadway network data from OSM +and output networks to csv files in the General Modeling Network Specification (GMNS) format. +It provides customized and practical functions to facilitate traffic modeling. Functions include +complex intersection consolidation, movement generation, traffic zone creation, short link + +Cloud Computing +aws +Data Source +Local Server Backup +MobileDevice +5 +DATA +Location Data +S3Online Bucket +Geospatial Maps +spark +Smart Location +AnnualAverage +Database +DailyTraffic +Amazon EMR +AmazonEC2 +DailyUpdate:1176.52to3401.80million +Annual VehicleMiles +OpenStreetMap +PySpark +points; 15.05 to 17.36 million devices. +ofTravel +Computation +Roadway Network +osm2gmns +WeightingandCalibration +ALGORITHM +Algorithms +1.Network parsing +RandomForestModel +Data Preprocessing +2. Missing value +3.ScalableacrossU.S. +8 +TripEnd Identification +County +networkx +Urban/Rural Status +Travel ModeImputation +1.Routing engine +#ofLanes,Speed Limit +MapMatchingandRouting +2. Short path algorithm +Weighting and Calibration +Built Environments +APPLICATION +VulnerableUserExposureRiskDashboard +Decision Support +Mobility Tracking +Safety Improvement7 + +combination, and network visualization. More details about osm2gmns can be found here: +https://osm2gmns.readthedocs.io/en/latest/ + +To match each location sighting to our OSM network, the OSM network is firstly parsed and +converted into the routable formats, where roadway segments are represented by links and nodes. +With the network topology, we use the networkX package to build a shortest path-based routing +engine. We then transform the latitude and longitude of the start node and end node for each link +to the plane coordinate (in meters), and then calculate link direction (degree) using the arctan value +between the two nodes (see Figure 3 for details). The travel direction between consecutive +sightings is also calculated. Similar to the method for link direction calculation, the coordinates of +each sighting are converted to plane coordinates, then the degree is calculated using the arctan +value between consecutive sightings. A spatial index structure, K-Dimensional Tree (K-D Tree), +is built using the link geometric nodes (i.e., link nodes). Then, for each sighting, we search all link +nodes that are within 100 meters. The 100-meter threshold is selected to balance the algorithm +efficacy and the computing speed. If we increase the value, more candidate links will be considered +but this will require more computing resources. If we decrease the value, we might not be able to +find a candidate link when the observation is sparse. To validate, we calculate the distance between +consecutive link nodes using the Maryland OSM network as an example. Results indicate that +more than 95% of the link nodes are within 100 meters of their neighbors, as shown in Figure 2. +Therefore, using the 100-meter value as the radius for searching candidate nodes is reasonable. + + +Figure 2. Distribution of Distance between Link Nodes in the OSM Network + +As the next step, for each sighting, we compare its travel direction to all candidate links. The +closest link with an absolute travel direction difference smaller than 30 degrees will be selected as +a valid matched link for the sighting. This 30-degree threshold is selected mainly to avoid the +sighting being matched to the link in the opposite direction. In common cases, the degree +difference between the travel direction and the link direction should be approximately 0. Here, we +use a 30-degree threshold to consider the uncertainty of location accuracy in MDLD. After the +matched link for each sighting is found, given the observed link sequence, the routing engine can +fill the gap between consecutively observed links and retrieve the complete route. Another layer +of reasonable checks is conducted at the routing stage. For each pair of consecutive sightings that + +Distribution of Distance between Link Nodes +40% +35% +30% +25% +20% +15% +10% +5% +0%8 + +are snapped to links, the routed distance is calculated by summing the link length of all the links +traveled between the two sightings. Two reasonableness checks are carried out: + +(1) If the routed distance is greater than the cumulative distance between the two sightings +snapped to links by 2,000 meters or more, we consider the route invalid. +(2) The travel time on these links will be calculated based on the timestamp difference between +the two sightings. With the routed distance and travel time, the average travel speed on +these links can be calculated. If the speed exceeds 50 m/s (i.e., 112 mph or 180 km/h), we +assume that one of the two sightings is matched to the wrong link. + +If either of these two violations is observed, we apply a trial-and-error process by removing the +latter sighting and performing the routing using the next sighting snapped to the network until it +does not violate the 2,000-meter threshold or the 50 m/s threshold (52). A simple example of the +map matching and routing method is illustrated in Figure 3. + + +Figure 3. Example of Map Matching and Routing. + +3.4. Weighting + +After map matching and routing, we collect routes for all vehicle trips and aggregate them by links +to obtain the observed vehicle volume for each link. Afterward, we develop a link-based weighting +method to match the AVMT in the region. We classify each link by county, urban/rural status, and +functional classes and calculate the link weight using the formula below: + +𝑤𝐶,𝑢,𝑓 = 𝐴𝑉𝑀𝑇𝐶,𝑢,𝑓 +∑ +𝑂𝐶,𝑢,𝑓,𝑖 +𝑁𝐶 + + +where 𝑤𝐶,𝑢,𝑓 represents the weight for links in county C, with urban/rural status of u, and with +functional class f; 𝐴𝑉𝑀𝑇𝑐,𝑢,𝑓 represent the AVMT; and 𝑂𝐶,𝑢,𝑓,𝑖 represents the observed vehicle +volume on link i; 𝑁𝐶 represents the total number of links in county C. For instance, if the study +area has 20 counties, 2 urban/rural status and 6 functional classes, then a total of 240 link-based + +>TravelDirection +o Link Centroid +Observation +Node +Matched Link +Degree +CandidateLink +Routed Link9 + +weights will be generated. Subsequently, the weighted vehicle volume for each link can be +calculated as: + +𝑉𝑐,𝑢,𝑓,𝑖 = 𝑤𝐶,𝑢,𝑓 × 𝑂𝑐,𝑢,𝑓,𝑖 + +where 𝑉𝑐,𝑢,𝑓,𝑖 represents the weighted vehicle volume on link i. + +3.5. Volume Calibration + +The weighted vehicle volume is further calibrated to match the ground truth AADT collected from +loop detectors at a limited number of locations. In this study, we use the random forest regression +to calibrate the weighted vehicle volume against the AADT to obtain the final vehicle volume. +During the calibration process, a 10-fold cross-validation (CV) process is used to fine-tune the +random forest regression hyperparameters with 90% training data. The fine-tuned models are then +applied to the 10% testing data. + +4. CASE STUDY: THE STATE OF MARYLAND + +4.1. Data + +4.1.1. Mobile Device Location Data and the Study Area + +This study used MDLD data obtained from Maryland Transportation Institute (MTI). MTI +integrated and cleaned the raw MDLD from multiple data vendors and built a national MDLD data +panel that consists of more than 270,000,000 Monthly Active Users (MAU) and represents +movements across the nation. (40-43, 51). Figure 4 shows the density of location sightings +covering locations within and outside of the boundaries of the state of Maryland. In this study, we +used all MDLD data that are observed in the state of Maryland for the entire year of 2019. The +MDLD is processed on a daily basis and the results are aggregated to produce an annual total result. + + +Figure 4. Mobile Device Location Data around the State of Maryland. + + +10 + +4.1.2. OpenStreetMap Network + +Using the osm2gmns package, we extracted a total of 634,516 drivable roadway segments within +the state of Maryland. Information about the number of lanes and speed limits was recorded for +only 111,835 roadway segments (17.6%) and 84,728 roadway segments (13.4%), respectively. As +shown on the left-hand side in Figure 5, the missing values for the number of lanes and speed +limits were estimated based on the corresponding values on nearby roadways in the same county, +and with the same urban/rural status, and road functional classes. These two variables are further +used as features in the vehicle volume calibration model. + + +Figure 5. Number of Lanes and Speed Limits in OSM + +4.1.3. Annual Vehicle Miles of Travel Data + +We use the vehicle miles traveled data from the Maryland Department of Transportation State +Highway Administration (MDOT SHA) as a control total number to weight observed vehicle +volume. Every year, MDOT SHA publishes an annual vehicle miles of travel (AVMT) report by +county and functional classification for the state, county, and municipal highway systems. This +AVMT report features the current FHWA Functional Classification Codes (1-7) and provides +additional classifications (i.e., Urban, Rural, Principal Arterial and Other Freeways and +Expressways, and Minor Collector). As discussed in the methodology section, the weights are +generated based on county, urban/rural status and functional classes. Here, 23 Maryland counties +plus Baltimore City, urban or rural, and two function classes (highway and non-highway) are +considered. We map the OSM link type to the FHWA Functional Classification Codes and +generated the highway and non-highway classes. More specifically, “motorway”, “trunk” and +“ramp” are classified as highway (i.e., 1, 2 in FHWA class), and the other types are classified as +non-highway (i.e., 3,4,5,6,7 in FHWA class). More details about the AVMT data can be found +here: https://www.roads.maryland.gov/mdotsha/Pages/index.aspx?PageId=302 + + +EstimatedNumberof +Lanes (lane) +1Lane +2Lanes +3or4Lanes +5or6Lanes +Morethan6Lanes +(a) +(b) +EstimatedSpeedLimits +(mph) +5-15 +15-25 +25-45 +45-60 +60-70 +(c) +(d)11 + +4.1.4. Annual Average Daily Traffic Data + +We use the AADT also from MDOT SHA to calibrate weighted vehicle volume against the ground +truth at a limited number of locations. The AADT data consists of linear and point geometric +features which represent the geographic locations and segments of roadway throughout the state +of Maryland that include traffic volume metrics such as AADT. More details about the AADT can +be found here:https://data.imap.maryland.gov/maps/77010abe7558425997b4fcdab02e2b64/about + +4.1.5. Smart Location Database and Features for Volume Calibration + +The Smart Location Database (SLD) is a nationwide geographic data resource for measuring +location efficiency. The SLD is produced by the U.S. Environmental Protection Agency (EPA)’s +Smart Growth Program. It provides more than 90 variables on land use and built environment +characteristics such as population and employment densities, land use diversity, urban design +attributes, destination accessibility, transit accessibility, and socioeconomic/sociodemographic +characteristics at the census block group level. Most attributes are available for every census block +group in the United States. In this study, we use SLD variables as features in the random forest +regression to calibrate weighted vehicle volume to account for the effects of the built environment. +The SLD variables used in this study include “TotEMP”, “Pct_AO0”, “D1A”, “D1C”, “D3AAO”, +“D3B”, and “D5AR”: +• TotEMP = total employment; +• Pct_AO0 = percent of zero-car households; +• D1A = gross residential density (housing units per acre) on unprotected land; +• D1C = gross employment density (jobs per acre) on unprotected land; +• D3AAO = network density in terms of facility miles of auto-oriented links per square +miles; +• D3B = street intersection density (weighted, auto-oriented intersections eliminated); +• D5AR = jobs within 45 minutes auto travel time, time decay (network travel time) +weighted +We also include urban/rural status, county code, link type, number of lanes, and speed limits as +features in the calibration process. + +4.2. Results + +4.2.1. Overall Comparison + +Figure 6 shows the weighting and calibration results for both training and testing sets. The blue +dots represent weighted volume comparisons and the green dots represent calibrated vehicle +volume comparisons with MDOT SHA AADT. Figure 6 (a) and (b) compares the weighted vehicle +volume and calibrated vehicle volume with the MDOT SHA AADT in the training set respectively; +Figure 6 (c) and (d) compares the weighted vehicle volume and calibrated vehicle volume with the +MDOT SHA AADT in the testing set respectively. As it can be seen from Figure 6 (a), for the +training set, the Pearson correlation value and the Root Mean Square Error (RMSE) between the +weighted vehicle volume and the ground truth AADT are 0.746 and 7,912, respectively. These +values are improved to 0.966 and 2,996 after calibration, as shown in Figure 6 (b). Similarly, for + +12 + +the testing set, the Pearson correlation and RMSE are improved from 0.764 and 7,548, to 0.854 +and 5,701 respectively after calibration. + + +Figure 6. (a) Weighted Vehicle Volume in Training Set; (b) Calibrated Vehicle Volume in Training Set; +(c) Weighted Vehicle Volume in Testing Set; (d) Calibrated Vehicle Volume in Testing Set. + +4.2.2. Vehicle Volume Validation by Link Types and Urban/Rural Status + +Figure 7 and Table 1 show the calibrated vehicle volume by link types for both the training and +testing sets. For all link types, a good correlation (i.e., over 0.80) can be observed between the +calibrated vehicle volume and the ground truth AADT, except for Local Roads and Highway +Ramps in the testing set. The results indicate that our proposed framework can accurately estimate +vehicle volume on higher-level roadways (i.e., Interstate Highways and Highways, Primary Roads, +Secondary Roads), while concurrently maintaining high correlations for lower-level roadways (i.e., +Tertiary Roads, Local Roads, Highway Ramps). The relatively weaker performance for the case +of lower-level roadways can be attributed to limitations in technology. The MDLD only capture +part of the daily trips of a device within the area with mobile network connections and higher-level +roadways usually have a better coverage compared to lower-level ones. This variability might also +result in capturing more travelers on highways and major arterials. In addition, the LBS data +sample is more likely to include the active travelers that make more trips and/or longer-duration + +140000 +140000 +Corr.=0.746 +MDOT SHA AADT (veh/day) +Corr.=0.966 +120000 +RMSE=7912 +120000 +RMSE=2996 +100000 +100000 +80000 +80000 +. +60000 +60000 +40000 +: +40000 +20000 +20000 +0 +0 +0 +20000 +40000 +60000 +80000100000120000 +140000 +0 +20000 +40000 +60000 +80000100000120000 +140000 +WeightedVehicleVolume(veh/day) +CalibratedVehicleVolume (veh/day) +140000 +140000 +Corr.= 0.764 + (veh/day) +Corr.=0.854 +MDOT SHA AADT (veh/day) +120000 +RMSE=7548 +120000 +RMSE=5701 +100000 +100000 +MDOT SHA AADT +80000 +80000 +60000 +60000 +40000 +40000 +20000 +20000 +0 +0 +0 +20000 +40000 +60000 +80000 +100000120000140000 +0 +20000 +40000 +60000 +80000 +100000 +120000 +140000 +Weighted Vehicle Volume (veh/day) +CalibratedVehicleVolume(veh/day)13 + +trips, such as long-distance travel for leisure or business purposes or long-distance commute which +usually happen on interstate highways. + + +Figure 7. Volume Calibration Results Comparison by Link Type. + +Figure 8 and Table 2 show the calibration of vehicle volume by urban/rural status for both the +training and testing sets. In summary, for both urban and rural roads, a good correlation (i.e., over + +100000 +100000 +100000 +100000 +50000 +50000 +50000 +50000 ++0 +-0 +50000100000 +0 +50000100000 +0 +50000100000 +0 +50000100000 +Ro +Road +60000 +60000 +60000 +60000 +40000 +40000 +40000 +40000 +20000 +20000 +20000 +20000 +0 +0- +0 +0- +0 +2500050000 +0 +2500050000 +0 +2500050000 +0 +2500050000 +yR +Roa +40000 +40000 +40000 +40000 +(veh/day) +(veh/day) +20000 +20000 +20000 +20000 +1 +0 +20000 40000 +2000040000 +2000040000 +0 +2000040000 +MDOT SHA AADT +AADT +Ro +load +SHA +60000 +60000 +60000 +60000 +40000 +40000 +MDOT : +40000 +40000 +20000 +20000 +20000 +20000 +0. +0 +50000 +0 +50000 +0 +50000 +50000 +Roa +ads +30000 +30000 +30000 +30000 +20000 +20000 +20000 +20000 +10000 +10000 +10000 +10000 +1 ++0 +20000 +0 +20000 +0 +20000 +20000 +Ral +amj +80000 +80000 +80000 +80000 +60000 +60000 +60000 +60000 +40000 +40000 +40000 +40000 +20000 +20000 +20000- +20000 +： +0 +0 +0 +50000 +0 +50000 +0 +50000 +0 +5000014 + +0.80) can be observed between the calibrated vehicle volume and the ground truth AADT, whereas +a higher correlation can be observed for urban roads. The relatively weaker performance in rural +roadways can also be attributed to the technology limitation mentioned above. + +Figure 8. Volume Calibration Results Comparison by Urban/Rural Status. + +Table 1. Volume Calibration Results Comparison by Link Type +Link Type +Training Set +Testing Set +Corr. +RMSE +Corr. +RMSE +Before +After +Before +After +Before +After +Before +After +All +0.746 +0.966 +7912 +2996 +0.764 +0.854 +7548 +5701 +Interstate Highways +and Highways +0.752 +0.975 +20081 +6559 +0.712 +0.775 +19633 +15246 +Primary Roads +0.699 +0.971 +7909 +2695 +0.721 +0.846 +8665 +6509 +Secondary Roads +0.627 +0.960 +4899 +1776 +0.617 +0.813 +3667 +2667 +Tertiary Roads +0.414 +0.959 +3486 +994 +0.511 +0.869 +3090 +1877 +Local Roads +0.374 +0.944 +2474 +853 +0.426 +0.742 +1701 +1083 +Highway Ramps +0.242 +0.866 +10426 +4722 +0.182 +0.402 +9119 +6846 + +Table 2. Volume Calibration Results by Urban/Rural Status. +Link Type +Training Set +Testing Set +Corr. +RMSE +Corr. +RMSE +Before +After +Before +After +Before +After +Before +After +All +0.746 +0.966 +7912 +2996 +0.764 +0.854 +7548 +5701 +Rural +0.769 +0.967 +3583 +1442 +0.727 +0.826 +4810 +4075 + +- +. +60000 +60000 +60000 +60000 +40000 +(veh/day) +40000 +20000 +20000 +20000 +20000 +E +200004000060000 +200004000060000 +200004000060000 +200004000060000 +MDOT SHA +125000 +125000 +2 +100000 +100000 +100000 +75000 +75000 +75000 +50000 +50000 +50000 +C +25000 +25000 +25000 +0 +0 +50000 +100000 +50000 +100000 +0 +50000 +100000 +50000 +10000015 + +Urban +0.738 +0.964 +8913 +3363 +0.764 +0.853 +8311 +6179 + +Figure 9 visualizes the calibrated vehicle volume averaged from the entire year of 2019 +(represented as AADT) on the all-street network in the state of Maryland. It can be seen that the +interstate highway and the highway skeletons can be clearly identified from the map. Major +arterials also stand out from the map. Figure 9 (b) zooms into the Washington D.C. area, where I- +495, I-270, I-95 and the Baltimore/Washington Parkway are clearly seen. Figure 9(c) zooms into +the Baltimore area, where I-395, I-695, I-795, I-95, and I-70 are all captured. Figure 9(d) zooms +into Hagerstown, MD, which is a city in Washington County, MD near the border of Pennsylvania. +The I-70, I-81, and MD-40 are all captured, demonstrating the ability of our proposed framework +to produce reliable results in rural areas. + + +Figure 9. Visualization of Calibrated Vehicle Volume. (a) the State of Maryland; (b) Washington D.C.; +(c) Baltimore City; (d) Hagerstown, MD. + +5. CONCLUSIONS AND DISCUSSIONS + +This paper presents a big-data driven framework that is able to ingest terabytes of MDLD and +estimate vehicle volume based on MDLD. The proposed framework first employs a series of +cloud-based computational algorithms to extract vehicle trajectories. A map-matching and routing +algorithm is then applied to snap and route vehicle trajectories to the road network. The observed +vehicle counts on each road segment are weighted and calibrated against the control total, i.e., +annual vehicle miles traveled (VMT), and data collected from real-world loop detectors. The +proposed framework is implemented and validated on the all-street network in the state of + +(a) +(b) +Calibrated VehicleVolume +(AADT) (veh/day) +<=5,000 +5,000-10,000 +10,00025,000 +25,000-50,000 +50,000-120,000 +(c) +(d)16 + +Maryland using MDLD data from 2019. After weighting and calibration processes, high +correlation and low RMSE values are observed between our vehicle volume estimates and the +ground truth data. + +The framework proposed in this study and the study findings have practical implications. For +instance, estimated vehicle volume based on MDLD can be leveraged in safety risk exposure +analysis. In particular, the proposed estimation method can particularly be beneficial for safety +risk exposure and crash analysis with respect to vulnerable road users (e.g., pedestrians and +bicyclists). Pedestrian and bicyclist exposure data have traditionally been collected through +surveys or count collections at sample locations (53, 54). In addition to being costly and labor- +intensive, these conventional data collection methods are susceptible to subjectivity and may yield +inaccurate data. Consequently, high-quality and readily-available pedestrian and bicyclist +exposure data are considered as a limitation in safety analysis (55). As exposure data are crucial +for contextualization of crash analysis and prioritization of safety countermeasures (53), utilization +of high-quality and consistent exposure data is imperative. When it comes to safety analysis, using +MDLD for volume estimation—as performed in this study—provides a tremendous advantage +over using data obtained from traditional volume estimation methods. This is due to the potential +of the MDLD to produce more reliable exposure data. Employment of such high-fidelity exposure +data (i.e., MDLD-estimated volumes) as input for safety and crash analyses can lead to more +accurate results and guide data-driven, evidence-based policy decision-making to improve the +safety of all road users including the most vulnerable ones. + +ACKNOWLEDGEMENTS +This study was conducted as part of a collaboration among the Maryland Department of +Transportation State Highway Administration (MDOT SHA), Maryland Transportation Institute +(MTI) at the University of Maryland College Park, and Shock, Trauma and Anesthesiology +Research (STAR) Center at the University of Maryland Baltimore through the sponsorship from +the Safety Data Initiative from the U.S. Department of Transportation (USDOT). + +CONFLICT OF INTEREST +The authors declare that they have no conflict of interest. + +AUTHOR CONTRIBUTION STATEMENT +The authors confirm contribution to the paper as follows: study conception and design: M.Y., W.L., +C.X.; data collection: M.Y., M.A., J.M., G.C., S.S.N.; analysis and interpretation of results: M.Y., +J.M., W.L.; methodology support (osm2gmns): J.L.; draft manuscript preparation: M.Y., W.L., +M.A., J.M., G.C., S.S.N., A.K.. + +REFERENCE + +1. 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FHWA, U.S. +Department of Transportation, (2017). + diff --git a/CNFAT4oBgHgl3EQfsx5b/content/tmp_files/load_file.txt b/CNFAT4oBgHgl3EQfsx5b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd3e60663e7ed8466ba92c71a79316cc0df2bacb --- /dev/null +++ b/CNFAT4oBgHgl3EQfsx5b/content/tmp_files/load_file.txt @@ -0,0 +1,1067 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf,len=1066 +page_content='1 A BIG-DATA DRIVEN FRAMEWORK TO ESTIMATING VEHICLE VOLUME BASED ON MOBILE DEVICE LOCATION DATA Mofeng Yang1, Weiyu Luo2, Mohammad Ashoori3, Jina Mahmoudi4, Chenfeng Xiong5*, Jiawei Lu6, Guangchen Zhao7, Saeed Saleh Namadi8, Songhua Hu9 and Aliakbar Kabiri10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Ph.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='edu) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Graduate Research Assistant (saeed@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='edu) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Graduate Research Assistant (hsonghua@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='edu) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Graduate Research Assistant (kabiri@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='edu) 1-4, 7-10: Maryland Transportation Institute (MTI), Department of Civil and Environmental Engineering, 1173 Glenn Martin Hall, University of Maryland, College Park MD 20742, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 5: Department of Civil and Environmental Engineering, College of Engineering, Villanova University, Villanova, PA 19085, USA 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ 85281, USA Words Count: 4,623 + 2 Tables (250*2) = 5,123 Submission Date: 07/31/2022 2 ABSTRACT Vehicle volume serves as a critical metric and the fundamental basis for traffic signal control, transportation project prioritization, road maintenance plans and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Traditional methods of quantifying vehicle volume rely on manual counting, video cameras, and loop detectors at a limited number of locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' These efforts require significant labor and cost for expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Researchers and private sector companies have also explored alternative solutions such as probe vehicle data, while still suffering from a low penetration rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In recent years, along with the technological advancement in mobile sensors and mobile networks, Mobile Device Location Data (MDLD) have been growing dramatically in terms of the spatiotemporal coverage of the population and its mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' This paper presents a big-data driven framework that can ingest terabytes of MDLD and estimate vehicle volume at a larger geographical area with a larger sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The proposed framework first employs a series of cloud-based computational algorithms to extract multimodal trajectories and trip rosters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' A scalable map matching and routing algorithm is then applied to snap and route vehicle trajectories to the roadway network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The observed vehicle counts on each roadway segment are weighted and calibrated against ground truth control totals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Annual Vehicle-Miles of Travel (AVMT), and Annual Average Daily Traffic (AADT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The proposed framework is implemented on the all-street network in the state of Maryland using MDLD for the entire year of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Results indicate that our proposed framework produces reliable vehicle volume estimates and also demonstrate its transferability and the generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Keywords: mobile device location data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' big data analytics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' vehicle volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' cloud computing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' map matching and routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' INTRODUCTION Vehicle volume measures the amount of traffic traveling through a roadway segment given a specific period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' It serves as a critical metric and the fundamental basis for various transportation applications including traffic signal control, transportation project prioritization and road maintenance plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Traditional methods to quantify vehicle volume rely on manual counting, video cameras, and loop detectors at a limited number of locations, a practice that requires significant human labor and a high cost for expansions (1-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Researchers and private sector companies have also explored alternative solutions such as probe vehicle data, while still suffering from the low penetration rate issue (6-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In the past two decades, along with the technological advancement in mobile sensors and mobile networks, mobile device location data (MDLD) have been growing dramatically in terms of coverage and size, with broader spatiotemporal coverage of the population and its mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' A series of research studies have demonstrated the usefulness of MDLD for enhancing the traditional travel survey and have revealed its potential to substitute surveys (11, 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' At the same time, obtaining travel statistics solely based on MDLD is also worth investigating to reduce human labor and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' However, MDLD do not include any ground truth information such as trip origins and destinations, travel modes, and trip purposes, which requires computational algorithms to be developed and validated against the existing travel surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' More importantly, unlike travel surveys which collect information from representative samples to obtain population-representative statistics, MDLD contain all available mobile devices with uneven data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' This study was conducted as part of the Vulnerable Road User Density Exposure Dashboard project (https://mti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='edu/sdi) - an interactive dashboard that utilizes MDLD to provide data and insights on multimodal volume and safety risk exposure of vulnerable road users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', pedestrians, bicycles) at intersections and roadway segments within Maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In this study, we present a big-data driven framework that ingests terabytes of MDLD and estimates vehicle volume for all roadway segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' First, a series of cloud-based computational algorithms are applied— including but not limited to—a trip and tour identification algorithm to mine travel behavior information and a travel mode imputation model that impute multimodal trajectories from MDLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' A map matching and routing algorithm is then applied to snap and route vehicle trajectories to the roadway network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The observed vehicle counts on each roadway segment are weighted to match the Annual Vehicle Miles of Travel (AVMT) by county, urban/rural status, and functional classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Further, a random forest regression model is used to calibrate the weighted vehicle volume against the Annual Average Daily Traffic (AADT) acquired from loop detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The proposed framework is implemented on the all-street network in the state of Maryland using MDLD data for the entire year of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' LITERATURE REVIEW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Application of Mobile Device Location Data in Transportation Research The appearance of MDLD in the transportation industry started in the 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Since the mid-1990s, researchers began installing Global Positioning System (GPS) data loggers in vehicles to supplement travel surveys (13-15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' With high-frequency in-vehicle GPS data, this approach can 4 significantly improve the accuracy of travel surveys by recording the exact origin and destination as well as the departure and arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' However, only a small number of vehicles can be sampled with this technique, a drawback limiting its capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Similarly, the wearable GPS, which was introduced in the early 2000s, allowed respondents to report non-vehicle travel modes while still suffering from small sample size issues (16, 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In the past decade, private sector entities such as INRIX and RITIS also started to incorporate the probe vehicle data into their commercial products (18-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Nonetheless, the low penetration rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', 2%-10%) of the commercial probe vehicle data remains the core challenge with respect to drawing the whole picture of travel patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' As mentioned above, despite having high precision, traditional MDLD usually suffer from small- sample-size issues, which significantly limits the usefulness of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Since mobile devices, such as smartphones and tablets, have become more popular, MDLD generated from these devices have a greater potential for being used in transportation applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' These new types of MDLD, namely cellular data and Location-Based Service (LBS) data, offer a more extensive spatiotemporal coverage and a larger sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The cellular data are generated through communication between cellphones and cell towers (22) and can be further categorized into Call Detail Record (CDR) and sightings (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The CDR data can only capture the cell tower location, whereas the sightings provide the exact latitude and longitude values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Both types of cellular data have been widely applied to research topics such as travel behavior, human mobility, and social networks in the past two decades (23-31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Despite the large volume of data, cellular data are limited by their spatial and temporal resolution, which is determined by the density of cell towers and users’ cellphone usage levels (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' On a positive note, however, cellular data require less advanced phones and can raise fewer user privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The LBS data provide the exact locations generated when a mobile application updates the device’s location with the most accurate sources, based on the existing location sensors such as Wi-Fi, Bluetooth, cellular tower, and GPS (11, 23- 25, 33, 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Many applications have been developed using the LBS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' For instance, a recent smartphone-enhanced travel survey conducted in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' used a mobile application, rMove developed by Resource Systems Group (RSG), to collect high-frequency location data and allow the respondents to recall their trips by showing the trajectories in rMove (35-38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Additionally, Airsage leveraged LBS data to develop a traffic platform that can estimate traffic flow, speed, congestion and road user sociodemographic for every road and time of day (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Further, the Maryland Transportation Institute (MTI) at the University of Maryland (UMD) developed the COVID-19 Impact Analysis Platform (https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='covid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='edu) to provide insight on COVID- 19’s impact on mobility, health, economy, and society across the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (40-43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Vehicle Volume Estimation Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Estimating Vehicle Volume with Loop Detectors Loop detectors are widely used to record traffic volumes and occupancy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' These sensors are usually buried under the pavements to detect the induction change from the presence of a vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2003 developed an algorithm using data from single loop detectors to estimate truck traffic volumes (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The results showed a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='7% error compared with the ground truth highway data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Loop detector data were also applied together with probe vehicle data to estimate queue length (44) and vehicle volume at a city-wide scale (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Although proven to be efficient in estimating vehicle 5 volume, the high installation and maintenance cost of loop detectors limit their capability of being scaled up to cover the entire transportation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Therefore, loop detector datasets are often incomplete and mostly unavailable at minor arterials and local streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Estimating Vehicle Volume with Probe Vehicle Data In the past two decades, MDLD have gained significant attention and have been utilized for estimating various traffic characteristics including vehicle volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' With the development of MDLD, estimating vehicle volume at the city scale became a reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Probe vehicles can record their trajectory data with high granularity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', 1Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Based on the trajectory data obtained from probe vehicles, a wide range of methods can be used by researchers to solve transportation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' proposed novel methods to estimate queue length and vehicle volume based on the probability theory without prior information about the penetration rate or queue length distribution (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' estimated vehicle volume and queue length at signalized intersections and proposed a new framework to optimize traffic signal control operations (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Sekuła et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' applied several machine learning and neural networks to estimate historical hourly vehicle volume between sparsely located sensors based on the probe vehicle data (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Shockwave theories were also applied to probe vehicle data by a few studies (9, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Estimating Vehicle Volume with Mobile Device Location Data Many studies have been conducted focusing on estimating traffic flow and detecting congestion using cellular data (46, 47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Xing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2019 utilized CDR with Time Difference of Arrival (TDOA) positioning technique in order to estimate multimodal traffic volumes on different types of urban roadways by identifying three modes of travel – namely, drive alone, carpooling, and bus (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The results showed that compared with the ground truth vehicle volume obtained from License Plate Recognition (LPR) cameras, the mean relative error was in the range of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1% to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='7%, depending on the roadway type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Despite significant advances in positioning techniques, cellular data still suffers from low accuracy issues, whereas LBS data have a noticeable advantage due to utilizing different sources to accurately locate the user – a feature that has resulted in an increased usage of this type of data by researchers and the private sector for estimating vehicle volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2019 developed a computing framework alongside a heuristic map matching algorithm to estimate Vehicle Miles of Travel (VMT) and AADT for the state of Maryland using INRIX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The results showed an R2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='878 when fitting the estimated AADT with the ground truth AADT (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Moreover, a number of state agencies conducted rigorous evaluations of vehicle volume obtained through traditional methods as well as from MDLD obtained by private sector companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' They found the latter to be a promising source for supplementing current surveys and traditional methods (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' THE BIG-DATA DRIVEN VEHICLE VOLUME ESTIMATION FRAMEWORK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Overview of the Framework In this study, we propose a big-data driven vehicle volume estimation framework, which offers the capability of efficiently estimating vehicle volume ingested from terabytes of MDLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 1 shows the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The proposed framework is built on Amazon Web Services (AWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' MDLD and all supporting data are stored in Simple Cloud Storage (S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' All algorithms 6 are developed based on Apache Spark, which uses Resilient Distributed Datasets (RDD), and are coded in PySpark using the Elastic MapReduce (EMR) services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In the cloud environment, MDLD are spliced into RDDs given the number of executors (43, 49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' At the same time, all external data sources (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', K-D Tree, network, routing engine) are broadcasted into all executors for master and core nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The same algorithms are applied to each RDD along with the broadcasted variables, and the results are aggregated and outputted into S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The Big-Data Driven Vehicle Volume Estimation Framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Trip End Identification and Travel Mode Imputation Trip is the basic unit of analysis for almost all transportation applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' However, MDLD usually do not contain any trip-related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Therefore, in this study, a trip end identification algorithm is used to extract trip-level information from the MDLD, including trip start location, trip end location, departure time, and arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Then, a travel mode imputation model is further applied to infer four travel modes–namely, the air, drive, rail, and nonmotorized modes based on heuristic rules and a random forest model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Detailed descriptions of the trip end identification algorithm and the travel mode imputation model can be found in the following references (12, 51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Map Matching and Routing To ensure flexibility and scalability of our map matching and routing method across the entire U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', we extract the drivable network from OpenStreetMap (OSM) using the latest open-source Python package osm2gmns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The osm2gmns package can parse roadway network data from OSM and output networks to csv files in the General Modeling Network Specification (GMNS) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' It provides customized and practical functions to facilitate traffic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Functions include complex intersection consolidation, movement generation, traffic zone creation, short link Cloud Computing aws Data Source Local Server Backup MobileDevice 5 DATA Location Data S3Online Bucket Geospatial Maps spark Smart Location AnnualAverage Database DailyTraffic Amazon EMR AmazonEC2 DailyUpdate:1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='52to3401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='80million Annual VehicleMiles OpenStreetMap PySpark points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='05 to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='36 million devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' ofTravel Computation Roadway Network osm2gmns WeightingandCalibration ALGORITHM Algorithms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='Network parsing RandomForestModel Data Preprocessing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Missing value 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='ScalableacrossU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 8 TripEnd Identification County networkx Urban/Rural Status Travel ModeImputation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='Routing engine #ofLanes,Speed Limit MapMatchingandRouting 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Short path algorithm Weighting and Calibration Built Environments APPLICATION VulnerableUserExposureRiskDashboard Decision Support Mobility Tracking Safety Improvement7 combination, and network visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' More details about osm2gmns can be found here: https://osm2gmns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='io/en/latest/ To match each location sighting to our OSM network, the OSM network is firstly parsed and converted into the routable formats, where roadway segments are represented by links and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' With the network topology, we use the networkX package to build a shortest path-based routing engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' We then transform the latitude and longitude of the start node and end node for each link to the plane coordinate (in meters), and then calculate link direction (degree) using the arctan value between the two nodes (see Figure 3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The travel direction between consecutive sightings is also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Similar to the method for link direction calculation, the coordinates of each sighting are converted to plane coordinates, then the degree is calculated using the arctan value between consecutive sightings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' A spatial index structure, K-Dimensional Tree (K-D Tree), is built using the link geometric nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', link nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Then, for each sighting, we search all link nodes that are within 100 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The 100-meter threshold is selected to balance the algorithm efficacy and the computing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' If we increase the value, more candidate links will be considered but this will require more computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' If we decrease the value, we might not be able to find a candidate link when the observation is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' To validate, we calculate the distance between consecutive link nodes using the Maryland OSM network as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Results indicate that more than 95% of the link nodes are within 100 meters of their neighbors, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Therefore, using the 100-meter value as the radius for searching candidate nodes is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Distribution of Distance between Link Nodes in the OSM Network As the next step, for each sighting, we compare its travel direction to all candidate links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The closest link with an absolute travel direction difference smaller than 30 degrees will be selected as a valid matched link for the sighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' This 30-degree threshold is selected mainly to avoid the sighting being matched to the link in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In common cases, the degree difference between the travel direction and the link direction should be approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Here, we use a 30-degree threshold to consider the uncertainty of location accuracy in MDLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' After the matched link for each sighting is found, given the observed link sequence, the routing engine can fill the gap between consecutively observed links and retrieve the complete route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Another layer of reasonable checks is conducted at the routing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' For each pair of consecutive sightings that Distribution of Distance between Link Nodes 40% 35% 30% 25% 20% 15% 10% 5% 0%8 are snapped to links, the routed distance is calculated by summing the link length of all the links traveled between the two sightings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Two reasonableness checks are carried out: (1) If the routed distance is greater than the cumulative distance between the two sightings snapped to links by 2,000 meters or more, we consider the route invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (2) The travel time on these links will be calculated based on the timestamp difference between the two sightings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' With the routed distance and travel time, the average travel speed on these links can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' If the speed exceeds 50 m/s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', 112 mph or 180 km/h), we assume that one of the two sightings is matched to the wrong link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' If either of these two violations is observed, we apply a trial-and-error process by removing the latter sighting and performing the routing using the next sighting snapped to the network until it does not violate the 2,000-meter threshold or the 50 m/s threshold (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' A simple example of the map matching and routing method is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Example of Map Matching and Routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Weighting After map matching and routing, we collect routes for all vehicle trips and aggregate them by links to obtain the observed vehicle volume for each link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Afterward, we develop a link-based weighting method to match the AVMT in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' We classify each link by county, urban/rural status, and functional classes and calculate the link weight using the formula below: 𝑤𝐶,𝑢,𝑓 = 𝐴𝑉𝑀𝑇𝐶,𝑢,𝑓 ∑ 𝑂𝐶,𝑢,𝑓,𝑖 𝑁𝐶 where 𝑤𝐶,𝑢,𝑓 represents the weight for links in county C, with urban/rural status of u, and with functional class f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 𝐴𝑉𝑀𝑇𝑐,𝑢,𝑓 represent the AVMT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and 𝑂𝐶,𝑢,𝑓,𝑖 represents the observed vehicle volume on link i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 𝑁𝐶 represents the total number of links in county C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' For instance, if the study area has 20 counties, 2 urban/rural status and 6 functional classes, then a total of 240 link-based >TravelDirection o Link Centroid Observation Node Matched Link Degree CandidateLink Routed Link9 weights will be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Subsequently, the weighted vehicle volume for each link can be calculated as: 𝑉𝑐,𝑢,𝑓,𝑖 = 𝑤𝐶,𝑢,𝑓 × 𝑂𝑐,𝑢,𝑓,𝑖 where 𝑉𝑐,𝑢,𝑓,𝑖 represents the weighted vehicle volume on link i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Volume Calibration The weighted vehicle volume is further calibrated to match the ground truth AADT collected from loop detectors at a limited number of locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In this study, we use the random forest regression to calibrate the weighted vehicle volume against the AADT to obtain the final vehicle volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' During the calibration process, a 10-fold cross-validation (CV) process is used to fine-tune the random forest regression hyperparameters with 90% training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The fine-tuned models are then applied to the 10% testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' CASE STUDY: THE STATE OF MARYLAND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Data 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Mobile Device Location Data and the Study Area This study used MDLD data obtained from Maryland Transportation Institute (MTI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' MTI integrated and cleaned the raw MDLD from multiple data vendors and built a national MDLD data panel that consists of more than 270,000,000 Monthly Active Users (MAU) and represents movements across the nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (40-43, 51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 4 shows the density of location sightings covering locations within and outside of the boundaries of the state of Maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In this study, we used all MDLD data that are observed in the state of Maryland for the entire year of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The MDLD is processed on a daily basis and the results are aggregated to produce an annual total result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Mobile Device Location Data around the State of Maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' OpenStreetMap Network Using the osm2gmns package, we extracted a total of 634,516 drivable roadway segments within the state of Maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Information about the number of lanes and speed limits was recorded for only 111,835 roadway segments (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='6%) and 84,728 roadway segments (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='4%), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' As shown on the left-hand side in Figure 5, the missing values for the number of lanes and speed limits were estimated based on the corresponding values on nearby roadways in the same county, and with the same urban/rural status, and road functional classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' These two variables are further used as features in the vehicle volume calibration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Number of Lanes and Speed Limits in OSM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Annual Vehicle Miles of Travel Data We use the vehicle miles traveled data from the Maryland Department of Transportation State Highway Administration (MDOT SHA) as a control total number to weight observed vehicle volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Every year, MDOT SHA publishes an annual vehicle miles of travel (AVMT) report by county and functional classification for the state, county, and municipal highway systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' This AVMT report features the current FHWA Functional Classification Codes (1-7) and provides additional classifications (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Urban, Rural, Principal Arterial and Other Freeways and Expressways, and Minor Collector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' As discussed in the methodology section, the weights are generated based on county, urban/rural status and functional classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Here, 23 Maryland counties plus Baltimore City, urban or rural, and two function classes (highway and non-highway) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' We map the OSM link type to the FHWA Functional Classification Codes and generated the highway and non-highway classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' More specifically, “motorway”, “trunk” and “ramp” are classified as highway (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', 1, 2 in FHWA class), and the other types are classified as non-highway (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', 3,4,5,6,7 in FHWA class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' More details about the AVMT data can be found here: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='gov/mdotsha/Pages/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='aspx?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='PageId=302 EstimatedNumberof Lanes (lane) 1Lane 2Lanes 3or4Lanes 5or6Lanes Morethan6Lanes (a) (b) EstimatedSpeedLimits (mph) 5 15 15 25 25 45 45 60 60 70 (c) (d)11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Annual Average Daily Traffic Data We use the AADT also from MDOT SHA to calibrate weighted vehicle volume against the ground truth at a limited number of locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The AADT data consists of linear and point geometric features which represent the geographic locations and segments of roadway throughout the state of Maryland that include traffic volume metrics such as AADT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' More details about the AADT can be found here:https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='imap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='gov/maps/77010abe7558425997b4fcdab02e2b64/about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Smart Location Database and Features for Volume Calibration The Smart Location Database (SLD) is a nationwide geographic data resource for measuring location efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The SLD is produced by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Environmental Protection Agency (EPA)’s Smart Growth Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' It provides more than 90 variables on land use and built environment characteristics such as population and employment densities, land use diversity, urban design attributes, destination accessibility, transit accessibility, and socioeconomic/sociodemographic characteristics at the census block group level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Most attributes are available for every census block group in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In this study, we use SLD variables as features in the random forest regression to calibrate weighted vehicle volume to account for the effects of the built environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The SLD variables used in this study include “TotEMP”, “Pct_AO0”, “D1A”, “D1C”, “D3AAO”, “D3B”, and “D5AR”: • TotEMP = total employment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' • Pct_AO0 = percent of zero-car households;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' • D1A = gross residential density (housing units per acre) on unprotected land;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' • D1C = gross employment density (jobs per acre) on unprotected land;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' • D3AAO = network density in terms of facility miles of auto-oriented links per square miles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' • D3B = street intersection density (weighted, auto-oriented intersections eliminated);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' • D5AR = jobs within 45 minutes auto travel time, time decay (network travel time) weighted We also include urban/rural status, county code, link type, number of lanes, and speed limits as features in the calibration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Overall Comparison Figure 6 shows the weighting and calibration results for both training and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The blue dots represent weighted volume comparisons and the green dots represent calibrated vehicle volume comparisons with MDOT SHA AADT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 6 (a) and (b) compares the weighted vehicle volume and calibrated vehicle volume with the MDOT SHA AADT in the training set respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 6 (c) and (d) compares the weighted vehicle volume and calibrated vehicle volume with the MDOT SHA AADT in the testing set respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' As it can be seen from Figure 6 (a), for the training set, the Pearson correlation value and the Root Mean Square Error (RMSE) between the weighted vehicle volume and the ground truth AADT are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='746 and 7,912, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' These values are improved to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='966 and 2,996 after calibration, as shown in Figure 6 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Similarly, for 12 the testing set, the Pearson correlation and RMSE are improved from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='764 and 7,548, to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='854 and 5,701 respectively after calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (a) Weighted Vehicle Volume in Training Set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (b) Calibrated Vehicle Volume in Training Set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (c) Weighted Vehicle Volume in Testing Set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (d) Calibrated Vehicle Volume in Testing Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Vehicle Volume Validation by Link Types and Urban/Rural Status Figure 7 and Table 1 show the calibrated vehicle volume by link types for both the training and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' For all link types, a good correlation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='80) can be observed between the calibrated vehicle volume and the ground truth AADT, except for Local Roads and Highway Ramps in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The results indicate that our proposed framework can accurately estimate vehicle volume on higher-level roadways (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Interstate Highways and Highways, Primary Roads, Secondary Roads), while concurrently maintaining high correlations for lower-level roadways (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Tertiary Roads, Local Roads, Highway Ramps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The relatively weaker performance for the case of lower-level roadways can be attributed to limitations in technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The MDLD only capture part of the daily trips of a device within the area with mobile network connections and higher-level roadways usually have a better coverage compared to lower-level ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' This variability might also result in capturing more travelers on highways and major arterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In addition, the LBS data sample is more likely to include the active travelers that make more trips and/or longer-duration 140000 140000 Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='746 MDOT SHA AADT (veh/day) Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='966 120000 RMSE=7912 120000 RMSE=2996 100000 100000 80000 80000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 60000 60000 40000 : 40000 20000 20000 0 0 0 20000 40000 60000 80000100000120000 140000 0 20000 40000 60000 80000100000120000 140000 WeightedVehicleVolume(veh/day) CalibratedVehicleVolume (veh/day) 140000 140000 Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='764 (veh/day) Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='854 MDOT SHA AADT (veh/day) 120000 RMSE=7548 120000 RMSE=5701 100000 100000 MDOT SHA AADT 80000 80000 60000 60000 40000 40000 20000 20000 0 0 0 20000 40000 60000 80000 100000120000140000 0 20000 40000 60000 80000 100000 120000 140000 Weighted Vehicle Volume (veh/day) CalibratedVehicleVolume(veh/day)13 trips, such as long-distance travel for leisure or business purposes or long-distance commute which usually happen on interstate highways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Volume Calibration Results Comparison by Link Type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 8 and Table 2 show the calibration of vehicle volume by urban/rural status for both the training and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In summary, for both urban and rural roads, a good correlation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=',' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='load ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='SHA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='40000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 0 50000 0 50000 0 50000 50000 Roa ads 30000 30000 30000 30000 20000 20000 20000 20000 10000 10000 10000 10000 1 +0 20000 0 20000 0 20000 20000 Ral amj 80000 80000 80000 80000 60000 60000 60000 60000 40000 40000 40000 40000 20000 20000 20000 20000 ： 0 0 0 50000 0 50000 0 50000 0 5000014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='80) can be observed between the calibrated vehicle volume and the ground truth AADT, whereas a higher correlation can be observed for urban roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The relatively weaker performance in rural roadways can also be attributed to the technology limitation mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Volume Calibration Results Comparison by Urban/Rural Status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Volume Calibration Results Comparison by Link Type Link Type Training Set Testing Set Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' RMSE Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' RMSE Before After Before After Before After Before After All 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='966 7912 2996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='854 7548 5701 Interstate Highways and Highways 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='975 20081 6559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='775 19633 15246 Primary Roads 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='971 7909 2695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='846 8665 6509 Secondary Roads 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='960 4899 1776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='813 3667 2667 Tertiary Roads 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='959 3486 994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='869 3090 1877 Local Roads 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='944 2474 853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='742 1701 1083 Highway Ramps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='866 10426 4722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='402 9119 6846 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Volume Calibration Results by Urban/Rural Status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Link Type Training Set Testing Set Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' RMSE Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' RMSE Before After Before After Before After Before After All 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='966 7912 2996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='854 7548 5701 Rural 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='967 3583 1442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='826 4810 4075 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 60000 60000 60000 60000 40000 (veh/day) 40000 20000 20000 20000 20000 E 200004000060000 200004000060000 200004000060000 200004000060000 MDOT SHA 125000 125000 2 100000 100000 100000 75000 75000 75000 50000 50000 50000 C 25000 25000 25000 0 0 50000 100000 50000 100000 0 50000 100000 50000 10000015 Urban 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='964 8913 3363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='853 8311 6179 Figure 9 visualizes the calibrated vehicle volume averaged from the entire year of 2019 (represented as AADT) on the all-street network in the state of Maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' It can be seen that the interstate highway and the highway skeletons can be clearly identified from the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Major arterials also stand out from the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 9 (b) zooms into the Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' area, where I- 495, I-270, I-95 and the Baltimore/Washington Parkway are clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 9(c) zooms into the Baltimore area, where I-395, I-695, I-795, I-95, and I-70 are all captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 9(d) zooms into Hagerstown, MD, which is a city in Washington County, MD near the border of Pennsylvania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The I-70, I-81, and MD-40 are all captured, demonstrating the ability of our proposed framework to produce reliable results in rural areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Visualization of Calibrated Vehicle Volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (a) the State of Maryland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (b) Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (c) Baltimore City;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (d) Hagerstown, MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSIONS This paper presents a big-data driven framework that is able to ingest terabytes of MDLD and estimate vehicle volume based on MDLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The proposed framework first employs a series of cloud-based computational algorithms to extract vehicle trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' A map-matching and routing algorithm is then applied to snap and route vehicle trajectories to the road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The observed vehicle counts on each road segment are weighted and calibrated against the control total, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', annual vehicle miles traveled (VMT), and data collected from real-world loop detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The proposed framework is implemented and validated on the all-street network in the state of (a) (b) Calibrated VehicleVolume (AADT) (veh/day) <=5,000 5,000 10,000 10,00025,000 25,000 50,000 50,000 120,000 (c) (d)16 Maryland using MDLD data from 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' After weighting and calibration processes, high correlation and low RMSE values are observed between our vehicle volume estimates and the ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' The framework proposed in this study and the study findings have practical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' For instance, estimated vehicle volume based on MDLD can be leveraged in safety risk exposure analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In particular, the proposed estimation method can particularly be beneficial for safety risk exposure and crash analysis with respect to vulnerable road users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', pedestrians and bicyclists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Pedestrian and bicyclist exposure data have traditionally been collected through surveys or count collections at sample locations (53, 54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In addition to being costly and labor- intensive, these conventional data collection methods are susceptible to subjectivity and may yield inaccurate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Consequently, high-quality and readily-available pedestrian and bicyclist exposure data are considered as a limitation in safety analysis (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' As exposure data are crucial for contextualization of crash analysis and prioritization of safety countermeasures (53), utilization of high-quality and consistent exposure data is imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' When it comes to safety analysis, using MDLD for volume estimation—as performed in this study—provides a tremendous advantage over using data obtained from traditional volume estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' This is due to the potential of the MDLD to produce more reliable exposure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Employment of such high-fidelity exposure data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', MDLD-estimated volumes) as input for safety and crash analyses can lead to more accurate results and guide data-driven, evidence-based policy decision-making to improve the safety of all road users including the most vulnerable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This study was conducted as part of a collaboration among the Maryland Department of Transportation State Highway Administration (MDOT SHA), Maryland Transportation Institute (MTI) at the University of Maryland College Park, and Shock, Trauma and Anesthesiology Research (STAR) Center at the University of Maryland Baltimore through the sponsorship from the Safety Data Initiative from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Department of Transportation (USDOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' CONFLICT OF INTEREST The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' AUTHOR CONTRIBUTION STATEMENT The authors confirm contribution to the paper as follows: study conception and design: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' data collection: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' analysis and interpretation of results: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' methodology support (osm2gmns): J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' draft manuscript preparation: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', J.' metadata={'source': 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Various methods for queue length and traffic volume estimation using probe vehicle trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies, 107, 70–91 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='trc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='008 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Guo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & (Jeff) Ban, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Urban traffic signal control with connected and automated vehicles: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In Transportation Research Part C: Emerging Technologies (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 101, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 313–334) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Elsevier Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='trc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='026 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Sekuła, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Marković, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', vander Laan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & Sadabadi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Estimating historical hourly traffic volumes via machine learning and vehicle probe data: A Maryland case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies, 97, 147–158 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='trc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='012 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Anuar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & Cetin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Estimating Freeway Traffic Volume Using Shockwaves and Probe Vehicle Trajectory Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Procedia, 22, 183–192 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='trpro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='025 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Tang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Real-Time Queue Length Estimation for Signalized Intersections Using Vehicle Trajectory Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Record, 2623(1), 49–59 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='3141/2623-06 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Susilo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. The promises of big data and small data for travel behavior (aka human mobility) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 68, 285-299, (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Darzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Ghader, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Xiong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. A data-driven travel mode share estimation framework based on mobile device location data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1-45 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Battelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Global Positioning Systems for Personal Travel Surveys: Lexington Area Travel Data Collection Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Final Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' FHWA, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Department of Transportation, (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2000–2001 California Statewide Household Travel Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Final Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' NuStats, Austin, Tex (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Kansas City Regional Travel Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Final Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' NuStats, Austin, Tex, (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Mid-Region Council of Governments 2013 Household Travel Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Final Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Westat, Rockville, Md, (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2014 Southern Nevada Household Travel Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Final Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Westat, Rockville, Md, (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' INRIX Traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='inrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='com/, (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Haghani, Ali, Masoud Hamedi, and Kaveh Farokhi Sadabadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' I-95 Corridor coalition vehicle probe project: Validation of INRIX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' I-95 Corridor Coalition 9, (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} 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Mobile phone data in transportation research: methods for benchmarking against other data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. On data processing required to derive mobility patterns from passively- generated mobile phone data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 87, 58-74, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & Ban, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Extracting trips from multi-sourced data for mobility pattern analysis: An app-based data example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 105, 183-202, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Puget Sound Regional Travel Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Report: Spring 2014 Household Travel Survey.' metadata={'source': 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Sound Regional Travel Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Report: 2015 Household Travel Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' RSG, (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2017 Puget Sound Regional Travel Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Draft Final Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' RSG, (2017).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Kabiri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. 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Mobile device data reveal the dynamics in a positive relationship between human mobility and COVID-19 infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 117(44), 27087-27089 (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 19 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Xiong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Younes, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Luo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Ghader, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. 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City-wide traffic volume inference with loop detector data and taxi trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In Proceedings of the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 1-10) (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Caceres, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Romero, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Benitez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and del Castillo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Traffic Flow Estimation Models Using Cellular Phone Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 1430-1441 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1109/TITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2189006 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Janecek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Valerio, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Hummel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Ricciato, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and Hlavacs, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. The Cellular Network as a Sensor: From Mobile Phone Data to Real-Time Road Traffic Monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 2551-2572, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='1109/TITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='2413215 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Xing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', & Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Traffic volume estimation in multimodal urban networks using cell phone location data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' IEEE Intelligent Transportation Systems Magazine, 11(3), 93- 104 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Fu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Stewart, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', and Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Using big GPS trajectory data analytics for vehicle miles traveled estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation research part C: emerging technologies 103 (2019): 298-307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Codjoe, Julius, Grace Ashley, and William Saunders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Evaluating cell phone data for AADT estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' FHWA/LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 18/591, LTRC Project Number: 16-3SA, State Project Number: DOTLT1000110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Louisiana Transportation Research Center, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Ghader, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Darzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Sun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Kabiri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and Zhao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Data analytics and modeling methods for tracking and predicting origin-destination travel trends based on mobile device data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Federal Highway Administration Exploratory Advanced Research Program (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Newson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' and Krumm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='. Hidden Markov map matching through noise and sparseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' In Proceedings of the 17th ACM SIGSPATIAL international conference on advances in geographic information systems (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 336-343) (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Sanders, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Frackelton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Gardner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Schneider, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', Hintze, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Ballpark Method for Estimating Pedestrian and Bicyclist Exposure in Seattle, Washington: Potential Option for Resource-constrained Cities in an Age of Big Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Transportation Research Record, 2605(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='32–44 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=', and Sener, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Emerging Data Mining for Pedestrian and Bicyclist Monitoring: A Literature Review Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' Safety through Disruption, National University Transportation Center (UTC) Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content=' https://safed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='vtti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf'} +page_content='edu/wp-content/uploads/2020/07/UTC- Safe-D_Emerging-Data-Mining-for-PedBike_TTI-Report_26Sep17_final.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 1, October 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 328~338 ISSN: 2502-4752, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='11591/ijeecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='v28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='pp328-338 \uf072 328 Journal homepage: http://ijeecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='iaescore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='com A systematic review of structural equation modeling in augmented reality applications Vinh The Nguyen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Chuyen Thi Hong Nguyen2 1Faculty of Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' TNU-University of Information and Communication Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Thai Nguyen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Vietnam 2Faculty of Primary Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Thai Nguyen University of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Thai Nguyen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Vietnam Article Info ABSTRACT Article history: Received Mar 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2022 Revised Jun 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2022 Accepted Jul 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2022 The purpose of this study is to present a comprehensive review of the use of structural equation modeling (SEM) in augmented reality (AR) studies in the context of the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' IEEE Xplore Scopus, Wiley Online Library, Emerald Insight, and ScienceDirect are the main five data sources for data collection from Jan 2020 to May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The preferred reporting items for systematic reviews and meta-analyses (PRISMA) approach was used to conduct the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' At the final stage, 53 relevant publications were included for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Variables such as the number of participants in the study, original or derived hypothesized model, latent variables, direct/indirect contact with users, country, limitation/suggestion, and keywords were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The results showed that a variety of external factors were used to construct the SEM models rather than using the parsimonious ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The reports showed a fair balance between the direct and indirect methods to contact participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Despite the COVID-19 pandemic, few publications addressed the issue of data collection and evaluation methods, whereas video demonstrations of the augmented reality (AR) apps were utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The current work influences new AR researchers who are searching for a theory-based research model in their studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Keywords: Augmented reality COVID-19 External factors Structural equation modeling Theory-based research This is an open access article under the CC BY-SA license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Corresponding Author: Vinh The Nguyen Faculty of Information Technology, TNU-University of Information and Communication Technology Z115 Street, Quyet Thang Commune, Thai Nguyen, Vietnam Email: vinhnt@ictu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='vn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' INTRODUCTION Augmented reality (AR) is a technology that has attracted a lot of attention in various domains [1]- [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Unlike virtual reality (VR) which allows users to be totally immersed in a virtual environment, AR enriches the real world with virtual artifacts [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The primary value of AR is that it allows digital objects to be blended more seamlessly into a person’s perception of the real world than simply displaying data on a screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Market research [5] anticipates that AR’s market will reach USD 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='4 billion, growing 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='5% from 2021 to 2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In addition, in response to the COVID-19 pandemic, more companies and organizations have adopted remote work and are utilizing augmented reality technology [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' What that means is that a huge number of AR applications are being developed, especially in electrical engineering and computer science [1]-[3], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Assessment is one of the key factors in ensuring the success of an AR application, especially when it is involved with end-users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' However, literature work reported that only a few studies afforded time for this type of evaluation (only 8% of published papers) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' One plausible explanation was that AR researchers/developers had to devote their time to solving technical issues [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Moreover, the lack of methods or theory-driven research on evaluating AR apps, considering end users’ involvement, contributed cC BY SAIndonesian J Elec Eng & Comp Sci ISSN: 2502-4752 \uf072 A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 329 to the scarcity of AR evaluation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In addition, after the COVID-19 outbreak, many conferences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', ISMAR) encouraged researchers to find alternative means of evaluating AR apps rather than canceling the submissions due to social distancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' There has been no study addressing this issue so far, thus it remains a gap in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' To close this gap, this paper-based on prior AR studies–provided an overview of theory- based methods that can effectively be used for AR assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Among many other end-user evaluation methods, the scope of the current study focused on structural equation modeling (SEM), a model commonly used in behavioral science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' SEM is a comprehensive statistical method that examines relationships between observed and latent factors [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' It has been widely used in confirmatory factory analysis in many topics and fields [13]-[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' A number of review studies on SEM applications have been conducted in various research domains, including ecology [16], social science [17], psychological research [18], and strategic management [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' It indicated that a review study would be valuable for new researchers to quickly acquire knowledge in the field effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Yet, it also implies that it would be important to look at SEM from AR’s perspective since AR is one of the emerging trends in the digital transformation era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' However, there is no study of SEM for AR applications other than previously mentioned review studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Thus, the current research is unique on its own by the AR’s topic and the outcomes of this study can be used as a referencguidene for researchers in similar studies, particularly in electrical engineering and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' More specifically, the present study tries to answer to following research questions: i) What are the preferred theory-driven models being used in prior AR studies amid the COVID-19 pandemic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' ii) What are the dimensions or variables being investigated by AR researchers so far?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' iii) How do researchers of prior AR studies communicate with end-users for evaluation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' vi) How many participants are typically involved in a study?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Would this number still be considered appropriate from the literature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' v) What are the main drawbacks of tR studies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Do they suffer from the COVID-19 pandemic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' METHOD This study involves a review of SEM in AR applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' thus, the preferred reporting items for systematic reviews and meta-analyses (PRISMA) statement was applied [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The PRISMA statement aims to assist scholars in improving the reporting of scientific reviews and meta-analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' It is an evidence-based minimum set of elements for systematic review reports that are intended to assist systematic reviewers in clearly explaining why the review was conducted and what the authors performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' It has previously been used to target comparable research objectives [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Source selection IEEE Xplore, Scopus, Wiley Online Library, Emerald Insight, and ScienceDirect databases were used to build the corpus, encompassing titles, abstracts, and keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' These five databases are regarded as essential and dependable sources of high-quality articles in the fields of computer science and engineering [21], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Although, some other indexing databases are available (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', Scholar) but they are out of scope in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Search criteria To add articles to our corpus, both of the following related criteria need to be fulfilled, i) Structural equation modeling search term: at least one SEM-related term must appear in an article’s title, abstract, or author keywords (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', structural equation modeling, SEM, planned behavior, theory of planned behaviour (TPB), motivational model, Michaelis–Menten (MM), reasoned action, theory of reasoned action (TRA), social cognitive, SCT, diffusion of innovation, IDT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' and ii) Augmented Reality search term: terms include augmented reality, AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Using the aforementioned criteria, 16 articles were discovered in IEEE Xplore, 107 articles in Scopus, 197 papers in Wiley Online Library, 68 papers in Emerald Insight, and 695 papers in ScienceDirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The corpus was collected between June 3, 2021, and June 12, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Eligibility assessment for the final analysis corpus To determine the acceptability of the obtained papers, the first researcher personally reviewed the entry criteria mentioned below by reviewing the titles and abstracts of the obtained publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' When a clear judgment could not be reached, other aspects of the publication, particularly the method and data acquisition descriptions, were discussed in conjunction with the second author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Only items that meet the following criteria are retained in the corpus: i) Peer-reviewed: The paper was peer-reviewed in the two indexing databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' This is due to the trustworthiness of peer-reviewed journals and the rigorous peer-review processes, only articles in these databases are considered for this review;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' ii) Topic relevant: The topic of an article is pertinent to the applications of SEM in AR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' iii) Language: Publication was reported in English;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' and vi) Duration: Paper was published between Jan 2020 and May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' \uf072 ISSN: 2502-4752 Indonesian J Elec Eng & Comp Sci, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 28, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 1, October 2022: 328-338 330 If the article meets any of the following criteria, it will be excluded from the corpus: i) Books and cover page, abstract only, poster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' ii) The paper was not written in English;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' iii) Application of SEM is not for AR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' and vi) Paper was published before Jan 2020 and after May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 1 depicts the flow of information through the different phases of the systematic review utilizing PRISMA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 1,083 records were found in all data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Duplications were removed based on the titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Each paper was screened individually to remove items that are out of scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Then 230 records were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As such, 309 candidates left for full-text retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Of these remaining items, 9 records cannot be retrieved due to access restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The authors examined each report for eligibility and removed 247 studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In the end, 53 items were included in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The remaining papers were examined individually to extract interesting variables such as the number of participants, original or derived hypothesized model, latent variables, direct/indirect contact with the user, country of origin, limitation/suggestion (if any), and keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The flow diagram represents the movement of information through the various stages of a systematic review 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Data coding and analysis To extract the data, all articles were loaded into NVivo software, and a coding scheme was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' NVivo is a program that facilitates qualitative analytical method research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' This tool enables researchers to organize, analyze and explore unstructured or qualitative data, including interviews, reviews, articles, social media, and web content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Codes included authors, journal name, year of publication, countries of authorship, title, abstract, author keywords, method, objectives, findings and limitations on how SEM was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' RESULTS AND DISCUSSION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' What are the preferred theory-based driven models being used in prior AR studies amid the COVID-19 pandemic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 2 depicts the distribution of papers over hypothesized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Most publications fall into the SEM category (accounted for 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='49%), followed by eTAM and TAM with 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='76% and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='32% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Although the UTAUT model was developed recently, the result shows less popularity of adopting this model (only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='77%), which is the same as the SOR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Technology acceptance model (TAM): originally developed by Davis [24], TAM is known as a theory of information systems that describes how consumers come to accept and use technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Real system usage is the point at which people interact with technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' People utilize technology because of their behavioral intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In this survey, 6 articles (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='32%) used original TAM for their research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Extended technology acceptance model (eTAM): In this category, 11 publications (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='75%) extended TAM with external variables such as perceived task-technology fit [25]-[28]–which asserted that Identification ofnew studies viadatabases Recordsidentifiedfromdatabases: Recordsremovedbeforescreening: N=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='083 N=544 Records screened: Records excluded: N=539 N=230 Reports soughtfor retrieval Reports not retrieved: N=309 N=9 Reports excluded:247 Reports assessed for eligibility: NotinvolveSEM(N=27) N=300 NotinvolveAR(N=127) ReviewOnly(N=93) Reports of included studies: N=53Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 \uf072 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 331 the technology must be utilized and a good fit with the tasks it supports to have positive impacts on individual performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' perceived visual design/appeal [25]-[31] which assumed that beauty is important,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' and it impacts decisions that should not be influenced by aesthetics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' perceived enjoyment [32]-[35]-which refers to the hedonic value of new technology and expresses how pleasurable a person finds its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Models distribution across prior studies The unified theory of acceptance and use of technology (UTAUT): Venkatesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [36] developed the UTAUT after reviewing and consolidating the components of eight previous models used to describe information system user behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In this review, several external variables were incorporated into the existing UTAUT model (eUTAUT) such as innovativeness, reward, trust, enjoyment, hedonic motivation, habit, and gamification [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=" Stimulus-organism-response (SOR): Mehrabian-stimulus Russell's model [39] depicts the occurrence of a person's response to environmental stimuli." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [40] decomposed stimulus into two external factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', Interactivity, Virtuality), Organism into 4 variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', Hedonic, Utilitarian, Informativeness, and Ease of Use), and Response into 2 factors including Attitude and Behavioral Intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Similarly in the scope of this review, Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [40] also included (critical mass, social interaction, information timelines, content richness) into stimulus, (attachment, conformity) into Organism, and (visiting intention, continue intention) into Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Structural equation modeling (SEM): This category contains the largest portion of the papers included in our investigation (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='49%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Authors in this group mainly adapted constructs, measures in the literature to form hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As such, PLS-SEM was utilized as an analytical method to conduct confirmatory factor analysis and path analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Confirmatory factor analysis, which originates in psychometrics, aims to quantify underlying psychological characteristics such as attitude and satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Path analysis, on the other hand, has its origins in biometrics and is intended to discover the causal link between variables by drawing a path diagram [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' What are dimensions or variables being investigated by AR researchers so far?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 3 depicts 77 unique constructs/latent variables from hypothesized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' There are 184 unique constructs found in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Behavioral intention, usefulness, ease of use, attitude, user behavior, and enjoyment are the most frequent items used in the hypothesized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Wordcloud depicts 77 unique constructs from all hypothesized models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Count of SEM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='eTAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='eUTAUT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='SEM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='SOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='TAMHedonic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='EaseOf Use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Novelty ConceptualUnderstanding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Performance Anxiety Quality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Technology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='SocialInteraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Enjoyment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Responses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='ntention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Interactivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='UseBehavior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Knowledge Gain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Immersion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Voluntariness TaskSatisfaction Aesthetics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Reievance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Sublective Norms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Behavioral Intentionsli Efiacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='TaskTechnologyFit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Environmental Motivation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Trust Fit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Playfulness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Behaviora ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Game ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Involvement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='BenefitJob ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Attitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='ExperienceVisual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Presence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Effort Richness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Perceived ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Informativeness Soclal ActualUsage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Engagement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Simulated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Purchase Intention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Behavior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Augmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Education ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Usefulness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Expectancy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='Image Entertainment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='\uf072 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='ISSN: 2502-4752 Indonesian J Elec Eng & Comp Sci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 28, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 1, October 2022: 328-338 332 Figure 4 captures the top 14 dominant keywords in the collection of papers in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Aside from “augmented reality”, TAM is the most popular term that the authors used for indexing their papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In total, this study extracted 319 keywords with 230 unique terms, indicating that there is a high variation of topics/techniques used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' However, in terms of their broad contents, the major theme of these collected papers can be categorized as the “social marketing” theme as they were mainly focused on “Intention to Purchase” or “Intention to Visit”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Frequency of keywords extracted from publications 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' How do researchers of prior AR studies communicate with end-users for evaluation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Table 1 reports the communication channels used to gather data from respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Results showed that there is a fair balance between the direct (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='28%) and indirect (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='94%) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Here, the indirect method means that the research teams did not contact participants directly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', lab setting, or field study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Instead, they contact users via online channels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', social network, email, discussion group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' On the other hand, the direct method requires subjects to be at the site of the study for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Communication channels to collect data from respondents Communication channel Count Percentage Indirect 27 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='94 Direct 24 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='28 Direct and Indirect 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='77 Total 53 100 Figure 5 depicts the spatial locations of authors researching AR utilizing the SEM method across the globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' It can be observed that most publications were conducted in the United States although this country was suffered heavily from the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' However, 8 out of 10 papers utilized the indirect research method to recruit and gather data, meaning that the study was conducted remotely, and opinions were collected through online tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=" 'SocialMarketing':AugmentedRealityappearsmostoften." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' AugmentedReality TechnologyAcceptance Model BehavioralIntentions Pokemon Go VirtualReality Social Marketing GeneralizedStructuredComponentAnalysis Presence TAM TechnologyAdoption Interactivity User Experience MobileAugmented Reality A-Frame MobileAugmented RealityApplications 0 5 10 15 20 25 30 35 40 Social MarketingIndonesian J Elec Eng & Comp Sci ISSN: 2502-4752 \uf072 A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 333 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Spatial locations of authors researching on AR utilizing SEM in 2020-2021 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' How many participants are typically involved in a study?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Would this number still be considered appropriate from the literature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 6 shows the distribution of sample size across peer-reviewed papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The whisker plot indicates that on average the sample size (the number of participants) who took part in the studies was approximately 300 subjects considering 4 extreme values (or outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The minimum sample size is 9 and the maximum is 1,566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The median indicates that most papers recruited around 200 users for their studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' When the four extreme values were not considered, the average sample size for direct communication with participant was 142 (median=113, range=340, min=24, max=364), and indirect method was 286 (median=302, range=710, min=9, max=719).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Distribution of sample size in the peer-reviewed papers Sample size is a debating subject in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As such, the determination of sample size varies from study to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Some researchers advocate a minimum sample size of 100–200 per a study, an acceptable sample size can range between 300 and 500, or with criteria such as acceptable of five cases per free parameter, moderate of ten cases per free parameter [12], and ideal of 20 instances per free parameter in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Kock and Hadaya [42] proposed a technique for determining an adequate sample size based on “inverse square root” and “gamma-exponential” approaches which were adapted by Nikhashemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [43] included in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' To some extent, Figure 6 reflects the balance of sample size recommendation in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Interestingly, the median sample size calculated in this study (Median=200) was aligned with the findings based on reviews of studies in different research areas, including operations management, education and psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' What are the main drawbacks of the AR studies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Do they suffer from the COVID-19 pandemic Table 2 reports the frequency of limitations addressed by authors in the collected publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The most common flaw that needs to be examined further in future studies is the failure to incorporate additional external components (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='62%) in the postulated model, followed by convenience sampling (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='85%), multi- level analysis (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='08%) and limited to one region (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='19%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In terms of convenience sampling drawback, United States 10 Germany n Taiwan 4 Greece 4 UnitedKingdom China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' SouthKorea Indonesia 2 Thailand 2 Romania 2 Vietnam 2 Australia 2 France 1 Italy 1 Spain 1 HongKong 1 Ireland 1 Turkey Netherlands 1 India 1 Oman 1 Portugal 1 Malaysia 1 Powered by Bing Tom,WikipediaDistribution of sample size across publications 1800 1600 ·1566 1400 1200 :1183 :1192 1000 800 719 600 400 412 X298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='8113208 200 200 68 \uf072 ISSN: 2502-4752 Indonesian J Elec Eng & Comp Sci, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 28, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 1, October 2022: 328-338 334 many authors acknowledged that they used the non-probability method to acquire sample data through their networks of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As such, their reports/findings cannot be generalized to the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Frequency of limitations addressed by the authors in the collected publications Limitations References Not consider other factors (21) [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [53],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [55],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [56],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [57] Convenience sampling (19) [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [53],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [55],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [57],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [59],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [60],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [61],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [62],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [63],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [64] Multi levels analysis (17) [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [55],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [56],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [59],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [62],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [65],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [66],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [67],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [68],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [69] Limited to one region (16) [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [56],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [59],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [63],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [68],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [70] Tailored to a specific AR product (14) [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [53],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [57],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [61],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [62],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [64],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [65],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [67],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [68],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [70] Small Sample Size (10) [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [60],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [71] Short term effect (10) [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [63],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [65],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [69],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [72] Not specified (9) [34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [73],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [74],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [75],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [76],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [77],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [78] Only Intention Model (6) [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [56],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [79] Lack of AR features (6) [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [63],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [71] Lack of functions (4) [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [32] Self-Administered Survey (3) [58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [66],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [79] Use Videos for demonstrations (3) [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [65] Technical challenges (2) [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [28] Standardized tools (2) [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [52] Single Analysis technique (2) [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [48] Lab setting (2) [55],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [64] Not consider privacy concerns (2) [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [60] Others (8) [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [56],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [59],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' [70] Along with convenience sampling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' limited study to one region is another shortcoming that is often mentioned with non-probability method limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Unlike convenience sampling drawback that subjects may come from different parts of the world, the regional issue was arising where the study was intentionally designed for a specific region through a case study or in the lab setting [55], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' A large portion of the published work was carried out with the help of pre-existing AR products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' This evaluation includes examples such as IKEA Place, YouCam Makeup, and Pokémon Go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Participants were asked if they had any experience with these AR apps, and if so, they were encouraged to take part in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=" Furthermore, the authors' capacity to extend the study to additional products/services was limited because they did not have control or flexibility over the AR apps." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The results show that though the sample size was a sufficiently addressed problem by the researchers, the proportion of this limitation was just 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='87%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Without considering publications that did not report limitations in their work (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=', not specified (9)), 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='27% (34/44 papers) of the research group justified their sample size using an analytical tool/method, a sample size recommendation in the literature, and the use of PLS-SEM, which can work with small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As a result, sample estimation was deemed sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Another issue worth mentioning is the short-term effect addressed by 10 author groups (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='87%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The short- term impact was explained by the fact that the experiments were only conducted for a limited period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As a result, the theorized models can only explain variables impacting user behavior at that point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The authors emphasized that because technology has evolved drastically over the years, the question of whether their proposed models stand up remained unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=" In addition, people's perspectives shift throughout time as they gain experience [36], as a consequence, long-term research was suggested to validate the models." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In terms of the indirect method to conduct an experiment with users, four studies administered their AR applications through video demonstrations [25], [26], [31], [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In this regard, instead of asking participants to download or use the AR apps directly, the authors created videos demonstrating the features of their studied AR apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Based on the evidence of previous studies using video depictions of AR prototypes [80], [81], these authors argued that the technology itself was not available for participants to interact with at the time, and the purpose of the hypothesized models was to examine the influential factors that affect behavioral intention before releasing the actual AR product to the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As such in this category, studies in [26], [29], [52] recommended that there is a need to have a tool or new evaluation method to overcome the current issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' In summary, compared with previous studies [16]-[19], this study has some similarities and differences as: First, it is the selection of model, our report also shows similar results, that is, many different types of models and variables are applied to the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' There has not yet been a general consensus set to guide new researchers to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The difference is that the variables in this study revolve around technology Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 \uf072 A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 335 rather than ecology, social science, psychology, and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Second is the issue of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' While similar studies only listed restrictions that exist in articles, our study quantified these limitations by specific numbers and arranges them in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As such, interested researchers can rely on it to cover the information more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The third consideration is the study’s time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' This investigation was carried out in the context of digital transformation and the influence of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Many new factors emerge and exert effect that have received little consideration in prior research (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Summarizing these factors will help researchers have more options instead of reading different articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' And finally, by synthesizing how the experiments were carried out during the pandemic, not only new researchers can adapt prior evaluation approach in the current situations but also improve them in the subsequent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' CONCLUSION This paper presented a systematic review of the use of SEM in AR studies during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The PRISMA model was adapted as a guideline for doing the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Five data sources were used for data retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' After a series of preprocessing steps, 53 publications were included in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The results showed that authors used a variety of external factors to form the generative hypothesized models (SEM), followed by the extension of TAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' The diversity of external factors indicated that there is no consensus among AR scholars for using common factors influencing AR adoption, thus opening a huge potential research gap for the AR community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Interestingly, United States was the most active country in conducting AR studies during the Covid-19 pandemic, however 80% of its studies were conducted through indirect communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Hence, they were not affected by the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' A large portion of AR studies focused on understanding factors influencing user behavioral toward using third-party AR apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' As such, participants were required to download and use the apps then answer the survey questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Sample size, in this regard, cannot be excused due to social distancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Only few studies examined user behavioral through developed AR apps and the corresponding authors suggested that there is a need to have an alternative approach to conduct user study rather than the traditional face-to-face fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Watching two separate videos (one with AR and one without AR) was currently be used as an alternative method to alleviate the issue but not a plausible approach in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Therefore, this research gap remains open and needs to be addressed in further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Thus, the outcomes of this study can be used as a reference guideline for researchers in similar studies where there is a lack of theoretical framework for assessment, particular in electrical engineering and computer science.' metadata={'source': 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mobile learning readiness in higher education based on the theory of planned behavior,” Computers & Education, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 1054–1064, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' 2012, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='compedu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' BIOGRAPHIES OF AUTHORS Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Vinh The Nguyen is currently a lecturer at the Faculty of Information Technology, University of Information and Communication Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' He is also a senior visiting lecturer at FPT University Greenwich, Hanoi branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=" He graduated with a master's degree in information systems management from Oklahoma State University, USA (under scholarship 322)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' He completed his PhD program under Project 911 in 2020 at Texas Tech University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' His main research interests are Computer Vision, Computer Visualization, and Computer in Human Behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' He has authored or coauthored more than 35 publications with 10 H-index and more than 250 citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' He can be contacted at email: vinhnt@ictu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Chuyen Thi Hong Nguyen is currently a lecturer at the Faculty of Primary Education, Thai Nguyen University of Education, Vietnam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=" She graduated with a master's degree in Theory and History of Education from Hanoi University of Education, Vietnam (2008)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' She completed her PhD program in 2016 at The Vietnam Institute of educational Sciences, Vietnam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' Her main research interests are method teaching, assessment in primary education, computational thinking, learning style, and augmented reality in education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content=' She can be contacted at email: chuyennh@tnue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf'} +page_content='vn.' metadata={'source': 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+1,1551 @@ +Strange Stars within Bosonic and Fermionic Admixed Dark Matter +Luiz L. Lopes1∗ and H. C. Das2,3† +1Centro Federal de Educac¸˜ao Tecnol´ogica de Minas Gerais Campus VIII; CEP 37.022-560, Varginha - MG - Brasil +2Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India and +3Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India +(Dated: January 3, 2023) +In this work, we study dark matter (DM) admixed strange quark stars exploring the different possibilities +about the nature of the DM and their effects on the macroscopic properties of strange stars, such as maximum +masses, radii, as well the dimensionless tidal parameter. We observe that the DM significantly affects the +macroscopic properties that depend on the DM mass, type, and fraction inside the star. +I. +INTRODUCTION +Recently, it was suggested that quarks probable appear in- +side the core of a massive neutron star (NS) due to a very high +density, where hadronic matter undergoes phase transitions to +a new phase of quarks and gluons [1]. Furthermore, several +exotic particles, such as hyperons production, dark matter ac- +cretions, kaon condensations, and so on, appeared primarily in +the core of the NS. As a result, in those regimes, the equation +of state (EoS) is the fundamental component that can describe +both micro/macroscopic properties. Another possibility is that +at least some of the observed pulsars are indeed stable quark +stars, or strange stars. Several theoretical works have pro- +posed the existence of strange quark stars (SQSs) [2], which +are made up of u, d, and s quarks in equilibrium in terms +of weak interactions. The Bodmen-Witten conjecture states +that strange quark matter (SQM) can have a lower energy per +baryon than pure nucleons because the exclusion principle +may be dominant at absolute zero pressure and temperature +[3, 4]. Hence, the SQM might be the true ground state of the +hadronic matter. Hence, it stands to reason that the SQS must +be more stable than the ordinary NS. +Various phenomenological models have been proposed to +explore the SQSs properties. +Among them, the MIT bag +model and Nambu-Jona-Lasinio (NJL) have been widely +used. In this study, we use the vector MIT (vMIT) bag model +to describe the quark matter interactions [5, 6]. In the MIT +bag model, it has been assumed that the quarks are bound +in a bag of finite dimensions. In contrast to their absolute +mass, which is very high, it is hypothesized that quarks in- +side the bag have a very low mass. The system is given a +bag constant B as a constant energy density to balance the +bag’s behavior and determine its size. The inward pressure +at the bag’s surface counterbalances the outward pressure the +quarks cause, which means the pressure between the true and +perturbative vacuum. Consequently, as B increases, the quark +pressure lowers, which impacts the star’s structure. The value +of B relies on the mass of the strange quark when u and d +quarks have very low masses. The values of B still need to +be established and are fully model-dependent. One can con- +strain its values with the help of observational results. For +∗ llopes@cefetmg.br +† harish.d@iopb.res.in +example, in the observational limit of GW170817, the pre- +dicted values of B1/4 = 134.1 − 141.4 MeV with low-spin +prior and B1/4 = 126.1 − 141.4 MeV with high spin prior +for SQSs [7]. In Ref. [8], they have predicted the range of +B1/4 = 133.68 − 222.5 MeV for SQSs. However, in the +vMIT bag model [5, 6], the value of B can be obtained by +including the stability window, as mentioned in Refs. [3, 4]. +Moreover, there are different phenomenological and macro- +scopic studies suggesting that the quark phases inside the +compact stars can undergo a phase transition into a color su- +perconducting state of 2-flavour superconducting (2SC), and +color-flavor locked (CFL) [9, 10]. They form Cooper pairs +at high density and low-temperature [11]. The gap parameter +(∆) determines the pairing strength of Cooper pairs influence +the formation of pure CFL stars [12–17] and CFL magnetars +[18, 19]. Recently, it has been suggested that with the proper +choice of ∆ and bag pressure B, the CFL stars and their EoS +can successfully reproduce various observational constraints +such as GW and NICER results [20–22]. In this study, we +want to explore the dark matter (DM) effects on the strange +stars with with and without CFL phases and try to constrain +the macroscopic properties with various observational data. +The compact objects such as NS, white dwarfs captures +some amount of DM inside it in their evolving time. The +amount of DM particles acrreted inside the star due to its im- +mense gravitational potential. Various theoretical predictions +provide us with the unknown nature of DM. Still, numerous +work has been fully dedicated to explaining its properties by +applying it to different systems such as white dwarf [23], NS +[24–28], and even our earth [29]. In the present study, we as- +sume that the SQSs might contain a certain amount of DM +in their life time. The types of DM particles may be either +bosonic or fermionic, and also the percentage of DM depends +on the (i) evolution time and (ii) types of accretions. How- +ever, the accreted DM particles interact directly or indirectly +with hadrons by exchanging other bosonic particles, mainly +depending on the model used. Here, we take different types +of possible scenarios for DM admixed SQS. +The direct detection experiments have already been estab- +lished, such as XENON100 [30], XENON1T[31], CDMS +[32], LUX [33], PANDAX-II [34] etc. to measure the scat- +tering cross-section of the DM and nucleons. +Although, +they provided some exclusions bound to the scattering cross- +section. Still, the null results provided by the experiments +alluded to an inconclusive nature of DM. However, the exclu- +sion bounds prescribed by such direct detection experiments +arXiv:2301.00567v1 [astro-ph.HE] 2 Jan 2023 + +2 +depend on the local DM density around the solar neighbor- +hood, which does not affect the density of DM in the NS/SQS +environment. After the accretion of DM inside NS/SQS, it +collides with nucleons or quarks by losing its kinetic energy, +and eventually, it is bound inside the star. When the accre- +tion ends, the DM particles finally reach thermal equilibrium +with one another due to their internal interactions. This ex- +plains why NSs with admixed DM have essentially constant +DM particle densities [25, 27, 28, 35]. Therefore, the accreted +DM particles are restricted to a narrow radius area inside the +star. In this study, we choose two types of DM and see their +effects on the SQS properties with the vMIT bag model and a +model with superconducting phases. +Recently, the fastest and heaviest Galactic NS named PSR +J0952-0607 (black widow) in the disk of the Milky Way has +been detected to have mass M = 2.35 ± 0.17 M⊙ in continu- +ation of the pulsars PSR J0740+6620 (M = 2.08 ± 0.07 M⊙ +[36, 37]). The simultaneous measurements of the M and R +for NS are done by neutron star interior composition explorer +(NICER) [38, 39] while the limit on the dimensionless tidal +deformability of Λ1.4 = 190+390 +−120 was provided in GW170817 +event [40]. We calculate the mass, radius, and tidal deforma- +bility for the DM admixed SQS and put constraints using +the observational results obtained from different x-ray/pulsars +data, GW170817 data. +II. +FORMALISM +A. +Vector MIT bag model +We use the thermodynamic consistent vector MIT bag +model introduced in Ref. [5, 6] to describe the quark matter. +In this model, the quark interaction is mediated by the vec- +tor channel V µ, analogous to the ω meson in QHD [41]. Its +Lagrangian reads: +LvMIT = +� +¯ψq +� +γµ(i∂µ − gqV Vµ) − mq +� +ψq +−B + 1 +2m2 +V V µVµ +� +Θ( ¯ψqψq), +(1) +where mq is the mass of the quark q of flavor u, d or s, ψq +is the Dirac quark field, B is the constant vacuum pressure, +and Θ( ¯ψqψq) is the Heaviside step function to assure that +the quarks exist only confined to the bag. Applying Euler- +Lagrange, we obtain the energy eigenvalue, which at T = 0 +K, is also the chemical potential: +Eq = µq = +� +m2q + k2 + gqV Vµ, +(2) +now, using Fermi-Dirac statistics, we can obtain the EoS in +mean field approximation. The energy density of the quarks +is: +ϵq = Nc +π2 +� kf +0 +Eqk2d3k, +(3) +where Nc = 3 is the number of colors and kf is the Fermi +momentum. The contribution of the bag and the mesonic mass +term is obtained with the Hamiltonian: H = −⟨L⟩. The total +quark energy density now reads: +ϵ = +� +q +ϵq + B − 1 +2m2 +vV 2 +0 . +(4) +To construct an electrically neutral, beta-stable matter, leptons +are added as a free Fermi gas. The pressure is obtained via the +relation: p = � µn − ϵ, where the sum runs over all the +fermions. +The parameters utilized in this work are the same as pre- +sented in Ref. [5]. We use mu = md = 4 MeV, and ms = 95 +MeV. We also assume a universal coupling of quarks with the +vector meson, i.e., guV = gdV = gsV = gV , and use a value +of GV = 0.3 fm2 as defined below: +GV = +� gV +mV +�2 += 0.3 fm2. +(5) +Now, the value of GV is somewhat arbitrary. To reproduce +stable strange matter, the value of GV combined with the bag +must lie in the range known as the stability window. The sta- +bility window is related to the so-called Bodmer-Witten con- +jecture [3, 4], which states that the true ground state of the +strongly interacting matter is not protons and neutrons but +consists of strange quark matter, which in turn is composed +of deconfined up, down, and strange quarks. For the SQM +hypothesis to be accurate, the energy per baryon of the decon- +fined phase (for p = 0 and T = 0) is lower than the nonstrange +infinite baryonic matter [3–5]. +Euds/A < 930 MeV, +(6) +at the same time, the nonstrange matter still needs to have +an energy per baryon higher than nonstrange infinite baryonic +one; otherwise, protons and neutrons would decay into u and +d quarks: +Eud/A > 930 MeV. +(7) +Therefore, both, Eqs. 6 and 7 must simultaneously satisfied. +For GV = 0.3 fm2 used in this work, the stability window +lies between 139 MeV < B1/4 < 146 MeV [5]. Here, we +assume the maximum allowed value: B1/4 = 146 MeV, as it +will produce the lower radius for the canonical star, as well the +lower value of the dimensionless tidal parameter Λ, while still +producing very massive strange quark stars, M > 2.40 M⊙. +B. +Superconducting CFL quark matter via analytical +approximation +Due to the low temperature and high densities reached in +the strange star interiors, the quark matter may be a color +superconductor, which is a degenerate Fermi gas of quarks +with a condensate of Cooper pairs near the Fermi surface +that induces color Meissner effects [11]. Among the vari- +ous possible configurations of superconducting matter, we can +cite two possibilities: The two-flavor color-superconducting + +3 +phase, where quarks with two out of three colors and two out +of three flavors pair in the standard BCS fashion. The flavors +with the most phase space near their Fermi surfaces, namely, +u and d, are the ones that pair, leaving the strange quark and +the remaining color unpaired. Such phase is expected at den- +sities around 2 < n/n0 < 4 [42]. Another one is the color- +flavor locked phase, where the up, down, and strange quarks +can be treated on an equal footing, and the disruptive effects +of the strange quark mass can be neglected. In this phase, +quarks of all three colors and all three flavors form conven- +tional spinless Cooper pairs. The CFL phase is expected at +n > 4n0 [42]. For additional discussion about 2SC, CFL, +and other color superconducting phases, see Ref. [11] and the +references therein. +The 2SC and the CFL phases were explored within the NJL +model in Ref. [43], while in Ref. [42], the authors show that +the color superconducting NJL EoS is very well fitted by an +analytical approximation, called constant-sound-speed (CSS) +parameterization, whose EoS reads [42, 44, 45]: +p = a(ϵ − ϵ∗), +n = n∗[(1 + a)p/(aϵ∗)]1/(1+a). +(8) +We have, therefore, three free parameters, the square of the +speed of sound (v2 +s = a), the energy density at p = 0 (ϵ∗), +which plays a role similar to the bag in the MIT base models, +and the number density at p = 0 (n∗). In Ref. [42], the authors +freely vary the value of a in the range 0.2 < a < 0.8 and +found that - depending on the NJL parametrization - the 2SC +phase is well described by a < 0.33 while the CFL phase is +described by a > 0.35. On the other hand, Ref. [44] uses the +extreme case a = 1. Here we consider that the quark matter +is in the CFL phase and use an intermediate value, a = 0.6 +(see the text and Fig. 4 from Ref. [42], as well Ref. [45]). +The value of ϵ∗ is chosen as 203 MeV/fm3 to match the value +coming from the vector MIT bag model. Finally, n∗ has to +be constrained, as we still need to reproduce strange quark +stars in accordance with the Bodmer-Witten conjecture. We +choose n∗ = 0.24 fm−3, which is very close to n0 = 0.23 fm−3 +coming from the vector MIT. Within this value, we have E/A += 906 MeV, with implies that the analytical approximation +of the CFL satisfies Eq. 6 and, therefore, the Bodmer-Witten +conjecture. +III. +RESULTS AND DISCUSSIONS +A. +Bosonic DM +This section briefly reviews the formalism of a bosonic DM +model initially proposed in Refs. [46, 47]. At very low tem- +peratures, all particles in a dilute Bose gas condense to the +same quantum ground state, forming a Bose-Einstein Con- +densate (BEC). Particles become correlated when their wave- +lengths overlap; that means the thermal wavelength is greater +than the mean inter-particle distance. Assuming T = 0 K +approximation, almost all the DM particles are in the con- +densate. Only binary collisions at low energy are relevant in +a dilute and cold gas. These collisions are characterized by +a single parameter, the s-wave scattering length la, indepen- +dently of the details of the two-body potential. Therefore, one +can replace the interaction potential with an effective repul- +sive interaction [48]: +V (⃗r − ⃗r′) = 4πla +mx +δ(⃗r − ⃗r′), +(9) +where mx is the mass of the bosonic DM. +The ground state properties of the DM are described by the +mean-field Gross-Pitaevskii (GP) equation, and the equation +of the state (EoS) has the form [27, 46, 47, 49]: +px = 2πla +m3x +ϵ2 +x.. +(10) +The scattering length la is assumed equal to 1 fm, as in the +Ref. [27, 46, 47, 49]. Moreover, the pressure strongly de- +pends on the bosonic DM’s mass due to the cubic dependence. +Therefore this parameter must be taken with care. Based on +the self-interaction cross-section of the DM constraint (see +Refs. [27, 49], the DM mass in the range 50 MeV < mx < +160 MeV. However, the original works from Ref. [46, 47] sug- +gest a mass of around 1 GeV. It is worth emphasizing that a +mass ten times larger imply in pressure 1000 times lower! In +Ref. [50], the authors use a slightly different model of bosonic +DM, where the self-interaction is based on a scalar quartic +term in the potential. +They use the same constraint based +on the self-interaction cross-section of the DM and suggest +a mass of 400 MeV. To explore the ambiguity relative to the +mass of the bosonic DM, we use here two values: 100 MeV, +which agrees with Ref. [27, 49] and 400 MeV, which is in +agreement with Ref. [50], and it is not so far from 1 GeV as +suggested in Ref. [46, 47]. With these settings, the pressure +for mx = 400 MeV is 64 times lower than for mx = 100 MeV. +The total EoS of the strange star is, therefore, the sum of +the contribution of the ordinary quark matter and the DM: +p = pq + px, +and +ϵ = ϵq + ϵx. +(11) +Another important quantity is the fraction of the DM. To solve +the TOV equations [51], we need to specify the central values +both for normal matter and for DM: pq(0), px(0) respectively. +Here, we follow Ref. [27, 49] and define: +fx = +px(0) +pq(0) + px(0), +(12) +and use three different values for fx = 0.05, 0.075 and 0.10. +As pointed out in Ref. [27, 49], these values agree with the +current DM constraints obtained from stars like the Sun. +1. Bosonic DM within vector MIT bag model +In Fig. 1, we plot the TOV solution for bosonic DM ad- +mixed strange stars with the mass of 100 MeV and 400 MeV. +As can be seen, for a bosonic DM mass of 100 MeV, we +have an increase in the maximum mass with the increase of the + +4 + 1 + 1.4 + 1.8 + 2.2 + 2.6 + 10 + 10.5 + 11 + 11.5 + 12 + 12.5 + 13 +mx = 100 MeV +M/M0 +R (km) +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 + 1 + 1.4 + 1.8 + 2.2 + 2.6 + 10 + 10.5 + 11 + 11.5 + 12 + 12.5 + 13 +mx = 400 MeV +M/M0 +R (km) +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 +FIG. 1. Mass-radius relation for bosonic DM admixed strange stars +with mx =100 MeV (left) and mx = 400 MeV (right). +fraction of DM. This result is coherent with those presented in +Ref. [27, 49] for the original, massless MIT. Moreover, as in +the case of the original massless MIT, with the massive vector +MIT, we also see that the presence of DM affects only massive +stars. Strange stars with M < 1.5 M⊙ reproduced essentially +the same radii. The maximum masses vary from 2.41 M⊙ for +pure strange stars to 2.51M⊙ for bosonic DM admixed with +a fraction of 0.10. This indicates that the PSR J0740+6620 +with a gravitational mass of 2.08 ± 0.07 M⊙ [37] can indeed +be a stable strange star with or without admixed bosonic DM. +Even the possible mass of 2.35 ± 0.17 M⊙ of the black widow +pulsar PSR J0952-0607 [52] can be explained as bosonic DM +matter admixed strange star. On the other hand, the radius +of the canonical star is in the narrow range of 11.37 km to +11.40 km. In the literature, there is no consensus about the +true value of the radius of the canonical star. For instance, in +ref. [53], the constraint on the radius of the canonical star is +10.1 − 11.1 km, which provides a very narrow range. If this +is true, neither of our results can fulfill such tight constraints. +In Ref. [54], an upper limit of 11.9 km was provided. In this +case, our results are in full agreement. However, recent results +from the NICER x-ray telescope point that the radius of the +canonical star is between 11.52 km and 13.85 km [39] and +between 11.96 km and 14.26 km as given in Ref. [38]. In +these cases, our radii are too small. + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 100 MeV +Λ +M/M0 +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 400 MeV +Λ +M/M0 +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 +FIG. 2. Dimensionless tidal parameter Λ for bosonic DM admixed +strange stars with mx = 100 MeV (top) and mx = 400 MeV (bot- +tom). +Now, we have opposite results for a mass mx = 400 MeV! +First, the maximum mass decrease with the increase of DM +fraction, dropping from 2.41 M⊙ to 2.29 M⊙ for a fraction fx +of 0.10. However, all values agree with the mass of the PSR +J0740+6620 [37] and the PSR J0952-0607 [52]. Secondly, +we see that even low-mass strange stars are already affected +by the DM and are significantly more compact. The radius +of the 1.4 M⊙ strange star can reach a value as low as 11.08 +km. Therefore, this result is in agreement with both Refs. [53, +54]. The polytropic EoS of Eq. 10 can easily explain these +results. A four times higher DM matter mass produces sixty- +four times smaller pressure! The reduction of the pressure +causes the reduction of the maximum mass and increases the +star compression. +Another essential quantity and constraint is the so-called +dimensionless tidal deformability parameter Λ. If we put an +extended body in an inhomogeneous external field, it will ex- +perience different forces throughout its surface. The result is a +tidal interaction. The tidal deformability of a compact object +is a single parameter λ that quantifies how easily the object +is deformed when subjected to an external tidal field. Larger +tidal deformability indicates that the object is easily deformed. +Conversely, a compact object with a small tidal deformability +parameter is more compact and more difficult to deform. The + +5 +TABLE I. Macroscopic properties of bosonic DM admixed strange +stars +mx (MeV) +fx +M/M⊙ R (km) R1.4 (km) Λ1.4 +100 +0.000 +2.41 +11.86 +11.37 +644 +100 +0.050 +2.46 +12.01 +11.37 +638 +100 +0.075 +2.48 +12.06 +11.38 +645 +100 +0.100 +2.51 +12.08 +11.40 +652 +400 +0.000 +2.41 +11.86 +11.37 +644 +400 +0.050 +2.31 +11.42 +11.16 +526 +400 +0.075 +2.30 +11.38 +11.12 +497 +400 +0.100 +2.29 +11.31 +11.08 +480 +tidal deformability is defined as: +Λ ≡ +λ +M 5 ≡ 2k2 +3C5 , +(13) +where M is the compact object mass and C = GM/R is +its compactness. The parameter k2 is called the second (or- +der) Love number. Additional discussion about the theory of +tidal deformability and the tidal Love numbers are beyond the +scope of this work and can be found in Refs. [22, 40, 55– +59] and references therein. Nevertheless, as pointed out in +Refs. [22, 58], the value of yR must be corrected since strange +stars are self-bound and present a discontinuity at the surface. +Therefore we must have +yR → yR − 4πR3∆ϵS +M +, +(14) +where R and M are the star radius and mass, respectively, +and ∆ϵS is the difference between the energy density at the +surface (p = 0) and the star’s exterior (which implies ϵ = 0). +The results for the dimensionless tidal parameter are displayed +in Fig. 2. +As we can be seen, some features present in the mass-radius +relation are also present here. For instance, for a mass mx = +100 MeV, the low masses of strange stars have similar tidal +parameters, despite their DM fraction. The tidal parameter for +the canonical mass lies between 638 and 644. These values +are in agreement with the constraint Λ < 800 [55], but fail to +fulfill the constraint 70 < Λ < 580 [40]. +In the case of mx = 400 MeV, the strange stars’ huge com- +pression due to an increase in the DM fraction reduces the +tidal parameter Λ. The tidal parameter now lies around 500. +This indicates that for mx = 400 MeV, we are able to explain +very massive neutron stars as the PSR J0952-0607 [52], and +simultaneously fulfills the constraints of Λ < 800 [55] and 70 +< Λ < 580 [40]. We summarize the results of this section in +Tab I. +2. Bosonic DM within CFL quark matter +In order to better understand the effects of the DM in +strange stars, we now assume that the quark matters are in + 1 + 1.5 + 2 + 2.5 + 3 + 10 + 11 + 12 + 13 + 14 +mx = 100 MeV +M/M0 +R (km) +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 + 1 + 1.5 + 2 + 2.5 + 3 + 10 + 11 + 12 + 13 + 14 +mx = 400 MeV +M/M0 +R (km) +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 +FIG. 3. Mass-radius relation for bosonic DM admixed CFL strange +stars with mx =100 MeV (left) and mx = 400 MeV (right). +the CFL superconducting phase via the analytical approxima- +tion EoS in Eq. 8. The mass-radius relations are presented in +Fig. 3. +As can be seen, for a bosonic DM mass of 100 MeV, we +have an increase in the maximum mass with the increase of the +fraction of DM. The qualitative results for CFL superconduct- +ing quark stars are analogous to both the original, massless +MIT as showed in Ref. [27, 49], as well for the massive vector +MIT bag model as presented in the last section. This indicates +a possible model-independent behavior about the effect of the +bosonic DM. Moreover, as in the case of the original mass- +less MIT and the massive vector MIT, in the CFL phase, we +also see that the presence of DM affects only massive stars. +CFL strange stars with M < 1.8 M⊙ reproduced essentially +the same radii. The maximum masses vary from 2.81 M⊙ for +pure strange stars to 2.88M⊙ for bosonic DM admixed with a +fraction of 0.10. This indicates that the PSR J0740+6620 with +M = 2.08 ± 0.07 M⊙ [37] can be a stable CFL strange star +with or without admixed bosonic DM. Even the possible mass +of 2.35 ± 0.17 M⊙ of the pulsar PSR J0952-0607 [52] can +be explained as bosonic DM matter admixed strange star. On +the other hand, the radius of the canonical star presents almost +no variation and is fixed at around 11.57 km. Such a value is +too low to reproduce the constraint range of 10.1 − 11.1 km, +shown in Ref. [53] while agreeing with Ref. [54], whose upper + +6 + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 100 MeV +Λ +M/M0 +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 400 MeV +Λ +M/M0 +fx = 0.000 +fx = 0.050 +fx = 0.075 +fx = 0.100 +FIG. 4. Dimensionless tidal parameter Λ for bosonic DM admixed +CFL superconducting strange stars with mx = 100 MeV (top) and +mx = 400 MeV (bottom). +limit is 11.9 km. About the NICER x-ray telescope, the con- +straint between 11.52 km and 13.85 km pointed in Ref. [39] +is fulfilled, but the bound in the range between 11.96 km and +14.26 km (Ref. [38]) is not. +For a mass mx = 400 MeV, the results for CLF super- +conducting strange stars are analogous to the massive MIT +bag model discussed in the last section. The maximum mass +decrease with the increase of DM fraction, dropping from +2.81 M⊙ to 2.61 M⊙ for a fraction fx of 0.10. However, all +values agree with the mass of the PSR J0740+6620 [37] and +the black widow pulsar PSR J0952-0607 [52]. Secondly, we +see that even low-mass strange stars are already affected by +the DM and are significantly more compact. The radius of +the 1.4 M⊙ for fx = 0.10 is about 11.29 km. Such a low ra- +dius fails to fulfill both NICER constraints [38, 39], but is in +agreement with Capano et al. [54]. The reduction of the CFL +strange star and its compression can again be explained by the +polytropic EoS of Eq. 10. A four times higher DM matter +mass produces sixty-four times smaller pressure! The reduc- +tion of the pressure causes the reduction of the maximum mass +and increases the star compression. +We also calculate the dimensionless tidal parameter Λ for +the CFL superconducting strange stars. The results are pre- +sented in Fig. 4. As we can be seen, the results are analogous +TABLE II. Macroscopic properties of bosonic DM admixed CFL su- +perconducting strange stars +mx (MeV) +fx +M/M⊙ R (km) R1.4 (km) Λ1.4 +100 +0.000 +2.81 +12.89 +11.57 +721 +100 +0.050 +2.83 +12.84 +11.57 +709 +100 +0.075 +2.86 +12.96 +11.58 +717 +100 +0.100 +2.88 +13.00 +11.58 +717 +400 +0.000 +2.81 +12.89 +11.57 +721 +400 +0.050 +2.63 +12.30 +11.37 +570 +400 +0.075 +2.62 +12.22 +11.32 +545 +400 +0.100 +2.61 +12.13 +11.29 +531 +to the vector MIT bag model. As in the case of the mass-radius +relation, for low mass stars there is very low variation in the +Λ. For instance, for a mass mx = 100 MeV, the low masses +strange stars have similar tidal parameters, despite their DM +fraction. The tidal parameter for the canonical mass lies be- +tween 709 and 721. These values are in agreement with the +constraint Λ < 800 [55], but fail to fulfill the constraint 70 +< Λ < 580 [40]. +In the case of mx = 400 MeV, the results for CFL super- +conducting strange stars are again analogous to vector MIT +strange stars. The stars’ huge compression as the DM fraction +increases reduce the tidal parameter Λ. The tidal parameter +now lies around 550. This indicates that for mx = 400 MeV, +we are able to explain very massive neutron stars as the PSR +J0952-0607 [52], and simultaneously fulfills the constraints +of Λ < 800 [55] and 70 < Λ < 580 [40]. We summarize the +results of this section in Tab II. +B. +Fermionic DM +The Lagrangian of the fermionic DM reads [22, 25, 35]: +LDM = ¯χ(iγµ∂µ − (mx − gHh))χ ++1 +2(∂µh∂µh − m2 +Hh2). +(15) +Here, we assume a dark fermion represented by the Dirac +field χ that self-interacts through the exchange of the Higgs +boson, whose mass is mH = 125 GeV. The coupling con- +stant is assumed to be gH = 0.1, which agrees with the con- +straints in Refs. [25, 27]. Within this prescription, the DM +self-interaction is very feeble and behaves as a free Fermi gas. +More explicitly, the strength of the interaction is: +GH = +� gH +mH +�2 += 2.492 × 10−8 +fm2. +(16) +The EoS is easily obtained in mean field approximation, +completely analogous to the QHD model [41]. The fermionic +DM is assumed to be the lightest neutralino, with mx = 200 +GeV, as done in Ref. [25, 35]. However, as pointed out in +Ref. [60], the lower limit for weakly interacting massive par- +ticles (WIMP) is 60 GeV. Therefore we also use mx = 60 GeV + +7 + 1 + 1.4 + 1.8 + 2.2 + 2.6 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 +mx = 200 GeV +M/M0 +R (km) +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV + 1 + 1.4 + 1.8 + 2.2 + 2.6 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 +mx = 60 GeV +M/M0 +R (km) +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV +FIG. 5. Mass-radius relation for fermionic DM admixed strange stars +with mx = 200 GeV (top) and mx = 60 GeV (bottom). +to better study the influence of the DM mass. As in the case +of the bosonic DM, we must fix the DM fraction. As we are +dealing here with fermionic DM, we follow ref. [25, 28, 35] +and use the Fermi momentum to fix the DM fraction, using +three different values: kDM +F += 0.02 GeV, 0.04 GeV, and 0.06 +GeV. +1. Fermionic DM within vector MIT bag model +We display in Fig. 5 the TOV solution for a fermionic DM +with a mass of 200 GeV and 60 GeV within the vector MIT +bag model. As can be seen, the results for fermionic DM are +significantly different when compared with bosonic DM. The +maximum masses are always reduced, and the star compres- +sion always increases, even for very low masses. Also, differ- +ent DM fractions always produce different mass-radius rela- +tions, affecting all the strange star families, unlike the bosonic +case, where we have very similar stars for different DM frac- +tions, which is easily understood by the different criteria of +the DM fraction. In the case of bosonic DM, the DM frac- +tion is dependent on the quark EoS via Eq. 12. In the case of +fermionic DM, the Fermi momentum is fixed and independent +of the quark EoS. +Qualitatively, the results for mx = 200 GeV and 60 GeV +TABLE III. Macroscopic properties of fermionic DM admixed +strange stars +mx (GeV) kDM +F +(GeV) M/M⊙ R (km) R1.4 (km) Λ1.4 +200 +0.000 +2.41 +11.86 +11.37 +644 +200 +0.02 +2.37 +11.75 +11.22 +586 +200 +0.04 +2.16 +10.70 +10.39 +346 +200 +0.06 +1.80 +8.72 +8.95 +108 +60 +0.000 +2.41 +11.86 +11.37 +644 +60 +0.02 +2.40 +11.84 +11.30 +625 +60 +0.04 +2.33 +11.46 +11.05 +524 +60 +0.06 +2.16 +11.31 +10.42 +351 +are the same. Increasing the DM fraction compress the star +and reduces the maximum mass. Quantitatively, we see that +a higher DM mass has a strong influence once it has a higher +increase in the energy density, and at the same time, that pro- +duces a lower contribution to the pressure. The maximum +mass drops from 2.41 M⊙ for kDM +F += 0.00 to only 1.80 M⊙ +for kDM +F += 0.06 GeV in the case of mx = 200 GeV and to 2.16 +M⊙ for mx = 60 GeV. In the same sense, the radius of the +canonical star drops from 11.37 km for kDM +F += 0.00 to only +8.95 km for kDM +F += 0.06 GeV in the case of mx = 200 GeV, +and to 10.42 km for mx = 60 GeV. As can be seen, the results +for kDM +F += 0.06 GeV with mx = 200 GeV can be ruled out +once it has a very low maximum mass in disagreement with +the NICER result of the PSR J0740+6620 with a gravitational +mass of 2.08 ± 0.07 M⊙ [37], and also a very low radius for +the canonical star, in disagreement even with the low limit of +10.1 km presented in Ref. [53]. +We plot in Fig. 6 the dimensionless parameter Λ for +fermionic DM admixed strange stars with mx = 200 GeV and +mx = 60 GeV within the vector MIT bag model. As we can +see, the strong compression due to the fermionic DM contri- +bution reduces the tidal parameter significantly. In the case +with mx = 200 GeV and kDM +F += 0.06 GeV, the tidal parame- +ter drops to only 108, which is six times lower than for kDM +F += +0.00, although, as we pointed out before, such parametrization +must be ruled out. +As can be seen, most of the parametrizations are able to +fulfill the main constraints for pulsar observations, i.e., M > +2.01M⊙ and 70 +< +Λ +< 580. Indeed, the presence of +DM improves the theoretical prediction and the observational +constraints, although it can be some debate about the radius of +the canonical star. They do not fulfill NICER results [38, 39] +but agree with Ref. [54]. +It is also worth to point the existence of almost degenerate +results. As can be seen, for mx = 200 GeV with kDM +f += 0.04 +GeV, the macroscopic are essentially the same for the mx = 60 +GeV and kDM +f += 0.06 GeV. The main results are summarized +in Tab. III. + +8 + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 200 GeV +Λ +M/M0 +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 60 GeV +Λ +M/M0 +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV +FIG. 6. Dimensionless tidal parameter Λ for fermionic DM admixed +strange stars with mx = 200 GeV (top) and mx = 60 GeV (bottom). +2. Fermionic DM within CFL quark matter +We now study the effect of Fermionic DM in CFL super- +conducting matter described by the analytical approximation +of Eq. 8. +We display in Fig. 7 the TOV solution for a fermionic DM +with a mass of 200 GeV and 60 GeV. As in the case of the vec- +tor MIT bag model, for CFL superconducting quark matter, +the results for fermionic DM are significantly different when +compared with bosonic DM. And again, the qualitative effect +of fermionic DM is the same for CFL as it is for the vector +MIT bag model. The maximum masses are always reduced, +and the star compression always increases, even for very low +masses. Again, different DM fractions always produce differ- +ent mass-radius relations, affecting all the strange star fami- +lies. +From the quantitative point of view, the maximum mass +drops from 2.81 M⊙ for kDM +F += 0.00 to 2.04 M⊙ for kDM +F += 0.06 GeV in the case of mx = 200 GeV and to 2.49 M⊙ +for mx = 60 GeV. In the same sense, the radius of the canon- +ical star drops from 11.57 km for kDM +F += 0.00 to 9.21 km for +kDM +F += 0.06 GeV in the case of mx = 200 GeV, and 10.66 +km for mx = 60 GeV. Now, unlike the case of the vector MIT, +none of the CFL superconducting strange stars can be ruled + 1 + 1.5 + 2 + 2.5 + 3 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 +mx = 200 GeV +M/M0 +R (km) +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV + 1 + 1.5 + 2 + 2.5 + 3 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 +mx = 60 GeV +M/M0 +R (km) +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV +FIG. 7. Mass-radius relation for fermionic DM admixed CFL super- +conducting strange stars with mx = 200 GeV (top) and mx = 60 +GeV (bottom). +out in the light of the PSR J0740+6620, M += 2.08 ± 0.07 +M⊙ [37], although for kDM +F += 0.06 GeV and mx = 200 GeV +the radius of the canonical star is below the lower limit of 10.1 +km presented in Ref. [53]. +We plot in Fig. 8 the dimensionless parameter Λ for +fermionic DM admixed superconducting strange stars with +mx = 200 GeV and mx = 60 GeV. The results are completely +analogous to the case of the vector MIT bag model; however, +the value of Λ here is always higher. The compression due to +the fermionic DM contribution reduces the tidal parameter. In +the case with mx = 200 GeV and kDM +F += 0.06 GeV, the tidal +parameter drops from 721 to 151. +It is also worth noting that some parametrizations can ful- +fill the main constraints for pulsar observations, 70 < Λ < +580, and yet produce a very high maximum mass, sometimes +reaching 2.50 M⊙. The presence of DM again improves the +theoretical prediction and the observational constraints, al- +though it can be some debate about the radius of the canonical +star. They do not fulfill NICER results [38, 39], but agree with +Ref. [54]. Moreover, most parametrizations can explain even +the black widow pulsar PSR J0952-0607 [52]. +Finally, even when we use a different model for the quark +matter, the existence of almost degenerate results is still +present: for mx = 200 GeV with kDM +f += 0.04 GeV and mx = + +9 + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 200 GeV +Λ +M/M0 +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV + 0 + 200 + 400 + 600 + 800 + 1000 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 +mx = 60 GeV +Λ +M/M0 +kf = 0.00 GeV +kf = 0.02 GeV +kf = 0.04 GeV +kf = 0.06 GeV +FIG. 8. Dimensionless tidal parameter Λ for fermionic DM admixed +CFL strange stars with mx = 200 GeV (top) and mx = 60 GeV +(bottom). +TABLE IV. Macroscopic properties of fermionic DM admixed color +superconducting quark stars +mx (GeV) kDM +F +(GeV) M/M⊙ R (km) R1.4 (km) Λ1.4 +200 +0.000 +2.81 +12.89 +11.57 +721 +200 +0.02 +2.75 +12.65 +11.43 +653 +200 +0.04 +2.50 +11.50 +10.67 +421 +200 +0.06 +2.04 +9.38 +9.21 +151 +60 +0.000 +2.81 +12.89 +11.57 +721 +60 +0.02 +2.78 +12.75 +11.53 +694 +60 +0.04 +2.69 +12.45 +11.28 +610 +60 +0.06 +2.49 +11.53 +10.66 +422 +60 GeV with kDM +f += 0.06 GeV. The main results are summa- +rized in Tab. IV. +3. Fermionic DM with a vector channel +Now we study if the presence of a dark, repulsive vector +channel affects the macroscopic properties of the fermionic +DM admixed strange stars. The new Lagrangian is the La- +grangian in Eq. 15 plus the repulsive channel and the respec- +tive meson mass, and reads [28]: +LVDM = gξ ¯χ(γµξµ)χ + 1 +2m2 +ξξµξµ − 1 +4V µνVµν. +(17) +The Lagrangian of Eq. 17 is analogous to the ω contribution +to the QHD Lagrangian [5, 41]. Indeed, the junction of Eq. 15 +and Eq. 17 makes this model of DM fully analogous to the +original σ −ω model of the QHD [41]. The coupling constant +gξ = 0.1 is fixed, and it is equal to the gH, while the mass of +the vector of the dark meson is assumed to be 34 MeV, follow- +ing Ref. [28]. As the mass of the dark vector meson is 3000 +times smaller than the mass of the Higgs boson, the repulsive +channel is much stronger than the attractive one. Indeed, we +have +Gξ = +� gξ +mξ +�2 += 0.337 +fm2, +(18) +which is stronger than the quark repulsion and millions of +times higher than the DM scalar coupling (Eq. 16). Never- +theless, despite the strong self-repulsion of the fermionic DM, +the numerical results are barely affected by the repulsive chan- +nel. For the vector MIT bag model, the only noticeable dif- +ference appears for mx = 200 GeV and kDM +F += 0.04 GeV. In +this case, the maximum mass increase from 2.16 M⊙ to 2.17 +M⊙. The radius of the canonical star also grows from 11.39 +km to 11.46 km. The tidal parameter Λ1.4 also increases from +346 to 358. It is worth noticing that all these variations are far +beyond the precision with which experimental measurements +are made. All the other parametrizations present even lower +(or none) differences. Herefore, we do not provide any figures +in this section since they would be visually indistinguishable +from those in the last paragraph. We only display the main +results in Tab. V. In the case of CFL superconducting quark +matter, the differences are even smaller! +The nature of the vector coupling can explain why the +differences are so small. The vector mesons couple to the +number density, and we are dealing with a very low-density +regime. Indeed, even kDM +F += 0.06 GeV implies a number den- +sity is around 9.6 × 10−4 fm−3. Of course, we could increase +the repulsion of the dark vector boson, but we believe this +would be very unrealistic since DM was proposed to explain +higher attraction in galaxy curves [61]. +IV. +CONCLUSIONS +In this work, we calculate the properties for the DM ad- +mixed for strange quark stars. We use two different mod- +els for the quark model: the vector MIT bag model, as pre- +sented in Refs. [5, 6] and the CFL color superconducting +quark matter via an analytical approximation, as discussed +in Refs. [42, 44, 45]; and two different kinds of dark mat- +ter: a bosonic as discussed in Refs. [27, 46, 47, 49] and for +fermionic [22, 25, 35]. For each kind of DM, we use two +different mass values, and the strange stars always agree with +the Bodmer-Witten conjecture [3, 4]. Our main conclusions +can be summarized as follows: + +10 +TABLE V. Macroscopic properties of dark vector boson fermionic +DM admixed strange stars within the vector MIT bag model. The +only significant differences are for kDM +F += 0.04 GeV +mx (GeV) kDM +F +(GeV) M/M⊙ R (km) R1.4 (km) Λ1.4 +200 +0.000 +2.41 +11.86 +11.37 +644 +200 +0.02 +2.37 +11.75 +11.22 +586 +200 +0.04 +2.17 +10.70 +10.46 +358 +200 +0.06 +1.80 +8.72 +9.01 +112 +60 +0.000 +2.41 +11.86 +11.37 +644 +60 +0.02 +2.40 +11.84 +11.30 +625 +60 +0.04 +2.33 +11.46 +11.10 +532 +60 +0.06 +2.16 +11.31 +10.43 +353 +• The qualitative results for DM admixed strange stars are +independent of the quark model utilized. This is true for +both bosonic and fermionic, as well it is independent of +the DM mass. +• For a bosonic DM with a mass of mx = 100 MeV, we +have an increase of the maximum mass, while the prop- +erties of low-mass strange stars are not significantly af- +fected. This is the only case in that we have an in- +crease in the star’s mass. Such a situation happens for +the vector MIT, the CFL superconducting quark mat- +ter, and also for the massless MIT, as pointed out in +Refs. [27, 49]. +• For a bosonic DM with a mass of mx = 400 MeV, we +have a decrease of the maximum mass, whilst the radii +of the low-mass strange stars, in this case, are also af- +fected. +• For a fermionic DM, the maximum mass always de- +creases. The higher the DM fraction, the lower the max- +imum mass, and the smaller the radii. Also, the higher +the DM mass, the higher the stellar compression and the +lower the maximum mass. +• Although we introduce a repulsive dark vector field +with a mass 3000 times smaller than the attractive scalar +field, we do not find significant variation in the stellar +macroscopic properties. +• There are almost degenerate results both for mx = 200 +GeV with kDM +f += 0.04 GeV and mx = 60 GeV with +kDM +f += 0.06 GeV, the maximum mass, as well the prop- +erties of the canonical star are essentially the same. +• About the observational constraints, we can see that the +mass of the PSR J0740+6620 pulsar, M = 2.08 ± 0.07 +M⊙ [37] is easily obtained. 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Educac¸˜ao Tecnol´ogica de Minas Gerais Campus VIII;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' CEP 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='022-560,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Varginha - MG - Brasil 2Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Sachivalaya Marg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Bhubaneswar 751005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' India and 3Homi Bhabha National Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Training School Complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Anushakti Nagar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Mumbai 400094,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' India (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 2023) In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' we study dark matter (DM) admixed strange quark stars exploring the different possibilities about the nature of the DM and their effects on the macroscopic properties of strange stars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' such as maximum masses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' radii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' as well the dimensionless tidal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We observe that the DM significantly affects the macroscopic properties that depend on the DM mass, type, and fraction inside the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' INTRODUCTION Recently, it was suggested that quarks probable appear in- side the core of a massive neutron star (NS) due to a very high density, where hadronic matter undergoes phase transitions to a new phase of quarks and gluons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Furthermore, several exotic particles, such as hyperons production, dark matter ac- cretions, kaon condensations, and so on, appeared primarily in the core of the NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As a result, in those regimes, the equation of state (EoS) is the fundamental component that can describe both micro/macroscopic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Another possibility is that at least some of the observed pulsars are indeed stable quark stars, or strange stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Several theoretical works have pro- posed the existence of strange quark stars (SQSs) [2], which are made up of u, d, and s quarks in equilibrium in terms of weak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The Bodmen-Witten conjecture states that strange quark matter (SQM) can have a lower energy per baryon than pure nucleons because the exclusion principle may be dominant at absolute zero pressure and temperature [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Hence, the SQM might be the true ground state of the hadronic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Hence, it stands to reason that the SQS must be more stable than the ordinary NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Various phenomenological models have been proposed to explore the SQSs properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Among them, the MIT bag model and Nambu-Jona-Lasinio (NJL) have been widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In this study, we use the vector MIT (vMIT) bag model to describe the quark matter interactions [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the MIT bag model, it has been assumed that the quarks are bound in a bag of finite dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In contrast to their absolute mass, which is very high, it is hypothesized that quarks in- side the bag have a very low mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The system is given a bag constant B as a constant energy density to balance the bag’s behavior and determine its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The inward pressure at the bag’s surface counterbalances the outward pressure the quarks cause, which means the pressure between the true and perturbative vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Consequently, as B increases, the quark pressure lowers, which impacts the star’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The value of B relies on the mass of the strange quark when u and d quarks have very low masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The values of B still need to be established and are fully model-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' One can con- strain its values with the help of observational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For ∗ llopes@cefetmg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='br † harish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='d@iopb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='in example, in the observational limit of GW170817, the pre- dicted values of B1/4 = 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 − 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 MeV with low-spin prior and B1/4 = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 − 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 MeV with high spin prior for SQSs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [8], they have predicted the range of B1/4 = 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='68 − 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 MeV for SQSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' However, in the vMIT bag model [5, 6], the value of B can be obtained by including the stability window, as mentioned in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Moreover, there are different phenomenological and macro- scopic studies suggesting that the quark phases inside the compact stars can undergo a phase transition into a color su- perconducting state of 2-flavour superconducting (2SC), and color-flavor locked (CFL) [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' They form Cooper pairs at high density and low-temperature [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The gap parameter (∆) determines the pairing strength of Cooper pairs influence the formation of pure CFL stars [12–17] and CFL magnetars [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Recently, it has been suggested that with the proper choice of ∆ and bag pressure B, the CFL stars and their EoS can successfully reproduce various observational constraints such as GW and NICER results [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In this study, we want to explore the dark matter (DM) effects on the strange stars with with and without CFL phases and try to constrain the macroscopic properties with various observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The compact objects such as NS, white dwarfs captures some amount of DM inside it in their evolving time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The amount of DM particles acrreted inside the star due to its im- mense gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Various theoretical predictions provide us with the unknown nature of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Still, numerous work has been fully dedicated to explaining its properties by applying it to different systems such as white dwarf [23], NS [24–28], and even our earth [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the present study, we as- sume that the SQSs might contain a certain amount of DM in their life time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The types of DM particles may be either bosonic or fermionic, and also the percentage of DM depends on the (i) evolution time and (ii) types of accretions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' How- ever, the accreted DM particles interact directly or indirectly with hadrons by exchanging other bosonic particles, mainly depending on the model used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Here, we take different types of possible scenarios for DM admixed SQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The direct detection experiments have already been estab- lished, such as XENON100 [30], XENON1T[31], CDMS [32], LUX [33], PANDAX-II [34] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' to measure the scat- tering cross-section of the DM and nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Although, they provided some exclusions bound to the scattering cross- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Still, the null results provided by the experiments alluded to an inconclusive nature of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' However, the exclu- sion bounds prescribed by such direct detection experiments arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00567v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='HE] 2 Jan 2023 2 depend on the local DM density around the solar neighbor- hood, which does not affect the density of DM in the NS/SQS environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' After the accretion of DM inside NS/SQS, it collides with nucleons or quarks by losing its kinetic energy, and eventually, it is bound inside the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' When the accre- tion ends, the DM particles finally reach thermal equilibrium with one another due to their internal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This ex- plains why NSs with admixed DM have essentially constant DM particle densities [25, 27, 28, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Therefore, the accreted DM particles are restricted to a narrow radius area inside the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In this study, we choose two types of DM and see their effects on the SQS properties with the vMIT bag model and a model with superconducting phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Recently, the fastest and heaviest Galactic NS named PSR J0952-0607 (black widow) in the disk of the Milky Way has been detected to have mass M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='17 M⊙ in continu- ation of the pulsars PSR J0740+6620 (M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='07 M⊙ [36, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The simultaneous measurements of the M and R for NS are done by neutron star interior composition explorer (NICER) [38, 39] while the limit on the dimensionless tidal deformability of Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 = 190+390 −120 was provided in GW170817 event [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We calculate the mass, radius, and tidal deforma- bility for the DM admixed SQS and put constraints using the observational results obtained from different x-ray/pulsars data, GW170817 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' FORMALISM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Vector MIT bag model We use the thermodynamic consistent vector MIT bag model introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [5, 6] to describe the quark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In this model, the quark interaction is mediated by the vec- tor channel V µ, analogous to the ω meson in QHD [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Its Lagrangian reads: LvMIT = � ¯ψq � γµ(i∂µ − gqV Vµ) − mq � ψq −B + 1 2m2 V V µVµ � Θ( ¯ψqψq), (1) where mq is the mass of the quark q of flavor u, d or s, ψq is the Dirac quark field, B is the constant vacuum pressure, and Θ( ¯ψqψq) is the Heaviside step function to assure that the quarks exist only confined to the bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Applying Euler- Lagrange, we obtain the energy eigenvalue, which at T = 0 K, is also the chemical potential: Eq = µq = � m2q + k2 + gqV Vµ, (2) now, using Fermi-Dirac statistics, we can obtain the EoS in mean field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The energy density of the quarks is: ϵq = Nc π2 � kf 0 Eqk2d3k, (3) where Nc = 3 is the number of colors and kf is the Fermi momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The contribution of the bag and the mesonic mass term is obtained with the Hamiltonian: H = −⟨L⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The total quark energy density now reads: ϵ = � q ϵq + B − 1 2m2 vV 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (4) To construct an electrically neutral, beta-stable matter, leptons are added as a free Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The pressure is obtained via the relation: p = � µn − ϵ, where the sum runs over all the fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The parameters utilized in this work are the same as pre- sented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We use mu = md = 4 MeV, and ms = 95 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We also assume a universal coupling of quarks with the vector meson, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=', guV = gdV = gsV = gV , and use a value of GV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='3 fm2 as defined below: GV = � gV mV �2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='3 fm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (5) Now, the value of GV is somewhat arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' To reproduce stable strange matter, the value of GV combined with the bag must lie in the range known as the stability window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The sta- bility window is related to the so-called Bodmer-Witten con- jecture [3, 4], which states that the true ground state of the strongly interacting matter is not protons and neutrons but consists of strange quark matter, which in turn is composed of deconfined up, down, and strange quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For the SQM hypothesis to be accurate, the energy per baryon of the decon- fined phase (for p = 0 and T = 0) is lower than the nonstrange infinite baryonic matter [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Euds/A < 930 MeV, (6) at the same time, the nonstrange matter still needs to have an energy per baryon higher than nonstrange infinite baryonic one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' otherwise, protons and neutrons would decay into u and d quarks: Eud/A > 930 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (7) Therefore, both, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 6 and 7 must simultaneously satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For GV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='3 fm2 used in this work, the stability window lies between 139 MeV < B1/4 < 146 MeV [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Here, we assume the maximum allowed value: B1/4 = 146 MeV, as it will produce the lower radius for the canonical star, as well the lower value of the dimensionless tidal parameter Λ, while still producing very massive strange quark stars, M > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='40 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Superconducting CFL quark matter via analytical approximation Due to the low temperature and high densities reached in the strange star interiors, the quark matter may be a color superconductor, which is a degenerate Fermi gas of quarks with a condensate of Cooper pairs near the Fermi surface that induces color Meissner effects [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Among the vari- ous possible configurations of superconducting matter, we can cite two possibilities: The two-flavor color-superconducting 3 phase, where quarks with two out of three colors and two out of three flavors pair in the standard BCS fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The flavors with the most phase space near their Fermi surfaces, namely, u and d, are the ones that pair, leaving the strange quark and the remaining color unpaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Such phase is expected at den- sities around 2 < n/n0 < 4 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Another one is the color- flavor locked phase, where the up, down, and strange quarks can be treated on an equal footing, and the disruptive effects of the strange quark mass can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In this phase, quarks of all three colors and all three flavors form conven- tional spinless Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The CFL phase is expected at n > 4n0 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For additional discussion about 2SC, CFL, and other color superconducting phases, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [11] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The 2SC and the CFL phases were explored within the NJL model in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [43], while in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [42], the authors show that the color superconducting NJL EoS is very well fitted by an analytical approximation, called constant-sound-speed (CSS) parameterization, whose EoS reads [42, 44, 45]: p = a(ϵ − ϵ∗), n = n∗[(1 + a)p/(aϵ∗)]1/(1+a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (8) We have, therefore, three free parameters, the square of the speed of sound (v2 s = a), the energy density at p = 0 (ϵ∗), which plays a role similar to the bag in the MIT base models, and the number density at p = 0 (n∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [42], the authors freely vary the value of a in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 < a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 and found that - depending on the NJL parametrization - the 2SC phase is well described by a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='33 while the CFL phase is described by a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' On the other hand, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [44] uses the extreme case a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Here we consider that the quark matter is in the CFL phase and use an intermediate value, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 (see the text and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 4 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [42], as well Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The value of ϵ∗ is chosen as 203 MeV/fm3 to match the value coming from the vector MIT bag model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Finally, n∗ has to be constrained, as we still need to reproduce strange quark stars in accordance with the Bodmer-Witten conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We choose n∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='24 fm−3, which is very close to n0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='23 fm−3 coming from the vector MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Within this value, we have E/A = 906 MeV, with implies that the analytical approximation of the CFL satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 6 and, therefore, the Bodmer-Witten conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Bosonic DM This section briefly reviews the formalism of a bosonic DM model initially proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' At very low tem- peratures, all particles in a dilute Bose gas condense to the same quantum ground state, forming a Bose-Einstein Con- densate (BEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Particles become correlated when their wave- lengths overlap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' that means the thermal wavelength is greater than the mean inter-particle distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Assuming T = 0 K approximation, almost all the DM particles are in the con- densate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Only binary collisions at low energy are relevant in a dilute and cold gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' These collisions are characterized by a single parameter, the s-wave scattering length la, indepen- dently of the details of the two-body potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Therefore, one can replace the interaction potential with an effective repul- sive interaction [48]: V (⃗r − ⃗r′) = 4πla mx δ(⃗r − ⃗r′), (9) where mx is the mass of the bosonic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The ground state properties of the DM are described by the mean-field Gross-Pitaevskii (GP) equation, and the equation of the state (EoS) has the form [27, 46, 47, 49]: px = 2πla m3x ϵ2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='. (10) The scattering length la is assumed equal to 1 fm, as in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 46, 47, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Moreover, the pressure strongly de- pends on the bosonic DM’s mass due to the cubic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Therefore this parameter must be taken with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Based on the self-interaction cross-section of the DM constraint (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 49], the DM mass in the range 50 MeV < mx < 160 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' However, the original works from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [46, 47] sug- gest a mass of around 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' It is worth emphasizing that a mass ten times larger imply in pressure 1000 times lower!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [50], the authors use a slightly different model of bosonic DM, where the self-interaction is based on a scalar quartic term in the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' They use the same constraint based on the self-interaction cross-section of the DM and suggest a mass of 400 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' To explore the ambiguity relative to the mass of the bosonic DM, we use here two values: 100 MeV, which agrees with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 49] and 400 MeV, which is in agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [50], and it is not so far from 1 GeV as suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' With these settings, the pressure for mx = 400 MeV is 64 times lower than for mx = 100 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The total EoS of the strange star is, therefore, the sum of the contribution of the ordinary quark matter and the DM: p = pq + px, and ϵ = ϵq + ϵx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (11) Another important quantity is the fraction of the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' To solve the TOV equations [51], we need to specify the central values both for normal matter and for DM: pq(0), px(0) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Here, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 49] and define: fx = px(0) pq(0) + px(0), (12) and use three different values for fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 49], these values agree with the current DM constraints obtained from stars like the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Bosonic DM within vector MIT bag model In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 1, we plot the TOV solution for bosonic DM ad- mixed strange stars with the mass of 100 MeV and 400 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As can be seen, for a bosonic DM mass of 100 MeV, we have an increase in the maximum mass with the increase of the 4 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 13 mx = 100 MeV M/M0 R (km) fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 13 mx = 400 MeV M/M0 R (km) fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Mass-radius relation for bosonic DM admixed strange stars with mx =100 MeV (left) and mx = 400 MeV (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' fraction of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This result is coherent with those presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 49] for the original, massless MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Moreover, as in the case of the original massless MIT, with the massive vector MIT, we also see that the presence of DM affects only massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Strange stars with M < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 M⊙ reproduced essentially the same radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The maximum masses vary from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 M⊙ for pure strange stars to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='51M⊙ for bosonic DM admixed with a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This indicates that the PSR J0740+6620 with a gravitational mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='07 M⊙ [37] can indeed be a stable strange star with or without admixed bosonic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Even the possible mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='17 M⊙ of the black widow pulsar PSR J0952-0607 [52] can be explained as bosonic DM matter admixed strange star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' On the other hand, the radius of the canonical star is in the narrow range of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 km to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='40 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the literature, there is no consensus about the true value of the radius of the canonical star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For instance, in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [53], the constraint on the radius of the canonical star is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 km, which provides a very narrow range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' If this is true, neither of our results can fulfill such tight constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [54], an upper limit of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='9 km was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In this case, our results are in full agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' However, recent results from the NICER x-ray telescope point that the radius of the canonical star is between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='52 km and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='85 km [39] and between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='96 km and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='26 km as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In these cases, our radii are too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 100 MeV Λ M/M0 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 400 MeV Λ M/M0 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Dimensionless tidal parameter Λ for bosonic DM admixed strange stars with mx = 100 MeV (top) and mx = 400 MeV (bot- tom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Now, we have opposite results for a mass mx = 400 MeV!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' First, the maximum mass decrease with the increase of DM fraction, dropping from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 M⊙ to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='29 M⊙ for a fraction fx of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' However, all values agree with the mass of the PSR J0740+6620 [37] and the PSR J0952-0607 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Secondly, we see that even low-mass strange stars are already affected by the DM and are significantly more compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The radius of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 M⊙ strange star can reach a value as low as 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Therefore, this result is in agreement with both Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The polytropic EoS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 10 can easily explain these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' A four times higher DM matter mass produces sixty- four times smaller pressure!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The reduction of the pressure causes the reduction of the maximum mass and increases the star compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Another essential quantity and constraint is the so-called dimensionless tidal deformability parameter Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' If we put an extended body in an inhomogeneous external field, it will ex- perience different forces throughout its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The result is a tidal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The tidal deformability of a compact object is a single parameter λ that quantifies how easily the object is deformed when subjected to an external tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Larger tidal deformability indicates that the object is easily deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Conversely, a compact object with a small tidal deformability parameter is more compact and more difficult to deform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The 5 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Macroscopic properties of bosonic DM admixed strange stars mx (MeV) fx M/M⊙ R (km) R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 (km) Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='86 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 644 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='46 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='01 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 638 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='48 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='38 645 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='51 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='40 652 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='86 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 644 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='31 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='42 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='16 526 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='30 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='38 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='12 497 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='29 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='31 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 480 tidal deformability is defined as: Λ ≡ λ M 5 ≡ 2k2 3C5 , (13) where M is the compact object mass and C = GM/R is its compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The parameter k2 is called the second (or- der) Love number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Additional discussion about the theory of tidal deformability and the tidal Love numbers are beyond the scope of this work and can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [22, 40, 55– 59] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Nevertheless, as pointed out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [22, 58], the value of yR must be corrected since strange stars are self-bound and present a discontinuity at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Therefore we must have yR → yR − 4πR3∆ϵS M , (14) where R and M are the star radius and mass, respectively, and ∆ϵS is the difference between the energy density at the surface (p = 0) and the star’s exterior (which implies ϵ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The results for the dimensionless tidal parameter are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As we can be seen, some features present in the mass-radius relation are also present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For instance, for a mass mx = 100 MeV, the low masses of strange stars have similar tidal parameters, despite their DM fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The tidal parameter for the canonical mass lies between 638 and 644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' These values are in agreement with the constraint Λ < 800 [55], but fail to fulfill the constraint 70 < Λ < 580 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the case of mx = 400 MeV, the strange stars’ huge com- pression due to an increase in the DM fraction reduces the tidal parameter Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The tidal parameter now lies around 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This indicates that for mx = 400 MeV, we are able to explain very massive neutron stars as the PSR J0952-0607 [52], and simultaneously fulfills the constraints of Λ < 800 [55] and 70 < Λ < 580 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We summarize the results of this section in Tab I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Bosonic DM within CFL quark matter In order to better understand the effects of the DM in strange stars, we now assume that the quark matters are in 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 3 10 11 12 13 14 mx = 100 MeV M/M0 R (km) fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 3 10 11 12 13 14 mx = 400 MeV M/M0 R (km) fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Mass-radius relation for bosonic DM admixed CFL strange stars with mx =100 MeV (left) and mx = 400 MeV (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' the CFL superconducting phase via the analytical approxima- tion EoS in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The mass-radius relations are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As can be seen, for a bosonic DM mass of 100 MeV, we have an increase in the maximum mass with the increase of the fraction of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The qualitative results for CFL superconduct- ing quark stars are analogous to both the original, massless MIT as showed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 49], as well for the massive vector MIT bag model as presented in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This indicates a possible model-independent behavior about the effect of the bosonic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Moreover, as in the case of the original mass- less MIT and the massive vector MIT, in the CFL phase, we also see that the presence of DM affects only massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' CFL strange stars with M < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 M⊙ reproduced essentially the same radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The maximum masses vary from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='81 M⊙ for pure strange stars to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='88M⊙ for bosonic DM admixed with a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This indicates that the PSR J0740+6620 with M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='07 M⊙ [37] can be a stable CFL strange star with or without admixed bosonic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Even the possible mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='17 M⊙ of the pulsar PSR J0952-0607 [52] can be explained as bosonic DM matter admixed strange star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' On the other hand, the radius of the canonical star presents almost no variation and is fixed at around 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='57 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Such a value is too low to reproduce the constraint range of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 km, shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [53] while agreeing with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [54], whose upper 6 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 100 MeV Λ M/M0 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 400 MeV Λ M/M0 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Dimensionless tidal parameter Λ for bosonic DM admixed CFL superconducting strange stars with mx = 100 MeV (top) and mx = 400 MeV (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' limit is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='9 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' About the NICER x-ray telescope, the con- straint between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='52 km and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='85 km pointed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [39] is fulfilled, but the bound in the range between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='96 km and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='26 km (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [38]) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For a mass mx = 400 MeV, the results for CLF super- conducting strange stars are analogous to the massive MIT bag model discussed in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The maximum mass decrease with the increase of DM fraction, dropping from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='81 M⊙ to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='61 M⊙ for a fraction fx of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' However, all values agree with the mass of the PSR J0740+6620 [37] and the black widow pulsar PSR J0952-0607 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Secondly, we see that even low-mass strange stars are already affected by the DM and are significantly more compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The radius of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 M⊙ for fx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='10 is about 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='29 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Such a low ra- dius fails to fulfill both NICER constraints [38, 39], but is in agreement with Capano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The reduction of the CFL strange star and its compression can again be explained by the polytropic EoS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' A four times higher DM matter mass produces sixty-four times smaller pressure!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The reduc- tion of the pressure causes the reduction of the maximum mass and increases the star compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We also calculate the dimensionless tidal parameter Λ for the CFL superconducting strange stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The results are pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As we can be seen, the results are analogous TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Macroscopic properties of bosonic DM admixed CFL su- perconducting strange stars mx (MeV) fx M/M⊙ R (km) R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 (km) Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='81 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='89 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='57 721 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='83 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='57 709 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='86 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='96 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='58 717 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='88 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='58 717 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='81 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='89 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='57 721 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='050 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='63 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='30 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 570 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='075 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='62 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='22 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='32 545 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='61 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='13 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='29 531 to the vector MIT bag model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As in the case of the mass-radius relation, for low mass stars there is very low variation in the Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For instance, for a mass mx = 100 MeV, the low masses strange stars have similar tidal parameters, despite their DM fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The tidal parameter for the canonical mass lies be- tween 709 and 721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' These values are in agreement with the constraint Λ < 800 [55], but fail to fulfill the constraint 70 < Λ < 580 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the case of mx = 400 MeV, the results for CFL super- conducting strange stars are again analogous to vector MIT strange stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The stars’ huge compression as the DM fraction increases reduce the tidal parameter Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The tidal parameter now lies around 550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This indicates that for mx = 400 MeV, we are able to explain very massive neutron stars as the PSR J0952-0607 [52], and simultaneously fulfills the constraints of Λ < 800 [55] and 70 < Λ < 580 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We summarize the results of this section in Tab II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Fermionic DM The Lagrangian of the fermionic DM reads [22, 25, 35]: LDM = ¯χ(iγµ∂µ − (mx − gHh))χ +1 2(∂µh∂µh − m2 Hh2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (15) Here, we assume a dark fermion represented by the Dirac field χ that self-interacts through the exchange of the Higgs boson, whose mass is mH = 125 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The coupling con- stant is assumed to be gH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1, which agrees with the con- straints in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Within this prescription, the DM self-interaction is very feeble and behaves as a free Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' More explicitly, the strength of the interaction is: GH = � gH mH �2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='492 × 10−8 fm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (16) The EoS is easily obtained in mean field approximation, completely analogous to the QHD model [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The fermionic DM is assumed to be the lightest neutralino, with mx = 200 GeV, as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [25, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' However, as pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [60], the lower limit for weakly interacting massive par- ticles (WIMP) is 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Therefore we also use mx = 60 GeV 7 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 6 7 8 9 10 11 12 13 mx = 200 GeV M/M0 R (km) kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 6 7 8 9 10 11 12 13 mx = 60 GeV M/M0 R (km) kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Mass-radius relation for fermionic DM admixed strange stars with mx = 200 GeV (top) and mx = 60 GeV (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' to better study the influence of the DM mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As in the case of the bosonic DM, we must fix the DM fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As we are dealing here with fermionic DM, we follow ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [25, 28, 35] and use the Fermi momentum to fix the DM fraction, using three different values: kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Fermionic DM within vector MIT bag model We display in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 5 the TOV solution for a fermionic DM with a mass of 200 GeV and 60 GeV within the vector MIT bag model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As can be seen, the results for fermionic DM are significantly different when compared with bosonic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The maximum masses are always reduced, and the star compres- sion always increases, even for very low masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Also, differ- ent DM fractions always produce different mass-radius rela- tions, affecting all the strange star families, unlike the bosonic case, where we have very similar stars for different DM frac- tions, which is easily understood by the different criteria of the DM fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the case of bosonic DM, the DM frac- tion is dependent on the quark EoS via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the case of fermionic DM, the Fermi momentum is fixed and independent of the quark EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Qualitatively, the results for mx = 200 GeV and 60 GeV TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Macroscopic properties of fermionic DM admixed strange stars mx (GeV) kDM F (GeV) M/M⊙ R (km) R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 (km) Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='86 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 644 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='22 586 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='70 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='39 346 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='80 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='72 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='95 108 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='86 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 644 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='40 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='30 625 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='33 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='05 524 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='16 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='42 351 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Increasing the DM fraction compress the star and reduces the maximum mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Quantitatively, we see that a higher DM mass has a strong influence once it has a higher increase in the energy density, and at the same time, that pro- duces a lower contribution to the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The maximum mass drops from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 M⊙ for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 to only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='80 M⊙ for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV in the case of mx = 200 GeV and to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='16 M⊙ for mx = 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the same sense, the radius of the canonical star drops from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 km for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 to only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='95 km for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV in the case of mx = 200 GeV, and to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='42 km for mx = 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As can be seen, the results for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV with mx = 200 GeV can be ruled out once it has a very low maximum mass in disagreement with the NICER result of the PSR J0740+6620 with a gravitational mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='07 M⊙ [37], and also a very low radius for the canonical star, in disagreement even with the low limit of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 km presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 6 the dimensionless parameter Λ for fermionic DM admixed strange stars with mx = 200 GeV and mx = 60 GeV within the vector MIT bag model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As we can see, the strong compression due to the fermionic DM contri- bution reduces the tidal parameter significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the case with mx = 200 GeV and kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV, the tidal parame- ter drops to only 108, which is six times lower than for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00, although, as we pointed out before, such parametrization must be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As can be seen, most of the parametrizations are able to fulfill the main constraints for pulsar observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=', M > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='01M⊙ and 70 < Λ < 580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Indeed, the presence of DM improves the theoretical prediction and the observational constraints, although it can be some debate about the radius of the canonical star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' They do not fulfill NICER results [38, 39] but agree with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' It is also worth to point the existence of almost degenerate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As can be seen, for mx = 200 GeV with kDM f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV, the macroscopic are essentially the same for the mx = 60 GeV and kDM f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The main results are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 8 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 200 GeV Λ M/M0 kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 60 GeV Λ M/M0 kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Dimensionless tidal parameter Λ for fermionic DM admixed strange stars with mx = 200 GeV (top) and mx = 60 GeV (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Fermionic DM within CFL quark matter We now study the effect of Fermionic DM in CFL super- conducting matter described by the analytical approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We display in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 7 the TOV solution for a fermionic DM with a mass of 200 GeV and 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As in the case of the vec- tor MIT bag model, for CFL superconducting quark matter, the results for fermionic DM are significantly different when compared with bosonic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' And again, the qualitative effect of fermionic DM is the same for CFL as it is for the vector MIT bag model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The maximum masses are always reduced, and the star compression always increases, even for very low masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Again, different DM fractions always produce differ- ent mass-radius relations, affecting all the strange star fami- lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' From the quantitative point of view, the maximum mass drops from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='81 M⊙ for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 M⊙ for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV in the case of mx = 200 GeV and to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='49 M⊙ for mx = 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the same sense, the radius of the canon- ical star drops from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='57 km for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='21 km for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV in the case of mx = 200 GeV, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='66 km for mx = 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Now, unlike the case of the vector MIT, none of the CFL superconducting strange stars can be ruled 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 3 6 7 8 9 10 11 12 13 14 mx = 200 GeV M/M0 R (km) kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 3 6 7 8 9 10 11 12 13 14 mx = 60 GeV M/M0 R (km) kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Mass-radius relation for fermionic DM admixed CFL super- conducting strange stars with mx = 200 GeV (top) and mx = 60 GeV (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' out in the light of the PSR J0740+6620, M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='07 M⊙ [37], although for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV and mx = 200 GeV the radius of the canonical star is below the lower limit of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 km presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 8 the dimensionless parameter Λ for fermionic DM admixed superconducting strange stars with mx = 200 GeV and mx = 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The results are completely analogous to the case of the vector MIT bag model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' however, the value of Λ here is always higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The compression due to the fermionic DM contribution reduces the tidal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the case with mx = 200 GeV and kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV, the tidal parameter drops from 721 to 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' It is also worth noting that some parametrizations can ful- fill the main constraints for pulsar observations, 70 < Λ < 580, and yet produce a very high maximum mass, sometimes reaching 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='50 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The presence of DM again improves the theoretical prediction and the observational constraints, al- though it can be some debate about the radius of the canonical star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' They do not fulfill NICER results [38, 39], but agree with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Moreover, most parametrizations can explain even the black widow pulsar PSR J0952-0607 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Finally, even when we use a different model for the quark matter, the existence of almost degenerate results is still present: for mx = 200 GeV with kDM f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV and mx = 9 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 200 GeV Λ M/M0 kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV 0 200 400 600 800 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 mx = 60 GeV Λ M/M0 kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='00 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV kf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Dimensionless tidal parameter Λ for fermionic DM admixed CFL strange stars with mx = 200 GeV (top) and mx = 60 GeV (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Macroscopic properties of fermionic DM admixed color superconducting quark stars mx (GeV) kDM F (GeV) M/M⊙ R (km) R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 (km) Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='81 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='89 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='57 721 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='75 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='65 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='43 653 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='50 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='50 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='67 421 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='21 151 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='81 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='89 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='57 721 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='78 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='53 694 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='69 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='45 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='28 610 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='49 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='53 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='66 422 60 GeV with kDM f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The main results are summa- rized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Fermionic DM with a vector channel Now we study if the presence of a dark, repulsive vector channel affects the macroscopic properties of the fermionic DM admixed strange stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The new Lagrangian is the La- grangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 15 plus the repulsive channel and the respec- tive meson mass, and reads [28]: LVDM = gξ ¯χ(γµξµ)χ + 1 2m2 ξξµξµ − 1 4V µνVµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' (17) The Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 17 is analogous to the ω contribution to the QHD Lagrangian [5, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Indeed, the junction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 15 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 17 makes this model of DM fully analogous to the original σ −ω model of the QHD [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The coupling constant gξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='1 is fixed, and it is equal to the gH, while the mass of the vector of the dark meson is assumed to be 34 MeV, follow- ing Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' As the mass of the dark vector meson is 3000 times smaller than the mass of the Higgs boson, the repulsive channel is much stronger than the attractive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Indeed, we have Gξ = � gξ mξ �2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='337 fm2, (18) which is stronger than the quark repulsion and millions of times higher than the DM scalar coupling (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Never- theless, despite the strong self-repulsion of the fermionic DM, the numerical results are barely affected by the repulsive chan- nel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For the vector MIT bag model, the only noticeable dif- ference appears for mx = 200 GeV and kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In this case, the maximum mass increase from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='16 M⊙ to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='17 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The radius of the canonical star also grows from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='39 km to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='46 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The tidal parameter Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 also increases from 346 to 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' It is worth noticing that all these variations are far beyond the precision with which experimental measurements are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' All the other parametrizations present even lower (or none) differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Herefore, we do not provide any figures in this section since they would be visually indistinguishable from those in the last paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We only display the main results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In the case of CFL superconducting quark matter, the differences are even smaller!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The nature of the vector coupling can explain why the differences are so small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The vector mesons couple to the number density, and we are dealing with a very low-density regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Indeed, even kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV implies a number den- sity is around 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='6 × 10−4 fm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Of course, we could increase the repulsion of the dark vector boson, but we believe this would be very unrealistic since DM was proposed to explain higher attraction in galaxy curves [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' CONCLUSIONS In this work, we calculate the properties for the DM ad- mixed for strange quark stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' We use two different mod- els for the quark model: the vector MIT bag model, as pre- sented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [5, 6] and the CFL color superconducting quark matter via an analytical approximation, as discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [42, 44, 45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' and two different kinds of dark mat- ter: a bosonic as discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 46, 47, 49] and for fermionic [22, 25, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For each kind of DM, we use two different mass values, and the strange stars always agree with the Bodmer-Witten conjecture [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Our main conclusions can be summarized as follows: 10 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Macroscopic properties of dark vector boson fermionic DM admixed strange stars within the vector MIT bag model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The only significant differences are for kDM F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV mx (GeV) kDM F (GeV) M/M⊙ R (km) R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 (km) Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='4 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='86 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 644 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='22 586 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='17 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='70 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='46 358 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='80 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='72 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='01 112 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='41 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='86 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='37 644 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='40 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='30 625 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='33 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='10 532 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='16 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='43 353 The qualitative results for DM admixed strange stars are independent of the quark model utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This is true for both bosonic and fermionic, as well it is independent of the DM mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For a bosonic DM with a mass of mx = 100 MeV, we have an increase of the maximum mass, while the prop- erties of low-mass strange stars are not significantly af- fected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' This is the only case in that we have an in- crease in the star’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Such a situation happens for the vector MIT, the CFL superconducting quark mat- ter, and also for the massless MIT, as pointed out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [27, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For a bosonic DM with a mass of mx = 400 MeV, we have a decrease of the maximum mass, whilst the radii of the low-mass strange stars, in this case, are also af- fected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' For a fermionic DM, the maximum mass always de- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The higher the DM fraction, the lower the max- imum mass, and the smaller the radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Also, the higher the DM mass, the higher the stellar compression and the lower the maximum mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Although we introduce a repulsive dark vector field with a mass 3000 times smaller than the attractive scalar field, we do not find significant variation in the stellar macroscopic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' There are almost degenerate results both for mx = 200 GeV with kDM f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='04 GeV and mx = 60 GeV with kDM f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='06 GeV, the maximum mass, as well the prop- erties of the canonical star are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' About the observational constraints, we can see that the mass of the PSR J0740+6620 pulsar, M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='07 M⊙ [37] is easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Even the mass range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='17 M⊙ of the black widow pulsar PSR J0952- 0607 [52] can be reached for some parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' The radius of the canonical star is still a matter of de- bate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Most of our results point to a radius between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='0 km to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content='5 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In general, our results are in agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [54] but are too low to reproduce the NICER results [38, 39] whilst at the same time are too high to agree with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Except for bosonic DM with a mass of mx = 100 MeV, in all other cases, the presence of the DM reduces the dimensionless tidal parameter Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' In most of these cases, the constraint 70 < Λ < 580 [40] is easily fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Annala, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Gorda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Kurkela, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} +page_content=' 90, 045002 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfrvmy/content/2301.00567v1.pdf'} diff --git a/I9FAT4oBgHgl3EQfux7Z/content/tmp_files/2301.08672v1.pdf.txt b/I9FAT4oBgHgl3EQfux7Z/content/tmp_files/2301.08672v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a7e53970137d2f1d4262227bbeb0acc3e0a2f05 --- /dev/null +++ b/I9FAT4oBgHgl3EQfux7Z/content/tmp_files/2301.08672v1.pdf.txt @@ -0,0 +1,1521 @@ +arXiv:2301.08672v1 [math.CT] 20 Jan 2023 +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +OLIVIA MONJON, J´ERˆOME SCHERER, AND FLORENCE STERCK +Abstract. The correspondence between the concept of conditional flatness and admissibil- +ity in the sense of Galois appears in the context of localization functors in any semi-abelian +category admitting a fiberwise localization. It is then natural to wonder what happens in +the category of crossed modules where fiberwise localization is not always available. In this +article, we establish an equivalence between conditional flatness and admissibility in the +sense of Galois (for the class of regular epimorphisms) for regular-epi localization functors. +We use this equivalence to prove that nullification functors are admissible for the class of +regular epimorphisms, even if the kernels of their localization morphisms are not acyclic. +Introduction +It is a natural question to ask whether the pullback of a nice extension inherits these +nice properties. When working with localization functors or reflections one particularly nice +feature for an extension is flatness. We say that an extension is L-flat, for a localization +functor L, if applying L to the extension yields another extension, see Definition 2.1. The +question is thus to understand when the pullback of an L-flat extension is again L-flat. +Such questions have been studied first in a homotopical context by Berrick and Farjoun, +[1]. For homotopical localization functors in the category of topological spaces (in the sense +of Bousfield, [5], see also Farjoun’s book [13]), preservation of L-flatness (for fiber sequences) +under pullbacks was shown to be equivalent for L to be a so-called nullification functor. The +situation is surprisingly more delicate in the category of groups. Farjoun and the second +author proved for example that all nilpotent quotient functors have this nice property, which +they called conditional flatness, see [14]. +The standard strategy to establish conditional flatness for a localization functor consists +in a few reduction steps culminating in a simpler form, which Gran identified as admissibility +in the sense of Galois for the class of regular epimorphisms [17, Proposition 3.3]. This shifted +the study of conditional flatness in homotopy theory to that of admissibility in semi-abelian +categories, see [15]. Admissibility had been introduced by Janelidze and Kelly in [17] and +has since then played a central role in the categorical study of extensions, let us mention for +example Everaert, Gran, and Van der Linden’s work in [12]. +In this article we study admissibility for localization functors in the category of crossed +modules (of groups), a category of interest to both topologists due to Whitehead’s work on +connected 2-types, [25], and algebraists since Brown and Spencer [7] proved the equivalence +between crossed modules and internal groupoids in the category of groups (a result that +they credit to Verdier). This equivalence relates two interesting notions and allows one to +deal with the concept of internal groupoid in an alternative way, that is useful for compu- +tations. Moreover, crossed modules form a semi-abelian category in the sense of Janelidze, +2020 Mathematics Subject Classification. 18G45, 55P60, 18E50, 55R70, 18E13. +Key words and phrases. Crossed modules, Localization functors, Admissibility, Regular epimorphisms, +Conditional flatness, Nullifications. +1 + +2 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +M´arki and Tholen, [18]. We adopt the algebraic point of view here and continue our work +started in [22]. Indeed, among the reduction steps we have mentioned above, the first one +calls on fiberwise localization techniques. For group theoretical localization and homotopy +localization functors, it allows one to reduce the study to extensions with local kernel (fiber). +Fiberwise localization techniques are available in the category of groups thanks to work of +Casacuberta and Descheemaeker, [10], but we proved in [22] that they are not at hand in +general for crossed modules. Our aim in this article is thus to modify the strategy to be able +to study admissibility in this setting. +We focus on localization functors such that the co-augmentation morphism ℓT: T → LT is +a regular epimorphism for all crossed modules T. We call them regular-epi localization and +notice that many examples of interest are provided by nullification functors, as defined in +Definition 1.10. Any crossed module A determines a nullification functor PA that “kills” all +morphisms from A and there are other regular-epi localization functors such as abelianization. +One first important observation which makes the reduction strategy viable is that, even +though fiberwise localization does not exist in general, even for nullification functors, we can +use this tool for certain extensions. +Lemma 2.5. Let L be a regular-epi localization. Let +(1) +T +Q +N +1 +1 +κ +α +be an L-flat exact sequence of crossed modules and g : Q′ → Q a morphism of crossed modules. +Then, we can construct the fiberwise localization of the pullback of (1) along g: +N +N +T′ +T +Q′ +Q +1 +1 +1 +1 +κ +πT +κ′ +g +πQ′ +α +This allows us to relate conditional flatness with admissibility, in the same spirit as what +was done in the category of groups, [14], or in the wider context of semi-abelian categories +where fiberwise localization exists, [15]. A localization functor L is said to be admissible for +the class of regular epimorphisms if it preserves any pullback of the form +LT +T′ +Q +LQ +πLT +ℓQ +πQ +α +where α is a regular epimorphism between L-local objects. +Theorem 3.4. +Let L be a regular-epi localization functor. Then the following statements +are equivalent +(1) L is conditionally flat; +(2) L is admissible for the class of regular epimorphisms. +One difference between groups and crossed modules and maybe the main source of com- +plication is highlighted by the behavior of kernels. This was already the reason why one + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +3 +cannot always construct fiberwise localization and we were also surprised to find examples of +nullification functors for which the kernel of the nullification morphism ℓT : T → PAT is not +always PA-acyclic, see [22, Proposition 4.6]. For groups and spaces, this property actually +characterizes nullification functors. +Still we prove here that acyclic kernels implies admissibility and in Proposition 4.3, that if +the kernels of the localization morphisms are Lf-acyclic, then Lf is a nullification functor. Well +behaved nullification functors are therefore admissible, but what about arbitrary nullification +functors, for which fiberwise localization does not necessarily exist and for which the kernel of +the nullification is not necessarily acyclic? By carefully looking at the inductive construction +of PAT we show our main result, namely that all nullification functors are admissible. +Theorem 5.5. Let A be any crossed module. The nullification functor PA is admissible for +the class of regular epimorphisms. +We end this introduction with a short outline. The first section consists of preliminaries +that we use in the rest of the article. Then in Section 2 we introduce L-flat exact sequences +and conditionally flat localization functors in the context of crossed modules. We show how to +construct fiberwise localization of L-flat exact sequences. The third section is essential in the +development of a simpler characterisation of conditional flatness: It provides an equivalence +with the notion of admissibility in the specific context of regular-epi localization functors. In +Section 4 the link between L-acyclicity and admissibility is established and the last section +is devoted to the proof that every nullification functor is admissible. +Acknowledgments. We would like to thank Marino Gran for sharing his insight about +admissibility. +1. Preliminaries +1.1. The semi-abelian category of crossed modules. In this subsection, following Norrie +[23] and Brown-Higgins [6], we provide the basic definitions and notation concerning crossed +modules. +Definition 1.1. [25] A crossed module of groups is a pair of groups T1 and T2, an action by +group automorphisms of T2 on T1, denoted by T2 × T1 → T1 : (b, t) �→ +bt, together with a +group homomorphism ∂T : T1 → T2 such that for any b in T2 and any t, s in T1, +(2) +∂T( bt) = b∂T(t)b−1, +(3) +∂T(t)s = tst−1. +Hence we often write a crossed module as a triple (T1, T2, ∂T), or simply T for short, and +we refer sometimes to ∂T as the connecting morphism. +Definition 1.2. Let N := (N1, N2, ∂N) and M := (M1, M2, ∂M) be two crossed modules. A +morphism of crossed modules α: N → M is a pair of group homomorphisms α1: N1 → M1 +and α2 : N2 → M2 such that the two following diagrams commute +N2 +N1 +M1 +M2 +∂N +∂M +α1 +α2 +M2 × M1 +N2 × N1 +N1 +M1. +(α2, α1) +α1 + +4 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +where the horizontal arrows in the diagram on the right are the respective group actions of +the two crossed modules. +We write XMod for the category of crossed modules of groups. +Remark 1.3. There is an embedding of the category of groups in this category via two +functors which are respectively left and right adjoint to the truncation functor Tr: XMod → +Grp that sends a crossed module T := (T1, T2, ∂T) to T2. The functor X: Grp → XMod which +sends a group G to the crossed module XG = (1, G, 1) reduced to the group G at level 2 is +the left adjoint functor and the functor R: Grp → XMod: G �→ (G, G, IdG) is the right ajoint +functor. This will help us to import group theoretical results into XMod. +There is an obvious notion of subcrossed module, see [23]. One simply requires the sub- +object to be made levelwise of subgroups, the connecting homomorphism and the action are +induced by the given connecting homomorphism and action. The notion of normality is less +obvious. +Definition 1.4. A subcrossed module N := (N1, N2, ∂N) of T := (T1, T2, ∂T) is normal if the +following three conditions hold +(1) N2 is a normal subgroup of T2; +(2) for any t2 ∈ T2 and n1 ∈ N1, we have t2n1 ∈ N1; +(3) [N2, T1] := ⟨ n2t1t−1 +1 +| t1 ∈ T1, n2 ∈ N2⟩ ⊆ N1. +In contrast to limits, which are built component-wise, colimits are generally more delicate +to construct. In particular, the construction of cokernels is not straightforward, but when +N is a normal subcrossed module of T the cokernel is simply the levelwise quotient by the +normal subgroups N1 ⊳ T1 and N2 ⊳ T2. +The category of crossed modules shares many nice properties with the category of groups. +The traditional homological lemmas, [2], the Split Short Five Lemma, [3], and the Noether +Isomorphism Theorems, [2], hold. +One can recognize pullbacks by looking at kernels or +cokernels, [2, Lemmas 4.2.4 and 4.2.5], and in fact Xmod is a semi-abelian category, as +introduced by Janelidze, M´arki, and Tholen in [18]. This is shown in [18]. There is one result +we will use several times in this article, namely [2, Lemma 4.2.4], which we recall now. +Proposition 1.5. Let C be a semi-abelian (or homological) category. Consider the following +diagram of exact rows: +T ′ +Q′ +N′ +T +Q +N +1 +1 +1 +(2) +w +u +v +κ +α +κ′ +α′ +Then the following statements hold. +(1) If u is an isomorphism then (2) is a pullback. +(2) If u and w are regular epimorphisms then v is also a regular epimorphism. +1.2. Localization functors. In this subsection we recall the definition of localization func- +tors in the category of crossed modules. We also recall some important properties of such +functor as well as some examples. + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +5 +Definition 1.6. A localization functor in the category of crossed modules is a coaugmented +idempotent functor L: XMod → XMod. The coaugmentation ℓ: Id → L is a natural transfor- +mation such that ℓLX and LℓX are isomorphisms. +In particular we have ℓLX = LℓX, see [9, Proposition 1.1]. +Definition 1.7. Let L be a localization functor. A crossed module T is L-local if ℓT : T → LT +is an isomorphism. A morphism f : N → M is an L-equivalence if Lf is an isomorphism. +We recall a few basic and useful closure properties of L-equivalences. +Lemma 1.8. +(1) The pushout of an L-equivalence is an L-equivalence. +(2) The composition of L-equivalences is an L-equivalence. +(3) A κ-filtered colimit of a diagram Tβ of L-equivalences Tβ → Tβ+1 for all successor +ordinals β + 1 < κ yields an L-equivalence T0 → Tκ = colimβ<κTβ. +(4) Let F be an I-indexed diagram of L-equivalences in the category of morphisms of +crossed modules. Then the colimit colimIF is an L-equivalence. +Sometimes a localization functor L is associated to a full reflexive subcategory L of XMod. +The pair of adjoint functors U: L ⇆ XMod: F provides a localization functor L = FU, as +Cassidy, H´ebert, and Kelly do in [11]. Some other times there is a morphism f one wishes +to invert so as to construct a localization functor often written Lf. +Definition 1.9. Let f be a morphism of crossed modules. A crossed module T is Lf-local if +Hom(f, T) is an isomorphism. A morphism g in XMod is an Lf-equivalence if Hom(g, T) is +an isomorphism for any Lf-local crossed module T. +Such localization functors exist in XMod, see for example Bousfield’s foundational work +[4]. Local objects and local equivalences coincide then with the notions introduced in Defini- +tion 1.7. Proposition 1.8 is the analogue of Hirschhorn’s [16, Proposition 1.2.20 and Propo- +sition 1.2.21]. +If the codomain of the morphism f is the trivial crossed module, the functor Lf is of +particular interest. +Definition 1.10. Let A be a crossed module and f be the morphism A → 1. The localization +functor Lf is then written PA and is called a nullification functor. An f-local object is called +A-null, or A-local and a crossed module T is A-acyclic if PAT = 1. The localization morphism +ℓT : T → PAT is written pT. +Proposition 1.11. Let A and T be crossed modules. Then there exists an ordinal λ depending +on A such that PAT is constructed as a transfinite filtered colimit of a diagram of the form +T = T0 → T1 → · · · → Tβ → . . . for β < λ where all morphisms are PA-equivalences and +regular epimorphims. +This inductive construction has been carefully described in [22, Proposition 2.8]. The rea- +son why each step is a PA-equivalence and a regular epimorphism is that Tβ+1 is constructed +from Tβ by taking the cokernel of all morphisms A → Tβ. We recall the details and use +them in Section 5. There is a larger class of localization functors we investigate in this se- +quel to [22]. They share with PA the property that the localization morphism is a regular +epimorphism. +Definition 1.12. A localization functor L is a regular-epi localization if for any crossed +module T the coaugmentation ℓT: T → LT is a regular epimorphism. + +6 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +Remark 1.13. In the category of crossed modules, a morphism α = (α1, α2) is a regular +epimorphism (a coequalizer of a pair of parallel arrows) if and only if both α1 and α2 are sur- +jective group homomorphisms [20, Proposition 2.2]. A surjective homomorphism of crossed +modules is an epimorphism but there exist epimorphisms that are not surjective. In a pointed +protomodular category such as XMod, regular epimorphisms and normal epimorphisms (the +cokernel of some morphism) coincide. +We present now some interesting examples of localization functors that will illustrate our +results in the rest of the article, see also the end of [22, Section 2]. +Example 1.14. The nullification functor PXZ with respect to the crossed module XZ is given +by: +PXZ + + + + + +N1 +N2 +∂ + + + + + = +N1/[N2, N1] +1 +Example 1.15. The abelianization functor Ab: XMod → XMod is already described in [24]. +It is defined by: +Ab + + + + + +N1 +N2 +∂ + + + + + = +N1/[N2, N1] +N2/[N2, N2] +˜∂ +Example 1.16. Our third and last example of localization functor of crossed modules is +I: XMod → XMod, see [22, Example 2.15]: +I + + + + + +N1 +N2 +∂N + + + + + = +N2 +N2 +IdN2 +This functor is induced by the adjunction between the truncation functor Tr: XMod → Grp, +defined by Tr(T1, T2, ∂T) = T2, see Remark 1.3, and its right adjoint R: Grp → XMod that +sends a group T to (T, T, IdT). +Remark 1.17. The functor considered in Example 1.14 is a regular-epi localization, since all +nullification functors are so. However regular-epi localizations are not nullification functors in +general as illustrated by the functor Ab in Example 1.15. Indeed, if Ab were a nullification PA, +then A = (A1, A2, ∂A) would be a perfect crossed module, i.e. one such that Ab(A) = (1, 1, Id). +In particular, the group A2 would be a perfect group. But then PA(XS3) = XS3 since there +are no non-trivial homomorphisms from a perfect group to the symmetric group S3. But we +know that Ab(XS3) = XC2, where C2 is the cyclic group of order two, so abelianization is not +a nullification. +We finally note that a localization functor Lf is a regular-epi localization functor if f itself +is a regular epimorphism, an analogous observation appears in [8] for groups. +To conclude these preliminaries, let us recall the notion of fiberwise localization. +We +introduced this for crossed modules in [22, Definition 3.1], but this is not new, for spaces a +good reference is [13, Section I.F]. + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +7 +Definition 1.18. Let L: XMod → XMod be a localization functor. An exact sequence +T +Q +N +1 +1 +κ +α +admits a fiberwise localization if there exists a commutative diagram of horizontal exact +sequences +T +Q +N +E +Q +LN +1 +1 +1 +1 +κ +j +ℓN +p +α +g +where g is an L-equivalence. +The following theorem is a fusion of two results from [22] namely Theorem 3.4 and Corollary +3.7. From now on, every localization functor that we consider is a regular-epi localization. +Theorem 1.19. Let L: XMod → XMod be a regular-epi localization functor. An exact se- +quence of crossed modules +(4) +T +Q +N +1 +1 +κ +α +admits a fiberwise localization if and only if we have the following inclusion +(5) +[κ2(ker(ℓN +2 )), T1] ⊆ κ1(ker(ℓN +1 )) +2. Fiberwise localization and flatness +In this section, we investigate the fiberwise localization of L-flat exact sequences and their +pullbacks in the context of regular-epi localization functors of crossed modules L: XMod → +XMod (even if this notion is not defined only for regular-epi functor as we will see in Propo- +sition 5.6). This section will be essential to study the link between conditionally flatness +and admissibility in Section 3. First, let us recall the definitions of L-flat and conditionally +flatness. +Definition 2.1. Let L be a localization functor, a short exact sequence +T +Q +N +1 +1 +κ +α +is called L-flat if the sequence +LT +LQ +LN +L(κ) +L(α) +is a short exact sequence. +Remark 2.2. We recall that limits are computed componentwise in the category of crossed +modules. In the case of pullbacks in XMod they are built as follows [19]. Let α: T → Q and +g : Q′ → Q be two morphisms of crossed modules. Then the pullback of α along g is given +by the following square +T +T′ +Q′ +Q +πT +g +πQ′ +α +The object part T′ of the pullback is built component-wise as in the case of groups +(T1 ×Q1 Q′ +1, T2 ×Q2 Q′ +2, ∂′), + +8 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +where ∂′ and the action are induced by the universal property of the pullbacks in Grp. The +projections are the natural ones, given also component-wise. +Following the terminology introduced in [14] for groups and spaces, we define the notion +of conditional flatness for localization functors in crossed modules. +Definition 2.3. Let L be a localization functor. We say that this functor is conditionally +flat if the pullback of any L-flat exact sequence is L-flat. +In Section 3 we provide a characterization of conditional flatness. To achieve this goal we +will use a similar strategy to the one applied to groups and topological spaces in [14]. The +authors exploit heavily the existence of fiberwise localization in the categories of groups and +spaces. However, in our article [22], we observed that fiberwise localization does not always +exist for a given localization functor and a given exact sequence in XMod. Fortunately, when +we work with L-flat exact sequences we can show that it is always possible to construct a +fiberwise localization. +Lemma 2.4. Let L be a regular-epi localization. Then any L-flat exact sequence of crossed +modules admits a fiberwise localization. +Proof. Let +T +Q +N +1 +1 +κ +α +be an L-flat exact sequence of +crossed modules. The L-flatness of the sequence implies in particular that Lκ is a monomor- +phism. Consider the following diagram of exact sequences: +1 +1 +ker(ℓT) +(1) +ker(ℓN) +N +T +LN +LT +κ +Lκ +ℓN +ℓT +We conclude from [2, Lemma 4.2.4.(1)] that (1) is a pullback since Lκ is a monomorphism. +Then we have that κ(ker(ℓN)) is a normal subcrossed module of T as it can be seen as the +intersection of the normal subcrossed modules N and ker(ℓT) of T. Therefore, we can apply +Theorem 1.19 +□ +To understand conditional flatness we must study the pullback of an L-flat exact sequence. +It will thus be very handy in Section 3 to know that any such pullback admits a fiberwise +localization. +Lemma 2.5. Let L be a regular-epi localization. Let +(6) +T +Q +N +1 +1 +κ +α +be an L-flat exact sequence of crossed modules and g : Q′ → Q a morphism of crossed modules. +Then, we can construct the fiberwise localization of the pullback of (6) along g +N +N +T′ +T +Q′ +Q +1 +1 +1 +1 +κ +πT +κ′ +g +πQ′ +α + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +9 +Remark 2.6. In the rest of the article, and in particular in the following proof, we identify +N with the normal subcrossed module κ(N) of T and with κ′(N), normal subcrossed module +of T′. We will therefore omit the us of κ and κ′. For example an element of the group N1 +that we want to consider in T′ +1 will be denoted (n1, 1) instead of κ′ +1(n1) = (κ1(n1), 1). +Proof of Lemma 2.5. We need to verify that ker(ℓN) is a normal crossed module of T′. Since +N is a subcrossed module of T′, we just need to verify (5) of Theorem 1.19. Let (t1, q1) be an +element in T ′ +1 and (x2, 1) be an element of ker(ℓN +2 ), then we have the following equality +(x2,1)(t1, q1)(t1, q1)−1 = ( x2t1t−1 +1 , q1q−1 +1 ) = ( x2t1t−1 +1 , 1). +Indeed, by Lemma 2.4 we know that the original sequence (6) admits a fiberwise localization +which then implies by Theorem 1.19 that [ker(ℓN +2 ), T1] ⊂ ker(ℓN +1 ) i.e for any x2 ∈ ker(ℓN +2 ) and +t1 ∈ T1 we have x2t1t−1 +1 +∈ ker(ℓN +1 ). But then, with the notation introduced in Remark 2.6, +this is equivalent to say that the element ( x2t1t−1 +1 , 1) belongs to ker(ℓN +1 ). +□ +This lemma is not trivial since the fiberwise localization of an exact sequence of crossed +modules does not always exist as we have proved in [22, Theorem 4.5]. If we want the strategy +for groups and spaces to be also viable in the study of conditional flatness for crossed modules, +we need a final ingredient, namely a commutation rule for the fiberwise localization and the +pullback operations. +Proposition 2.7. Let us consider an L-flat exact sequence where L is a regular-epi localization +functor. +Then, the pullback of its fiberwise localization is the fiberwise localization of its +pullback. +Proof. Let +N +N +T′ +T +Q′ +Q +1 +1 +1 +1 +κ +πT +κ′ +g +πQ′ +α +be the pullback of an L-flat exact sequence. Then we construct the fiberwise localizations of +the two sequences by quotienting out the kernel of the localization morphism ℓN as follows. +N +1 +T′ +Q′ +1 +N +1 +T +Q +1 +LN +1 +T′/ker(ℓN) +Q′ +1 +LN +1 +T/ker(ℓN) +Q +1 +κ′ +πQ′ +κ +α +g +πT +j +j′ +p +p′ +g +f ′ +f +ℓN +ℓN +We complete the diagram by defining a morphism δ: T′/ker(ℓN) → T/ker(ℓN) via the +universal property of the cokernel since f ◦ πT ◦ κ′|ker(ℓN) = 1, where κ′|ker(ℓN) : ker(ℓN) → T′ is + +10 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +the inclusion of the kernel of ℓN. +ker(ℓN) +ker(ℓN) +T′ +T +T ′/ker(ℓN) +T/ker(ℓN) +1 +1 +1 +1 +κ|ker(ℓN) +πT +κ′|ker(ℓN) +f ′ +f +δ +N +1 +T′ +Q′ +1 +N +1 +T +Q +1 +LN +1 +T′/ker(ℓN) +Q′ +1 +LN +1 +T/ker(ℓN) +Q +1 +κ′ +πQ′ +κ +α +g +πT +j +j′ +p +p′ +g +δ +f ′ +f +lN +lN +We can check that δ makes the two front faces commute. Indeed, the right and left faces +commute by using the fact that ℓN and f ′ are epimorphisms respectively. +The commutativity of the above diagram and Proposition 1.5 implies that +LN +T′/ker(ℓN) +Q′ +1 +1 +j′ +p′ +is the pullback of +T/ker(ℓN) +Q +LN +1 +1 +j +p +along g. +□ +Remark 2.8. In [14], the construction of the fiberwise localization in the category of groups +was functorial, therefore from the morphism T′ → T between the pullback sequence and the +sequence itself we have directly a morphism between the fiberwise localization of the pullback +sequence and the fiberwise localization of the original sequence. In other words the map δ +comes for free in contrast to the category of crossed modules where we have to build the map +δ explicitly. +3. Conditional flatness and admissibility +In this section, we develop a simpler characterisation of conditional flatness, thanks to +the results of the previous section. We introduce the notion of admissibility for the class +of regular epimorphisms and show that it is equivalent to conditional flatness. With this +equivalence, we can easily establish conditional flatness for a given localization functor. We +observe that some properties of localization functors, such as right-exactness, imply directly +admissibility for the class of regular epimorphism. +The first step allows us to restrict the definition of conditional flatness (Definition 2.3) to +fiberwise localizations of L-flat exact sequences (Lemma 3.1). More precisely, we show that +the pullback of an L-flat exact sequence is L-flat if and only if the pullback of its fiberwise +localization is so. + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +11 +Lemma 3.1. Let L be a regular-epi localization functor. Then L is conditionally flat if and +only if for any L-flat exact sequence +T +Q +N +1 +1 +κ +α +with N +an L-local crossed module, the pullback sequence along any morphism Q′ → Q is L-flat. +Proof. This is clear since f ′ and ℓN are L-equivalences in this diagram: +LN +N +T′ +T′/ker(ℓN) +Q′ +Q′ +1 +1 +1 +1 +j′ +f ′ +ℓN +κ′ +πQ′ +p′ +The top row is thus L-flat if and only if so is the bottom row and we conclude by Proposi- +tion 2.7. +□ +The previous lemma allows us to follow the approach introduced in [14]. For the sake of +completeness, we give an explicit proof of the following results even if the arguments are +similar to the group theoretical ones. +Proposition 3.2. Let L be a regular-epi localization functor. Then L is conditionally flat if +and only if the pullback of any exact sequence of L-local objects is L-flat. +Proof. By the previous lemma it is sufficient to consider exact sequence with an L-local kernel +LN. Consider thus an L-flat exact sequence +T +Q +LN +1 +1 +j +p +. +We build the following diagram where g : Q′ → Q is any morphism of crossed modules and +(1) is a pullback. +LN +LN +LN +T′ +(1) +(2) +T +LT +Q′ +Q +LQ +1 +1 +1 +1 +1 +1 +j +L(j) +ℓT +ℓQ +πT +j′ +g +πQ′ +L(p) +p +We observe that since each row is exact, (2) is a pullback by Proposition 1.5, and then +(1) + (2) is also a pullback. Hence, the top row is the pullback of the bottom exact sequence +of L-local objects along the map ℓQ ◦ g, which shows the claim. +□ +Definition 3.3. A localization functor L is said to be admissible for the class of regular +epimorphisms if it preserves any pullback of the form +LT +T′ +Q +LQ +πLT +ℓQ +πQ +α + +12 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +where α is a regular epimorphism. +Theorem 3.4. Let L be a regular-epi localization functor. Then the following statements are +equivalent +(1) L is conditionally flat; +(2) L is admissible for the class of regular epimorphisms. +Proof. The implication (1) ⇒ (2) is trivial, so let us prove (2) ⇒ (1). Consider any exact +sequence of L-local objects +LT +LQ +LN +1 +1 +α +and any morphism g : A → LQ. By Proposition 3.2 conditional flatness is established if we +prove that the pullback of the exact sequence along g is L-flat. Let us first observe that this +morphism g factors through LA via the universal property of the localization: +A +LA +LQ +ℓA +g +˜g +Hence, we can first construct the pullback of +LT +LQ +LN +1 +1 +α +along +˜g and then pullback the resulting sequence along ℓA: +LN +LN +LN +T′′ +T′ +LT +A +LA +LQ +1 +1 +1 +1 +1 +1 +g +πLT +˜g +ℓA +πA +α +πLA +Since the category of L-local objects is closed under pullbacks, T′ is L-local and we can +apply condition (2) to conclude that the upper row is L-flat. This observation implies that +the pullback of +LT +LQ +LN +1 +1 +α +along g is an L-flat sequence as +desired. +□ +The above theorem gives an easier characterisation of conditionally flatness in the category +of crossed modules. It will be useful in rest of the article. +Remark 3.5. Admissibility for the class of regulars epimorphisms in the context of semi- +abelian categories is studied in [15]. Similar results are proven for functors of localizations that +admit a functorial fiberwise localization. Note that their result does not imply Theorem 3.4 +since localization functors of crossed modules do not admit functorial fiberwise localizations +in general. However, the implication “(1) implies (2)”, in Theorem 3.4, holds even for not +necessarily regular-epi localization functors. + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +13 +Proposition 3.6. If L: XMod → XMod is a localization functor that is right exact in XMod, +then L is admissible for the class of regular epimorphisms. +Proof. Let us consider the following pullback of an L-flat exact sequence of crossed modules +along a morphism g : Q′ → Q. +T′ +Q′ +N +T +Q +N +1 +1 +1 +1 +(1) +g +πT +κ +f +κ′ +πQ′ +By applying L to this diagram, we obtain (since L is right exact) the following diagram +LT′ +LQ′ +LN +LT +LQ +LN +1 +1 +1 +L(g) +L(πT) +L(κ) +L(f) +L(κ′) +L(πQ′) +Since L(κ) = L(πT)◦L(κ′) is a (normal) monomorphism, we conclude that L(κ′) is a monomor- +phism. Normality follows then by right-exactness and we conclude by Theorem 3.4. +□ +Note that this proof holds in any semi-abelian category. +Corollary 3.7. The functor of abelianization Ab: XMod → XMod is admissible for the class +of regular epimorphisms. +Proof. The functor of abelianization Ab: XMod → XMod is right exact. Since the exactness +can be shown component-wise, the result follows. +□ +Sometimes it is handy to rely on our group theoretical knowledge to construct simple +examples of localization functors and how they behave on crossed modules. The proof of the +following proposition is based on a counter-example coming from groups via the functor X +defined in Remark 1.3. +Proposition 3.8. There are regular-epi localization functors L: XMod → XMod that are not +admissible for the class of regular epimorphisms. +Proof. We export via X: Grp → XMod the example in [14, Theorem 5.1] of a localization +functor in groups that is not admissible for the class of regular epimorphisms. +Let Lφ be the localization functor induced by the projection φ: C4 → C2, where Cn denotes +a cyclic group of order n. It gives rise to a localization functor LXφ : XMod → XMod. In +particular, if we apply X to the extension of Lφ-local groups considered in [14], we obtain an +exact sequence of LXφ-local crossed modules: +(1, Z) +(1, C2) +(1, Z) +1 +1 +If we pullback along the morphism of crossed modules Xφ, we obtain the following exact +sequence +(1, Z × C2) +(1, C4) +(1, Z) +1 +1 + +14 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +We conclude from [22, Lemma 1.4] that this exact sequence is not LXφ-flat. Indeed, if it was +the case we would have a contradiction with the group theoretical observation in [14]. +□ +4. Admissibility and acyclicity +In the categories of groups and topological spaces, the localization functor L is a nullification +functor if and only if the kernels of the localization morphisms are L-acyclic (which means +that Lker(ℓM) is trivial for any M ∈ XMod). This characterisation implies in particular that +any nullification functor is admissible for the class of regular epimorphisms. It is interesting +to notice that even if nullification functors of crossed modules do not have acyclic kernels, +we have a similar result in XMod: the L-acyclicity of the kernels of localization morphisms +implies the admissibility. +Proposition 4.1. Let L: XMod → XMod be a regular-epi localization functor such that +ker(ℓM : M → LM) is L-acyclic for any M ∈ XMod. Then L is admissible for the class of +regular epimorphisms. +Proof. Consider the pullback of +LT +LQ +LN +1 +1 +κ +f +along ℓQ: Q → +LQ: +LN +LN +T′ +LT +Q +LQ +1 +1 +1 +1 +κ +πLT +κ′ +ℓQ +πQ +f +We need to prove that πLT is an L-equivalence. Since XMod is a pointed protomodular +category and ℓQ is a regular epi by assumption, we know that πLT is the cokernel of ker(ℓQ) ∼= +ker(πLT) → T′. Let Y be a local object, for any g : T′ → Y we have the following diagram: +ker(ℓQ) +Lker(ℓQ) = 1 +T′ +LT +Y +g′ +πLT +g +˜g +By the universal property of the localization there exists g′: 1 → Y that makes the left square +commute. Hence, by the universal property of the cokernel there exists a unique ˜g: LT → Y +such that the triangle commutes and we conclude that πLT is an L-equivalence. +□ +However, localization functors of crossed modules do not behave like localization functors +of groups. As explained above, in the category of groups (but also of topological spaces), the +kernels of the localization morphisms are L-acyclic if and only if L is a nullification functor +[14]. In the context of crossed modules, we do not have such a characterization of nullification +functors. +Remark 4.2. We know by [22, Proposition 4.6] that there are nullification functors, for +example PXZ defined in Example 1.14, such that the kernels of their localization morphisms +are not acyclic in general. Still, in the next proposition, we prove that if the kernel of the +localization morphism is L-acyclic, as in Proposition 4.1, then the localization functor is a +nullification. + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +15 +The cardinal in the next proof is chosen exactly as in Bousfield’s [5, Theorem 4.4] for +spaces. +Proposition 4.3. Let f : B → C be a morphism of crossed modules and Lf : XMod → XMod +be a regular-epi localization functor. +If the kernels of the localization morphisms are Lf- +acyclic, then Lf is a nullifcation functor. +Proof. Our strategy is to construct a crossed module A such that we can compare the functor +Lf with the nullification functor PA (Definition 1.10) via a natural transformation ψ. We +choose κ to be the first infinite ordinal greater than the number of chosen generators of B +and C, i.e., generators of the groups B1, B2, C1 and C2. We construct the crossed module +A := � Aα, where Aα are all the Lf-acyclic crossed modules with less than 2κ generators, see +[5, Theorem 4.4]. +The first step of this proof is to show that if a crossed module X is Lf-local then it is A-local. +Let φ be a morphism in Hom(A, X) and construct by naturality the following commutative +diagram +1 = LfA +A +X +LfX +Lfφ +∼= +φ +By hypothesis, we have an isomorphism between X and LfX and by construction of A, we +obtain LfA = 1. Therefore, φ factors through the zero object and hence Hom(A, X) = 1, +which is equivalent to say that X is A-local. Now consider the PA-equivalence pT: T → PAT +and the Lf-local object LfT. By the above observation, we have that LfT is A-local and by +the universal property we have the desired morphism ψT +T +PAT +LfT +pT +ℓT +ψT +We construct next the fiberwise A-nullification of the following exact sequence +T +LfT +ker(ℓT) +1 +1 +ℓT +By assumption ker(ℓT) is Lf-acyclic, hence also PA-acyclic by design. +This implies that +ker +� +pT: ker(ℓT) → PAker(ℓT) +� +is equal to ker(ℓT). Hence, the exact sequence satisfies condi- +tion (5) of Theorem 1.19 and we obtain the following fiberwise nullification +T +LfT +ker(ℓT) +T/ker(ℓT) +LfT +1 +1 +1 +1 +1 +pT +f +∼= +ℓT + +16 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +Since f is a PA-equivalence, so is ℓT. Hence, we obtain a morphism ϕT in the following +commutative diagram: +T +LfT +PAT +ψT +ℓT +pT +ϕT +By universal property, we can conclude that the two compositions of ψT and ϕT are isomorphic +to identities so that LfT ∼= PAT. A similar argument shows the naturality of ψ and ϕ and +therefore Lf is a nullification functor, namely PA. +□ +5. Nullification functors and admissibility +In the category of groups, the fact that kernels of localization morphisms are L-acyclic +was fundamental to prove that nullification functors are admissible for the class of regular +epimorphisms. This fact is not true in general for nullification functors in the category of +crossed modules as shown in [22, Proposition 4.6], it is thus natural to ask whether nullifica- +tion functors are admissible. We provide an affirmative answer in this final section, but let +us first prove that our counter-example PXZ is admissible. +Proposition 5.1. The nullification functor PXZ is admissible for the class of regular epimor- +phisms. +Proof. Theorem 5.1 in [15] implies that PXZ is admissible provided that the reflective category +of PXZ-local objects is a Birkhoff subcategory, i.e., it is closed under regular quotients and +subobjects. Here PXZ-local objects are crossed modules of the form A → 1 where A is any +abelian group and the connecting homomorphism is the trivial homomorphism. Therefore it +is clearly closed under subobjects. Moreover, the quotient of A → 1 by a normal subcrossed +modules N → 1 is the crossed module A/N → 1 that is PXZ-local. +□ +The remaining part of the section is devoted to the proof that all nullification functors are +admissible for the class of regular epmorphisms. Consider a nullification functor PA where +A = (A1, A2, ∂) is a crossed module. To show the admissibility, it is enough to prove that the +pullback of an exact sequence of PA-local crossed modules along the coaugmentation map +is PA-flat, in other words that the map f in the following commutative diagram of crossed +modules is a PA-equivalence +W +Q +PAN +PAT +PAQ +PAN +1 +1 +1 +1 +(1) +pQ +f +h +g +where (1) is a pullback and g and h are regular epimorphisms. To do so we follow step by +step the inductive construction of PAQ = colimQβ as presented in [22, Proposition 2.8], see +also Proposition 1.11. For each successor ordinal β + 1 we obtain Qβ+1 from Qβ by killing all +morphisms out of A so let us start with the construction of Q1 from Q0 = Q. + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +17 +Remark 5.2. Let ϕ : A → Q be a morphism of crossed modules. The crossed module Q1 is +the quotient of Q by the normal closure KQ in Q of the image of +ev: +� +ϕ∈Hom(A,Q) +A = M −→ Q +which is defined by ϕ on the copy of A indexed by ϕ. The idea behind the construction we +perform next is that we do not need to kill all morphisms from A to the extension W in order +to construct its nullification PAW, it is sufficient to take care of those factoring through Q. +Beware that given an extension N → T → Q with N an A-acyclic crossed module, it is not +true in general that all morphisms from A to T factor through Q. +By definition of pQ we have the following equality for the composition pQ ◦ ϕ = 1 = h ◦ 1 +as below. Therefore, any morphism from A to Q induces one from A to W: +(7) +PAT +W +Q +PAQ +A +h +pQ +f +g +1 +ϕ +∃!ψ +We call ψ the morphism determined by ϕ and it makes sense now to consider KW, the normal +closure in W of the image of M → W. +Lemma 5.3. With the same notation as in Remark 5.2, we have an isomorphism KW ∼= KQ. +Proof. Limits are computed levelwise for crossed modules, so the pullback W consists of +compatible pairs (x, q) for x ∈ (PAT)i and q ∈ Qi for i = 1, 2. By construction of ψ we have +ψ(a) = (1, ϕ(a)). +Now, we compute the kernels of the cokernels of ev: M → Q and (1, ev): M → W. We +have the two following descriptions of the kernels. +KQ = +� +ev1(M1)Q2[ev2(M2)Q2, Q1], ev2(M2)Q2, ∂ +� +KW = +� +(1, ev1)(M1)W2[(1, ev2)(M2)W2, W1], (1, ev2)(M2)W2, ∂′� +The second group of the crossed module KW is the easier one: +(1, ev2)(M2)W2 = {(t2,q2)(1, ev2(m2)) | (t2, q2) ∈ W2, m2 ∈ M2} += {(1, q2ev2(m2)) | q2 ∈ Q2, m2 ∈ M2} += 1 × ev2(M2)Q2. +where the second equality holds since h is surjective. Via similar computations, we see that +(1, ev1)(M1)W1 = 1 × ev1(M1)Q1, so we are left with proving that +[(1, ev2)(M2)W2, W1] = 1 × [ev2(M2)Q2, Q1] + +18 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +This we do via the following equalities: +[(1, ev2)(M2)2, W1] = [(1 × ev2(M2)Q2), W1] += {(1,x2)(t1, q1)(t1, q1)−1 | x2 ∈ ev2(M2)Q2, (t1, q1) ∈ W1} += {(1,x2 q1q−1 +1 +| x2 ∈ ev2(M2)Q2, q1 ∈ Q1} += 1 × [(ev2(M2)Q2, Q1] +So finally we can conclude that KW = 1 × KQ, in particular KW and KQ are isomorphic. +□ +Proposition 5.4. For any ordinal β, we have a commutative diagram +PAT +Wβ +Qβ +PAQ +W +Q +(2) +hβ +g +fβ +pQ +β +h +where (2) is a pullback square, the maps fβ : W → Wβ and pQ +β : Q → Qβ are PA-equivalences, +and hβ is a regular epimorphism. +Proof. We prove it by induction. Since the nullification uses possibly a transfinite construc- +tion we have to initialize the induction, but the case β = 0 holds by assumption, and then +check the statement for successor and limit ordinals. +The successor case Suppose that for an ordinal β the lemma is proved. Then we consider +the kernels KW +β and KQ +β of the cokernels of the evaluation maps ev : � +Hom(A,Qβ) A −→ Qβ and +ev : � +Hom(A,Qβ) A −→ Wβ respectively. They fit in the following diagram of exact rows: +Wβ+1 +Wβ +Qβ +Qβ+1 +KW +β +KQ +β +(2) +pQ +(β→β+1) +f(β→β+1) +hβ +∼= +iW +iQ +∃!hβ+1 +Lemma 5.3 applies here and gives us the isomorphism between KW +β and KQ +β . The composition +pQ +(β→β+1) ◦ hβ ◦ iW : KW +β → Qβ+1 +is zero by commutativity, yielding by the universal property of the cokernel the morphism +hβ+1: Wβ+1 → Qβ+1. The isomorphism between the kernels implies that (2) is a pullback (see +Proposition 1.5). By induction hypothesis hβ is a regular epimorphism and the composition +pQ +(β→β+1) ◦ hβ : Wβ → Qβ+1 is also a regular epimorphism, hence so is hβ+1. We show now +that pQ +(β→β+1) and f(β→β+1) are PA-equivalences. + +ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES +19 +For the first one we write the cokernel Qβ+1 as the pushout along the evaluation morphism: +1 +� A +Qβ +Qβ+1 +pQ +(β→β+1) +1 +ϕ +inc +where the coproduct is taken over Hom(A, Q). The trivial map A → 1 is a PA-equivalence, +thus so is the pushout pQ +(β→β+1) : Qβ → Qβ+1 by Lemma 1.8 (1). By composing with the +PA-equivalence Q → Qβ we see that pQ +β+1 : Q → Qβ+1 is a PA-equivalence as well. The same +argument shows that fβ+1 : W → Wβ+1 is also a PA-equivalence. By the universal property +of the localization, we obtain two maps, one from Wβ+1 to PAT and the other from Qβ+1 to +PAQ such that (2) commutes: +PAT +Wβ+1 +Qβ+1 +PAQ +W +Q +(2) +(1) +hβ+1 +g +fβ+1 +pQ +β+1 +h +f +pQ +Since (1) and the outer rectangle are pullbacks and hβ+1 is a regular epimorphism, we can +conclude by Proposition 4.1.4 in [2] that (2) is a pullback. +The limit case To prove the statement for a general transfinite induction we need to prove +it for a limit ordinal as well. Let γ be a limit ordinal and +Qγ = colimα<γQα +Wγ = colimα<γWα +We have shown that pQ +(α−1→α) : Qα−1 → Qα is a PA-equivalence for all α < γ. Hence the +composition pQ +α : Q → Qα is also a PA-equivalence and Lemma 1.8 (3), implies that pQ +γ : Q → +Qγ is a PA-equivalence. The same reasoning holds for fγ : W → Wγ. The existence of the +maps f : W → PAT and pQ : Q → PAQ give us two maps Wγ → PAT and Qγ → PAQ as +shown on the diagram below (8). +The nullification PAQ is constructed as filtered colimit of the Qα, see Proposition 1.11. +Filtered colimits commutes with finite limits, in particular with kernels. Therefore +KQ +γ := ker(Q → Qγ) ∼= colimα<γker(Q → Qα) +where ker(Q → Qα) will be denoted KQ +α. The category XMod is a variety of algebras (also +called algebra category of fixed type). Hence, by [21, Proposition IX.1.2], we know that the +forgetful functor U : XMod → Set creates filtered colimits. In other words we have : +U(colimα<γKQ +α) = colimα<γUKQ +α = +� +α<γ +UKQ +α +where the colimit in the first term lies in the category of crossed modules and the second +colimit in the category of sets. This means that we know the structure of colimα<γKQ +α as a + +20 +OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK +set. Now since KQ +α ∼= KW +α for all α < γ and KQ +γ can be written as a union of KQ +α (as well as +KW +γ ) we conclude that KQ +γ ∼= KW +γ . We consider now the diagram: +(8) +PAT +Wγ +Qγ +PAQ +W +Q +(1) +(2) +hγ +g +fγ +pQ +γ +h +f +pQ +Since the kernels of fγ and pQ +γ are isomorphic we deduce that (2) is a pullback. +As we +have shown that every map pQ +(α→α+1) : Qα → Qα+1 is a regular epimorphism, the morphism +pQ +α : Q → Qα is also a regular epimorphism, being a composition of regular epimorphisms in +a regular category. The colimit functor being a left adjoint functor, it preserves colimits and +in particular cokernels. In a pointed protomodular category, any regular epimorphism is a +cokernel, therefore +pQ +γ : Q → Qγ +is a regular epimorphism. The composition pQ +γ ◦ g is also a regular epimorphism, and we +conclude that so is hγ. With the same argument as for the successor step, we get that (1) is +a pullback, which ends the induction proof. +□ +We are ready now for the main result of this section. +Theorem 5.5. Let A be any crossed module. The nullification functor PA is admissible for +the class of regular epimorphisms. +Proof. Let W be the pullback of a regular epimorphism h: PAT → PAQ between PA-local +crossed modules along the localization morphism pQ : Q → PAQ. Let λ be the ordinal such +that Qλ ∼= PAQ (see Proposition 1.11). By Proposition 5.4 we have a diagram: +PAT +Wλ +Qλ +PAQ +W +Q +(2) +∼= +hλ +g +h +fλ +pQ +λ +where the outer rectangle is a pullback, the morphisms fλ and pQ +λ are PA-equivalences, and +(2) is a pullback. Since isomorphisms are stable under pullbacks, we have an isomorphism +Wλ ∼= PAT. We have thus proved that the map f : W → PAT is a PA-equivalence, which +means that the functor PA is admissible. +□ +In this article we have focused on regular-epi localization functors because they appear nat- +urally when studying conditional flatness and admissibility in the category of groups, crossed +modules, or more general semi-abelian categories. We conclude this section by observing that +the notion of conditional flatness can also be defined for non regular-epi localization functor. +The next proposition gives an example of such a localization functor which is conditionally + +REFERENCES +21 +flat. Let us stress that we will not a priori have an equivalence with admissiblity, as was +the case for regular-epi localization functors by Theorem 3.4. In the proof of the following +proposition we have thus to verify the more general condition for conditional flatness, as in +Definition 2.3. +Proposition 5.6. There exists a non regular-epi localization functor which is nevertheless +conditionally flat and therefore admissible for the class of regular epimorphisms. +Proof. We consider the functor I defined in Example 1.16 which sends any crossed module +(N1, N2, ∂N) to (N2, N2, IdN2). This functor is not regular-epi because if we consider a crossed +module for which the connecting morphism is not surjective then the localization morphism +will not be a regular epimorphism. +We prove now that I is conditional flat. Let +T +Q +N +1 +1 +κ +α +be any exact sequence of crossed modules. +We see that I((N1, N2, ∂N)) = (N2, N2, IdN2) +is a normal subcrossed module of (T2, T2, IdT2) = I((T1, T2, ∂T) and that I((Q1, Q2, ∂Q)) = +(Q2, Q2, IdQ2) is the cokernel of κ: N → T. Therefore any exact sequence of crossed modules +is I-flat. In particular any pullback along any morphism of crossed modules of an I-flat exact +sequence is I-flat, hence I is conditionally flat. +□ +References +[1] +A. J. Berrick and E. Dror Farjoun. “Fibrations and nullifications”. In: Israel J. Math. +135 (2003), pp. 205–220. +[2] +F. Borceux and D. Bourn. Mal’cev, protomodular, homological and semi-abelian cate- +gories. Vol. 566. Springer Science & Business Media, 2004. +[3] +D. Bourn. “Normalization Equivalence, Kernel Equivalence, and Affine Categories”. In: +2006, pp. 43–62. +[4] +A. K. Bousfield. “Constructions of factorization systems in categories”. In: J. Pure +Appl. Algebra 9.2 (1976), pp. 207–220. +[5] +A. K. Bousfield. “Homotopical localizations of spaces”. In: Amer. J. Math. 119.6 (1997), +pp. 1321–1354. issn: 0002-9327. url: http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.6bousfield.pdf. +[6] +R. Brown and P. Higgins. “On the connection between the second relative homotopy +groups of some related spaces”. In: Proceedings of The London Mathematical Society +(1978), pp. 193–212. +[7] +R. Brown and C. Spencer. “G-groupoids, crossed modules and the fundamental groupoid +of a topological group”. In: Indag. Math. (Proceedings) 79.4 (1976), pp. 296–302. +[8] +C. Casacuberta. “Anderson localization from a modern point of view”. In: Contemp. +Math. 181 (1995), pp. 35–46. +[9] +C. Casacuberta. “On structures preserved by idempotent transformations of groups +and homotopy types”. In: Crystallographic groups and their generalizations (Kortrijk, +1999). Vol. 262. Contemp. Math. Amer. Math. Soc., Providence, RI, 2000, pp. 39–68. +[10] +C. Casacuberta and A. Descheemaeker. “Relative group completions”. In: Journal of +Algebra 285.2 (2005), pp. 451–469. +[11] +C. Cassidy, M. H´ebert, and G. M. Kelly. “Reflective subcategories, localizations and +factorization systems”. In: J. Austral. Math. Soc. Ser. A 38.3 (1985), pp. 287–329. +[12] +T. Everaert, M. Gran, and T. Van der Linden. “Higher Hopf formulae for Homology +via Galois Theory”. In: Adv. Math. 217 (2008), pp. 2231–2267. + +22 +REFERENCES +[13] +E. Dror Farjoun. Cellular spaces, null spaces and homotopy localization. Vol. 1622. +Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996, pp. xiv+199. +[14] +E. Dror Farjoun and J. Scherer. “Conditionally flat functors on spaces and groups”. In: +Collect. Math. 66.1 (2015), pp. 149–160. +[15] +M. Gran and J. Scherer. “Conditional flatness and admissibility of a reflector in a +semi-abelian category”. In: preprint (2022), 15 pages. +[16] +Philip S. Hirschhorn. Model categories and their localizations. Vol. 99. Mathemati- +cal Surveys and Monographs. American Mathematical Society, Providence, RI, 2003, +pp. xvi+457. +[17] +G. Janelidze and G. M. Kelly. “Galois theory and a general notion of central ex- +tension”. In: J. Pure Appl. Algebra 97.2 (1994), pp. 135–161. issn: 0022-4049. doi: +10.1016/0022-4049(94)90057-4. url: https://doi.org/10.1016/0022-4049(94)90057-4. +[18] +G. Janelidze, L. M´arki, and W. Tholen. “Semi-abelian categories”. In: J. Pure Appl. +Algebra 168 (2002), pp. 367–386. +[19] +M. Ladra and A.R. Grandjean. “Crossed modules and homology”. In: Journal of Pure +and Applied Algebra 95.1 (1994), pp. 41–55. +[20] +M. Ladra, M.P. L´opez L´opez, and E. Rodeja. “Epimorphisms of crossed modules”. In: +Southeast Asian Bulletin of Mathematics 28 (Jan. 2004). +[21] +S. Mac Lane. Categories for the Working Mathematicians. 2nd ed. Springer, 1997. +[22] +O. Monjon, J. Scherer, and F. Sterck. Non-existence of fiberwise localization for crossed +modules. arxiv:2207.09702, to appear in Israel J. Math. 2022. +[23] +K. Norrie. “Actions and automorphisms of crossed modules”. In: Bulletin de la Soci´et´e +Math´ematique de France 118.2 (1990), pp. 129–146. +[24] +K. Norrie. “Crossed modules and analogues of group theorems”. PhD thesis. King’s +College, University of London, 1987. +[25] +J. H. C. Whitehead. “Combinatorial homotopy II”. In: Bull. Amer. Math. Soc 55 (1949), +pp. 453–496. +Mathematics, Ecole Polytechnique F´ed´erale de Lausanne, EPFL, Switzerland +Email address: olivia.monjon@gmail.com +Mathematics, Ecole Polytechnique F´ed´erale de Lausanne, EPFL, Switzerland +Email address: jerome.scherer@epfl.ch +Institut de Recherche en Math´ematique et Physique, Universit´e catholique de Louvain, +Belgium +Email address: +florence.sterck@uclouvain.be + diff --git a/JdAyT4oBgHgl3EQffviy/content/tmp_files/2301.00347v1.pdf.txt b/JdAyT4oBgHgl3EQffviy/content/tmp_files/2301.00347v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9860f5a57a4dbb2c06417f46eda7f0cb0c9b749b --- /dev/null +++ b/JdAyT4oBgHgl3EQffviy/content/tmp_files/2301.00347v1.pdf.txt @@ -0,0 +1,692 @@ +JWST high redshift galaxy observations have a strong tension with Planck CMB +measurements +Deng Wang∗ and Yizhou Liu +National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China +JWST high redshift galaxy observations predict a higher star formation efficiency that the stan- +dard cosmology, which poses a new tension to ΛCDM. We find that the situation is worse than +expected. +The true situation is that the Planck CMB measurement has a strong tension with +JWST high redshift galaxy observations. Specifically, we make a trial to alleviate this tension by +considering alternative cosmological models including dark matter-baryon interaction, f(R) gravity +and dynamical dark energy. Within current cosmological constraints from Planck-2018 CMB data, +we find that these models all fail to explain such a large tension. A possible scenario to escape +from cosmological constraints is the extended Press-Schechter formalism, where we consider the +local environmental effect on the early formation of massive galaxies. Interestingly, we find that an +appropriate value of nonlinear environmental overdensity of a high redshift halo can well explain +this tension. +I. +INTRODUCTION +Since the cosmic acceleration is discovered by Type Ia supernovae (SNe Ia) [1, 2] and confirmed by two independent +probes cosmic microwave background (CMB) [3–5] and baryon acoustic oscillations (BAO) [6, 7], the standard 6- +parameter cosmological model, Λ-cold dark matter (ΛCDM) has achieved great success in characterizing the physical +phenomena across multiple scales at the background and perturbation levels. However, the validity of ΛCDM is +challenged by various kinds of new observations for a long time, and consequently new puzzles emerge such as the +so-called Hubble constant (H0) tension (see [8, 9] for recent reviews). It is noteworthy that, so far, we can not study +effectively the correctness of ΛCDM around redshift z ∼ 10, since currently mainstream probes BAO and SNe Ia +can not give direct observations at high redshifts. The lack of stable high redshift observations will prevent us from +testing ΛCDM more completely during the early stage of the evolution of our universe. +Very excitingly, the recent released high redshift galaxy observations [10–13] in the range z ∈ [7, 11] by JWST, +which contains a population of surprisingly massive galaxy candidates with stellar masses of order of 109M⊙, can +help explore whether ΛCDM is valid at high redshifts. +In the literature, Refs.[10, 11, 14, 15] have reported the +cumulative stellar mass density (CSMD) estimated from early JWST data is higher than that predicted by ΛCDM +within z ∈ [7, 11]. Ref.[16] points out that dynamical dark energy (DDE) can explain this anomalous signal and the +corresponding constraint on DDE is displayed. Subsequently, if the nature of dark matter (DM) is fuzzy, this high +SMD can be recovered [17]. Furthermore, Ref.[18] discusses under which circumstances primordial non-Gaussianity +can act as a solution. +Since these high redshift galaxy observations from JWST have important implications on cosmology and astro- +physics, we attempt to probe whether early JWST data indicates any possible signal of new physics. Specifically, we +study three classes of beyond ΛCDM cosmological models, i.e., DM-baryon interaction (DMBI), modified gravity (MG) +and DDE. In addition, we consider the case of the extended halo mass function (HMF). We find that Within current +cosmological constraints from Planck-2018 CMB obervations, these three models all fail to explain this large tension. +A possibly successful scenario to escape from cosmological constraints is the extended Press-Schechter formalism. +This study is outlined in the following manner. In the next section, we introduce the basic formula of CSMD. In +Section III, we review briefly the alternative cosmological models and extended Press-Schechter HMF. In Section IV, +numerical results are displayed. The discussions and conclusions are presented in the final section. +II. +BASIC FORMULA +As shown in Ref.[10], the CSMD from early JWST data has a large excess relative to that predicted by ΛCDM. +To explain this excess, we shall briefly introduce the basic formula of the cumulative SMD. The HMF for a given +∗Electronic address: cstar@nao.cas.cn +arXiv:2301.00347v1 [astro-ph.CO] 1 Jan 2023 + +2 +cosmological model reads as +dn +dM = F(ν) ρm +M 2 +���� +d ln σ +d ln M +���� , +(1) +where the function F(ν) for the Press-Schechter HMF [19] is expressed as +F(ν) = +� +2 +π νe− ν2 +2 , +(2) +and ρm denotes the average background matter density, M the halo mass, σ the variance of smoothed linear matter +density field and reads as +σ2(R) = +1 +2π2 +� ∞ +0 +k2P(k)W 2(kR)dk, +(3) +where k is the comoving wavenumber, P(k) the matter power spectrum, W(kR) = 3(sin kR − kR cos kR)/(kR)3 the +Fourier transformation of a spherical top-hat filter with radius R = [3M/(4π¯ρ0)]1/3, ν = δc/[D(z) σ] [20] (δc = 1.686 +is the critical collapsed density) and D(z) = g(z)/[g(0)(1 + z)] the linear growth factor for a specific cosmological +model, where g(z) for ΛCDM reads as +g(z) = 5 +2Ωm(z) +� +Ωm(z) +4 +7 − ΩΛ(z) + +� +1 + Ωm(z) +2 +� � +1 + ΩΛ(z) +70 +��−1 +, +(4) +where Ωm(z) and ΩΛ(z) are energy densities of matter and dark energy (DE) at a given redshift, respectively. +An effective quantity to study the validity of the ΛCDM model is the CSMD ρ⋆, which can be characterized by a +fraction of baryon mass contained within a given DM halo above a certain mass scale M⋆ and reads as +ρ⋆(> M⋆, z) = ϵfb +� z2 +z1 +� ∞ +M⋆ +ϵfb +dn +dM MdM dV +dz +dz +V (z1, z2), +(5) +where ϵ is the star formation efficiency, fb the baryon fraction and V (z1, z2) the comoving volume in the redshift range +z ∈ [z1, z2]. +III. +ALTERNATIVE MODELS +A. +Dark matter-baryon interaction +Up to now, the standard cosmological paradigm indicates that DM is cold, collisionless and only participates in +gravitational interactions [9]. In light of the lack of experimental detections of DM and emergent cosmological tensions +in recent years, the scenario beyond the standard DM assumption becomes more and more attractive. An interesting +category is interactions between DM and the Standard Model particles such as baryons, photons and neutrinos. In +this study, we consider the case of DMBI. +The interaction between DM and baryons produces a momentum exchange proportional to momentum transfer +cross section, which can be shown as +σT = +� +(1 − cos θ)dΩ d¯σ +dΩ, +(6) +In the weakly coupled theory, σT can just depend on even powers of DM-baryon relative velocity v and, in general, it is +a power law function of v. Here we adopt σT = σDM−bvnb and denote the DMBI cross section as σDM−b. Specifically, +we study the mini-charged DM (DM particle with a fractional electric charge) corresponding to the case of nb = −4, +which has been used to explain the anomalous 21 cm signal from EDGES [21]. +For this model, we introduce two basic assumptions: (i) DM and baryons obey the Maxwell velocity distribution; (ii) +both species are non-relativistic. As a consequence, the Euler equation of DM can obtain an extra term ΓDM−b(θb − +θDM), where ΓDM−b is the conformal DM-baryon momentum exchange rate, and θDM and θb represent the velocities +of DM and baryons, respectively. At leading order, ΓDM−b is expressed in terms of DM bulk velocity and reads as +[22] +ΓDM−b = aρbfHeσDM−bc−4 +mDM + mb +� TDM +mDM ++ Tb +mb ++ V 2 +RMS +3 +�−1.5 +, +(7) + +3 +where a is the scale factor, ρb the average baryon energy density, fHe ≃ 0.76, c−4 = 0.27 the integration constant +(see [22, 23] for details), and Ti and mi denote the temperature and average mass of species i, respectively. The bulk +velocity dispersion can be shown as [24] +V 2 +RMS = +� +� +� +10−8, +z > 103 +(1 + z)2 +10 +, +z ≤ 103 . +(8) +The interaction between DM and baryons can produce the energy and momentum exchange. It is clear that DMBI +reduces to ΛCDM when σDM−b = 0. There is a possibility that DMBI can increase the baryon fraction and conse- +quently give a large star formation efficiency. This indicates that DMBI can act as a potential solution to the recent +puzzle from JWST data. +B. +Modified gravity +Since general relativity (GR) can not explain current cosmic expansion in the absence of cosmological constant, the +modifications in the gravity sector on cosmic scales has inspired a broad interest in order to describe this anomalous +phenomenon. Here we shall consider the simplest extension to GR, f(R) gravity, where the modification is a function +of Ricci scalar R. f(R) gravity was firstly introduced by Buchdahl [25] in 1970 and more detailed information can be +found in recent reviews [26, 27]. Its action is written as +S = +� +d4x√−g +�f(R) +2 ++ Lm +� +, +(9) +where Lm and g denote the matter Lagrangian and the trace of a given metric, respectively. +For the late-time universe, a viable f(R) gravity scenario should explain the cosmic expansion, pass the local +gravity test and satisfy the stability conditions. To investigate whether MG can explain the high redshift galaxy +data from JWST, in this study, we consider the so-called Hu-Sawicki f(R) model (hereafter HS model) [28], which is +characterized by +f(R) = R − +2ΛR¯n +R¯n + µ2¯n , +(10) +where ¯n and µ are two free parameters characterizing this model. By taking R ≫ µ2, the approximate f(R) function +can be expressed as +f(R) = R − 2Λ − fR0 +¯n +R¯n+1 +0 +R¯n , +(11) +where R0 is the present-day value of Ricci scalar and fR0 = −2Λµ2/R2 +0. Note that HS f(R) gravity reduces to ΛCDM +when fR0 = 0. +An intriguing question is whether recent JWST anomaly is a signal of beyond GR. We will carefully analyze this +possibility in this study. +C. +Dynamical dark energy +Although Ref.[16] has claimed that DDE can explain the large CSMD from JWST, we think their method is +inappropriate and consequently their result maybe incorrect. We need to reanalyze the case of DDE. +As is well known, the equation of state (EoS) of DE w = −1 in the standard cosmological model. However, starting +from observations, the doubt about the correctness of ΛCDM stimulates the community to explore whether DE is +dynamical over time or not. In general, one depicts the DDE model by a simple Taylor expansion of DE EoS, i.e., +ω(a) = ω0 + (1 − a)ωa [29, 30], where ωa characterizes the time evolution of DE EoS. The dimensionless Hubble +parameter is expressed as +EDDE(z) = +� +Ωm(1 + z)3 + (1 − Ωm)(1 + z)3(1+ω0+ωa)e +−3ωaz +1+z +� 1 +2 +. +(12) +Note that this model is a two-parameter extension to ΛCDM and it reduces to ΛCDM when ω0 = −1 and ωa = 0. + +4 +D. +Extended halo mass function +When applied into a complicated gravity system, the function of Press-Schechter HMF is limited, since it does +not consider the nonlinear environmental effects. To overcome this shortcoming, the extended Press-Schechter (EPS) +HMF is proposed in Ref.[31] and reads as +dn(M1, z|M2, δ2) +dM1 += M2 +M1 +fm(S1, δ1|S2, δ2) +���� +dS1 +dM1 +���� , +(13) +where the mass variance S1 = σ2(M1) and S2 = σ2(M2) (see Eq.(3)), and one can obtain the average number of +progenitors at time t1 in the mass range (M1, M1 + dM1) which by time t2 (t2 > t1) have merged to form a large halo +of mass M2. The multiplicity function fm is expressed as +fm(S1, δ1|S2, δ2) = +1 +√ +2π +δ1 − δ2 +(S1 − S2)3/2 exp +� +− (δ1 − δ2)2 +2(S1 − S2) +� +dS1. +(14) +δ1 and δ2 are, respectively, the linear overdensities in spherical regions of masses M1 and M2. To study the environ- +mental impacts on the high redshift HMF, we choose M2 as a present-day halo corresponding to current overdensity +δ2. To compute δ2, one should transform the nonlinear overdensity δnl at redshift z in Eulerian space into the linear +overdensity in Lagrangian space. The corresponding analytic fitting formula based on spherical collapse model is +[32, 33] +δ2(δnl, z) = +δ1 +1.68647 +� +1.68647 − +1.35 +(1 + δnl)2/3 − +1.12431 +(1 + δnl)1/2 + +0.78785 +(1 + δnl)0.58661 +� +. +(15) +Since there is a possibility that the excessively high CSMD from JWST is caused by nonlinear environmental effect, +we attempt to explain it using the EPS formalism. +IV. +METHODS AND RESULTS +At first, we employ the best fits from current cosmological constraints as our baseline values for four models. Since +we hope that the following calculations can be permitted by present-day observations, our discussions and results +will mainly focus on the allowed parameter space. Then, for different models, we use different Boltzmann codes to +calculate their background evolution, growth factors and matter power spectrum at different redshifts. Specifically, +we take CLASS [22, 23, 34] for DMBI and use modified CAMB [35, 36] for f(R) gravity, DDE and EPS scenarios. Note +that ΛCDM is adopted in the EPS scenario. Subsequently, we compute the HMF at different redshifts for the above +four models. Finally, we work out the maximal CSMD for each model according to the permitted parameter space, +and check whether these scenarios are consistent with the latest JWST data. Notice that Eq.(4) is only used in the +EPS model and the growth factors of the other three models are obtained from the corresponding software package. +Our numerical analysis results are presented in Figs.1-3. At first, we display the CSMD of ΛCDM in the redshift +range z ∈ [7, 9] and see its performance. In general, the SFE ϵ is about 10% according to current observational +constraints [11]. Nonetheless, one can see that in the top left panel of Fig.1, 10% is nowhere near enough to reach +the lower bounds of JWST data points in ΛCDM. One needs the star formation rate in galaxies to be at least 50% +in order to explain the inconsistency. In the meanwhile, one can easily find that ϵ = 0.8 can successfully explain two +data points but 100% SFE can not. Except for ΛCDM, we all calculate the maximal CSMD in the other models, i.e., +assuming ϵ = 1. +In the second place, we make a trial to explore whether alternative cosmological models can alleviate even solve +the tension between JWST and Planck CMB observations. In the DMBI case, we attempt to acquire a higher baryon +fraction by the coupling between DM and baryons, and consequently explain this discrepancy occurred in ΛCDM. +However, we find that varying coupling strength σDM−b hardly affects the CSMD, and only the variation of interaction +DM fraction Ωidm affects significantly the CSMD. When assuming the DM particle mass mDM = 100 GeV, the cross +section σDM−b = 10−42 cm2 and choosing the fraction Ωidm = 0.01, 0.03 and 0.05, this tension can be efficiently +relieved but it seems that this model is difficult to explain both data points. However, if considering the current +cosmological constraint that gives a very small Ωidm [24], DMBI still behaves like ΛCDM and can not resolve this +discrepancy. We have also studied the impacts of mDM and find different DM particle masses also can not explain +JWST data. +In f(R) gravity, we find small fR0 such as 0.1 and 1 can not expalin the anomaly but a very large value fR0 = 10 +can do. This implies that one needs a large deviation from GR to be responsible for JWST data. Unfortunately, the + +5 +109 +1010 +1011 +M +102 +103 +104 +105 +106 +107 +108 +( > M )[M +Mpc +3] +CDM, = 1 +CDM, = 0.8 +CDM, = 0.5 +CDM, = 0.1 +109 +1010 +1011 +M +103 +104 +105 +106 +107 +108 +( > M )[M +Mpc +3] +CDM +idm=0.01, mDM=100 GeV +idm=0.03, mDM=100 GeV +idm=0.05, mDM=100 GeV +109 +1010 +1011 +M +103 +104 +105 +106 +107 +108 +( > M )[M +Mpc +3] +CDM +fR0 = 0.1 +fR0 = 1 +fR0 = 10 +109 +1010 +1011 +M +102 +103 +104 +105 +106 +107 +108 +( > M )[M +Mpc +3] +nl = 1 +nl = 0.7 +nl = 0.6 +nl = 0.5 +nl = 0.4 +nl = 0.1 +CDM +FIG. 1: The CSMDs for the ΛCDM, DMBI, f(R) gravity and EPS models are shown from top to bottom and left to right, +respectively. Note that for ΛCDM, we compute the CSMDs in the redshift range z ∈ [7, 9] by choosing different values of the +SFE ϵ. For the other models, we calculate the CSMDs in the redshift range z ∈ [9, 11] when ϵ = 1. +latest cosmological constraint gives log10 fR0 < −6.32 at the 2 σ confidence level [37], which is much smaller than 10. +Therefore, similar to DMBI, f(R) gravity also fails to alleviate this tension. Interestingly, this gives us a hint that, if +two galaxies observed by JWST are located in the low density region of the universe where MG effect is very large, +the data can be appropriately explained. +Furthermore, we are interested in whether the nature an simple extension to ΛCDM, DDE, can explain the incon- +sistency. As mentioned above, Ref.[16] claimed that JWST data can clearly constrain DDE. However, within current +constraining precision, we query this conclusion. To ensure the validity of our conclusion, we constrain ΛCDM and +DDE models using the Planck-2018 CMB temperature and polarization data (see Fig.2), and then obtain the best +fitting values of parameters of these two models. One can easily find the constrained values of model parameters +of ΛCDM in Ref.[5]. For DDE, we obtain current baryon and CDM densities Ωbh2 = 0.0225 and Ωch2 = 0.1184, +the ratio between angular diameter distance and sound horizon at the redshift of last scattering θMC = 1.04109, +the optical depth due to the reionization τ = 0.06, the amplitude and spectral index of primordial power spectrum +As = 2.114 × 10−9 and ns = 0.9698, and two DE EoS parameters ω0 = −0.38 and ωa = −4.8. Same as DMBI +and f(R) gravity models, we use the same method to work out the CSMD of DDE, and find that the variation of +the CSMD is largely dominated by the values of six basic parameters Ωbh2, Ωch2, θMC, τ, As and ns. Although +ω0 and ωa is loosely constrained by CMB data (constrained ω0-ωa parameter space is large), different values of ω0 +and ωa hardly affect the CSMD. For instance, in the left panel of Fig.3, ω0 = −0.38 and ωa = −4.8 plus the ΛCDM +and DDE best fits gives completely different CSMDs. Choosing the ΛCDM best fit, (ω0, ωa) = (−0.38, −4.8) and + +6 +0.0220 +0.0224 +0.0228 +bh2 +0.7 +0.8 +0.9 +1.0 +1.1 +8 +0.2 +0.3 +0.4 +0.5 +m +50 +60 +70 +80 +90 +H0 +8 +6 +4 +2 +0 +2 +4 +wa +2 +1 +0 +w +3.00 +3.05 +3.10 +ln(1010As) +0.96 +0.97 +0.98 +ns +0.02 +0.04 +0.06 +0.08 +0.10 +1.0400 +1.0405 +1.0410 +1.0415 +1.0420 +100 +MC +0.114 +0.116 +0.118 +0.120 +0.122 +0.124 +ch2 +0.114 +0.117 +0.120 +0.123 +ch2 +1.040 +1.041 +1.042 +100 +MC +0.02 +0.04 +0.06 +0.08 +0.10 +0.96 +0.97 +0.98 +ns +3.00 +3.05 +3.10 +ln(1010As) +2 +1 +0 +w +8 +4 +0 +4 +wa +50 +60 +70 +80 +90 +H0 +0.2 +0.3 +0.4 +0.5 +m +0.7 +0.8 +0.9 +1.0 +1.1 +8 +DDE +CDM +FIG. 2: The marginalized posterior probability distributions of the ΛCDM and DDE models from the Planck-2018 CMB +constraints are shown. +(ω0, ωa) = (−1, −1) gives very similar results in the logarithmic space. In the medium and right panels of Fig.3, we +verifies that taking same best fits of ΛCDM and DDE, respectively, choosing different DE EoS parameter pair just +produces very limited differences. After scanning the DDE parameter space, we find clearly that DDE also can not +explain this tension, but its best fit can help increase the value of CSMD and become closer to JWST data points (see +the left panel of Fig.3). The reason that the result in Ref.[16] is different from ours is that they do not implement an +appropriate cosmological constraint based on the Planck CMB data. +The result from f(R) gravity prompts us to study the environmental effect of JWST galaxies on the CSMD. The +most straightforward method is replacing the Press-Schechter HMF with the EPS formalism in the framework of +ΛCDM, where the sole parameter δnl characterizes the nonlinear environmental effect of a high redshift halo. In the +bottom right panel, we calculate the maximal CSMDs in the redshift range z ∈ [9, 11] for the EPS model. We find +that neither overlarge (δnl = 1) nor too small (δnl = 0.1) explain JWST observations and that the larger δnl is, the +larger the CSMD is. Since the total sky area covered by the JWST initial observation is large enough (∼ 40 armin2) + +7 +109 +1010 +1011 +M +103 +104 +105 +106 +107 +108 +( > M )[M +Mpc +3] +CDM +w0=-0.38, wa=-4.8 + DDE best fit +w0=-0.38, wa=-4.8 + CDM best fit +w0=-1, wa=-1 + CDM best fit +3.60 +3.62 +3.64 +3.66 +3.68 +3.70 +M +1e10 +12000 +12200 +12400 +12600 +12800 +13000 +( > M )[M +Mpc +3] +DDE w0=-0.38, wa=-4.8 +DDE w0=-1, wa=-5 +DDE w0=-1, wa=-1 +DDE w0=-1, wa=0 +3.60 +3.62 +3.64 +3.66 +3.68 +3.70 +M +1e10 +42000 +42500 +43000 +43500 +44000 +44500 +45000 +( > M )[M +Mpc +3] +DDE w0=-0.38, wa=-4.8 +DDE w0=-1, wa=-5 +DDE w0=-1, wa=-1 +DDE w0=-1, wa=0 +FIG. 3: The CSMDs of the DDE model computed at in the redshift range z ∈ [9, 11] are shown when assuming ϵ = 1. Left: +Different combinations of parameter values and best fits from constraints, respectively. Medium: Only the ΛCDM best fit; +Right: Only the DDE best fit. +[10], we can not rule out this possibly local environmental effect. However, unfortunately, there is no δnl passing two +data points simultaneously. +V. +DISCUSSIONS AND CONCLUSIONS +Recently, the early data release of JWST reveals the possible existence of high redshift galaxies. What is interesting +is these galaxies in the redshift range z ∈ [7, 11] exhibit the overlarge star formation rate, which is incompatible with +the prediction of the standard cosmology. This may indicate that JWST data contain the signal of new physics. +In this study, we try to resolve this tension with alternative cosmological models including DMBI, f(R) gravity +and DDE. We find that in light of the precision of current cosmological constraint from Planck-2018 CMB data, these +models all fail to explain this large tension. Specifically, for DMBI, the coupling strength σDM−b between DM and +baryons hardly affects the CSMD. For f(R) gravity, the effect of varying fR0 on the CSMD is too small to relive the +tension. For DDE, although the constrained DE EoS parameter space is large, different parameter pair (ω0, ωa) just +produces very limited differences in the CSMD. Interestingly, a large interacting DM fraction and a large deviation +from Einstein’s gravity can both generate a large CSMD. +A possible scenario to escape from current cosmological constraints is the EPS formalism, where we consider the +local environmental effect on the CSMD. We find that an appropriate value of nonlinear environmental overdensity +of a high redshift halo can well explain the CSMD discrepancy. However, we do not find an EPS model that can +simultaneously explain two data points. +In the near future, JWST will bring more useful data to human beings, so that we can extract more physical +information to uncover the mysterious veil of nature. +Acknowledgments +DW warmly thanks Liang Gao, Jie Wang and Qi Guo for helpful discussions. We thank Hang Yang for letting us +notice the JWST related works. 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D 106, no.6, 6 (2022). + diff --git a/JdAyT4oBgHgl3EQffviy/content/tmp_files/load_file.txt b/JdAyT4oBgHgl3EQffviy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e98562b8bb28f5343e1c1f5f52040392602f28c9 --- /dev/null +++ b/JdAyT4oBgHgl3EQffviy/content/tmp_files/load_file.txt @@ -0,0 +1,547 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf,len=546 +page_content='JWST high redshift galaxy observations have a strong tension with Planck CMB measurements Deng Wang∗ and Yizhou Liu National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China JWST high redshift galaxy observations predict a higher star formation efficiency that the stan- dard cosmology, which poses a new tension to ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We find that the situation is worse than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The true situation is that the Planck CMB measurement has a strong tension with JWST high redshift galaxy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Specifically, we make a trial to alleviate this tension by considering alternative cosmological models including dark matter-baryon interaction, f(R) gravity and dynamical dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Within current cosmological constraints from Planck-2018 CMB data, we find that these models all fail to explain such a large tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' A possible scenario to escape from cosmological constraints is the extended Press-Schechter formalism, where we consider the local environmental effect on the early formation of massive galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Interestingly, we find that an appropriate value of nonlinear environmental overdensity of a high redshift halo can well explain this tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' INTRODUCTION Since the cosmic acceleration is discovered by Type Ia supernovae (SNe Ia) [1, 2] and confirmed by two independent probes cosmic microwave background (CMB) [3–5] and baryon acoustic oscillations (BAO) [6, 7], the standard 6- parameter cosmological model, Λ-cold dark matter (ΛCDM) has achieved great success in characterizing the physical phenomena across multiple scales at the background and perturbation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' However, the validity of ΛCDM is challenged by various kinds of new observations for a long time, and consequently new puzzles emerge such as the so-called Hubble constant (H0) tension (see [8, 9] for recent reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' It is noteworthy that, so far, we can not study effectively the correctness of ΛCDM around redshift z ∼ 10, since currently mainstream probes BAO and SNe Ia can not give direct observations at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The lack of stable high redshift observations will prevent us from testing ΛCDM more completely during the early stage of the evolution of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Very excitingly, the recent released high redshift galaxy observations [10–13] in the range z ∈ [7, 11] by JWST, which contains a population of surprisingly massive galaxy candidates with stellar masses of order of 109M⊙, can help explore whether ΛCDM is valid at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the literature, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [10, 11, 14, 15] have reported the cumulative stellar mass density (CSMD) estimated from early JWST data is higher than that predicted by ΛCDM within z ∈ [7, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [16] points out that dynamical dark energy (DDE) can explain this anomalous signal and the corresponding constraint on DDE is displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Subsequently, if the nature of dark matter (DM) is fuzzy, this high SMD can be recovered [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Furthermore, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [18] discusses under which circumstances primordial non-Gaussianity can act as a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Since these high redshift galaxy observations from JWST have important implications on cosmology and astro- physics, we attempt to probe whether early JWST data indicates any possible signal of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Specifically, we study three classes of beyond ΛCDM cosmological models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=', DM-baryon interaction (DMBI), modified gravity (MG) and DDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In addition, we consider the case of the extended halo mass function (HMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We find that Within current cosmological constraints from Planck-2018 CMB obervations, these three models all fail to explain this large tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' A possibly successful scenario to escape from cosmological constraints is the extended Press-Schechter formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' This study is outlined in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the next section, we introduce the basic formula of CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In Section III, we review briefly the alternative cosmological models and extended Press-Schechter HMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In Section IV, numerical results are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The discussions and conclusions are presented in the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' BASIC FORMULA As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [10], the CSMD from early JWST data has a large excess relative to that predicted by ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' To explain this excess, we shall briefly introduce the basic formula of the cumulative SMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The HMF for a given ∗Electronic address: cstar@nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='00347v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='CO] 1 Jan 2023 2 cosmological model reads as dn dM = F(ν) ρm M 2 ���� d ln σ d ln M ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (1) where the function F(ν) for the Press-Schechter HMF [19] is expressed as F(ν) = � 2 π νe− ν2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (2) and ρm denotes the average background matter density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' M the halo mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' σ the variance of smoothed linear matter density field and reads as σ2(R) = 1 2π2 � ∞ 0 k2P(k)W 2(kR)dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (3) where k is the comoving wavenumber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' P(k) the matter power spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' W(kR) = 3(sin kR − kR cos kR)/(kR)3 the Fourier transformation of a spherical top-hat filter with radius R = [3M/(4π¯ρ0)]1/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' ν = δc/[D(z) σ] [20] (δc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='686 is the critical collapsed density) and D(z) = g(z)/[g(0)(1 + z)] the linear growth factor for a specific cosmological model, where g(z) for ΛCDM reads as g(z) = 5 2Ωm(z) � Ωm(z) 4 7 − ΩΛ(z) + � 1 + Ωm(z) 2 � � 1 + ΩΛ(z) 70 ��−1 , (4) where Ωm(z) and ΩΛ(z) are energy densities of matter and dark energy (DE) at a given redshift, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' An effective quantity to study the validity of the ΛCDM model is the CSMD ρ⋆, which can be characterized by a fraction of baryon mass contained within a given DM halo above a certain mass scale M⋆ and reads as ρ⋆(> M⋆, z) = ϵfb � z2 z1 � ∞ M⋆ ϵfb dn dM MdM dV dz dz V (z1, z2), (5) where ϵ is the star formation efficiency, fb the baryon fraction and V (z1, z2) the comoving volume in the redshift range z ∈ [z1, z2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' ALTERNATIVE MODELS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Dark matter-baryon interaction Up to now, the standard cosmological paradigm indicates that DM is cold, collisionless and only participates in gravitational interactions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In light of the lack of experimental detections of DM and emergent cosmological tensions in recent years, the scenario beyond the standard DM assumption becomes more and more attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' An interesting category is interactions between DM and the Standard Model particles such as baryons, photons and neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In this study, we consider the case of DMBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The interaction between DM and baryons produces a momentum exchange proportional to momentum transfer cross section, which can be shown as σT = � (1 − cos θ)dΩ d¯σ dΩ, (6) In the weakly coupled theory, σT can just depend on even powers of DM-baryon relative velocity v and, in general, it is a power law function of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Here we adopt σT = σDM−bvnb and denote the DMBI cross section as σDM−b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Specifically, we study the mini-charged DM (DM particle with a fractional electric charge) corresponding to the case of nb = −4, which has been used to explain the anomalous 21 cm signal from EDGES [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' For this model, we introduce two basic assumptions: (i) DM and baryons obey the Maxwell velocity distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (ii) both species are non-relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' As a consequence, the Euler equation of DM can obtain an extra term ΓDM−b(θb − θDM), where ΓDM−b is the conformal DM-baryon momentum exchange rate, and θDM and θb represent the velocities of DM and baryons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' At leading order, ΓDM−b is expressed in terms of DM bulk velocity and reads as [22] ΓDM−b = aρbfHeσDM−bc−4 mDM + mb � TDM mDM + Tb mb + V 2 RMS 3 �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='5 , (7) 3 where a is the scale factor, ρb the average baryon energy density, fHe ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='76, c−4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='27 the integration constant (see [22, 23] for details), and Ti and mi denote the temperature and average mass of species i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The bulk velocity dispersion can be shown as [24] V 2 RMS = � � � 10−8, z > 103 (1 + z)2 10 , z ≤ 103 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (8) The interaction between DM and baryons can produce the energy and momentum exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' It is clear that DMBI reduces to ΛCDM when σDM−b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' There is a possibility that DMBI can increase the baryon fraction and conse- quently give a large star formation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' This indicates that DMBI can act as a potential solution to the recent puzzle from JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Modified gravity Since general relativity (GR) can not explain current cosmic expansion in the absence of cosmological constant, the modifications in the gravity sector on cosmic scales has inspired a broad interest in order to describe this anomalous phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Here we shall consider the simplest extension to GR, f(R) gravity, where the modification is a function of Ricci scalar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' f(R) gravity was firstly introduced by Buchdahl [25] in 1970 and more detailed information can be found in recent reviews [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Its action is written as S = � d4x√−g �f(R) 2 + Lm � , (9) where Lm and g denote the matter Lagrangian and the trace of a given metric, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' For the late-time universe, a viable f(R) gravity scenario should explain the cosmic expansion, pass the local gravity test and satisfy the stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' To investigate whether MG can explain the high redshift galaxy data from JWST, in this study, we consider the so-called Hu-Sawicki f(R) model (hereafter HS model) [28], which is characterized by f(R) = R − 2ΛR¯n R¯n + µ2¯n , (10) where ¯n and µ are two free parameters characterizing this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' By taking R ≫ µ2, the approximate f(R) function can be expressed as f(R) = R − 2Λ − fR0 ¯n R¯n+1 0 R¯n , (11) where R0 is the present-day value of Ricci scalar and fR0 = −2Λµ2/R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Note that HS f(R) gravity reduces to ΛCDM when fR0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' An intriguing question is whether recent JWST anomaly is a signal of beyond GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We will carefully analyze this possibility in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Dynamical dark energy Although Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [16] has claimed that DDE can explain the large CSMD from JWST, we think their method is inappropriate and consequently their result maybe incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We need to reanalyze the case of DDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' As is well known, the equation of state (EoS) of DE w = −1 in the standard cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' However, starting from observations, the doubt about the correctness of ΛCDM stimulates the community to explore whether DE is dynamical over time or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In general, one depicts the DDE model by a simple Taylor expansion of DE EoS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=', ω(a) = ω0 + (1 − a)ωa [29, 30], where ωa characterizes the time evolution of DE EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The dimensionless Hubble parameter is expressed as EDDE(z) = � Ωm(1 + z)3 + (1 − Ωm)(1 + z)3(1+ω0+ωa)e −3ωaz 1+z � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (12) Note that this model is a two-parameter extension to ΛCDM and it reduces to ΛCDM when ω0 = −1 and ωa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Extended halo mass function When applied into a complicated gravity system, the function of Press-Schechter HMF is limited, since it does not consider the nonlinear environmental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' To overcome this shortcoming, the extended Press-Schechter (EPS) HMF is proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [31] and reads as dn(M1, z|M2, δ2) dM1 = M2 M1 fm(S1, δ1|S2, δ2) ���� dS1 dM1 ���� , (13) where the mass variance S1 = σ2(M1) and S2 = σ2(M2) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (3)), and one can obtain the average number of progenitors at time t1 in the mass range (M1, M1 + dM1) which by time t2 (t2 > t1) have merged to form a large halo of mass M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The multiplicity function fm is expressed as fm(S1, δ1|S2, δ2) = 1 √ 2π δ1 − δ2 (S1 − S2)3/2 exp � − (δ1 − δ2)2 2(S1 − S2) � dS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (14) δ1 and δ2 are, respectively, the linear overdensities in spherical regions of masses M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' To study the environ- mental impacts on the high redshift HMF, we choose M2 as a present-day halo corresponding to current overdensity δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' To compute δ2, one should transform the nonlinear overdensity δnl at redshift z in Eulerian space into the linear overdensity in Lagrangian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The corresponding analytic fitting formula based on spherical collapse model is [32, 33] δ2(δnl, z) = δ1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='68647 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='68647 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='35 (1 + δnl)2/3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='12431 (1 + δnl)1/2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='78785 (1 + δnl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='58661 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (15) Since there is a possibility that the excessively high CSMD from JWST is caused by nonlinear environmental effect, we attempt to explain it using the EPS formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' METHODS AND RESULTS At first, we employ the best fits from current cosmological constraints as our baseline values for four models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Since we hope that the following calculations can be permitted by present-day observations, our discussions and results will mainly focus on the allowed parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Then, for different models, we use different Boltzmann codes to calculate their background evolution, growth factors and matter power spectrum at different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Specifically, we take CLASS [22, 23, 34] for DMBI and use modified CAMB [35, 36] for f(R) gravity, DDE and EPS scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Note that ΛCDM is adopted in the EPS scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Subsequently, we compute the HMF at different redshifts for the above four models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Finally, we work out the maximal CSMD for each model according to the permitted parameter space, and check whether these scenarios are consistent with the latest JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (4) is only used in the EPS model and the growth factors of the other three models are obtained from the corresponding software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Our numerical analysis results are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' At first, we display the CSMD of ΛCDM in the redshift range z ∈ [7, 9] and see its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In general, the SFE ϵ is about 10% according to current observational constraints [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Nonetheless, one can see that in the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1, 10% is nowhere near enough to reach the lower bounds of JWST data points in ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' One needs the star formation rate in galaxies to be at least 50% in order to explain the inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the meanwhile, one can easily find that ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 can successfully explain two data points but 100% SFE can not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Except for ΛCDM, we all calculate the maximal CSMD in the other models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=', assuming ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the second place, we make a trial to explore whether alternative cosmological models can alleviate even solve the tension between JWST and Planck CMB observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the DMBI case, we attempt to acquire a higher baryon fraction by the coupling between DM and baryons, and consequently explain this discrepancy occurred in ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' However, we find that varying coupling strength σDM−b hardly affects the CSMD, and only the variation of interaction DM fraction Ωidm affects significantly the CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' When assuming the DM particle mass mDM = 100 GeV, the cross section σDM−b = 10−42 cm2 and choosing the fraction Ωidm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='03 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='05, this tension can be efficiently relieved but it seems that this model is difficult to explain both data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' However, if considering the current cosmological constraint that gives a very small Ωidm [24], DMBI still behaves like ΛCDM and can not resolve this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We have also studied the impacts of mDM and find different DM particle masses also can not explain JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In f(R) gravity, we find small fR0 such as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1 and 1 can not expalin the anomaly but a very large value fR0 = 10 can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' This implies that one needs a large deviation from GR to be responsible for JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Unfortunately, the 5 109 1010 1011 M 102 103 104 105 106 107 108 ( > M )[M Mpc 3] CDM, = 1 CDM, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 CDM, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='5 CDM, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1 109 1010 1011 M 103 104 105 106 107 108 ( > M )[M Mpc 3] CDM idm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='01, mDM=100 GeV idm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='03, mDM=100 GeV idm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='05, mDM=100 GeV 109 1010 1011 M 103 104 105 106 107 108 ( > M )[M Mpc 3] CDM fR0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1 fR0 = 1 fR0 = 10 109 1010 1011 M 102 103 104 105 106 107 108 ( > M )[M Mpc 3] nl = 1 nl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='7 nl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='6 nl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='5 nl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='4 nl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1 CDM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' 1: The CSMDs for the ΛCDM, DMBI, f(R) gravity and EPS models are shown from top to bottom and left to right, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Note that for ΛCDM, we compute the CSMDs in the redshift range z ∈ [7, 9] by choosing different values of the SFE ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' For the other models, we calculate the CSMDs in the redshift range z ∈ [9, 11] when ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' latest cosmological constraint gives log10 fR0 < −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='32 at the 2 σ confidence level [37], which is much smaller than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Therefore, similar to DMBI, f(R) gravity also fails to alleviate this tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Interestingly, this gives us a hint that, if two galaxies observed by JWST are located in the low density region of the universe where MG effect is very large, the data can be appropriately explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Furthermore, we are interested in whether the nature an simple extension to ΛCDM, DDE, can explain the incon- sistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' As mentioned above, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [16] claimed that JWST data can clearly constrain DDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' However, within current constraining precision, we query this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' To ensure the validity of our conclusion, we constrain ΛCDM and DDE models using the Planck-2018 CMB temperature and polarization data (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='2), and then obtain the best fitting values of parameters of these two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' One can easily find the constrained values of model parameters of ΛCDM in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' For DDE, we obtain current baryon and CDM densities Ωbh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0225 and Ωch2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1184, the ratio between angular diameter distance and sound horizon at the redshift of last scattering θMC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='04109, the optical depth due to the reionization τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='06, the amplitude and spectral index of primordial power spectrum As = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='114 × 10−9 and ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='9698, and two DE EoS parameters ω0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='38 and ωa = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Same as DMBI and f(R) gravity models, we use the same method to work out the CSMD of DDE, and find that the variation of the CSMD is largely dominated by the values of six basic parameters Ωbh2, Ωch2, θMC, τ, As and ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Although ω0 and ωa is loosely constrained by CMB data (constrained ω0-ωa parameter space is large), different values of ω0 and ωa hardly affect the CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' For instance, in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='3, ω0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='38 and ωa = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 plus the ΛCDM and DDE best fits gives completely different CSMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Choosing the ΛCDM best fit, (ω0, ωa) = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='38, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8) and 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0228 bh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='5 m 50 60 70 80 90 H0 8 6 4 2 0 2 4 wa 2 1 0 w 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='10 ln(1010As) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='98 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0405 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0410 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0415 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0420 100 MC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='116 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='123 ch2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='042 100 MC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='06 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='10 ln(1010As) 2 1 0 w 8 4 0 4 wa 50 60 70 80 90 H0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='5 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1 8 DDE CDM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' 2: The marginalized posterior probability distributions of the ΛCDM and DDE models from the Planck-2018 CMB constraints are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' (ω0, ωa) = (−1, −1) gives very similar results in the logarithmic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the medium and right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='3, we verifies that taking same best fits of ΛCDM and DDE, respectively, choosing different DE EoS parameter pair just produces very limited differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' After scanning the DDE parameter space, we find clearly that DDE also can not explain this tension, but its best fit can help increase the value of CSMD and become closer to JWST data points (see the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The reason that the result in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [16] is different from ours is that they do not implement an appropriate cosmological constraint based on the Planck CMB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The result from f(R) gravity prompts us to study the environmental effect of JWST galaxies on the CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' The most straightforward method is replacing the Press-Schechter HMF with the EPS formalism in the framework of ΛCDM, where the sole parameter δnl characterizes the nonlinear environmental effect of a high redshift halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the bottom right panel, we calculate the maximal CSMDs in the redshift range z ∈ [9, 11] for the EPS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We find that neither overlarge (δnl = 1) nor too small (δnl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='1) explain JWST observations and that the larger δnl is, the larger the CSMD is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Since the total sky area covered by the JWST initial observation is large enough (∼ 40 armin2) 7 109 1010 1011 M 103 104 105 106 107 108 ( > M )[M Mpc 3] CDM w0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='38, wa=-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 + DDE best fit w0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='38, wa=-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 + CDM best fit w0=-1, wa=-1 + CDM best fit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='70 M 1e10 12000 12200 12400 12600 12800 13000 ( > M )[M Mpc 3] DDE w0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='38, wa=-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 DDE w0=-1, wa=-5 DDE w0=-1, wa=-1 DDE w0=-1, wa=0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='70 M 1e10 42000 42500 43000 43500 44000 44500 45000 ( > M )[M Mpc 3] DDE w0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='38, wa=-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='8 DDE w0=-1, wa=-5 DDE w0=-1, wa=-1 DDE w0=-1, wa=0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' 3: The CSMDs of the DDE model computed at in the redshift range z ∈ [9, 11] are shown when assuming ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Left: Different combinations of parameter values and best fits from constraints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Medium: Only the ΛCDM best fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Right: Only the DDE best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' [10], we can not rule out this possibly local environmental effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' However, unfortunately, there is no δnl passing two data points simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' DISCUSSIONS AND CONCLUSIONS Recently, the early data release of JWST reveals the possible existence of high redshift galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' What is interesting is these galaxies in the redshift range z ∈ [7, 11] exhibit the overlarge star formation rate, which is incompatible with the prediction of the standard cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' This may indicate that JWST data contain the signal of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In this study, we try to resolve this tension with alternative cosmological models including DMBI, f(R) gravity and DDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We find that in light of the precision of current cosmological constraint from Planck-2018 CMB data, these models all fail to explain this large tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Specifically, for DMBI, the coupling strength σDM−b between DM and baryons hardly affects the CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' For f(R) gravity, the effect of varying fR0 on the CSMD is too small to relive the tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' For DDE, although the constrained DE EoS parameter space is large, different parameter pair (ω0, ωa) just produces very limited differences in the CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Interestingly, a large interacting DM fraction and a large deviation from Einstein’s gravity can both generate a large CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' A possible scenario to escape from current cosmological constraints is the EPS formalism, where we consider the local environmental effect on the CSMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We find that an appropriate value of nonlinear environmental overdensity of a high redshift halo can well explain the CSMD discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' However, we do not find an EPS model that can simultaneously explain two data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' In the near future, JWST will bring more useful data to human beings, so that we can extract more physical information to uncover the mysterious veil of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' Acknowledgments DW warmly thanks Liang Gao, Jie Wang and Qi Guo for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' We thank Hang Yang for letting us notice the JWST related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content=' This study is supported by the National Nature Science Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='11988101 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf'} +page_content='11851301.' metadata={'source': 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b/KdFLT4oBgHgl3EQfLC8K/content/tmp_files/2301.12010v1.pdf.txt @@ -0,0 +1,1811 @@ +Leveraging Google’s Tensor Processing Units +for tsunami-risk mitigation planning in the +Pacific Northwest and beyond +Ian Madden1, Simone Marras2, and Jenny Suckale1,3 +1Institute for Computational and Mathematical Engineering, Stanford +University +2Department of Mechanical Engineering & Center for Applied Mathematics and +Statistics, New Jersey Institute of Technology +3Department of Geophysics, Doerr School of Sustainability, Stanford University +January 31, 2023 +Abstract +Tsunami-risk and flood-risk mitigation planning has particular importance for commu- +nities like those of the Pacific Northwest, where coastlines are extremely dynamic and a +seismically-active subduction zone looms large. The challenge does not stop here for risk +managers: mitigation options have multiplied since communities have realized the viability +and benefits of nature-based solutions. To identify suitable mitigation options for their +community, risk managers need the ability to rapidly evaluate several different options +through fast and accessible tsunami models, but may lack high-performance computing +infrastructure. The goal of this work is to leverage the newly developed Google’s Ten- +sor Processing Unit (TPU), a high-performance hardware accessible via the Google Cloud +framework, to enable the rapid evaluation of different tsunami-risk mitigation strategies +available to all communities. We establish a starting point through a numerical solver of +the nonlinear shallow-water equations that uses a fifth-order Weighted Essentially Non- +Oscillatory method with the Lax-Friedrichs flux splitting, and a Total Variation Diminish- +ing third-order Runge-Kutta method for time discretization. We verify numerical solutions +through several analytical solutions and benchmarks, reproduce several findings about one +particular tsunami-risk mitigation strategy, and model tsunami runup at Crescent City, +California whose topography comes from a high-resolution Digital Elevation Model. The +direct measurements of the simulations performance, energy usage, and ease of execution +show that our code could be a first step towards a community-based, user-friendly virtual +laboratory that can be run by a minimally trained user on the cloud thanks to the ease of +use of the Google Cloud Platform. +1 +Introduction +The coast of the Pacific Northwest, from Cape Mendocino in California to Northern Vancouver +Island in Canada as depicted in Fig. 1, is located on the seismically active Cascadia subduction +zone [Heaton and Hartzell, 1987, Petersen et al., 2002]. Along the 1200-km-long Cascadia sub- +duction zone, there have been no large, shallow subduction earthquakes over the approximately +1 +arXiv:2301.12010v1 [physics.geo-ph] 27 Jan 2023 + +200 years of modern-data monitoring, but large historic earthquakes have left an unambiguous +imprint on the coastal stratigraphy [Clague, 1997]. Sudden land level change in tidal marshes +and low-lying forests provide testimony of 12 earthquakes over the last 6700 years [Witter et al., +2003], including one megathrust event that ruptured the entirety of the current Cascadia sub- +duction zone in 1700 BC [Nelson et al., 1995, Wang et al., 2013]. The event created a massive +tsunami that swept across the entire Pacific Ocean devastating communities as far away as +Japan [Satake et al., 1996, Atwater et al., 2011]. Current seismic-hazard models estimate that +the probability of another magnitude 9+ earthquake happening within the next 50 years is +about 14% [Petersen et al., 2002]. +Figure 1: Map of the Cascadia Subduction Zone in the Pacific Northwest of the United States. +Relative location of Crescent City with respect to the Megathrust Fault Line, with a more +detailed picture of the Crescent City coastline. Esri provided access to the satellite imagery. +Crescent City map at high resolution provided by Maxar. Pacific Northwest Map provided by +Earthstar Geographics. +A magnitude 9+ Cascadia earthquake and tsunami occurring during modern times would +devastate many low-lying communities along the Pacific Northwest. A recent assessment sug- +gests that deaths and injuries could exceed tens of thousands and entails economic damages +in the order of several billions of dollars for Washington and Oregon State [see, e.g., Knudson +and Bettinardi, 2013, Gordon, 2012], with potentially severe repercussions for the entire Pacific +coast and country as a whole. The tsunami itself would put tens of thousands at risk of in- +undation, and threaten the low-lying coastal communities specifically in the Pacific Northwest +with very little warning time for evacuation Gordon [2012]. But how to confront this risk? +Traditionally, the most common approach to reducing tsunami risk is the construction of sea +walls, but this hardening of the shoreline comes at a staggering price in terms of the economic +construction costs (e.g., 245 miles of sea walls in Japan cost $12.74 billion) and in terms of +2 + +Cascadia +WA +Megathrust Fault +OR +Crescent +City +CAlong-term negative impact on coastal ecosystems [Peterson and Lowe, 2009, Dugan and Hub- +bard, 2010, Bulleri and Chapman, 2010] and shoreline stability [Dean and Dalrymple, 2002, +Komar, 1998]. +A potentially appealing alternative to sea walls are so-called hybrid approaches. Hybrid +risk mitigation combines nature-based elements and traditional engineering elements to reduce +risk while also providing co-benefits to communities and ecosystems. +An example of a hy- +brid approach to tsunami-risk mitigation is a coastal mitigation park: A landscape unit on +the shoreline built specifically to protect communities or critical infrastructure and provide +vertical evacuation space, in the styles of Fig. 2. Communities across the Pacific Northwest +are increasingly considering these nature-based or hybrid options [Freitag et al., 2011], but +many important science questions regarding protective benefits and optimal design remain +open [Lunghino et al., 2020, Mukherjee et al., 2023]. This gap is particularly concerning given +that existing models show that a careful design is necessary to avoid potential adverse effects +[Lunghino et al., 2020]. The design of current mitigation parks, such as the one being built +in Constituci´on, Chile, is not yet underpinned by an in-depth quantification of how different +design choices affect risk-reduction benefits, partly because numerical simulations of tsunami +impacts are computationally expensive. +Figure 2: Map view (top) and side view (bottom) of a proposed tsunami-mitigation berm as +designed by Project Safe Havens. The berm provides vertical evacuation space for the adjacent +community and could also lower the onshore energy flux that drives the damage created by +tsunami impact. We show this design as one example of a hybrid approach to tsunami-risk +mitigation as it combines an engineered hill and ramp with natural vegetation. Sketches adapted +from Freitag et al. [2011]. +The goal of this paper is to leverage Google’s Tensor Process Units (TPUs) for enabling +a fast evaluation of different mitigation park design and ultimately advancing evidence-based +tsunami-risk mitigation planning rooted in quantitative assessments. TPUs are a new class of +hardware accelerators developed by Google with the primary objective of accelerating machine +learning computations. They are accessible via the Google Cloud Platform [Jouppi et al., 2017, +Google.com] and have recently been used for many different applications in computational +physics and numerical analysis [Hu et al., 2022, Lu et al., 2020b,a, Belletti et al., 2020]. We +build upon and extend an existing implementation [Hu et al., 2022] to simulate the impact +of idealized tsunamis on the coastline. Our implementation of the shallow water equations +includes a non-linear advective term, not considered in [Hu et al., 2022], within the software +framework based on Google’s TensorFlow necessary to execute code on the TPU. We discretize +the new equations using the Weighted Essentially Non-Ocillatory (WENO) method [Liu et al., +3 + +1994] and a third-order Runge-Kutta in time. +Numerical simulations of tsunamis have contributed to our understanding of and ability +to mitigate wave impacts for many decades now, starting from the early work by Isozaki and +Unoki [1964] for Tokyo Bay, and Ueno [1960] for the Chilean coast. The ability to capture the +rupture mechanism that generates the initial condition for tsunami propagation then enabled +the reproduction of many historical tsunamis [Aida, 1969, 1974]. Since then, many numerical +models have been developed to simulate tsunami generation [Borrero et al., 2004, Pelties et al., +2012, L´opez-Venegas et al., 2014, Galvez et al., 2014, Ulrich et al., 2019], propagation [Titov +et al., 2005, LeVeque et al., 2011, Chen et al., 2014, Allgeyer and Cummins, 2014, Abdolali +and Kirby, 2017, Bonev et al., 2018, Abdolali et al., 2019], and inundation [Lynett, 2007, Park +et al., 2013, Leschka and Oumeraci, 2014, Chen et al., 2014, Marsooli and Wu, 2014, Maza +et al., 2015, Oishi et al., 2015, Prasetyo et al., 2019, Lunghino et al., 2020] by solving different +variations of the shallow water and Navier-Stokes equations. +The list of existing numerical models is long and was recently reviewed in Marras and +Mandli [2021] and Horrillo et al. [2015]. Some commonly used ones are FUNWAVE [Kennedy +et al., 2000, Shi et al., 2012], pCOULWAVE [Lynett et al., 2002, Kim and Lynett, 2011], Delft3D +[Roelvink and Van Banning, 1995], GeoCLAW [Berger et al., 2011], NHWAVE [Ma et al., 2012], +Tsunami-HySEA [Mac´ıas et al., 2017, Mac´ıas et al., 2020b,a], FVCOM [Chen et al., 2003, 2014]. +Our work here relies on well-known numerical techniques to solve idealized tsunami problems. +Its novelty lies in demonstrating the capability and efficiency of TPUs to solve the non-linear +shallow water equations to model tsunamis. +We intentionally use a hardware infrastructure that is relatively easy to use without specific +training in high-performance computing (See the Google Cloud TPU page at Google.com), and +may become a standard hardware on which physics-based machine-learning algorithms will be +built [Rasp et al., 2018, Mao et al., 2020, Wessels et al., 2020, Fauzi and Mizutani, 2020, Liu +et al., 2021, Kamiya et al., 2022]. We propose that our implementation is one step towards +a community-based, user-friendly virtual laboratory that can be run by a minimally trained +user on the cloud thanks to the ease of use of the Google Cloud Platform. The tool, which is +freely available on Github at [tsunamiTPUlab, 2023] under an Apache License, Version 2.0 for +collaborative open source software development, can be modified to include machine learning +capabilities and, eventually, extended to coupled models for earth-quake generation, inundation, +and human interaction. +Methods +Numerical approximation +We model tsunami propagation and runup with the 2D non-linear shallow water equations in +the conservative formulation with a source term in a Cartesian coordinate system. Letting +x = (x, y) denote position, we define u(x, t) and v(x, t) as the flow velocities in the x and y +directions, respectively. In our implementation, we solve for h, hu, and hv. We define h(x, t) +as the dynamic water height and b(x) as the imposed bathymetry, meaning that the quantity +h + b represents the water surface level. This leads to the following system of equations, a set +4 + +very similar to that suggested by Xing and Shu [2005] +∂ +∂th + +∂ +∂x(hu) + ∂ +∂y(hv) = 0 +(1) +∂ +∂t(hu) + +∂ +∂x +� +(hu)2 +h ++ 1 +2g(h2 − b2) +� ++ ∂ +∂y(huv) = −g(h + b) ∂b +∂x − +gn2√ +(hu)2+(hv)2 +h7/3 +(hu) +(2) +∂ +∂t(hv) + +∂ +∂x(huv) + ∂ +∂y +� +(hv)2 +h ++ 1 +2g(h2 − b2) +� += −g(h + b) ∂b +∂x − +gn2√ +(hu)2+(hv)2 +h7/3 +(hv) , (3) +where g = 9.81 ms−2 is the acceleration of gravity, and n is the Manning friction coefficient. +For ease of future notation, we let u = +�h +hu +hv�T , and we rewrite the above equations +in a vector form, namely: +∂u +∂t + ∂F +∂x + ∂G +∂y = S +(4) +where F and G are the fluxes in the x and the y directions for the vector u, and S is a source +term arising from variations in topography and Manning coefficient. +We implement these shallow-water equations using the finite volume method whereby the +half-step flux and height values are determined through a 5th-order WENO scheme [Liu et al., +1994, Jiang and Shu, 1996]. +We approximate solutions to cell-wise Riemann problems by +formulating fluxes using the Lax-Friedrichs method as in [LeVeque, 2011]. We formulate the +bed source term as suggested by Xing and Shu [2005], and formulate the friction term explicitly +rather than using the implicit process suggested by Xia and Liang [2018]. We use a 3rd-order, +Total Variation Diminishing Runge-Kutta scheme to step the numerical solution forward in +time. +We begin the discretization of the equation in continuous variables t, x, and y, using re- +spective step sizes ∆t, ∆x, and ∆y, which indicate the distance between consecutive integral +steps in the discrete indices n, i, and j, respectively. We use the 5th-order WENO scheme in +the x and y directions, where two values of each quantity h, hu, and hv are determined at each +half-step of x and y. These two values correspond to a positive and negative characteristic, +due to the nature of the footprint that is chosen at a given point. In other words, given the +conservative form with relevant variable u, ui,j centered on a finite volume cell, we label these +outputs of WENO: +u+ +i+ 1 +2 ,j, u− +i+ 1 +2 ,j +for WENO in x, +or +u+ +i,j+ 1 +2 , u− +i,j+ 1 +2 for WENO in y. +(5) +From here, we use the Lax-Friedrichs method to approximate flux values that serve as solutions +to the Riemann problem; i.e. we approximate +Fi+ 1 +2 ,j = 1 +2 +� +F(u+ +i+ 1 +2 ,j) + F(u− +i+ 1 +2 ,j) − αu +� +F(u+ +i+ 1 +2 ,j) − F(u− +i+ 1 +2 ,j +�� +(6) +where αu is the associated Lax-Frierichs global maximum characteristic speed. Now, we dis- +cretize Eq. 4 explicitly as: +un+1 +i,j +− un +i,j +∆t ++ +Fn +i+ 1 +2 ,j − Fn +i− 1 +2 ,j +∆x ++ +Gn +i,j+ 1 +2 − Gn +i,j− 1 +2 +∆y += S(un +i,j) +(7) +Note that in our case, we also choose to formulate the source term S(un +i,j) explicitly and centered +at the grid point. Since we use an entirely explicit formulation, we can rewrite Eq. 7 as a +time stepping operator for un+1, namely un+1 = T(un). Because Runge-Kutta uses multiple +stages within each time step, we reassign the output of the T operator to be u(n+1) = T(un), +5 + +where(n + 1) indicates an intermediate full time step forward. This means a full Runge-Kutta +step progresses as follows: +u(n+2) = T(T(un)) +−→ +u(n+ 3 +2 ) = T(0.25u(n+2)+0.75un) +−→ +un+1 = 2 +3u(n+ 3 +2 )+ 1 +3un +(8) +The process outlined by Eq. 8 outputs a final un+1 representing a full-step forward in simulation +time. +TPU implementation +To leverage the TPU’s several cores, we divide the domain into multiple subdomains and in- +dependently compute the numerical solution to the governing equations on each core. While +a lot of the computation can take place independently, each subdomain remains dependent on +the others via their boundaries and the Lax-Friedrichs global maximum in characteristic speed. +We determine global maximum characteristic speed by sharing and reducing the Lax-Friedrich +maximum characteristic speed calculated on each core. We transfer subdomain boundary infor- +mation with further care by using a halo exchange. The data transfer behavior and computation +structure is summarized in Fig. 3. +Figure 3: Left: Initialization of implementation takes advantage of CPU to allocate initial +conditions and topography. Center: Regular computation period occurring on each subdomain, +run independently on TPU cores with some data sharing coordinated by CPU. Right: CPU +Gather to write results to output files. +Our implementation is inspired by Hu et al. [2022], who chose halo exchange as an instrument +for the TPU to communicate information across subdomain boundaries in their formulation of +the shallow water equations. In the halo exchange process, we transfer slices of the domain +from one core to the others immediately adjacent. While Hu et al. [2022]’s methodology only +involved the exchange of a single slice from one core to the other, we transfer several slices in +order to take full advantage of the high accuracy and larger footprint of the WENO scheme. +These halo exchanges are then performed in every stage of the Runge-Kutta scheme, meaning +that they occur multiple times in a single time step. +The initial conditions and results are communicated from the remote program, which resides +on the CPU, to the TPU workers by means of tpu.replicate which sends TensorFlow code +to each TPU. We refer to Hu et al. [2022] for further details on the TPU implementation. +6 + +TPU Core +TPU Core +TPU Core +TPU Core +TPU Core +TPU Core +TPU Core +TPU Core +TPU Core +Calculate initial condition, +Perform Calculations +Gather from TPU Cores and +CPU +partitiondomain,and disperse +CPU +Write Output File +subdomainstoTPUCores +CPU +CPU Coordinates Exchange of +Information2 +Model verification and validation +We differentiate between model verification and validation in the manner suggested by Carson. +Specifically, we check for model and implementation error by quantifying the extent to which +numerical solutions compare to correct analytical solutions [Carson]: wet dam break (Section +2.1), oscillations in a parabolic bowl (Section 2.2), and steady state flow down a slope with +friction (Section 2.3). Following this, we validate by checking how well numerical solutions +reflect the real system and apply to the context [Carson]. To do this, we compare against an +existing numerical benchmark from the Inundation Science and Engineering Cooperative [ISEC, +2004] and results from an investigation of nature-based solutions [Lunghino et al., 2020] in +Section 2.4, and consider the propagation of a computed tsunami over the observed topography +of Crescent City in Section 2.5. To quantify the accuracy of the solutions, we test our numerical +solver against some classical analytical solutions to the shallow water equations. We assess the +model’s ability to capture key physical processes relevant to inundation, including steep wave +propagation, friction, and topography dependence. We use relative errors in the L∞ and L2 +sense as the metric to determine model accuracy. These are approximated in this paper in the +following manner: +L∞ = maxΩ |hc − ha| +maxΩ |ha| +, +L2 = +�� +Ω(hc − ha)2∆Ω +� +Ω(ha)2∆Ω +, +(9) +where hc is the computed solution at the discretized cells, ha is the analytical solution at the +corresponding cells, and Ω denotes the computational domain. +2.1 +Wet dam break +Figure 4: On the left, several instances in time of the computed (purple) water heights to wet +dam break compared with the analytical (orange, dashed) water heights. The rightmost figure +plots the L2 and L∞ relative norms of the error between the analytical and computed solutions. +The classical one-dimensional Wet Dam Break [Stoker, 1957] provides us an opportunity to +test the ability of our code to capture shock propagation and advection. In this case, there is +no friction (n = 0) and the topography is flat (b(x) = 0). The boundaries are set at a constant +height with zero flux. We impose the following initial condition: +(hu) = 0, (hv) = 0, h(x) = +� +hl +x ≤ x0 +hr +x > x0 +, +(10) +where hl and hr are the constant water heights on either side of a shock front x0. We compare +our numerical solution for water height against the dynamic analytical solution from Delestre +7 + +t = o.0 seconds +t = 2.5 seconds +t = 9.0 seconds +Relative Error in Norm +10 +Computed +10 +Computed +10 +Computed +Lo +Analytical +9 +Analytical +9 +Analytical +9- +L2 +8 +8 +8 +10-2 +N +N +7 +6 - +6 +9 +5 +10° +1200125013001350140014501500 +1200125013001350140014501500 +1200125013001350140014501500 +2 +6 +8 +10 +X [-] +X [-] +x[-] +Time [s]et al. [2013]: +h(x, t) = +� +� +� +� +� +� +� +� +� +� +� +hl +x ≤ x1 +4 +9g +�√ghl − x−x0 +2t +�2 +x1(t) < x ≤ x2(t) +c2 +m +g +x2(t) < x ≤ x3(t) +hr +x > x3(t) +, +(11) +x1(t) = x0 − t +� +ghl , +(12) +x2(t) = x0 + t(2 +� +ghl − 3cm) , +(13) +x3(t) = x0 + t2c2 +m(√ghl − cm) +c2m − ghr +, +and +(14) +cmis the solution to − 8ghrc2 +m( +� +ghl − cm)2 + (c2 +m − ghr)2(c2 +m + ghr) = 0 . +(15) +A qualitative comparison of the computed and analytical solutions for times t =0, 2.5, and +9 seconds is shown in the left plots of Fig. 4. The relative error between the analytical and +computed solutions in the infinity and 2-norms at a small distance away from the shock front +are plotted on the right. We interpret the converging relative error norms to a low magnitude +as verification of our implementation to sufficiently capture shock propagation and advection. +2.2 +Planar parabolic bowl +Figure 5: On the left, several instances in time of the computed (purple) water heights to the +one-dimensional parabolic bowl compared with the analytical (orange, dashed) water heights. +The rightmost figure plots the L2 and L∞ relative norms of the error between the analytical +and computed solutions. +The classical one-dimensional planar parabolic bowl originally suggested by [Thacker, 1981], +is an oscillating solution allowing us to test the source term for topography without friction +(n = 0). We enforce homogeneous Dirichlet conditions in both flux and water height, at a +resolution of 1 m. Once again, we take the test directly from Delestre et al. [2013], resulting in +the following description of the base topography: +b(x) = h0 +� +1 +a2 +� +x − L +2 +�2 +− 1 +� +, +(16) +corresponding with the following initial condition: +(hu) = 0, (hv) = 0, h(x) = +� +−h0 +�� 2x−L+1 +2a +�2 − 1 +� +1−2a+L +2 +< x < 1+2a+L +2 +0 +otherwise +. +(17) +8 + +t = 3.5 seconds +t = 12.5 seconds +Relative Error in Norm +10.0 +10.0 +7.5 +7.5 +10-3, +5.0 - +5.0 +10-4 +2.5 +2.5 +N 0.0 +N 0.0 +10-5 +2.5 - +Computed Solution +2.5 - +Computed Solution +10-6 +Lo +Analytical Solution +Analytical Solution +5.0 +5.0 +L2 +Topography +Topography +10-7 +0 +5 +10 +15 +20 +25 +30 +35 +40 +0 +5 +10 +15 +20 +25 +30 +35 +40 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +X [-] +X[-] +Time [s]This leads to the following dynamic analytical solution for the water height: +h(x, t) = +� +� +� +−h0 +�� +2x−L +2a ++ 1 +2a cos +� √2gh0t +a +��2 +− 1 +� +x1(t) < x < x1(t) + 2a +0 +otherwise +, +(18) +where x1(t) = 1 +2 cos +� √2gh0t +a +� +− a + L +2 . A qualitative comparison of the parabolic bowl solution +at the time instances t =3.5 sand t =12.5 s can be seen on the left of Fig. 5. The analytical +and computed solutions appear to correspond to one another well. For a more quantitative +analysis,the relative error-norms of the solutions are depicted on the right of Fig. 5. We interpret +the converging relative error norms to a low magnitude as verification of our implementation +to sufficiently capture the source term of the shallow water equations induced by topography. +2.3 +Steady flow down a slope with friction +Figure 6: On the left, several instances in time of the computed (purple) fluxes given a water +level and a slope, compared to the analytical (orange, dashed) flux. The rightmost figure plots +the L2 and L∞ relative norms of the error between the analytical and computed solutions. +We do a short test in order to assess the correctness the discretized friction source term, +focusing on a relatively simple flow down a slope with finite friction (n = 0.033) as tested by +Xia and Liang [2018]. The steady state flow down a slope then becomes +(hu) = +√bx +n h +5 +3 +(19) +where bx is the slope. In this test, we initialize the problem close to the steady state solution +for a wave height of 0.5 and a slope of +1 +20. This specific example and its convergence toward +steady state is shown in Fig. 6. The left plots show the flux at t = 100 s and t = 200 s. The +results are almost identical, indicating that a steady state has been reached. On the right plot, +the error norm of the steady state flux takes some time to reach steady state, but reaches a +very small level. Because we approach the appropriate steady state solution and achieve a very +small error norm, our implementation is verified in capturing a manning friction law. +2.4 +Validation for tsunami simulations +To assess the ability of the code to capture tsunami propagation, we start with a popular +numerical benchmark from the Inundation and Science Engineering Cooperative (ISEC) [ISEC, +2004] that represents tsunami runup over an idealized planar beach that provides solutions for +9 + +t = 100.0 seconds +t = 200.0 seconds +Relative Error in Norm +2.120 +2.120 +Computed +Computed +Loo +Analytical +Analytical +10-2 - +-2.125 +2.125 +L2 +6 × 10-3 +Flux +-2.130 - +2.130 +4×10-3 +3 ×10-3 +2.135 +2.135 +2×10-3 +0 +100 +200 +300 +400 +0 +100 +200 +300 +400 +50 +100 +150 +200 +X [-] +X[-] +Time [s]tsunami runup at times t =, 180 s, 195 s, 220 s. We formulate the initial condition for water +height using Lunghino et al. [2020]. The solutions from the benchmark (dashed, orange) are +qualitatively compared with the numerical solution produced by our code (solid, purple) in +Fig. 7. We take the qualitative agreement as validation of the model’s ability to model the +runup of a Carrier N-Wave [Carrier et al., 2003]. +Figure 7: Qualitative comparison of the computed solution with resolution 1 m compared to +the ISEC benchmark at three different time instances. +Since we are interested in leveraging TPUs for tsunami-risk mitigation planning, we take a +look at the ability of our shallow water equation code to reproduce a few particular results by +Lunghino et al. [2020] who investigated the effects of hills on a tsunami running up on a planar +beach. The tsunami is initialized as Carrier’s N-wave [Carrier et al., 2003]: +η = 2(a1 exp{−ˆk1(x − ˆx1)2} − a2 exp{ˆk2(x − ˆx2)2}), +(20) +where η = h + z, ˆx1 = 1000 + 0.5151125λ, ˆx2 = 1000 + 0.2048λ, ˆk1 = 28.416/λ2, ˆk2 = 256/λ2, +a1 = A, and a2 = A/3. While this is the analytically correct form, the flow origin in the +code is not the shoreline, so there are some effective shifts ˆx1 and ˆx2 that we need to do. An +example of the Carrier wave initial condition and offshore propagation behavior for A = 15 +and λ = 2000 is shown in Figure 8. We apply free slip, no-penetration boundary conditions +to the four domain boundaries, which means that that the component of the boundary-normal +component of the velocity vector is zero whereas its tangential component is unaltered. The +Figure 8: Several snapshots in time of the tsunami’s propagation over a modeled ellipsoidal hill +on a slope. From left to right, the formulation of the initial Carrier N-wave at t = 0, followed +by the propagation of a wave front toward the hill at t = 50, collision of the wave front of the +hill at t = 95, and the formation of a reflected wave at t = 135. +shallow water equation model presented in this study is able to reproduce the wave reflection +10 + +Water Height at t = 160.0 seconds +Water Height at t = 175.0 seconds +Water Height at t = 220.0 seconds +5040 +Computed +5040 +Computed +5040 +Computed +ISEC Benchmark +ISEC Benchmark +ISEC Benchmark +5020 +Topography +5020 +Topography +5020 +Topography +N +N +N +4980 +4980- +4980 +4960 +4960 +4960 +48500 +49000 +49500 +50000 +48500 +49000 +49500 +50000 +48500 +49000 +49500 +50000 +X [m] +X [m] +X [m]Water Height at t = o.o seconds +Water Height at t = 50.0 seconds +Water Height at t = 95.0 seconds +Water Height at t = 135.0 seconds +OE +OE +20 +20 +20 +20 +N +N +N +N +0 +0 +-10 +-10 - +-10 +-10 +-4000 +-3000 +-2000 +-1000 +1000 +-4000 +-3000 +-2000 +-1000 +1000 +-4000 +-3000 +-2000 +-1000 +0 +1000 +-4000 +-3000 +-1000 +1000 +X [m] +X [m] +X [m] +X [m]provided by a hill, consistent with results from Lunghino et al. [2020]. Because this simulation +is possible by the implementation, other further analysis can be conducted to understand the +mitigative benefit of other nature-based solutions. +2.5 +Real-world scenario +Past tsunamis impacting the West Coast of the United States have caused more damage around +the harbor of Crescent City in California than elsewhere along the Pacific Coast [Arcas and +Uslu, 2010]. +For this reason, we chose an area of approximately 105 km2 around Crescent +City to demonstrate the code’s ability to capture the impact of an idealized tsunami event +for a real location at high resolution. To approximate the actual bathymetry and topography, +we use a Digital Elevation Model for this area with uniform grid spacing of 4 m provided by +NOAA [NOAA National Geophysical Data Center, 2010, Grothe et al., 2011]. For the sake of +providing a proof of concept, we initialize the tsunami with the idealized waveform described +above, with slightly adjusted parameters: A = 10, λ = 2000, ˆx1 = 6000 + 0.5151125λ, and +ˆx2 = 6000 + 0.2048λ. The chosen parameters lead to maximum inundation patterns similar to +that seen in one modeled extreme scenario from Arcas and Uslu [2010]. In Fig. 9, we start with +an absence of any nearshore wave (including at t = 50s) and then a development of a tsunami +front that is visible to the shoreline by t = 140s. That front penetrates the harbor by t = 220s, +and is soon followed by the inundation of the coastline as well as reflection of wave energy back +to the ocean. We also observe that the mountain range on the upper part of the figure clearly +provides a significant protective benefit to the land beyond it. +Figure 9: Several snapshots of modeled tsunami propagation over terrain and geological features +of Crescent City, CA, where any level of blue indicates water cover and green depicts a stylized +map of the topography above surface level. From left to right, then top to bottom, we have +steady state near shore at t = 50; followed by the propagation of a wave front at t = 100 and +140; contact with Crescent City harbor at t = 180; inundation of the harbor and some of the +coastline at t = 220 and 260; and tsunami reflection and inundation at t = 300 and 340. +11 + +0- +0 +0 +20.0 +006 +006 +006 + 006 +1800 +1800- +1800 +1800 +17.5 +2700 +2700 +2700 +2700 +> +> +15.0 +4500 - +4500 +4500 - +4500 +5400 - +5400 +5400 +5400 +6300- +12.5 +6300 +t = 50 s +t=100 s +t = 140 s +t = 180 s +7200 + +7200 + +7200 + +7200 + +0 +1496 +2992 +4488 +5984 +1496 +2992 +4488 +5984 +！ +1496 +2992 +4488 +5984 +1496 +2992 +4488 +5984 +X [m] +X [m] +X [m] +X [m] +0 +006 + 006 + 006 + 006 +7.5 +1800 +1800 +1800 +1800 +2700 +2700 +2700 +2700 + 5.0 +> +> +> +> +4500 - +4500 +4500 - +4500 + 2.5 +5400 +5400 +5400 +5400 +6300 +0.0 +t = 220 s +t = 260 s +t = 300 s +t = 340 s +7200 + +7200 + +7200 + +7200 +1496 +2992 +4488 +5984 +0 +1496 +2992 +4488 +5984 +1496 +0 +0 +4488 +5984 +1496 +4488 +5984 +X [m] +X [m] 3 +Performance analysis on TPU +3.1 +Number of TPU cores +In communities where users may not have access to high performance computing facilities, +the Cloud TPU Platform provides a unique ability to perform large-scale computations, and +perform them rapidly. To address this potential benefit, we first measure the wall-clock time of +a simulation of a tsunami reaching Crescent City using a varying number of TPU cores on one +TPU device. As shown in Table 1, the problem size posed by the realistic scenario is sufficient +to see rapid improvements in runtime based on the number of cores. +Number of Cores +1 +2 +4 +8 +TPU Runtime [s] +6894 +3876 +2000 +1036 +Table 1: Approximate TPU Runtimes (in seconds) with varying numbers of TPU cores for +Crescent City Configuration using time step of ∆t = 5·10−3, Total Array Size of approximately +1802 by 3984 elements (4 meter resolution), cores all aligned in the y-direction as suggested in +[Hu et al., 2022]. This runtime excludes transfer times between the CPU and TPU. +3.2 +Geophysical problem resolution +The typical realistic scenario that decision-makers will face will involve large problem sizes +due to both the extent of their spatial domain, but also the level of resolution necessary to +model tsunami propagation and inundation over complex topography. +Therefore, we check +convergence and runtime under varying degrees of resolution for the current realistic scenario +as well, depicted graphically in 10 and in table 2. +Figure 10: Graphical depiction of the TPU runtimes (purple) and relative error norms (blue) +under varying resolutions for computing the tsunami propagation over the Crescent City DEM. +We finally perform the same analysis under varying degrees of resolution using the bench- +mark from the Inundation Science and Engineering Cooperative [ISEC, 2004] that we previously +12 + +0.35 +3500 +0.30 +ODOE +Relative Error Norm [-] +0.25 +2500 +Runtime [s] +0.20 +1500 +0.15 +L2 +0.10 +TPURuntime +500 +0.05 +2 +4 +- +B +10 +12 +Resolution [m]Resolution [m] +2 +4 +6 +8 +10 +12 +Runtime [s] +3805.167 +1035.719 +506.0652 +337.9506 +241.0804 +187.3016 +Number of Elements +28713068 +7179168 +3192512 +1794792 +1149274 +798128 +Efficiency +0.000133 +0.000144 +0.000159 +0.000188 +0.00021 +0.000235 +L2 Error Norm +* +0.062475 +0.092618 +0.115671 +0.093901 +0.132735 +L∞ Error Norm +* +0.212536 +0.340507 +0.291389 +0.265669 +0.259101 +Table 2: Approximate TPU Runtimes (in seconds) with varying resolutions for Crescent City +Configuration using time step of ∆t = 5 · 10−3, using the 2 m resolution as a benchmark for +correctness. Ran on a single TPU with 8 cores. +validated in Fig. 7. A qualitative comparison of the tsunami propagation under different reso- +lutions are graphically depicted in Fig. 11 in the top two and bottom left figure. In the bottom +right plot of Fig. 11, we see the expected fall in runtime based on coarser resolution (purple), +and a rise in relative error (with the exception of the highest resolution 1 m). The corresponding +values are documented in 3. +Figure 11: Graphical depiction of the TPU runtimes (purple) and relative error norms (blue) +under varying resolutions for computing the tsunami propagation in the ISEC Benchmark. +13 + +ISECBenchmark +ISEC Benchmark +5020 +5020 +Computed atResolution1m +ComputedatResolution1m +Computed atResolution2m +Computedat Resolution2m +5010 +ComputedatResolution4m +5010- +Computedat Resolution4 m +ComputedatResolution8m +ComputedatResolution8m +三 +Topography +E +Topography +N +5000 +N +5000 +4990 +4990- +4980- +4980- +49000 +49200 +49400 +49600 +49800 +50000 +50200 +49000 +49200 +49400 +49600 +49800 +50000 +50200 +X [m] +X [m] +4.0 +5020 +140 +3.5 +5010 +3.0 +120 +Runtime [s] +三 +2.5 +5000 +1008 +N +ISECBenchmark +ComputedatResolution1m +2.0 +lative +80 +4990 +Computed at Resolution 2 m +1.5 +TPURuntime +Computed atResolution4m +Rel +Computed at Resolution 8 m +60 +4980 +1.0 +Topography +49000 +49200 +49400 +49600 +49800 +50000 +50200 +1 +2 +3 +4 +5 +6 +7 +8 +X [m] +Resolution[m]Resolution [m] +1 +2 +4 +8 +Runtime [s] +150.0193 +81.74382 +56.74928 +47.3934 +Number of Elements +1060521 +277761 +75756 +18939 +Efficiency +0.000141 +0.000294 +0.000749 +0.002502 +L2 Error +0.000206 +7.19E-05 +0.00031 +0.000282 +L∞ Error Norm +0.000305 +0.000155 +0.000398 +0.000326 +Table 3: Approximate TPU Runtimes (in seconds) with varying resolutions for the ISEC +Tsunami Benchmark using time step of ∆t = 5 · 10−3. Ran on a single TPU with 8 cores. +3.3 +Comparison with GeoClaw +Figure 12: TPU solution (top row) at several time instances compared to the GeoClaw solution +(bottom row). The arrival of the tsunami front (t = 100, 180), the inundation of the harbor +(t = 260), and coastal inundation and reflection is depicted, and relatively comparable. +We used GeoClaw [Clawpack Development Team, 2020, Mandli et al., 2016, Berger et al., 2011] +ran on a single thread of a CPU (Intel i7-8650 with a base frequency of 1.9 GHz) to compare +numerical solutions to a tsunami propagation in order to assess performance enhancements +provided by the TPU. While our TPU-based code completes the 400 second simulation in +approximately 17 minutes of wall-clock time, the GeoClaw implementation on the CPU takes +approximately 630 minutes. Comparisons of the two numerical solutions can be seen in Fig. 12, +where the top row includes several instances in time of the TPU numerical solution, and the +bottom row depicts the GeoClaw numerical solution at the same instances in time. Although +some differences can be seen in the geometry of inundation by t = 380s in the rightmost plots, +the solutions do generally appear similar over time, lending credibility to the validity of the +numerical solution. +14 + +tsunamiTPUlab +tsunamiTPUlab +tsunamiTPUlab +tsunamiTPUlab +tsunamiTPUlab +20 + 40 +900 - + 006 +- 006 + 006 +900 +1800- +1800 +1800 +1800 +1800 +2700 - +2700 +2700 +2700 +2700 +15 +4500 +4500 +4500 +4500 - +4500 - +30 +00ts +5400 +5400 - +5400 - +5400 - +6300 +6300 ++0059 +- 009 +- 009 +[m] +t = 100 s +t = 180 s +t= 260 s +t = 340 s +t= 380 s +7200 + +7200+ +7200 + +7200 + +7200 + +1296 +2592 +3888 +5184 +2592 +3888 +5184 +1296 +0 +2592 +3888 +5184 +2592 +3888 +5184 +0 +1296 +2592 +3888 +5184 +X [m] +X [m] +X [m] +X [m] +X [m] +GeoClaw +GeoClaw +GeoClaw +GeoClaw +GeoClaw + 006 +- 006 +- 006 + 006 + 006 +1800 - +1800 +1800 +1800 +1800- +2700 - +2700 +2700 + +2700- +2700 - +10 +三 3600 +三 3600- +4500 - +4500 +4500 +4500 +4500 - +5400 +5400 +5400 - +5400 - +5400 - +6300 - +6300 +6300 +6300 +6300 +t = 100 s +t = 180 s +t= 260 s +t = 340 s +t= 380 s +7200 + +7200 +7200 + +7200 +7200 + +1296 +3888 +5184 +1296 +2592 +3888 +5184 +0 +1296 +2592 +3888 +5184 +0 +1296 +,2592 +3888 +5184 +1296 +2597 +3888 +5184 +X [m] +X [m] +X [m] +X [m] 3.4 +Energy utilization +Estimates of energy efficiency of computing operations are becoming increasingly popular, es- +pecially in response to progressing climate change [Fuhrer et al., 2018, Fourestey et al., 2014]. +To address this potential benefit provided by TPUs, we conduct a heuristic analysis of the +energy savings of running these operations on a TPU rather than a CPU. We base an order- +of-magnitude estimate on the claims of Google in the maximum power efficiency of the TPU, 2 +trillion operations per second (TOPS) per Watt [Google.com]. We use this heuristic for power +efficiency, consider the tsunami propagation problem under a 8 m resolution for Crescent City, +and manually approximate the number of floating point operations that our implementation +performs in each quadrature step at this resolution. We estimate that the TPU performs ap- +proximately 11.9 million floating-point operations for each simulated time step, and we see that +each time-step takes an average of about 4.2 milliseconds, corresponding to about 2.83 TOPS. +Assuming constant power efficiency regardless of capacity, this translates to 1.41 Watts and +an energy usage of 1.18 J for each simulated second given our current time-step configuration. +Combining with the total runtime for a 400 modeled-second simulation of Crescent City, we +see a total cost of approximately 4.76 × 102 J = 1.32 × 10−1 Wh of energy. At a price of 21 +cents/kWh in the U.S. at the time of writing this article, this simulation has a monetary cost +of 2.7 × 10−3 cents. +We use the aforementioned GeoClaw run as our CPU comparison on energy utilization in +order to get an order of magnitude estimate of the power savings. The particular processor for +this CPU comparison, the Intel i7-8650 with a base frequency of 1.9 GHz, has a Thermal Design +Power of 15 W for 8 total threads. Each simulated second took approximately 96.4 seconds +of CPU runtime, which translates to approximately 181 Joules for each simulated second of +a grid with 8 m resolution. Given the order of magnitude estimates, we note that the 400 +modeled-second simulation would see a total cost of approximately 7.24 × 104 J = 2.01 × 101 +Wh of energy, or a monetary cost of 0.04 cents, a cost multiplicative factor of 20. We linearly +extrapolate to estimate the number of CPU threads needed to match the runtime speed of our +model using the TPU, and find that approximately 37 CPU threads would be needed. With +this in mind, we find that the cost multiplicative factor for a CPU simulation of performance +equivalent to that of a TPU would be closer to an order of 700. TPU energy savings for high +performance are clearly substantial, and not ignorable. +4 +Discussion +Sustainable tsunami-risk mitigation in the Pacific Northwest is a challenging task. Some chal- +lenges come from beneath, because previous large subduction zone earthquakes at Cascadia led +to 0.5 − 1 m of co-seismic subsidence, the sudden sinking of land during an earthquake [Wang +et al., 2013]. Strong shaking can also lead to liquefaction [Atwater, 1992, Takada and Atwater, +2004]. Other challenges come from the ocean, where sea-level rise [Church and White, 2006, +Bindoff et al., 2007] and intensifying winter storms [Graham and Diaz, 2001] have increased +wave heights [Ruggiero et al., 2010, Ruggiero, 2013] and accelerated coastal erosion [Ruggiero, +2008]. A recent USGS report documented rapid shoreline changes at an average rate of almost 1 +m/yr across 9,087 individual transects [Ruggiero et al., 2013], suggesting the possibility that the +shoreline might change significantly during the century-long return-period of large earthquakes +in Cascadia [Witter et al., 2003]. +The picture that emerges is that of a highly dynamic coastline – maybe too dynamic for +an entirely static approach. Nature is not only continuing to shape the coastline, but is also a +fundamental component of the region’s cultural heritage, identity and local economy. So, it is +15 + +maybe not surprising that the Pacific Northwest is a thought-leader when it comes to designing +hybrid approaches to sustainable climate adaptation through the Green Shores program [Dalton +et al., 2013] and to vertical tsunami evacuation through Project Safe Haven [Freitag et al., 2011]. +Project Safe Havens is a grass-roots approach to reducing tsunami risk mostly by providing +accessible vertical-evacuation options for communities. Many proposed designs entail reinforced +hillscapes like the one shown in figure 2, intended to dissipate wave energy and provide vertical +evacuation space during tsunami inundation. To build confidence in such a solution and its +mitigation effects, risk managers must be able to quickly and precisely forecast a tsunami +inundation, preferably via a publicly available, centralized modeling infrastructure. +This paper is meant to be a first step towards a major community based infrastructure +that will allow local authorities around the world to readily execute tsunami simulation once +a tsunami in their proximity has been detected. We aim to provide a proof-of-concept rather +than a complete implementation. As such, we used a very similar base framework used by Hu +et al. [2022] of halo exchange in combination with a WENO and Runge-Kutta scheme, which +may not be optimally taking advantage of the TPU’s computing structure and capabilities. We +originally chose these schemes to maintain higher order accuracy and ease of implementation +but, eventually, a convolution-based implementation of the quadrature of the shallow water +equations should be tested for maximum performance utilization of the TPUs. +Because our code is specifically an implementation of the shallow water equations for the +TPU, it is currently unable to model tsunami initiation, or any fluid structure interactions +that may be desired to accompany analysis of nature-based solutions. Instead, it requires an +initial condition for wave heights and fluxes, meaning a full tsunami simulation would require +coupling the results of a tsunami initiation model as an input. While our implementation is +relatively limited in scope, the model is certainly able to provide a starting point for a more +complete software package for communities as they evaluate nature-based options for tsunami +mitigation. +Though just a starting point for a remotely-available package, we successfully replicated +some of the results found about coastal mitigation parks posed by Lunghino et al. [2020], and +we were able to model tsunami runup over with the real bathymetry around Crescent City, +in California, included in the code by means of a DEM file from the USGS digital elevation +database; for comparison purposes, we ran the same test using the popular open-source solver +GeoClaw [Clawpack Development Team, 2020, Berger et al., 2011]. +The results of the two +models are in good agreement, as shown in Fig. 12. Our contribution lies in demonstrating +an enhanced ability to run high quality simulations using the TPU available remotely using +Google Cloud Platform. As a result, high quality tsunami simulations are available to remote +communities for rapidly evaluating mitigation or evacuation options when faced with coastal +flooding. +We argue that TPUs are preferable to large, heavily parallel simulations on CPU or GPUs, +because the TPU-based simulations we show here do not require access to large computing +clusters. These are usually made available to scientists and engineers by supercomputing centers +around the world by means of competitive grants for computing time or by use of the cloud +offered by private companies. +However, an expert user knowledge of these systems from a +scientific computing perspective is necessary to design, run, and interpret model results, and +the compute infrastructure itself may not be available to early warning centers in many parts +of the world. In contrast, our code is available on Github and fully implemented in Python, +can be ran through a web browser, and visualized through a simple notebook file using Google +Colab. While performance can be enhanced with some knowledge about TPU architectures, +community risk managers do not need this knowledge to run high quality tsunami simulations +rapidly for real, physical domains with associated DEMs. +16 + +Finally, in the face of rising energy costs in both a monetary and climate sense, the TPU +infrastructure allows the support of more energy-efficient simulations over those of CPU-based +clusters. This means that those coastal flood risk managers in remote communities immediately +are able to support design decisions with model results in a climate-friendly manner. +Although not our focus here, we note our approach may also contribute to early tsunami +warning. Once triggered, tsunamis move fast; this fact makes it necessary to model and as- +sess their potential for damage ahead of time once they have been detected offshore. For a +sufficiently fast early warning and prompt evacuation, the tsunami modeling infrastructure +has an important time constraint [Giles et al., 2021] to be considered, and Faster Than Real +Time (FTRT) simulations are necessary [Behrens et al., 2021, Løvholt et al., 2019]. To make +FTRT simulations a reality, tsunami models are being rewritten or adapted to run on Graphical +Processing Units (GPUs) [Løvholt et al., 2019, Behrens and Dias, 2015, Satria et al., 2012]. +A TPU-based implementation as proposed here might be another meaningful step into that +direction. +Author contributions +Ian Madden: Software, Analysis, Writing. Simone Marras, PI: Conceptualization, Method- +ology, Writing, Supervision. Jenny Suckale: Conceptualization, Methodology, Writing, Su- +pervision. +Competing interests +The authors declare that they have no conflict of interest. +Data availability statement +Our work is available as a GitHub release at https://github.com/smarras79/tsunamiTPUlab/ +releases/tag/v1.0.0 or on archive at 10.5281/zenodo.7574655. +Acknowledgements +This work was supported by the National Science Foundation’s Graduate Research Fellowships +Program (GRFP) awarded to Ian Madden. +References +A. Abdolali and J. T. Kirby. Role of compressibility on tsunami propagation. J. Geophys. Res.: +Oceans, 122:9780–9794, 2017. +A. Abdolali, U. Kadri, and J. T. Kirby. Effect of water compressibility, sea-floor elasticity, and +field gravitational potential on tsunami phase speed. Scientific Reports, 9:1–8, 2019. +I. Aida. Numercal experiments for the tsunami propagation of the 1964 Niigata tsunami and +1968 Tokachi-Oki tsunami. Bull. Earthquake Res. Int. Univ. Tokyo, 47:673–700, 1969. +I. Aida. 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URL https://doi.org/10.1016/j.jcp.2005. +02.006. +5 +Appendix +5.1 +Running the code +Due to the restrictions of the TPU using Google Cloud Storage, Google’s buckets will need to be +used in order to run the notebooks. With a computing project setup on Google Cloud, users can +quickly run any of the example notebooks or design their own simulation. Any of the example +notebooks available on GitHub (with the exclusion of tpu tsunami.ipynb, which contains the +full implementation with all of the different scenarios; and Create Scenarios.ipynb, which can +aid users in generating a custom DEM file) can be quickly ran by going through the notebook +after a few early setup steps. +1. Download the TPU-Tsunami Repository from https://github.com/smarras79/tsunamiTPUlab/ +releases/tag/v1.0.0 to your local machine. Create a project on Google Cloud Platform +and associate a publicly available bucket with the project. +2. Modify the user constants.py file to specify the PROJECT ID and BUCKET with the +specifics of your Google Cloud Project. If you wish to change some simulation constants, +modify the beginning of the tpu simulation utilities.py file. +3. Navigate to https://colab.research.google.com/ and open the example notebook (or +your own notebook) from the TPU-Tsunami Repository using Colab’s open from Github +tool. +4. Navigate to Runtime ¿ Change runtime type, and verify that the TPU option is chosen +as the Hardware Accelerator. +24 + +5. Upload your user constants.py and tpu simulation utilities.py files to your note- +book session using the drag-and-drop feature under Files. +Upload any corresponding +DEM files to the session as well. +6. Specify a function corresponding to an initial condition for your DEM file (or use one +example initial condition). +7. Set initial conditions, boundary conditions as clarified in the bottom of any example +notebook run. Set last simulation parameters defining numerical resolution (resolution), +time step size (dt), output file times, TPU core configuration (currently only capable of +variation of cy), and DEM file name on bucket (dem bucket filename). +8. Run the simulation. +9. Analyze results. +25 + diff --git a/KdFLT4oBgHgl3EQfLC8K/content/tmp_files/load_file.txt b/KdFLT4oBgHgl3EQfLC8K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f366444d10c56f298843a69241632afc65db693 --- /dev/null +++ b/KdFLT4oBgHgl3EQfLC8K/content/tmp_files/load_file.txt @@ -0,0 +1,1673 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf,len=1672 +page_content='Leveraging Google’s Tensor Processing Units for tsunami-risk mitigation planning in the Pacific Northwest and beyond Ian Madden1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Simone Marras2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' and Jenny Suckale1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='3 1Institute for Computational and Mathematical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Stanford University 2Department of Mechanical Engineering & Center for Applied Mathematics and Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' New Jersey Institute of Technology 3Department of Geophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Doerr School of Sustainability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Stanford University January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2023 Abstract Tsunami-risk and flood-risk mitigation planning has particular importance for commu- nities like those of the Pacific Northwest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' where coastlines are extremely dynamic and a seismically-active subduction zone looms large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The challenge does not stop here for risk managers: mitigation options have multiplied since communities have realized the viability and benefits of nature-based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To identify suitable mitigation options for their community, risk managers need the ability to rapidly evaluate several different options through fast and accessible tsunami models, but may lack high-performance computing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The goal of this work is to leverage the newly developed Google’s Ten- sor Processing Unit (TPU), a high-performance hardware accessible via the Google Cloud framework, to enable the rapid evaluation of different tsunami-risk mitigation strategies available to all communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We establish a starting point through a numerical solver of the nonlinear shallow-water equations that uses a fifth-order Weighted Essentially Non- Oscillatory method with the Lax-Friedrichs flux splitting, and a Total Variation Diminish- ing third-order Runge-Kutta method for time discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We verify numerical solutions through several analytical solutions and benchmarks, reproduce several findings about one particular tsunami-risk mitigation strategy, and model tsunami runup at Crescent City, California whose topography comes from a high-resolution Digital Elevation Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The direct measurements of the simulations performance, energy usage, and ease of execution show that our code could be a first step towards a community-based, user-friendly virtual laboratory that can be run by a minimally trained user on the cloud thanks to the ease of use of the Google Cloud Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 1 Introduction The coast of the Pacific Northwest, from Cape Mendocino in California to Northern Vancouver Island in Canada as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 1, is located on the seismically active Cascadia subduction zone [Heaton and Hartzell, 1987, Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Along the 1200-km-long Cascadia sub- duction zone, there have been no large, shallow subduction earthquakes over the approximately 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='12010v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='geo-ph] 27 Jan 2023 200 years of modern-data monitoring, but large historic earthquakes have left an unambiguous imprint on the coastal stratigraphy [Clague, 1997].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Sudden land level change in tidal marshes and low-lying forests provide testimony of 12 earthquakes over the last 6700 years [Witter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2003], including one megathrust event that ruptured the entirety of the current Cascadia sub- duction zone in 1700 BC [Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 1995, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The event created a massive tsunami that swept across the entire Pacific Ocean devastating communities as far away as Japan [Satake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 1996, Atwater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Current seismic-hazard models estimate that the probability of another magnitude 9+ earthquake happening within the next 50 years is about 14% [Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Figure 1: Map of the Cascadia Subduction Zone in the Pacific Northwest of the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Relative location of Crescent City with respect to the Megathrust Fault Line, with a more detailed picture of the Crescent City coastline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Esri provided access to the satellite imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Crescent City map at high resolution provided by Maxar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Pacific Northwest Map provided by Earthstar Geographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A magnitude 9+ Cascadia earthquake and tsunami occurring during modern times would devastate many low-lying communities along the Pacific Northwest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A recent assessment sug- gests that deaths and injuries could exceed tens of thousands and entails economic damages in the order of several billions of dollars for Washington and Oregon State [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', Knudson and Bettinardi, 2013, Gordon, 2012], with potentially severe repercussions for the entire Pacific coast and country as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The tsunami itself would put tens of thousands at risk of in- undation, and threaten the low-lying coastal communities specifically in the Pacific Northwest with very little warning time for evacuation Gordon [2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' But how to confront this risk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Traditionally, the most common approach to reducing tsunami risk is the construction of sea walls, but this hardening of the shoreline comes at a staggering price in terms of the economic construction costs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 245 miles of sea walls in Japan cost $12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='74 billion) and in terms of 2 Cascadia WA Megathrust Fault OR Crescent City CAlong-term negative impact on coastal ecosystems [Peterson and Lowe, 2009, Dugan and Hub- bard, 2010, Bulleri and Chapman, 2010] and shoreline stability [Dean and Dalrymple, 2002, Komar, 1998].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A potentially appealing alternative to sea walls are so-called hybrid approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Hybrid risk mitigation combines nature-based elements and traditional engineering elements to reduce risk while also providing co-benefits to communities and ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' An example of a hy- brid approach to tsunami-risk mitigation is a coastal mitigation park: A landscape unit on the shoreline built specifically to protect communities or critical infrastructure and provide vertical evacuation space, in the styles of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Communities across the Pacific Northwest are increasingly considering these nature-based or hybrid options [Freitag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011], but many important science questions regarding protective benefits and optimal design remain open [Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020, Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' This gap is particularly concerning given that existing models show that a careful design is necessary to avoid potential adverse effects [Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The design of current mitigation parks, such as the one being built in Constituci´on, Chile, is not yet underpinned by an in-depth quantification of how different design choices affect risk-reduction benefits, partly because numerical simulations of tsunami impacts are computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Figure 2: Map view (top) and side view (bottom) of a proposed tsunami-mitigation berm as designed by Project Safe Havens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The berm provides vertical evacuation space for the adjacent community and could also lower the onshore energy flux that drives the damage created by tsunami impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We show this design as one example of a hybrid approach to tsunami-risk mitigation as it combines an engineered hill and ramp with natural vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Sketches adapted from Freitag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The goal of this paper is to leverage Google’s Tensor Process Units (TPUs) for enabling a fast evaluation of different mitigation park design and ultimately advancing evidence-based tsunami-risk mitigation planning rooted in quantitative assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' TPUs are a new class of hardware accelerators developed by Google with the primary objective of accelerating machine learning computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' They are accessible via the Google Cloud Platform [Jouppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2017, Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='com] and have recently been used for many different applications in computational physics and numerical analysis [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2022, Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020b,a, Belletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We build upon and extend an existing implementation [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2022] to simulate the impact of idealized tsunamis on the coastline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Our implementation of the shallow water equations includes a non-linear advective term, not considered in [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2022], within the software framework based on Google’s TensorFlow necessary to execute code on the TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We discretize the new equations using the Weighted Essentially Non-Ocillatory (WENO) method [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 3 1994] and a third-order Runge-Kutta in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Numerical simulations of tsunamis have contributed to our understanding of and ability to mitigate wave impacts for many decades now, starting from the early work by Isozaki and Unoki [1964] for Tokyo Bay, and Ueno [1960] for the Chilean coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The ability to capture the rupture mechanism that generates the initial condition for tsunami propagation then enabled the reproduction of many historical tsunamis [Aida, 1969, 1974].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Since then, many numerical models have been developed to simulate tsunami generation [Borrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2004, Pelties et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2012, L´opez-Venegas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2014, Galvez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2014, Ulrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2019], propagation [Titov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2005, LeVeque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2014, Allgeyer and Cummins, 2014, Abdolali and Kirby, 2017, Bonev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2018, Abdolali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2019], and inundation [Lynett, 2007, Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2013, Leschka and Oumeraci, 2014, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2014, Marsooli and Wu, 2014, Maza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2015, Oishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2015, Prasetyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2019, Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020] by solving different variations of the shallow water and Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The list of existing numerical models is long and was recently reviewed in Marras and Mandli [2021] and Horrillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Some commonly used ones are FUNWAVE [Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2000, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2012], pCOULWAVE [Lynett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2002, Kim and Lynett, 2011], Delft3D [Roelvink and Van Banning, 1995], GeoCLAW [Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011], NHWAVE [Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2012], Tsunami-HySEA [Mac´ıas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2017, Mac´ıas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020b,a], FVCOM [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2003, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Our work here relies on well-known numerical techniques to solve idealized tsunami problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Its novelty lies in demonstrating the capability and efficiency of TPUs to solve the non-linear shallow water equations to model tsunamis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We intentionally use a hardware infrastructure that is relatively easy to use without specific training in high-performance computing (See the Google Cloud TPU page at Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='com), and may become a standard hardware on which physics-based machine-learning algorithms will be built [Rasp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2018, Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020, Wessels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020, Fauzi and Mizutani, 2020, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2021, Kamiya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We propose that our implementation is one step towards a community-based, user-friendly virtual laboratory that can be run by a minimally trained user on the cloud thanks to the ease of use of the Google Cloud Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The tool, which is freely available on Github at [tsunamiTPUlab, 2023] under an Apache License, Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 for collaborative open source software development, can be modified to include machine learning capabilities and, eventually, extended to coupled models for earth-quake generation, inundation, and human interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Methods Numerical approximation We model tsunami propagation and runup with the 2D non-linear shallow water equations in the conservative formulation with a source term in a Cartesian coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Letting x = (x, y) denote position, we define u(x, t) and v(x, t) as the flow velocities in the x and y directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In our implementation, we solve for h, hu, and hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We define h(x, t) as the dynamic water height and b(x) as the imposed bathymetry, meaning that the quantity h + b represents the water surface level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' This leads to the following system of equations, a set 4 very similar to that suggested by Xing and Shu [2005] ∂ ∂th + ∂ ∂x(hu) + ∂ ∂y(hv) = 0 (1) ∂ ∂t(hu) + ∂ ∂x � (hu)2 h + 1 2g(h2 − b2) � + ∂ ∂y(huv) = −g(h + b) ∂b ∂x − gn2√ (hu)2+(hv)2 h7/3 (hu) (2) ∂ ∂t(hv) + ∂ ∂x(huv) + ∂ ∂y � (hv)2 h + 1 2g(h2 − b2) � = −g(h + b) ∂b ∂x − gn2√ (hu)2+(hv)2 h7/3 (hv) , (3) where g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='81 ms−2 is the acceleration of gravity, and n is the Manning friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' For ease of future notation, we let u = �h hu hv�T , and we rewrite the above equations in a vector form, namely: ∂u ∂t + ∂F ∂x + ∂G ∂y = S (4) where F and G are the fluxes in the x and the y directions for the vector u, and S is a source term arising from variations in topography and Manning coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We implement these shallow-water equations using the finite volume method whereby the half-step flux and height values are determined through a 5th-order WENO scheme [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 1994, Jiang and Shu, 1996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We approximate solutions to cell-wise Riemann problems by formulating fluxes using the Lax-Friedrichs method as in [LeVeque, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We formulate the bed source term as suggested by Xing and Shu [2005], and formulate the friction term explicitly rather than using the implicit process suggested by Xia and Liang [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We use a 3rd-order, Total Variation Diminishing Runge-Kutta scheme to step the numerical solution forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We begin the discretization of the equation in continuous variables t, x, and y, using re- spective step sizes ∆t, ∆x, and ∆y, which indicate the distance between consecutive integral steps in the discrete indices n, i, and j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We use the 5th-order WENO scheme in the x and y directions, where two values of each quantity h, hu, and hv are determined at each half-step of x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' These two values correspond to a positive and negative characteristic, due to the nature of the footprint that is chosen at a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In other words, given the conservative form with relevant variable u, ui,j centered on a finite volume cell, we label these outputs of WENO: u+ i+ 1 2 ,j, u− i+ 1 2 ,j for WENO in x, or u+ i,j+ 1 2 , u− i,j+ 1 2 for WENO in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' (5) From here, we use the Lax-Friedrichs method to approximate flux values that serve as solutions to the Riemann problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' we approximate Fi+ 1 2 ,j = 1 2 � F(u+ i+ 1 2 ,j) + F(u− i+ 1 2 ,j) − αu � F(u+ i+ 1 2 ,j) − F(u− i+ 1 2 ,j �� (6) where αu is the associated Lax-Frierichs global maximum characteristic speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Now, we dis- cretize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 4 explicitly as: un+1 i,j − un i,j ∆t + Fn i+ 1 2 ,j − Fn i− 1 2 ,j ∆x + Gn i,j+ 1 2 − Gn i,j− 1 2 ∆y = S(un i,j) (7) Note that in our case, we also choose to formulate the source term S(un i,j) explicitly and centered at the grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Since we use an entirely explicit formulation, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 7 as a time stepping operator for un+1, namely un+1 = T(un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Because Runge-Kutta uses multiple stages within each time step, we reassign the output of the T operator to be u(n+1) = T(un), 5 where(n + 1) indicates an intermediate full time step forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' This means a full Runge-Kutta step progresses as follows: u(n+2) = T(T(un)) −→ u(n+ 3 2 ) = T(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='25u(n+2)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='75un) −→ un+1 = 2 3u(n+ 3 2 )+ 1 3un (8) The process outlined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 8 outputs a final un+1 representing a full-step forward in simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' TPU implementation To leverage the TPU’s several cores, we divide the domain into multiple subdomains and in- dependently compute the numerical solution to the governing equations on each core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' While a lot of the computation can take place independently, each subdomain remains dependent on the others via their boundaries and the Lax-Friedrichs global maximum in characteristic speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We determine global maximum characteristic speed by sharing and reducing the Lax-Friedrich maximum characteristic speed calculated on each core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We transfer subdomain boundary infor- mation with further care by using a halo exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The data transfer behavior and computation structure is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Figure 3: Left: Initialization of implementation takes advantage of CPU to allocate initial conditions and topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Center: Regular computation period occurring on each subdomain, run independently on TPU cores with some data sharing coordinated by CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Right: CPU Gather to write results to output files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Our implementation is inspired by Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2022], who chose halo exchange as an instrument for the TPU to communicate information across subdomain boundaries in their formulation of the shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In the halo exchange process, we transfer slices of the domain from one core to the others immediately adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' While Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2022]’s methodology only involved the exchange of a single slice from one core to the other, we transfer several slices in order to take full advantage of the high accuracy and larger footprint of the WENO scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' These halo exchanges are then performed in every stage of the Runge-Kutta scheme, meaning that they occur multiple times in a single time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The initial conditions and results are communicated from the remote program, which resides on the CPU, to the TPU workers by means of tpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='replicate which sends TensorFlow code to each TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We refer to Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2022] for further details on the TPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 6 TPU Core TPU Core TPU Core TPU Core TPU Core TPU Core TPU Core TPU Core TPU Core Calculate initial condition, Perform Calculations Gather from TPU Cores and CPU partitiondomain,and disperse CPU Write Output File subdomainstoTPUCores CPU CPU Coordinates Exchange of Information2 Model verification and validation We differentiate between model verification and validation in the manner suggested by Carson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Specifically, we check for model and implementation error by quantifying the extent to which numerical solutions compare to correct analytical solutions [Carson]: wet dam break (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1), oscillations in a parabolic bowl (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2), and steady state flow down a slope with friction (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Following this, we validate by checking how well numerical solutions reflect the real system and apply to the context [Carson].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To do this, we compare against an existing numerical benchmark from the Inundation Science and Engineering Cooperative [ISEC, 2004] and results from an investigation of nature-based solutions [Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2020] in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='4, and consider the propagation of a computed tsunami over the observed topography of Crescent City in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To quantify the accuracy of the solutions, we test our numerical solver against some classical analytical solutions to the shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We assess the model’s ability to capture key physical processes relevant to inundation, including steep wave propagation, friction, and topography dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We use relative errors in the L∞ and L2 sense as the metric to determine model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' These are approximated in this paper in the following manner: L∞ = maxΩ |hc − ha| maxΩ |ha| , L2 = �� Ω(hc − ha)2∆Ω � Ω(ha)2∆Ω , (9) where hc is the computed solution at the discretized cells, ha is the analytical solution at the corresponding cells, and Ω denotes the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1 Wet dam break Figure 4: On the left, several instances in time of the computed (purple) water heights to wet dam break compared with the analytical (orange, dashed) water heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The rightmost figure plots the L2 and L∞ relative norms of the error between the analytical and computed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The classical one-dimensional Wet Dam Break [Stoker, 1957] provides us an opportunity to test the ability of our code to capture shock propagation and advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In this case, there is no friction (n = 0) and the topography is flat (b(x) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The boundaries are set at a constant height with zero flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We impose the following initial condition: (hu) = 0, (hv) = 0, h(x) = � hl x ≤ x0 hr x > x0 , (10) where hl and hr are the constant water heights on either side of a shock front x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We compare our numerical solution for water height against the dynamic analytical solution from Delestre 7 t = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 seconds t = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds Relative Error in Norm 10 Computed 10 Computed 10 Computed Lo Analytical 9 Analytical 9 Analytical 9- L2 8 8 8 10-2 N N 7 6 - 6 9 5 10° 1200125013001350140014501500 1200125013001350140014501500 1200125013001350140014501500 2 6 8 10 X [-] X [-] x[-] Time [s]et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2013]: h(x, t) = � � � � � � � � � � � hl x ≤ x1 4 9g �√ghl − x−x0 2t �2 x1(t) < x ≤ x2(t) c2 m g x2(t) < x ≤ x3(t) hr x > x3(t) , (11) x1(t) = x0 − t � ghl , (12) x2(t) = x0 + t(2 � ghl − 3cm) , (13) x3(t) = x0 + t2c2 m(√ghl − cm) c2m − ghr , and (14) cmis the solution to − 8ghrc2 m( � ghl − cm)2 + (c2 m − ghr)2(c2 m + ghr) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' (15) A qualitative comparison of the computed and analytical solutions for times t =0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5, and 9 seconds is shown in the left plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The relative error between the analytical and computed solutions in the infinity and 2-norms at a small distance away from the shock front are plotted on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We interpret the converging relative error norms to a low magnitude as verification of our implementation to sufficiently capture shock propagation and advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2 Planar parabolic bowl Figure 5: On the left, several instances in time of the computed (purple) water heights to the one-dimensional parabolic bowl compared with the analytical (orange, dashed) water heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The rightmost figure plots the L2 and L∞ relative norms of the error between the analytical and computed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The classical one-dimensional planar parabolic bowl originally suggested by [Thacker, 1981], is an oscillating solution allowing us to test the source term for topography without friction (n = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We enforce homogeneous Dirichlet conditions in both flux and water height, at a resolution of 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Once again, we take the test directly from Delestre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2013], resulting in the following description of the base topography: b(x) = h0 � 1 a2 � x − L 2 �2 − 1 � , (16) corresponding with the following initial condition: (hu) = 0, (hv) = 0, h(x) = � −h0 �� 2x−L+1 2a �2 − 1 � 1−2a+L 2 < x < 1+2a+L 2 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' (17) 8 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 seconds t = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 seconds Relative Error in Norm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 10-3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 10-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 10-5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 - Computed Solution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 - Computed Solution 10-6 Lo Analytical Solution Analytical Solution 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 L2 Topography Topography 10-7 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 X [-] X[-] Time [s]This leads to the following dynamic analytical solution for the water height: h(x, t) = � � � −h0 �� 2x−L 2a + 1 2a cos � √2gh0t a ��2 − 1 � x1(t) < x < x1(t) + 2a 0 otherwise , (18) where x1(t) = 1 2 cos � √2gh0t a � − a + L 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A qualitative comparison of the parabolic bowl solution at the time instances t =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 sand t =12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 s can be seen on the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The analytical and computed solutions appear to correspond to one another well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' For a more quantitative analysis,the relative error-norms of the solutions are depicted on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We interpret the converging relative error norms to a low magnitude as verification of our implementation to sufficiently capture the source term of the shallow water equations induced by topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='3 Steady flow down a slope with friction Figure 6: On the left, several instances in time of the computed (purple) fluxes given a water level and a slope, compared to the analytical (orange, dashed) flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The rightmost figure plots the L2 and L∞ relative norms of the error between the analytical and computed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We do a short test in order to assess the correctness the discretized friction source term, focusing on a relatively simple flow down a slope with finite friction (n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='033) as tested by Xia and Liang [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The steady state flow down a slope then becomes (hu) = √bx n h 5 3 (19) where bx is the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In this test, we initialize the problem close to the steady state solution for a wave height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 and a slope of 1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' This specific example and its convergence toward steady state is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The left plots show the flux at t = 100 s and t = 200 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The results are almost identical, indicating that a steady state has been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' On the right plot, the error norm of the steady state flux takes some time to reach steady state, but reaches a very small level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Because we approach the appropriate steady state solution and achieve a very small error norm, our implementation is verified in capturing a manning friction law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='4 Validation for tsunami simulations To assess the ability of the code to capture tsunami propagation, we start with a popular numerical benchmark from the Inundation and Science Engineering Cooperative (ISEC) [ISEC, 2004] that represents tsunami runup over an idealized planar beach that provides solutions for 9 t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds t = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds Relative Error in Norm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='120 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='120 Computed Computed Loo Analytical Analytical 10-2 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='125 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='125 L2 6 × 10-3 Flux 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='130 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='130 4×10-3 3 ×10-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='135 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='135 2×10-3 0 100 200 300 400 0 100 200 300 400 50 100 150 200 X [-] X[-] Time [s]tsunami runup at times t =, 180 s, 195 s, 220 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We formulate the initial condition for water height using Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The solutions from the benchmark (dashed, orange) are qualitatively compared with the numerical solution produced by our code (solid, purple) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We take the qualitative agreement as validation of the model’s ability to model the runup of a Carrier N-Wave [Carrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Figure 7: Qualitative comparison of the computed solution with resolution 1 m compared to the ISEC benchmark at three different time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Since we are interested in leveraging TPUs for tsunami-risk mitigation planning, we take a look at the ability of our shallow water equation code to reproduce a few particular results by Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2020] who investigated the effects of hills on a tsunami running up on a planar beach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The tsunami is initialized as Carrier’s N-wave [Carrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2003]: η = 2(a1 exp{−ˆk1(x − ˆx1)2} − a2 exp{ˆk2(x − ˆx2)2}), (20) where η = h + z, ˆx1 = 1000 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5151125λ, ˆx2 = 1000 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2048λ, ˆk1 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='416/λ2, ˆk2 = 256/λ2, a1 = A, and a2 = A/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' While this is the analytically correct form, the flow origin in the code is not the shoreline, so there are some effective shifts ˆx1 and ˆx2 that we need to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' An example of the Carrier wave initial condition and offshore propagation behavior for A = 15 and λ = 2000 is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We apply free slip, no-penetration boundary conditions to the four domain boundaries, which means that that the component of the boundary-normal component of the velocity vector is zero whereas its tangential component is unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The Figure 8: Several snapshots in time of the tsunami’s propagation over a modeled ellipsoidal hill on a slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' From left to right, the formulation of the initial Carrier N-wave at t = 0, followed by the propagation of a wave front toward the hill at t = 50, collision of the wave front of the hill at t = 95, and the formation of a reflected wave at t = 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' shallow water equation model presented in this study is able to reproduce the wave reflection 10 Water Height at t = 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds Water Height at t = 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds Water Height at t = 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds 5040 Computed 5040 Computed 5040 Computed ISEC Benchmark ISEC Benchmark ISEC Benchmark 5020 Topography 5020 Topography 5020 Topography N N N 4980 4980- 4980 4960 4960 4960 48500 49000 49500 50000 48500 49000 49500 50000 48500 49000 49500 50000 X [m] X [m] X [m]Water Height at t = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='o seconds Water Height at t = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds Water Height at t = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds Water Height at t = 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 seconds OE OE 20 20 20 20 N N N N 0 0 10 10 - 10 10 4000 3000 2000 1000 1000 4000 3000 2000 1000 1000 4000 3000 2000 1000 0 1000 4000 3000 1000 1000 X [m] X [m] X [m] X [m]provided by a hill, consistent with results from Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Because this simulation is possible by the implementation, other further analysis can be conducted to understand the mitigative benefit of other nature-based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 Real-world scenario Past tsunamis impacting the West Coast of the United States have caused more damage around the harbor of Crescent City in California than elsewhere along the Pacific Coast [Arcas and Uslu, 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' For this reason, we chose an area of approximately 105 km2 around Crescent City to demonstrate the code’s ability to capture the impact of an idealized tsunami event for a real location at high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To approximate the actual bathymetry and topography, we use a Digital Elevation Model for this area with uniform grid spacing of 4 m provided by NOAA [NOAA National Geophysical Data Center, 2010, Grothe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' For the sake of providing a proof of concept, we initialize the tsunami with the idealized waveform described above, with slightly adjusted parameters: A = 10, λ = 2000, ˆx1 = 6000 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5151125λ, and ˆx2 = 6000 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2048λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The chosen parameters lead to maximum inundation patterns similar to that seen in one modeled extreme scenario from Arcas and Uslu [2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 9, we start with an absence of any nearshore wave (including at t = 50s) and then a development of a tsunami front that is visible to the shoreline by t = 140s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' That front penetrates the harbor by t = 220s, and is soon followed by the inundation of the coastline as well as reflection of wave energy back to the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We also observe that the mountain range on the upper part of the figure clearly provides a significant protective benefit to the land beyond it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Figure 9: Several snapshots of modeled tsunami propagation over terrain and geological features of Crescent City, CA, where any level of blue indicates water cover and green depicts a stylized map of the topography above surface level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' From left to right, then top to bottom, we have steady state near shore at t = 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' followed by the propagation of a wave front at t = 100 and 140;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' contact with Crescent City harbor at t = 180;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' inundation of the harbor and some of the coastline at t = 220 and 260;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' and tsunami reflection and inundation at t = 300 and 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 11 0- 0 0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 006 006 006 006 1800 1800- 1800 1800 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 2700 2700 2700 2700 > > 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 4500 - 4500 4500 - 4500 5400 - 5400 5400 5400 6300- 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 6300 t = 50 s t=100 s t = 140 s t = 180 s 7200 + 7200 + 7200 + 7200 + 0 1496 2992 4488 5984 1496 2992 4488 5984 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 1496 2992 4488 5984 1496 2992 4488 5984 X [m] X [m] X [m] X [m] 0 006 006 006 006 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 1800 1800 1800 1800 2700 2700 2700 2700 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 > > > > 4500 - 4500 4500 - 4500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 5400 5400 5400 5400 6300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 t = 220 s t = 260 s t = 300 s t = 340 s 7200 + 7200 + 7200 + 7200 1496 2992 4488 5984 0 1496 2992 4488 5984 1496 0 0 4488 5984 1496 4488 5984 X [m] X [m] 3 Performance analysis on TPU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1 Number of TPU cores In communities where users may not have access to high performance computing facilities, the Cloud TPU Platform provides a unique ability to perform large-scale computations, and perform them rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To address this potential benefit, we first measure the wall-clock time of a simulation of a tsunami reaching Crescent City using a varying number of TPU cores on one TPU device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' As shown in Table 1, the problem size posed by the realistic scenario is sufficient to see rapid improvements in runtime based on the number of cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Number of Cores 1 2 4 8 TPU Runtime [s] 6894 3876 2000 1036 Table 1: Approximate TPU Runtimes (in seconds) with varying numbers of TPU cores for Crescent City Configuration using time step of ∆t = 5·10−3, Total Array Size of approximately 1802 by 3984 elements (4 meter resolution), cores all aligned in the y-direction as suggested in [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' This runtime excludes transfer times between the CPU and TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2 Geophysical problem resolution The typical realistic scenario that decision-makers will face will involve large problem sizes due to both the extent of their spatial domain, but also the level of resolution necessary to model tsunami propagation and inundation over complex topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Therefore, we check convergence and runtime under varying degrees of resolution for the current realistic scenario as well, depicted graphically in 10 and in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Figure 10: Graphical depiction of the TPU runtimes (purple) and relative error norms (blue) under varying resolutions for computing the tsunami propagation over the Crescent City DEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We finally perform the same analysis under varying degrees of resolution using the bench- mark from the Inundation Science and Engineering Cooperative [ISEC, 2004] that we previously 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='35 3500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='30 ODOE Relative Error Norm [-] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='25 2500 Runtime [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='20 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='15 L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='10 TPURuntime 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='05 2 4 B 10 12 Resolution [m]Resolution [m] 2 4 6 8 10 12 Runtime [s] 3805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='167 1035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='719 506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0652 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='9506 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0804 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='3016 Number of Elements 28713068 7179168 3192512 1794792 1149274 798128 Efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='00021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000235 L2 Error Norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='062475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='092618 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='115671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='093901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='132735 L∞ Error Norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='212536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='340507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='291389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='265669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='259101 Table 2: Approximate TPU Runtimes (in seconds) with varying resolutions for Crescent City Configuration using time step of ∆t = 5 · 10−3, using the 2 m resolution as a benchmark for correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Ran on a single TPU with 8 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' validated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A qualitative comparison of the tsunami propagation under different reso- lutions are graphically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 11 in the top two and bottom left figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In the bottom right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 11, we see the expected fall in runtime based on coarser resolution (purple), and a rise in relative error (with the exception of the highest resolution 1 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The corresponding values are documented in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Figure 11: Graphical depiction of the TPU runtimes (purple) and relative error norms (blue) under varying resolutions for computing the tsunami propagation in the ISEC Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 13 ISECBenchmark ISEC Benchmark 5020 5020 Computed atResolution1m ComputedatResolution1m Computed atResolution2m Computedat Resolution2m 5010 ComputedatResolution4m 5010- Computedat Resolution4 m ComputedatResolution8m ComputedatResolution8m 三 Topography E Topography N 5000 N 5000 4990 4990- 4980- 4980- 49000 49200 49400 49600 49800 50000 50200 49000 49200 49400 49600 49800 50000 50200 X [m] X [m] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 5020 140 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 5010 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 120 Runtime [s] 三 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 5000 1008 N ISECBenchmark ComputedatResolution1m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 lative 80 4990 Computed at Resolution 2 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 TPURuntime Computed atResolution4m Rel Computed at Resolution 8 m 60 4980 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 Topography 49000 49200 49400 49600 49800 50000 50200 1 2 3 4 5 6 7 8 X [m] Resolution[m]Resolution [m] 1 2 4 8 Runtime [s] 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0193 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='74382 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='74928 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='3934 Number of Elements 1060521 277761 75756 18939 Efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='002502 L2 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000206 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='19E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='00031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000282 L∞ Error Norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='000326 Table 3: Approximate TPU Runtimes (in seconds) with varying resolutions for the ISEC Tsunami Benchmark using time step of ∆t = 5 · 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Ran on a single TPU with 8 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='3 Comparison with GeoClaw Figure 12: TPU solution (top row) at several time instances compared to the GeoClaw solution (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The arrival of the tsunami front (t = 100, 180), the inundation of the harbor (t = 260), and coastal inundation and reflection is depicted, and relatively comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We used GeoClaw [Clawpack Development Team, 2020, Mandli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2016, Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011] ran on a single thread of a CPU (Intel i7-8650 with a base frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='9 GHz) to compare numerical solutions to a tsunami propagation in order to assess performance enhancements provided by the TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' While our TPU-based code completes the 400 second simulation in approximately 17 minutes of wall-clock time, the GeoClaw implementation on the CPU takes approximately 630 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Comparisons of the two numerical solutions can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 12, where the top row includes several instances in time of the TPU numerical solution, and the bottom row depicts the GeoClaw numerical solution at the same instances in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Although some differences can be seen in the geometry of inundation by t = 380s in the rightmost plots, the solutions do generally appear similar over time, lending credibility to the validity of the numerical 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='6300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='6300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='6300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='t = 100 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='t = 180 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='t= 260 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='t = 340 s ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2592 3888 5184 1296 2597 3888 5184 X [m] X [m] X [m] X [m] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='4 Energy utilization Estimates of energy efficiency of computing operations are becoming increasingly popular, es- pecially in response to progressing climate change [Fuhrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2018, Fourestey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To address this potential benefit provided by TPUs, we conduct a heuristic analysis of the energy savings of running these operations on a TPU rather than a CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We base an order- of-magnitude estimate on the claims of Google in the maximum power efficiency of the TPU, 2 trillion operations per second (TOPS) per Watt [Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='com].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We use this heuristic for power efficiency, consider the tsunami propagation problem under a 8 m resolution for Crescent City, and manually approximate the number of floating point operations that our implementation performs in each quadrature step at this resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We estimate that the TPU performs ap- proximately 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='9 million floating-point operations for each simulated time step, and we see that each time-step takes an average of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2 milliseconds, corresponding to about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='83 TOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Assuming constant power efficiency regardless of capacity, this translates to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='41 Watts and an energy usage of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='18 J for each simulated second given our current time-step configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Combining with the total runtime for a 400 modeled-second simulation of Crescent City, we see a total cost of approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='76 × 102 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='32 × 10−1 Wh of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' At a price of 21 cents/kWh in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' at the time of writing this article, this simulation has a monetary cost of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='7 × 10−3 cents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We use the aforementioned GeoClaw run as our CPU comparison on energy utilization in order to get an order of magnitude estimate of the power savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The particular processor for this CPU comparison, the Intel i7-8650 with a base frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='9 GHz, has a Thermal Design Power of 15 W for 8 total threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Each simulated second took approximately 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='4 seconds of CPU runtime, which translates to approximately 181 Joules for each simulated second of a grid with 8 m resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Given the order of magnitude estimates, we note that the 400 modeled-second simulation would see a total cost of approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='24 × 104 J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='01 × 101 Wh of energy, or a monetary cost of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='04 cents, a cost multiplicative factor of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We linearly extrapolate to estimate the number of CPU threads needed to match the runtime speed of our model using the TPU, and find that approximately 37 CPU threads would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' With this in mind, we find that the cost multiplicative factor for a CPU simulation of performance equivalent to that of a TPU would be closer to an order of 700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' TPU energy savings for high performance are clearly substantial, and not ignorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 4 Discussion Sustainable tsunami-risk mitigation in the Pacific Northwest is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Some chal- lenges come from beneath, because previous large subduction zone earthquakes at Cascadia led to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5 − 1 m of co-seismic subsidence, the sudden sinking of land during an earthquake [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Strong shaking can also lead to liquefaction [Atwater, 1992, Takada and Atwater, 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Other challenges come from the ocean, where sea-level rise [Church and White, 2006, Bindoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2007] and intensifying winter storms [Graham and Diaz, 2001] have increased wave heights [Ruggiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2010, Ruggiero, 2013] and accelerated coastal erosion [Ruggiero, 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A recent USGS report documented rapid shoreline changes at an average rate of almost 1 m/yr across 9,087 individual transects [Ruggiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2013], suggesting the possibility that the shoreline might change significantly during the century-long return-period of large earthquakes in Cascadia [Witter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The picture that emerges is that of a highly dynamic coastline – maybe too dynamic for an entirely static approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Nature is not only continuing to shape the coastline, but is also a fundamental component of the region’s cultural heritage, identity and local economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' So, it is 15 maybe not surprising that the Pacific Northwest is a thought-leader when it comes to designing hybrid approaches to sustainable climate adaptation through the Green Shores program [Dalton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2013] and to vertical tsunami evacuation through Project Safe Haven [Freitag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Project Safe Havens is a grass-roots approach to reducing tsunami risk mostly by providing accessible vertical-evacuation options for communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Many proposed designs entail reinforced hillscapes like the one shown in figure 2, intended to dissipate wave energy and provide vertical evacuation space during tsunami inundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To build confidence in such a solution and its mitigation effects, risk managers must be able to quickly and precisely forecast a tsunami inundation, preferably via a publicly available, centralized modeling infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' This paper is meant to be a first step towards a major community based infrastructure that will allow local authorities around the world to readily execute tsunami simulation once a tsunami in their proximity has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We aim to provide a proof-of-concept rather than a complete implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' As such, we used a very similar base framework used by Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2022] of halo exchange in combination with a WENO and Runge-Kutta scheme, which may not be optimally taking advantage of the TPU’s computing structure and capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We originally chose these schemes to maintain higher order accuracy and ease of implementation but, eventually, a convolution-based implementation of the quadrature of the shallow water equations should be tested for maximum performance utilization of the TPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Because our code is specifically an implementation of the shallow water equations for the TPU, it is currently unable to model tsunami initiation, or any fluid structure interactions that may be desired to accompany analysis of nature-based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Instead, it requires an initial condition for wave heights and fluxes, meaning a full tsunami simulation would require coupling the results of a tsunami initiation model as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' While our implementation is relatively limited in scope, the model is certainly able to provide a starting point for a more complete software package for communities as they evaluate nature-based options for tsunami mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Though just a starting point for a remotely-available package, we successfully replicated some of the results found about coastal mitigation parks posed by Lunghino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' [2020], and we were able to model tsunami runup over with the real bathymetry around Crescent City, in California, included in the code by means of a DEM file from the USGS digital elevation database;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' for comparison purposes, we ran the same test using the popular open-source solver GeoClaw [Clawpack Development Team, 2020, Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' The results of the two models are in good agreement, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Our contribution lies in demonstrating an enhanced ability to run high quality simulations using the TPU available remotely using Google Cloud Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' As a result, high quality tsunami simulations are available to remote communities for rapidly evaluating mitigation or evacuation options when faced with coastal flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' We argue that TPUs are preferable to large, heavily parallel simulations on CPU or GPUs, because the TPU-based simulations we show here do not require access to large computing clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' These are usually made available to scientists and engineers by supercomputing centers around the world by means of competitive grants for computing time or by use of the cloud offered by private companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' However, an expert user knowledge of these systems from a scientific computing perspective is necessary to design, run, and interpret model results, and the compute infrastructure itself may not be available to early warning centers in many parts of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' In contrast, our code is available on Github and fully implemented in Python, can be ran through a web browser, and visualized through a simple notebook file using Google Colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' While performance can be enhanced with some knowledge about TPU architectures, community risk managers do not need this knowledge to run high quality tsunami simulations rapidly for real, physical domains with associated DEMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 16 Finally, in the face of rising energy costs in both a monetary and climate sense, the TPU infrastructure allows the support of more energy-efficient simulations over those of CPU-based clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' This means that those coastal flood risk managers in remote communities immediately are able to support design decisions with model results in a climate-friendly manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Although not our focus here, we note our approach may also contribute to early tsunami warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Once triggered, tsunamis move fast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' this fact makes it necessary to model and as- sess their potential for damage ahead of time once they have been detected offshore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' For a sufficiently fast early warning and prompt evacuation, the tsunami modeling infrastructure has an important time constraint [Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2021] to be considered, and Faster Than Real Time (FTRT) simulations are necessary [Behrens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2021, Løvholt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' To make FTRT simulations a reality, tsunami models are being rewritten or adapted to run on Graphical Processing Units (GPUs) [Løvholt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2019, Behrens and Dias, 2015, Satria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A TPU-based implementation as proposed here might be another meaningful step into that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Author contributions Ian Madden: Software, Analysis, Writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Simone Marras, PI: Conceptualization, Method- ology, Writing, Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Jenny Suckale: Conceptualization, Methodology, Writing, Su- pervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Competing interests The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Data availability statement Our work is available as a GitHub release at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='com/smarras79/tsunamiTPUlab/ releases/tag/v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 or on archive at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='7574655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Acknowledgements This work was supported by the National Science Foundation’s Graduate Research Fellowships Program (GRFP) awarded to Ian Madden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' References A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Abdolali and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' T.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Hemphill-Haley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Great Cascadia earthquakes and tsunamis of the past 6700 years, Coquille River estuary, southern coastal Oregon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Geological Society of America Bulletin, 115(10):1289–1306, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Xia and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' A new efficient implicit scheme for discretising the stiff friction terms in the shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Advances in Water Resources, 117:87–97, July 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Xing and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Shu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' High order finite difference WENO schemes with the exact conservation property for the shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Journal of Computational Physics, 208(1):206–227, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 5 Appendix 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='1 Running the code Due to the restrictions of the TPU using Google Cloud Storage, Google’s buckets will need to be used in order to run the notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' With a computing project setup on Google Cloud, users can quickly run any of the example notebooks or design their own simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Any of the example notebooks available on GitHub (with the exclusion of tpu tsunami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='ipynb, which contains the full implementation with all of the different scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' and Create Scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='ipynb, which can aid users in generating a custom DEM file) can be quickly ran by going through the notebook after a few early setup steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Download the TPU-Tsunami Repository from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='com/smarras79/tsunamiTPUlab/ releases/tag/v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='0 to your local machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Create a project on Google Cloud Platform and associate a publicly available bucket with the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Modify the user constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='py file to specify the PROJECT ID and BUCKET with the specifics of your Google Cloud Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' If you wish to change some simulation constants, modify the beginning of the tpu simulation utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='py file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Navigate to https://colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='com/ and open the example notebook (or your own notebook) from the TPU-Tsunami Repository using Colab’s open from Github tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Navigate to Runtime ¿ Change runtime type, and verify that the TPU option is chosen as the Hardware Accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Upload your user constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='py and tpu simulation utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content='py files to your note- book session using the drag-and-drop feature under Files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Upload any corresponding DEM files to the session as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Specify a function corresponding to an initial condition for your DEM file (or use one example initial condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Set initial conditions, boundary conditions as clarified in the bottom of any example notebook run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Set last simulation parameters defining numerical resolution (resolution), time step size (dt), output file times, TPU core configuration (currently only capable of variation of cy), and DEM file name on bucket (dem bucket filename).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Run the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' Analyze results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf'} diff --git a/PNE3T4oBgHgl3EQfxguY/content/tmp_files/2301.04712v1.pdf.txt b/PNE3T4oBgHgl3EQfxguY/content/tmp_files/2301.04712v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..92118c5e7273bc03485b39840a3cee5cc8e74291 --- /dev/null +++ b/PNE3T4oBgHgl3EQfxguY/content/tmp_files/2301.04712v1.pdf.txt @@ -0,0 +1,2457 @@ +Generated using the official AMS LATEX template v6.1 two-column layout. This work has been submitted for +publication. Copyright in this work may be transferred without further notice, and this version may no longer be +accessible. +An analytic formula for entraining CAPE in mid-latitude storm environments +John M. Petersa , Daniel R. Chavasb , Chun-Yian Sua , Hugh Morrisonc , and Brice E. Cofferd +a Department of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, PA +b Department of Earth, Atmospheric, and Planetary Sciences, Purdue University, West Lafayette, IN +c National Center for Atmospheric Research, Boulder, CO +d Department of Marine, Earth and Atmospheric Sciences, North Carolina State University, Raleigh, NC +ABSTRACT: This article introduces an analytic formula for entraining convective available potential energy (ECAPE) with an entrainment +rate that is determined directly from the storm environment. Extending previous formulas derived in Peters et al. (2020a), entrainment +is connected to the background environment via an analytic manipulation of the equations of motion that yields a direct correspondence +between the storm relative flow and the updraft radius, and an inverse scaling between the updraft radius squared and entrainment rate. +These concepts, combined with the assumption of adiabatic conservation of moist static energy, yield an explicit analytic equation for +ECAPE that depends entirely on state variables in an atmospheric profile and a few constant parameters with values that are established in +past literature. Using a simplified Bernoulli-like equation, a second formula is derived that accounts for updraft enhancement via kinetic +energy extracted from the cloud’s background environment. CAPE and ECAPE can be viewed as predictors of the maximum vertical +velocity 𝑤𝑚𝑎𝑥 in an updraft. Hence, these formulas are evaluated using 𝑤𝑚𝑎𝑥 from past numerical modeling studies. Both of the new +formulas improve predictions of 𝑤𝑚𝑎𝑥 substantially over undiluted CAPE, ECAPE with a prescribed entrainment rate, and the ECAPE +formula from Peters et al. (2020a). The formula that incorporates environmental kinetic energy contribution to the updraft correctly predicts +instances of exceedance of +√ +2CAPE by 𝑤𝑚𝑎𝑥 in simulations, and provides a conceptual explanation for why such exceedance is rare +among past simulations. These formulas are potentially useful in nowcasting and forecasting thunderstorms and as thunderstorm proxies +in climate change studies. +SIGNIFICANCE STATEMENT: +Substantial mixing +occurs between the upward moving air currents in thun- +derstorms (updrafts) and the surrounding comparatively +dry environmental air, through a process called entrain- +ment. Entrainment controls thunderstorm intensity via its +diluting effect on the buoyancy of air within updrafts. A +challenge to representing entrainment in forecasting and +predictions of the intensity of updrafts in future climates is +to determine how much entrainment will occur in a given +thunderstorm environment without a computationally ex- +pensive high resolution simulation. To address this gap, +this article derives a new formula that computes entrain- +ment from the properties of an updraft’s background envi- +ronment. This formula is shown to predict updraft vertical +velocity more accurately than past diagnostics, and can +be used in forecasting and climate prediction to improve +predictions of thunderstorm behavior and impacts. +1. Introduction +Middle-to-upper1 tropospheric vertical velocities in +deep convective updrafts influence a variety of storm- +related societal impacts, including precipitation (e.g., Jo +Corresponding author: John M. Peters, John.M.Peters@psu.edu +1We contrast middle-to-upper tropospheric vertical velocities, which +are primarily buoyantly driven, with lower tropospheric vertical veloc- +ities which are often dynamically driven in squall lines (e.g., Bryan +and Rotunno 2014; Jeevanjee and Romps 2015) and supercells (e.g., +Weisman and Rotunno 2000; Peters et al. 2019). +and Lasher-Trapp 2022), hail (e.g., Danielsen et al. 1972; +Lin and Kumjian 2022), electrification (e.g., Romps et al. +2014; Stolz et al. 2015), downdraft and cold pool inten- +sity (e.g., Marion and Trapp 2019), tropospheric convec- +tive mass flux (e.g., Peters et al. 2020b), and the flux of +mass, aerosols, and water vapor across the tropopause (e.g., +Mullendore et al. 2013). The magnitude of vertical veloc- +ities in the upper reaches of deep convective updrafts are +strongly influenced by updraft buoyancy (e.g., Morrison +and Peters 2018; Peters et al. 2019; Jeevanjee 2017). It is +well known that entrainment-driven dilution of deep con- +vective updrafts substantially influences updraft buoyancy +and vertical velocity (e.g., Zipser 2003; Romps and Kuang +2010a,b). For instance, weakly sheared deep convective +updrafts with large fractional entrainment rates are sub- +stantially diluted and often only realize a small fraction +(e.g., 20-30 %) of their convective available potential en- +ergy (CAPE) as updraft kinetic energy KE (Romps and +Kuang 2010a). In contrast, more organized modes of deep +convection such as squall lines and supercells with smaller +fractional entrainment rates and less dilution can realize +much larger fractions of their CAPE as KE (i.e., 80-100 +% Lebo and Morrison 2015; Peters et al. 2019; Mulhol- +land et al. 2021b). Hence, storm-to-storm variations in +entrainment substantially alter how much CAPE a storm +is able to process, and consequently its updraft kinetic en- +ergy and vertical velocity. These storm-to-storm variations +in entrainment also generally supersede the influences of +variations in other updraft processes and environment fac- +1 +arXiv:2301.04712v1 [physics.ao-ph] 11 Jan 2023 + +2 +tors on vertical velocity that receive substantial attention +in the literature (e.g., Lebo 2018; Grabowski and Morrison +2021), such as aerosol effects, pressure perturbations, and +precipitation behavior. Hence, the atmospheric sciences +community would benefit from an accurate representation +of entrainment in research and forecasting diagnostic pa- +rameters, such as CAPE, so that the parameters can more +accurately characterize the intensity of convective updrafts +that might form in a given environment. +CAPE calculations that include entrainment effects are +referred to as entraining CAPE, or ECAPE. Whereas CAPE +is often viewed as the theoretical maximum kinetic energy +that can be extracted by an isolated parcel from its envi- +ronment, ECAPE makes additional assumptions about up- +draft steadiness and mixing to estimate how the efficiency +of this kinetic energy extraction is affected by entrain- +ment. Various ECAPE-like calculations have been used +for the better part of the last century, primarily in the cli- +mate, tropical meteorology, and cumulus parameterization +communities. For instance, simple plume models (e.g., +Squires and Turner 1962) for moist convective updrafts +predict profiles of buoyancy that include entrainment ef- +fects, which can be vertically integrated to obtain ECAPE. +The “cloud work function”, which is an essential element +of many cumulus parameterizations (Arakawa and Schu- +bert 1974), uses the buoyancy of a diluted parcel within +its calculation, and yields a quantity that is analogous to +ECAPE. ECAPE is used as diagnostic tool in the research +of tropical environments to explain the sensitivity of deep +convection initiation to free tropospheric moisture (Brown +and Zhang 1997), and in the closure formulation of cumu- +lus parameterizations (Zhang 2009). The zero-buoyancy +plume model, in which buoyancy is assumed to be exactly +extinguished by entrainment, yields analytic solutions for +the mean state thermal structure of the tropical atmosphere +(Singh and O’Gorman 2013). The range of fractional en- +trainment rates in the tropics is typically smaller than that +of the mid latitudes (e.g., Takahashi et al. 2021). Hence, +using an ECAPE calculated with an empirically obtained +constant fractional entrainment rate provides reasonably +accurate predictions of deep convective updraft character- +istics in the tropics (e.g., Gregory 2001) +There are also a few scattered applications of ECAPE +in the weather forecasting community. For instance, the +spatial distribution of ECAPE has been shown to better +identify the tornadic regions of tropical (Sueki and Niino +2016) and extratropical cyclones (Tochimoto et al. 2019) +than undiluted CAPE. ECAPE has also been used to pre- +dict vertical velocities in supercells more accurately than +standard CAPE calculations (Peters et al. 2020a). There +is substantially larger variability in fractional entrainment +in the continental mid-latitudes (e.g., Peters et al. 2020c; +Takahashi et al. 2021; Lasher-Trapp et al. 2021) than in +the tropics, meaning that ECAPE computed with a sin- +gle fractional entrainment rate cannot accurately describe +all midliatude convective environments (e.g., Peters et al. +2020c). This makes using ECAPE in midlatudes more dif- +ficult than in the tropics, because it is not always clear what +entrainment rate should be used in the calculation. +To address the issue over what choice of fractional en- +trainment rate to use in the midlatitudes, Peters et al. +(2020a) (hereafter P20) developed an analytic formula +for maximum updraft vertical velocity (which is equal to +√ +2𝐸𝐶𝐴𝑃𝐸) that calculated entrainment from attributes of +a storm’s background environment, rather than requiring +that the user specify an entrainment rate. The connection +between entrainment and the background environment in +this formula was based on the previously-established nega- +tive correspondence between vertical wind shear and frac- +tional entrainment (e.g., Peters et al. 2019, 2020c, 2022a,b). +That is, mature deep convective updrafts tend to be wider +in environments with strong vertical wind shear and have +accordingly smaller fractional entrainment rates. This for- +mula more accurately predicted maximum updraft vertical +velocities than standard ECAPE computed with constant +pre-specified fractional entrainment rate. +There are several shortcomings of the P20 study that +warrant a revisit of the concepts contained therein. First, +the expression derived in the paper uses a hodgepodge of +formulas from previous studies, such as Morrison (2017) +and Peters et al. (2019) as a starting point2. The assump- +tions underlying these formulas from previous studies are +not explicitly discussed in P20, nor are they even thor- +oughly scrutinized in their source articles. Because of this +rooting in past studies, a few of the terms that end up in the +P20 equation are complicated and lack obvious physical +underpinning, which is challenging for end users of this +formula. +Second, the end formula for maximum updraft vertical +velocity is a third-order polynomial equation that must +either be solved explicitly with the complicated quartic +equation, or with a numerical root finding procedure. End +users of the formula found this quartic solution difficult +to efficiently incorporate into software routines. This 3rd +order polynomial equation results from the assumption that +fractional entrainment 𝜀 scales with the inverse of updraft +radius 𝑅−1. However, there is now evidence that 𝜀 ∼ 𝑅−2 +is a more realistic scaling (Peters et al. 2019; Morrison +et al. 2022; Mulholland et al. 2021b). Re-formulating the +P20 equation with 𝜀 ∼ 𝑅−2 yields a 2nd-order polynomial +equation that is much easier to solve, as will be shown in +the present study. +Third, the title of that paper, which is “A formula for +the maximum vertical velocity in supercell updrafts”, ob- +scures the take-home messages. The title does not contain +2Note a litany of constants are carried over into P20 from these past +formulas, and some of the symbols used (such as 𝐻𝑣 for the latent heat +of vaporization) are inconsistent with the symbols used in some of our +more recent articles (e.g., 𝐿𝑣 for the latent heat of vaporization Peters +and Chavas 2021; Peters et al. 2022c,a)). + +3 +the terms entrainment or CAPE, so it is not obvious that +the parameter derived in the paper essentially modifies +CAPE to account for the effects of entrainment (which is +by definition ECAPE). The concepts contained within the +paper apply to any isolated deep convective updraft exist- +ing within moderate to strong vertical wind shear – they are +not limited to supercells. There is no assumption about up- +draft rotation within the mathematical framework. Hence, +the inclusion of the term supercell in the title made the +application of the formula sound unnecessarily restrictive. +Our goal in this article is to revisit the concepts of P20 to +derive ECAPE formulas (Sections 2-3) that improve upon +the concepts in the P20 study in the following ways: +1. The buoyancy formula in the present study is de- +rived directly from the assumed conservation of moist +static energy, which differs from the P20 formula +which used the supersaturation tendency equation +from Politovich and Cooper (1988) as a starting point. +This methodological alteration requires less severe +assumptions and results in formulas with greater ac- +curacy in the present study. +2. The new formula uses the 𝜀 ∼ 𝑅−2 scaling, with fur- +ther improves accuracy over the P20 formula. +3. We also account for additional processes that were +not considered by P20, such as the contribution to +updraft kinetic energy from the kinetic energy an up- +draft extracts from its inflow via pressure gradient +accelerations. +The new ECAPE formulas are evaluated with output from +four past numerical modeling studies that included 141 +simulations (Section 4). +The formulas and their con- +stituent terms, along with recommended parameter values, +are summarized in the discussion and conclusions (Section +5). +2. Derivation of analytic ECAPE formula +The derivation relies on three underlying concepts: a +scaling between entrainment and updraft radius (section +2a), an analytic relationship between ECAPE and entrain- +ment (section 2b), and an analytic relationship between +updraft radius and state variables within an atmospheric +sounding (sections 2c-d). Combining these components +allows us to eliminate entrainment and updraft radius to +express ECAPE as a function of the state variables within +a sounding. +We will need to make numerous approximations through +the course of the derivation. To evaluate the accuracy of +these approximations, we will first establish a benchmark +calculation of both buoyancy and ECAPE computed with +as few approximations as possible. This benchmark cal- +culation uses the adiabatic unsaturated and saturated lapse +rate equations derived in Peters et al. (2022c), eqs. 19 and +24 from that article respectively, with a mixed-phase layer +in the parcel temperature range of 273.5 K to 233.15 K +(see that study for details on the mixed-phase calculation), +and the bulk plume entrainment approximation for the mix- +ing of individual state variables with that of a horizontally +invariant background environment (see eq. 36-38 in that +study). +The formulas are evaluated using the severe weather +proximity sounding dataset of Thompson et al. (2003). +This dataset includes 1028 atmospheric profiles taken near +severe weather events that ranged from disorganized deep +convection to tornadic supercells. In each profile, the par- +cel with the largest undiluted CAPE in lowest 5 km of +the atmosphere is lifted to calculate buoyancy, CAPE, and +ECAPE. +a. Connecting fractional entrainment to updraft radius +Our first step is to establish a relationship between up- +draft radius and the fractional entrainment rate 𝜀. We ac- +complish this by deriving an expression for passive tracer +dilution in the cloud core assuming that entrained air has +a tracer value of zero, and assuming that detrained air has +a tracer value equal to that locally in the cloud core. Here +𝜀 is the fractional entrainment rate needed to produce a +vertical profile of cloud core passive tracer consistent with +the dilution it undergoes. +The derivation closely follows that of Morrison (2017) +(hereafter M17), section 2a therein. We first consider a +passive tracer 𝐶, whose mixing ratio (in kg kg-1) is 1 in +a cloud’s effective inflow layer (i.e., the layer of nonzero +CAPE Thompson et al. 2007; Nowotarski et al. 2020), and +0 above this layer. Conceptually, the passive tracer value +represents the degree to which a parcel has been diluted +via entrainment, with 𝐶 ≈ 1 indicating undiluted air, and +𝐶 << 1 indicating highly diluted air. +The anelastic Lagrangian tendency equation for 𝐶 may +be written in cylindrical coordinates as: +𝑑𝐶 +𝑑𝑡 = 𝜕𝐶 +𝜕𝑡 + 1 +𝑟 +𝜕𝑟𝑢𝐶 +𝜕𝑟 ++ 1 +𝑟 +𝜕𝑣𝐶 +𝜕𝜙 + 1 +𝜌0 +𝜕𝜌0𝑤𝐶 +𝜕𝑧 += 0, +(1) +where 𝑟, 𝜙, and 𝑧 are the radial, azimuthal, and vertical +coordinates, 𝑢, 𝑣, and 𝑤 are the corresponding radial, az- +imuthal, and vertical velocities, and 𝜌0(𝑧) is a reference +density profile. Azimuthally averaging this equation, and +then Reynolds averaging, yields: +𝑑𝐶 +𝑑𝑡 = −1 +𝑟 +𝜕𝑟𝑢′𝐶′ +𝜕𝑟 +− 1 +𝜌0 +𝜕𝜌0𝑤′𝐶′ +𝜕𝑧 +(2) +where overbar denotes a spatial average with a filter scale +similar to that of the updraft width (tyically on the order of +1-2 km), primes denote deviations smaller than the filter +scale, and 𝑑𝐶 +𝑑𝑡 = 𝜕𝐶 +𝜕𝑡 + 𝑢 𝜕𝐶 +𝜕𝑟 + 𝑣 𝜕𝐶 +𝜕𝜙 + 𝑤 𝜕𝐶 +𝜕𝑧 . Physically, the +overbar terms correspond to updraft-scale flow patterns, + +4 +whereas the ′ terms correspond to turbulent fluxes. We +neglect the vertical turbulent flux term since recent large +eddy simulations have supported a dominant role of lateral +mixing in entrainment (Böing et al. 2014). All quantities +are valid at the updraft horizontal center unless explicitly +stated otherwise. +Following M17 and De Rooy and Siebesma (2010), we +assume that 𝑢′𝐶′ varies linearly over a turbulent mixing +length scale 𝐿𝑚𝑖𝑥 and vanishes at the updraft center, such +that 𝑢′𝐶′(𝑟) = 𝑢′𝐶′ +��� +𝐿𝑚𝑖𝑥 +� +𝑟 +𝐿𝑚𝑖𝑥 +� +, where the 𝑢′𝐶′ +��� +𝐿𝑚𝑖𝑥 de- +notes the value of 𝑢′𝐶′ at distance 𝐿𝑚𝑖𝑥 from the updraft +center. Finally, we use the chain rule to write +𝑑 +𝑑𝑡 = 𝑤 𝑑 +𝑑𝑧 , +where +𝑑 +𝑑𝑧 is the rate of change of a quantity as the parcel +changes height. Making these approximations allows us to +write eq. 2 as: +𝑑𝐶 +𝑑𝑧 = −2 +𝑢′𝐶′ +��� +𝐿𝑚𝑖𝑥 +𝑤𝐿𝑚𝑖𝑥 +. +(3) +In the eddy diffusivity approximation (e.g., Kuo 1962), we +assume that turbulent fluxes act to diffuse a quantity down- +gradient. Using this approach, we may write 𝑢′𝐶′ +��� +𝐿𝑚𝑖𝑥 ≈ +− +𝑘2𝐿2 +𝑚𝑖𝑥 +𝑃𝑟 +�� 𝜕𝑤 +𝜕𝑟 +�� 𝜕𝐶 +𝜕𝑟 (eqs. 5-6 in M17) and eq. 3 as: +𝑑𝐶 +𝑑𝑧 = 2 𝑘2𝐿𝑚𝑖𝑥 +𝑤𝑃𝑟 +���� +𝜕𝑤 +𝜕𝑟 +���� +𝜕𝐶 +𝜕𝑟 , +(4) +where 𝑘2 is the von Karman constant and 𝑃𝑟 is the turbulent +Prandtl number. Finally, we use linear approximations to +the lateral gradients in 𝐶 and 𝑤, such that 𝜕𝐶 +𝜕𝑟 = 𝐶0−𝐶 +𝑅 +and +�� 𝜕𝑤 +𝜕𝑟 +�� = |𝑤0−𝑤 | +𝑅 +, and assume that 𝑤0 = 0 and 𝐶0 = 0 to write: +𝑑𝐶 +𝑑𝑧 = −𝜀𝐶, +(5) +where +𝜀 = 2𝑘2𝐿𝑚𝑖𝑥 +𝑃𝑟 𝑅2 +. +(6) +Equation 5 takes the form of a classical steady-state plume +equation (Squires and Turner 1962; Betts 1975), where 𝜀 is +the fractional entrainment inverse length scale. This term +represents the rate at which 𝐶 is diluted with height by +entrainment. There is some debate in past literature over +how 𝐿𝑚𝑖𝑥 should be interpreted. For instance, in Morrison +et al. (2020), P20, and Peters et al. (2020b), we simply +set 𝐿𝑚𝑖𝑥 ∼ 𝑅, which from Equation 6 results in a 𝜀 ∼ 𝑅−1 +scaling. However, analysis of large eddy simulations (LES) +in our more recent work (e.g., Mulholland et al. 2021b; +Morrison et al. 2022) indicates that 𝜀 ∼ 𝑅−2, suggesting +from Equation 6 that 𝐿𝑚𝑖𝑥 should be viewed as a constant. +Hence, we set 𝐿𝑚𝑖𝑥 to a fixed value following Morrison +et al. (2022). +The eddy diffusivity approximation for lateral mixing +implicitly neglects the entrainment of air occurring within +organized updraft-scale flow, which is known as dynamic +entrainment (e.g., De Rooy et al. 2013). However, our past +work has shown that dynamic entrainment primarily affects +updraft properties below the height of maximum 𝑤 where +flow is laterally convergent into the updraft (e.g., Morrison +2017; Morrison et al. 2020, 2022). Hence, it is reasonable +to neglect dynamic entrainment in our present objective +of deriving an expression for ECAPE, which pertains to +the maximum kinetic energy achieved by the updraft that +coincides with the position of maximum 𝑤. +b. Derivation of analytic expressions for the buoyancy and +𝐸𝐶𝐴𝑃𝐸 of an entraining parcel +Our next step is to express ECAPE as an analytic func- +tion of 𝜀, wherein 𝜀 is not contained within integrals or +differentials. +We begin with the first law of thermody- +namics for a rising parcel, which may be written as (e.g., +Emanuel 1994; Romps 2015; Peters et al. 2022c): +𝑐 𝑝𝑚 +𝑑𝑇 +𝑑𝑧 − 1 +𝜌 +𝑑𝑝 +𝑑𝑧 + 𝐿𝑣 +𝑑𝑞𝑣 +𝑑𝑧 − 𝐿𝑖 +𝑑𝑞𝑖 +𝑑𝑧 = 𝑄 +(7) +where 𝑐 𝑝𝑚 is the moist heat capacity that depends on water +vapor and condensates, 𝑇 is temperature, 𝜌 is density, 𝑝 +is pressure, 𝐿𝑣 is the temperature dependent latent heat +of vaporization, 𝑞𝑣 is the water vapor mass fraction, 𝐿𝑖 is +the temperature dependent latent heat of freezing, 𝑞𝑖 is the +ice mass fraction, 𝑄 represents all diabatic effects, and +𝑑 +𝑑𝑧 +represents the rate at which a quantity changes as a parcel +changes its vertical position. +We simplify this equation by making a series of approx- +imations. First, we replace the moist heat capacity 𝑐 𝑝𝑚 +with the constant dry-air heat capacity 𝑐 𝑝𝑑. Second, we +use the hydrostatic equation to write 1 +𝜌 +𝑑𝑝 +𝑑𝑧 = −𝑔, where 𝑔 is +the acceleration of gravity. Third, we neglect ice (𝑞𝑖 = 0). +Fourth, we replace the temperature-dependent latent heat +of vaporization with its reference value at the triple point +temperature 𝐿𝑣,𝑟. Fifth, we assume that the only diabatic +effect is the mixing of a parcel with its far-field environmen- +tal profile. Using these approximations, we may re-write +eq. 7 as: +𝑑ℎ +𝑑𝑧 = −𝜀 (ℎ − ℎ0) , +(8) +where ℎ is the moist static energy, defined as +ℎ = 𝑐 𝑝𝑑𝑇 + 𝐿𝑣,𝑟𝑞 +𝑔𝑧, +(9) +ℎ0 is the moist static energy of the background environ- +ment, defined as: +ℎ0 = 𝑐 𝑝𝑑𝑇0 + 𝐿𝑣,𝑟𝑞0 +𝑔𝑧, +(10) + +5 +the subscripts 0 denote the height-dependent background +environmental profile, and we have dropped the 𝑣 subscript +on 𝑞 for simplicity. The −𝜀 (ℎ − ℎ0) term represents dilu- +tion of ℎ with height due to entrainment, and is expressed in +a manner consistent with a classical plume updraft model +(e.g., Betts 1975). Note that for an adiabatic parcel (i.e., +𝜀 → 0), ℎ is conserved. Hence, ℎ is analogous to equivalent +potential temperature (𝜃𝑒). It will also be useful later to +define the saturated moist static energy of the environment +ℎ∗ +0 as: +ℎ∗ +0 = 𝑐 𝑝𝑑𝑇0 + 𝐿𝑣,𝑟𝑞∗ +0 +𝑔𝑧, +(11) +where 𝑞∗ is the saturation mass fraction defined via eq. 10 +in Bolton (1980). Finally, we define the buoyancy 𝐵 of an +updraft air parcel as: +𝐵 = 𝑔𝑇 −𝑇0 +𝑇0 +, +(12) +which neglects the effects of water vapor and condensate +loading on buoyancy. +To evaluate the accuracy of these approximate equa- +tions, we integrate eq. 8 upward using a forward Euler +integration scheme with a vertical grid spacing of 100 m, +and solve for 𝑇 at each height using a numerical nonlinear +equation solver. We use 𝑑𝑞 +𝑑𝑧 = −𝜀 (𝑞 − 𝑞0) during the un- +saturated part of parcel ascent, and set 𝑞 = 𝑞∗ during the +saturated part of parcel ascent. Quantities such as buoy- +ancy and ECAPE computed with 8 and eq. 12 are referred +to as “approximate”. The vertical distributions of ℎ0 and +ℎ∗ +0 in a typical deep convective environment are shown +in Fig. +1a. +Much like the typical vertical distribution +of 𝜃𝑒, ℎ has a local maximum in the lower troposphere +when nonzero CAPE is present, a local minimum in the +middle troposphere, and becomes large again in the lower +stratosphere. An undiluted parcel lifted from the surface +has larger ℎ than its surroundings until it reaches the lower +stratosphere. In an entraining parcel, ℎ gradually relaxes to +that of the background environment as the parcel ascends. +Profiles of approximate buoyancy are compared to bench- +mark buoyancy, calculated from equations in Peters et al. +(2022c) as described earlier in this section, for undiluted +and diluted parcels in Fig. 1b. Despite the assumptions +made thus far, the approximate and benchmark buoyancy +profiles are comparable, having similar profile shapes and +magnitudes at all heights. +Combining eqs. 9, 10, and eq. 11 yields: +𝐵 = +𝑔 +𝑐 𝑝𝑑𝑇0 +�ℎ − ℎ∗ +0 +� − 𝑔𝐿𝑣,𝑟 +𝑐 𝑝𝑑𝑇0 +�𝑞∗ − 𝑞∗ +0 +� , +(13) +where we have assumed that the updraft parcel is saturated, +such that 𝑞 = 𝑞∗. The second term on the RHS of eq. 13 +is often small relative to the first. Hence, eq. 13 suggests +that 𝐵 > 0 when ℎ > ℎ∗ +0. +This agrees with Fig. +1a-b, +which shows approximate coincidence between the vertical +extent of ℎ > ℎ∗ +0 (Fig. 1a) and the vertical extent of 𝐵 > 0 +(Fig. 1b). An entrainment term (i.e., 𝜀) does not show up +explicitly in eq. 13, but is included implicitly via the moist +static energy of the updraft parcel ℎ, which is affected by +entrainment. To make 𝜀 show up explicitly, we find the +particular solution to eq. 8 with ℎ = ℎ0 at 𝑧 = 0, which may +be written as: +ℎ = 𝑒−𝜀𝑧 +� +ℎ𝑢𝑑 + +∫ +𝜉=𝑧 +𝜉=0 +𝜀𝑒𝜀 𝜉 ℎ0𝑑𝜉 +� +, +(14) +where ℎ𝑢𝑑 is the moist static energy of an undiluted parcel +(or equivalently the moist static energy of the entraining +parcel at its origin height since we assume ℎ is conserved +for undilute ascent), 𝜉 is a dummy variable of integration, +and we defined the parcel starting height as 𝑧 = 0 for sim- +plicity. Combining eq. 14 with eq. 13 yields the following: +𝐵 = +𝑔 +𝑐 𝑝𝑑𝑇0 +� +𝑒−𝜀𝑧 +� +ℎ𝑢𝑑 + +∫ +𝜉=𝑧 +𝜉=0 +𝜀𝑒𝜀 𝜉 ℎ0𝑑𝜉 +� +− ℎ∗ +0 +� +− 𝑔𝐿𝑣,𝑟 +𝑐 𝑝𝑑𝑇0 +�𝑞∗ − 𝑞∗ +0 +� . +(15) +The term 𝜀 now shows up explicitly in the equation, but +is contained within integrals. We will need to make some +additional approximations to bring this term out of the +integrals to obtain our desired analytic solution. +Eq. 15 can be re-arranged to express 𝐵 as a modification +to the undiluted buoyancy 𝐵𝑢𝑑 using eq. 13 evaluated with +ℎ = ℎ𝑢𝑑 and 𝑞 = 𝑞𝑢𝑑: +𝐵 = 𝐵𝑢𝑑𝑒−𝜀𝑧 + +𝑔 +𝑐 𝑝𝑑𝑇0 +� +𝑒−𝜀𝑧 +∫ +𝜉=𝑧 +𝜉=0 +𝜀𝑒𝜀 𝜉 ℎ0𝑑𝜉 − (1− 𝑒−𝜀𝑧) ℎ∗ +0 +� +− 𝑔𝐿𝑣,𝑟 +𝑐 𝑝𝑑𝑇0 +�𝑞∗ − 𝑞∗ +0 +� + 𝑒−𝜀𝑧 𝑔𝐿𝑣,𝑟 +𝑐 𝑝𝑑𝑇0 +�𝑞∗ +𝑢𝑑 − 𝑞∗ +0 +� . +(16) +This re-arrangement provides us with the opportunity to +use the the undiluted buoyancy computed with the bench- +mark parcel to calculate 𝐵𝑢𝑑 rather than the approximate +𝐵𝑢𝑑 when evaluating eq. 16 (i.e., the black line in Fig. 1 in- +stead of the red line). This substitution generally improves +the accuracy of the formula, and is used in all subsequent +calculations. +We note that the two terms on the RHS of eq. 16 will +cancel each other in the limit of 𝜀 → 0. In the opposite limit +of 𝜀 → ∞, each of these terms individual vanish because +𝑞∗ → 𝑞∗ +0 and 𝑒−𝜀𝑧 → 0. We assume these terms are small +in the intermediary range of 𝜀, and consequently neglect +them to simplify the equation. Using integration by parts +and neglecting the aforementioned terms, we may re-write +eq. 16 as: +𝐵 = 𝐵𝑢𝑑𝑒−𝜀𝑧 + +𝑔 +𝑐 𝑝𝑑𝑇0 +� +𝜀𝑧 �ℎ0 + 𝑒−𝜀𝑧𝜀2 +∫ +𝜉=𝑧 +𝜉=0 +�ℎ0𝜉𝑒𝜀 𝜉 𝑑𝜉 − (1− 𝑒−𝜀𝑧) ℎ∗ +0 +� +. +(17) + +6 +Fig. 1. Panel a: profiles of environmental ℎ0, ℎ∗ +0, and ℎ of an undiluted parcel, and the ℎ of a diluted parcel with 𝜀 = 1×10−4 m-1 (“h dil.”), +computed using the tornadic supercell composite profile from Parker (2014). Moist static energies have been divided by 𝑐𝑝𝑑 to yield “energy +temperature" with units of K. Panel b: buoyancy of the diluted (dashed lines) and undiluted (solid lines) parcels, computed using the benchmark +parcel (black, described in the beginning of this section) and from the approximate formula for ℎ calculated by numerically integrating eq.8 as +described in the text (red). +where �ℎ0(𝜉) ≡ 1 +𝜉 +∫ 𝜉 ∗=𝜉 +𝜉 ∗=0 ℎ0𝑑𝜉∗ is the average of ℎ0 below +height 𝜉 and �ℎ0 in the first term in the parentheses on the +RHS is evaluated at 𝜉 = 𝑧. If we assume that �ℎ0 is approxi- +mately constant with height3 in the integral term in eq. 17, +the equation simplifies dramatically to the following: +𝐵 = 𝐵𝑢𝑑𝑒−𝜀𝑧 + +𝑔 +𝑐 𝑝𝑑𝑇0 +(1− 𝑒−𝜀𝑧) +� +�ℎ0 − ℎ∗ +0 +� +. +(18) +This equation is an analytic function of 𝐵𝑢𝑑, 𝜀, and the +state variables within a sounding. The first term on the +RHS represents the direct dilution of the updraft’s temper- +ature perturbation via entrained air with no temperature +perturbation, whereas the second term encapsulates the +reduced condensation rate resulting from the entrainment +of unsaturated air by the updraft, relative to an undiluted +parcel. +Before moving on to an analytic formula for ECAPE, we +evaluate the accuracy of this analytic buoyancy formula by +comparing the average buoyancy 𝐵 between the level of +free convection (LFC) and the level of neutral buoyancy +(LNB) to that of the benchmark buoyancy profile and the +formula from P20 (eqs. 4-5 therein4). Here, the LFC is +the highest instance of zero buoyancy below the height of +maximum buoyancy, and the LNB is the highest instance +of zero buoyancy in the profile. We define three metrics +for evaluation: Pearson correlation coefficient 𝐶𝐶 among +soundings of 𝐵 from eq. 18 with 𝐵 from the more accurate +benchmark lapse rate formula; the fractional reduction in +undiluted 𝐵 by entrainment; and normalized root-mean- +3This assumption is reasonable, given that vertical variations in � +ℎ0 +are on the order of 1×104 J kg-1, whereas the typical magnitude of this +quantity is on the order of 1×106 J kg-1. +4We also use the 𝐵𝑢𝑑 computed with the benchmark parcel in the +P20 formula to maximize this formula’s accuracy. +square-error (NRMSE), defined as the the average over all +soundings of the squared difference between 𝐵 from eq. 18 +and 𝐵 from the benchmark lapse rate formula, divided by +the magnitude of 𝐵 from the benchmark formula. These +metrics are plotted as a function of 𝜀 and updraft radius 𝑅 +on the 𝑥 axis. We relate 𝑅 to 𝜀 using eq. 6, with 𝑘2 = 0.18, +𝑃𝑟 = 1 +3, and 𝐿𝑚𝑖𝑥 = 120 m following Morrison et al. (2022). +The 𝐶𝐶 of the new formula with the benchmark calcu- +lation is very close to 1 (Fig. 2a) for all 𝑅 > 750 m and for +fractional reductions in CAPE of < 0.9 (i.e., updrafts that +realize 10 % or more of their CAPE; Fig. 2c), which is +the range of fractional reductions expected in midlatitude +deep convection (e.g., Peters et al. 2020c; Lasher-Trapp +et al. 2021). For 𝑅 less than 750 m and when fractional re- +ductions approach 1, 𝐶𝐶 begins to drop, suggesting that the +formula is less accurate for strongly entraining weak con- +vection. The story is similar for NRMSE (Fig. 2e), which +is relatively small in magnitude (i.e. < 0.1) for 𝑅 > 750 +m, but increases when 𝑅 falls below 750 m. Compared to +the P20 formula, the new formula derived here has smaller +NRMSE Fig. 2e) and larger 𝐶𝐶 Fig. 2a), indicating that +we have made an improvement in accuracy in the present +derivation. This improvement over the P20 formula is pri- +marily due to an over-estimation of the fractional reduction +in buoyancy via entrainment in the P20 formula that does +not occur in the one derived here (Fig. 2c). This difference +is particularly noticeable when we restrict our analysis to +soundings with less than 1000 J kg−1 of undiluted CAPE +(Fig. 2b,d,f). In this low CAPE regime, the NRMSE (Fig. +2f) and 𝐶𝐶 (Fig. 2b) of the new formula are comparable +to the errors for the whole sounding data set, whereas the +P20 formula performs considerably worse with respect to +both 𝐶𝐶 and errors in the low CAPE regime. + +7 +Fig. 2. Comparison of vertically-averaged buoyancy 𝐵 calculated using the formula from the present study (eq. 18, red), the P20 buoyancy +formula (gray), and the benchmark parcel (black). Panels a,b show 𝐶𝐶, c,d the fractional reduction in 𝐵, and e,f the normalized error NRMSE. +𝐶𝐶 and NRMSE are calculated relative to the benchmark parcel. Left panels show results from all Thompson et al. (2003) soundings, and right +panels show results from only soundings with < 1000 J kg-1 undiluted CAPE. +Our next task is to use eq. 18 to obtain an expression for +ECAPE. We formally define ECAPE as: +ECAPE = +∫ +𝑧=𝐿𝑁 𝐵 +𝑧=𝐿𝐹𝐶 +𝐵𝑑𝑧. +(19) +Vertically integrating eq. 18 from the LFC to the LNB and +combining with eq. 19 yields: +ECAPE = +∫ +𝑧=𝐿𝑁 𝐵 +𝑧=𝐿𝐹𝐶 +𝐵𝑢𝑑𝑒−𝜀𝑧𝑑𝑧+ +∫ +𝑧=𝐿𝑁 𝐵 +𝑧=𝐿𝐹𝐶 +𝑔 +𝑐 𝑝𝑑𝑇0 +(1− 𝑒−𝜀𝑧) +� +�ℎ0 − ℎ∗ +0 +� +𝑑𝑧. +(20) +It will become advantageous later to have the integral +bounds on the RHS of eq. 20 extend to the equilibrium +level for an undiluted parcel5 𝐻, rather than to the LNB. +We note that the integral of the first term from the LNB to +the 𝐻 will always be positive, since 𝐵𝑢𝑑 is positive below +the 𝐻 by definition. On the other hand, the integral of +the second term over this range is typically negative (as +will be discussed shortly), and at least partially cancels the +contribution of the integral of the first term over this range. +Hence, we extend the upper bounds of these integrals to +the 𝐻, assuming that the partial cancellation between the +terms mitigates the resulting errors. +5The equilibrium level is typically denoted with the acronym EL. We +instead use the symbol 𝐻 for compactness in equations. +To pull 𝜀 out of the integrals in eq. 20, we use integration +by parts and these integral definitions to write the first term +on the RHS of eq. 20 as: +∫ +𝑧=𝐻 +𝑧=𝐿𝐹𝐶 +𝐵𝑢𝑑𝑒−𝜀𝑧𝑑𝑧 = 𝑒−𝜀𝐻CAPE+𝜀 +∫ +𝑧=𝐻 +𝑧=𝐿𝐹𝐶 +𝑒−𝜀𝑧𝐵𝑢𝑑𝑑𝑧 +(21) +where +CAPE = +∫ +𝑧=𝐻 +𝑧=𝐿𝐹𝐶 +𝐵𝑢𝑑𝑑𝑧. +(22) +We make the approximation that 𝐵𝑢𝑑 is linear with height +on the RHS of eq. 21: +𝐵𝑢𝑑 ≈ � +𝐵𝑢𝑑 (𝑧 − 𝐿𝐹𝐶) , +(23) +where � +𝐵𝑢𝑑 is the average undilute 𝐵 between the 𝐿𝐹𝐶 +and 𝐻. We then vertically integrate eq. 21, assume that +𝐿𝐹𝐶 << 𝐻 and hence 𝐻 − 𝐿𝐹𝐶 ≈ 𝐻, and neglect entrain- +ment below the LFC such that 𝑒−𝜀𝐿𝐹𝐶 ≈ 1. +We apply +analogous assumptions to the 2nd term on the RHS of eq. +20. Modifying eq. 20 with these assumptions yields: +ECAPE = +�1− 𝑒−𝜀𝐻 +𝜀𝐻 +� +CAPE− +� +1− 1− 𝑒−𝜀𝐻 +𝜀𝐻 +� +NCAPE +(24) + +8 +where +NCAPE = − +∫ +𝑧=𝐻 +𝑧=𝐿𝐹𝐶 +𝑔 +𝑐 𝑝𝑑𝑇0 +� +�ℎ0 − ℎ∗ +0 +� +𝑑𝑧. +(25) +NCAPE represents the buoyancy dilution potential of the +free troposphere: the potential buoyancy loss that could +be induced by entrainment mixing due principally to the +saturation deficit of the environment. It is a purely environ- +mental quantity that does not depend on parcel properties. +As defined here with �ℎ0, it specifically measures the energy +difference between the saturation MSE at a given level and +the mean MSE of the free troposphere below it. The latter +captures the environment through which a parcel would +have to rise, and potentially mix with, prior to reaching a +particular level. Because ℎ∗ +0 is comparable to or larger than +�ℎ0 (Fig. 3a), NCAPE is typically (but not always) positive +(Fig. 3b). The difference term in the integral �ℎ0 − ℎ∗ +0 (Fig. +3a) and hence the magnitude of NCAPE (Fig. 3b) will +be larger when the free troposphere is dry and �ℎ0 is far +smaller than ℎ∗ +0, compared to when the free troposphere is +moist and �ℎ0 is closer in magnitude to ℎ∗ +0. A warm free tro- +posphere at a given RH generally increases the difference +between ℎ∗ +0 and �ℎ0 (Fig. 3c) compared to a situation when +the free troposphere is cool at the same RH. For a fixed +RH, this makes NCAPE larger when the free troposphere is +warm, relative to when it is cool (Fig. 3d). Hence, NCAPE +generally encapsulates the effects of tropospheric dryness +and temperature on buoyancy via entrainment. +Eq. 24 achieves the stated purpose of this derivation, +since 𝜀 is now outside of the integral terms. It will become +advantageous in the next sub-section to further simplify the +exponential terms in eq. 24. One may consider making +first order Taylor series approximations for the exponential +terms. For instance 1−𝑒−𝜀𝐻 +𝜀𝐻 +≈ 1 − 𝜀𝐻. However, the ex- +ponential functions in eq. 24 are strongly nonlinear with +respect to 𝜀𝐻 in the range of 0 < 𝜀𝐻 < 10, which is the +typical range we would encounter in our analysis, mak- +ing the first order Taylor series approximation inaccurate +(compare the blue and black lines in Fig. 4a). Instead, we +invert the exponential term 1−𝑒−𝜀𝐻 +𝜀𝐻 +, approximate its inverse +with a first order Taylor series, and then invert the result. +For instance: +𝜀𝐻 +1− 𝑒𝜀𝐻 ≈ 1+ 𝜀𝐻 +2 . +(26) +and consequently: +1− 𝑒𝜀𝐻 +𝜀𝐻 +≈ +1 +1+ 𝜀𝐻 +2 +. +(27) +This approximation is far more accurate (compare the red +and black lines in Fig. 4a). Substituting these approxima- +tions into eq. 24 and re-arranging yields: +ECAPE = CAPE− 𝜀𝐻 +2 NCAPE +1+ 𝜀𝐻 +2 +. +(28) +As a sanity check, examine the behavior of eq. 28 under +limiting scenarios. For instance, in the limit of no entrain- +ment where 𝜀 → 0, ECAPE → CAPE, which makes sense +given that ECAPE for an undiluted parcel intuitively con- +verges to the CAPE. In the converse limit of 𝜀 → ∞, we may +use L’Hôpital’s rule to deduce that ECAPE → NCAPE, +which is inconsistent with the definition of CAPE as a +quantity greater than or equal to zero. However, this situa- +tion is easily remedied by simply setting ECAPE to 0 if eq. +28. Finally, the case NCAPE = 0 yields ECAPE = 𝐶 𝐴𝑃𝐸 +1+ 𝜀𝐻 +2 , +indicating that ECAPE is still smaller than CAPE when +𝜀 ≠ 0 and hence dilution still reduces buoyancy in this +situation. Indeed, for a saturated parcel to be positively +buoyant in the first place requires ℎ > ℎ∗ +0 (Eq. 13), and +since ℎ∗ +0 ≥ ℎ0 by definition, then ℎ > ℎ0 and entrainment +will dilute ℎ (e.g. Eq. 8; and by extension, 𝐵). One spe- +cific example of this situation is an adiabatic atmosphere +(dry or saturated; constant ℎ0), in which a parcel must be +warmed in order to become positively buoyant and have +non-zero CAPE, but in doing so the parcel will also have +higher energy than the environment at all levels through +which it rises. +The analytic formula for ECAPE in eq. 28 loses a bit +of accuracy relative to the numerically integrated analytic +buoyancy equation at larger values of 𝜀 (i.e., smaller up- +draft radii; Fig. 4b-d), but remains more accurate than the +formula for maximum updraft vertical velocity 𝑤𝑚𝑎𝑥 from +P20 (Eq. 18 therein), which is converted to ECAPE via +𝑤2 +𝑚𝑎𝑥 +2 +. These errors stem from a slight underestimation of +the fractional reduction in undiluted CAPE at large 𝜀 val- +ues (Fig. 4c) that results from our changing of the integral +bounds in eq. 20 from the LNB to 𝐻. Despite these errors, +this formula is quite accurate over the range of 𝑅 and 𝜀 that +typify deep moist convection (i.e., fractional reductions of +no greater than 0.8, Fig. 4c). +c. Relating fractional entrainment to environmental vari- +ables +It will be convenient later in the derivation to manipulate +a nondimensional form of eq. 28. We define the nondimen- +sional ECAPE as �𝐸 ≡ ECAPE +CAPE , the nondimensional NCAPE +as �𝑁 ≡ NCAPE +CAPE , and the nondimensional fractional entrain- +ment rate �𝜀 ≡ 𝜀𝐻. Using these definitions, we re-write eq. +28 as: +�𝐸 = 1− �𝜀 +2 �𝑁 +1+ �𝜀 +2 +. +(29) +Our next task is to eliminate �𝜀 from eq. 28 by expressing +this term as function of other updraft and environmental + +9 +Fig. 3. Demonstrations of the sensitivities of NCAPE to relative humidity (RH) and free tropospheric temperature. Panel a: profiles of ℎ∗ +0 (red, +divided by 𝑐𝑝𝑑 to yield units of K), and � +ℎ0 (blue, K) for the baseline sounding (solid), RH increased by 20 % (dashed blue), and RH decreased by +20 % (dotted blue). Panel b: profiles of NCAPE (J kg-1) corresponding to panel a. Panels c-d: analogous to panels a-b, but showing differences in +ℎ∗ +0 and � +ℎ0 resulting from an increase in 𝑇 by 2 K with RH held constant (dashed), and a decrease in 𝑇 of 2 K with RH held constant (dotted). +attributes. We proceed by defining �𝑅 ≡ 𝑅 +𝐻 and use eq. 6 to +write: +�𝜀 = 𝜖 �𝑅−2, +(30) +where +𝜖 = 2𝑘2𝐿𝑚𝑖𝑥 +𝐻𝑃𝑟 +. +(31) +Combining eq. 30 with eq. 29 yields: +�𝐸 = +1− +𝜖 +2 �𝑅2 �𝑁 +1+ +𝜖 +2 �𝑅2 +. +(32) +Following P20 and Peters et al. (2022a), we may express +�𝑅 as a function of updraft and environmental attributes by +making the following assumptions about updraft geometry +and inflow: +1. Updrafts are cylindrical. +2. Updraft radius 𝑅 is constant with height. Numerous +previous studies show this to be approximately valid +(e.g., Sherwood et al. 2013; Hernandez-Deckers and +Sherwood 2016; Morrison et al. 2021). +3. We assume that all environmental storm-relative wind +V𝑆𝑅 that encounters the cross-sectional area of the +updraft on the upstream side becomes inflow. Past +studies also show this assumption to be reasonable +(e.g., Peters et al. 2019, 2022b). +4. The updraft maximum vertical velocity 𝑤𝑚𝑎𝑥 is pro- +portional to the horizontally averaged vertical velocity +< 𝑤 > at the same height, such that < 𝑤 >= 𝛼𝑤𝑚𝑎𝑥, +where 0 < 𝛼 < 1 (e.g., Morrison 2017; Morrison and +Peters 2018). +5. The updraft maximum vertical velocity is primarily +determined by updraft buoyancy, such that 𝑤𝑚𝑎𝑥 = +√ +2ECAPE. This assumption is supported by (Mor- +rison and Peters 2018; Jeevanjee 2017; Peters et al. +2019, 2020a). +6. The maximum vertical velocity occurs at height 𝐻. + +10 +Fig. 4. Panel a: comparison of the scale factor in eq. 24 (solid black) with its first order Taylor series approximation (blue dashed), and the first +order Taylor series approximation of its inverse (dashed red). Panels b-d: analogous to Fig. 2a,b,c, but evaluating ECAPE from eq. 28 (red, the +present article), ECAPE from P20 (gray), and ECAPE from numerically integrating eq. 18 (black), all relative to the benchmark calculation. +With these assumptions at hand, we start by writing the +anelastic continuity equation in cylindrical coordinates as: +𝜌0 +𝜕𝑟𝑢 +𝜕𝑟 + 𝜌0 +𝜕𝑣 +𝜕𝜙 +𝑟 𝜕𝜌0𝑤 +𝜕𝑧 += 0. +(33) +Azimuthally integrating from 𝜙 = 0 to 𝜙 = 2𝜋, radially +integrating from 𝑟 = 0 to the updraft radius at 𝑟 = 𝑅, and +vertically integrating from the surface to 𝐻 (assuming 𝑤 = +0 at 𝑧 = 0) and dividing by 2𝜋 yields: +𝐻 �𝜌0�𝑢𝑅 + 𝑅 𝜌0,𝐻 < 𝑤𝐻 > +2 += 0. +(34) +where +�𝑢𝑅 = 1 +2𝜋 +∫ 𝑧=𝐻 +𝑧=0 +𝜌0 +∫ 𝜙=2𝜋 +𝜙=0 +𝑢𝑑𝜙𝑑𝑧 +∫ 𝑧=𝐻 +𝑧=0 +𝜌0𝑑𝑧 +(35) +is the density-weighted vertical average of 𝑢 at radius 𝑅, +and between the surface and height 𝐻, and represents the +average inflow speed, +< 𝑤 >= +1 +𝜋𝑅2 +∫ 𝑟=𝑅 +𝑟=0 +∫ +𝜙=2𝜋 +𝜙=0 +𝑟𝑤𝑑𝜙𝑑𝑟 +(36) +is the area average of 𝑤 within radius 𝑅, �𝜌0 is the vertical +average of 𝜌0 between the surface and height 𝐻, and 𝜌0,𝐻 +is 𝜌0 valid at height 𝐻. Making use of < 𝑤 >= 𝛼𝑤𝑚𝑎𝑥 (as- +sumption 4) at height H and 𝑤2 +𝑚𝑎𝑥 +2 += 𝐸𝐶𝐴𝑃𝐸 (assumption +5), and re-arranging eq. 34 yields: +�𝑅 = −2𝜎 +𝛼 +�𝑢𝑅 +√ +2ECAPE +, +(37) +where 𝜎 = +� +𝜌0 +𝜌0,𝐻 > 1. We may relate �𝑢𝑅 to the horizontal +storm-relative wind speed 𝑉𝑆𝑅 = |V𝑆𝑅|, where V𝑆𝑅 is the +storm-relative wind vector, by first defining the upstream +flank of the updraft as the range from 𝜙 = − 𝜋 +2 to 𝜙 = 𝜋 +2 . We +next assume that all inflow is accomplished by the cloud- +relative wind entering the upstream updraft flank, and the +radial component of the environmental cloud-relative wind +at the updraft edge is 𝑢 = −𝑉𝑆𝑅 cos𝜙 and 𝑢 = 0 m s-1 on the +downstream edge. These assumptions allow us to re-write +eq. 35 as: +�𝑢𝑅 = − 1 +2𝜋 +∫ 𝑧=𝐻 +𝑧=0 +∫ 𝜙= 𝜋 +2 +𝜙=− 𝜋 +2 𝜌0𝑉𝑆𝑅 cos𝜙𝑑𝜙𝑑𝑧 +∫ 𝑧=𝐻 +𝑧=0 +𝜌0𝑑𝑧 += +� +𝑉𝑆𝑅 +𝜋 , +(38) +where � +𝑉𝑆𝑅 is the density weighted vertical average of 𝑉𝑆𝑅 +below height 𝐻. In defining �𝑣 ≡ +� +𝑉𝑆𝑅 +√ +2CAPE, combining eqs. +37 and 38 and the definition of 𝜖, and squaring and inverting +the result, we obtain +�𝑅−2 = 𝛼2𝜋2 +4𝜎2 +�𝐸 +�𝑣2 . +(39) + +11 +combining eq. 39 with eq. 32 to eliminate 𝑅 yields: +�𝐸2 𝜓 +�𝑣2 + �𝐸 +� +1+ 𝜓 +�𝑣2 �𝑁 +� +−1 = 0, +(40) +where +𝜓 = 𝑘2𝛼2𝜋2𝐿𝑚𝑖𝑥 +4𝑃𝑟𝜎2𝐻 +. +(41) +Solving for �𝐸 using the quadratic formula gives: +�𝐸 = +−1− 𝜓 +�𝑣2 �𝑁 + +√︂� +1+ 𝜓 +�𝑣2 �𝑁 +�2 ++4 𝜓 +�𝑣2 +2 𝜓 +�𝑣2 +, +(42) +where we have neglected the negative quadratic root that +yields an imaginary solution. Solutions for �𝐸, which rep- +resent the fractional reduction of undiluted CAPE by en- +trainment, are contoured in Fig. +5a as a function of �𝑣 +(non-dimensional storm-relative flow speed) and �𝑁 (non- +dimensional NCAPE). In general, �𝐸 increases from left-to- +right in the figure as �𝑣 becomes large, indicating stronger +storm-relative inflow, wider updrafts, and hence smaller +fractional entrainment. From bottom-to-top on the figure, +�𝐸 decreases as �𝑁 increases. This trend occurs because +larger �𝑁 implies a drier and/or warmer mean free tropo- +sphere, both of which amplify entrainment-driven dilution +relative to situations with a cooler and/or moister free tro- +posphere. +In dimensional form, eq 42 is: +ECAPE = +−1− 2𝜓 +𝑉 2 +𝑆𝑅 NCAPE+ +√︂� +1+ 2𝜓 +𝑉 2 +𝑆𝑅 NCAPE +�2 ++ 8𝜓 +𝑉 2 +𝑆𝑅 CAPE +4 𝜓 +𝑉 2 +𝑆𝑅 +. +(43) +Solutions for ECAPE from eq. 43 as a function of 𝑉𝑆𝑅 +and CAPE are shown in Fig. 5b,c,d for NCAPE=500 J +kg-1, 1000 J kg-1, and 5000 J kg-1 respectively. In general, +curves of ECAPE take on hyperbolic shapes with respect +to the 𝑥 and 𝑦 axes, with contours of ECAPE parallelling +the 𝑥 axis for large 𝑉𝑆𝑅, and the 𝑦 axis for small 𝑉𝑆𝑅 and +large CAPE, and with the largest values coinciding with the +largest 𝑉𝑆𝑅 and undiluted CAPE in the upper-right corners +of the figures. This pattern means that different combi- +nations of 𝑉𝑆𝑅 and undiluted CAPE may result in similar +ECAPE. For instance, an environment with 1000 J kg-1 of +undiluted CAPE, a 𝑉𝑆𝑅 of 30 m s-1, and an NCAPE of +-5000 J kg-1, has an ECAPE of roughly 1000 J kg-1 (Fig. +5d). Mature isolated deep convective updrafts in this en- +vironment will be sufficiently wide, due to their large 𝑉𝑆𝑅, +such that their cores are approximately undiluted and they +realize nearly all of their undiluted CAPE. A contrasting +environment with 6000 J kg-1 of undiluted CAPE and an +NCAPE of -5000 J kg-1, but with a 𝑉𝑆𝑅 of only 5 m s-1 will +have a similar ECAPE of 1000 J kg-1. Despite the large +undiluted CAPE in the second environment, updrafts are +narrow and substantially diluted by entertainment because +of small 𝑉𝑆𝑅. +Consistent with the dependence of �𝐸 on �𝑁 seen in Fig. +5a, the fractional reduction in undiluted CAPE by ECAPE +increases as NCAPE increases, particularly for smaller val- +ues of undiluted CAPE. This is most evident as a movement +to the right of the contours of �𝐸 (black) in Fig. 5b-d as +NCAPE increases, indicating that an updraft with a given +combination of undiluted CAPE and 𝑉𝑆𝑅 will realize less +of its CAPE when NCAPE is large, compared to when +NCAPE is small. +d. Accounting for kinetic energy the storm derives from its +environment +While it is somewhat infrequent, past studies have doc- +umented instances in supercells where the maximum up- +draft 𝑤 exceeds +√ +2CAPE for extended periods of time +(e.g., Fiedler 1994). +Hence, there are factors, such as +vertical pressure gradient accelerations, that can explain +why updrafts are sometimes more intense than buoyancy +alone would suggest. This section introduces a simple ad- +justment factor to the ECAPE formula to represent of how +such pressure effects redirect environmental kinetic energy +into the updraft. To derive this adjustment factor, we must +make the following assumptions: +1. The Lagrangian evolution of kinetic energy following +an air parcel is well described by the Boussinesq ap- +proximation, meaning that 𝜌0 is constant. Past studies +have shown that errors related to an over-estimation +of 𝜌0 aloft in deep convective environments have a +small effect on analytic solutions for vertical velocity, +(e.g., Morrison 2016a,b). +2. Perturbation pressure accelerations in the middle-to- +upper troposphere are neglected. +Pressure pertur- +bations aloft may be large, but they typically oc- +cur within the toroidal circulations of moist thermals +(e.g., Romps and Charn 2015; Morrison and Peters +2018; Peters and Chavas 2021). As parcels ascend +through these thermals, they experience an upward +acceleration below the minimum in 𝑝′, and then a +commensurate downward acceleration above the min- +imum in 𝑝′. Hence, any temporary 𝐾𝐸 gained by the +interaction of a parcel with these pressure perturba- +tions is quickly lost. We therefore neglect pressure +perturbations at the height of maximum 𝑤. +3. Direct dilution of 𝐾𝐸 via entrainment is negligible. +This assumption is also supported by past studies +(e.g., Sherwood et al. 2013). Note that entrainment +will still indirectly affect KE via the entrainment- +driven dilution of updraft buoyancy. + +12 +Fig. 5. Panel a: � +𝐸 (shading) as a function of �𝑣 (𝑥 axis) and � +𝑁 (𝑦 axis), with 𝐻 set to 12,000 m, 𝐿 = 120 m, 𝛼 = 0.8, 𝜎 = 1.131, 𝑘2 = 0.18, and +𝑃𝑟 = 1 +3 . Panels b-d: ECAPE (shading, J kg-1) as a function of 𝑉𝑆𝑅 (𝑥 axis, m s-1) and undiluted CAPE (𝑦 axis, J kg-1), and � +𝐸 (black contours), +with NCAPE = 500 J kg-1 (panel a), NCAPE = 1000 J kg-1 (panel b), and NCAPE = 5000 J kg-1 (panel c). In panels b-d, 𝐻 is determined via +𝐻 = 5808+96.12 +√ +2𝐶 𝐴𝑃𝐸, based on a linear regression between these variables among the soundings. All other parameters are the same as in +panel a. +4. Updrafts are approximately steady, such that +𝜕 +𝜕𝑡 of +quantities are small. +5. The magnitude of convective inhibition (CIN) is neg- +ligable relative to the magnitude of ECAPE. +6. Horizontal storm-relative flow vanishes at the height +of 𝑤𝑚𝑎𝑥. +We may use the first assumption to write eq. +15 in +Peters and Chavas (2021), which describes the Lagrangian +tendency for 𝐾𝐸, as as: +𝑑𝐾𝐸 +𝑑𝑡 += V· ∇ +� 𝑝′ +𝜌0 +� ++ 𝑤𝐵 +(44) +where 𝑝′ is a pressure perturbation. We define 𝐾𝐸 here +in an updraft relative sense, such that 𝐾𝐸 = +𝑢2 +𝐶𝑅+𝑣2 +𝐶𝑅+𝑤2 +2 +, +where 𝑢𝐶𝑅 and 𝑣𝐶𝑅 are the 𝑢 and 𝑣 cloud-relative wind +components. Because of the steady state assumption, we +may substitute +𝑑 +𝑑𝑡 +� +𝑝′ +𝜌0 +� += V · ∇ +� +𝑝′ +𝜌0 +� +. We further use the +chain rule to write +𝑑 +𝑑𝑡 = 𝑤 𝑑 +𝑑𝑧 , where +𝑑 +𝑑𝑧 is the rate of +change of a quantity as a parcel changes height. Making +these assumptions and substitutions, and integrating from +a parcel starting position (defined as 𝑧 = 0) to an ending +position at the height of 𝑤𝑚𝑎𝑥 yields the following form of +the classical Bernoulli equation: +𝐾𝐸𝐿𝑁 𝐵 − 𝐾𝐸0 = +𝑝′ +𝐿𝑁 𝐵 +𝜌 +− +𝑝′ +0 +𝜌 + +∫ +𝑧=𝐿𝑁 𝐵 +𝑧=0 +𝐵𝑑𝑧. +(45) +If a parcel originates within an updraft’s unmodified +background environmental flow then 𝑝′ = 0, 𝑤 = 0, and +𝐾𝐸0 = +𝑉 2 +𝑆𝑅 +2 . We may also neglect +𝑝′ +𝐿𝑁 𝐵 +𝜌 +because of as- +sumption (2) above. Finally, we note that +∫ 𝑧=𝐿𝑁 𝐵 +𝑧=0 +𝐵𝑑𝑧 = +ECAPE + ECIN, where ECIN is the convective inhibi- +tion for an entraining parcel (ECAPE here is defined via +eq. +43). +Combining all these assumptions and substi- +tutions, neglecting ECIN, and assuming that horizontal +storm-relative flow vanishes at the height of 𝑤𝑚𝑎𝑥 gives: +ECAPE𝐴 = 𝑤2 +𝑚𝑎𝑥 +2 += +𝑉2 +𝑆𝑅 +2 ++ECAPE +(46) +where the subscript 𝐴 indicates “adjusted”. According to +this equation, the role of low-level pressure perturbations is +to preserve the incoming cloud-relative horizontal kinetic +energy, deflecting it into the vertical. Further, the maxi- +mum updraft kinetic energy at the height of 𝑤𝑚𝑎𝑥 consists + +13 +of the sum of the kinetic energy gained from the release +of ECAPE and the kinetic energy of the redirected inflow. +Nondimensionalizing by the undiluted CAPE yields: +�𝐸𝐴 = �𝑣2 + �𝐸, +(47) +where �𝐸𝐴 is the nondimensional analogy to ECAPE𝐴. +Recall that in the derivation in the previous sub-section, +we neglected pressure effects and assumed that ECAPE = +𝑤2 +𝑚𝑎𝑥 +2 +when deriving the expression for 𝑅−2 in eq. 39. Now +we must account for the influence of the added contribution +to 𝑤𝑚𝑎𝑥 from velocity from environmental kinetic energy +on updraft radius. Hence, we set ECAPE𝐴 = 𝑤2 +𝑚𝑎𝑥 +2 +, and +adjust eq. 39 using eq. 47 to: +�𝑅−2 = 𝛼2𝜋2 +4𝜎2 +𝑤2 +𝑚𝑎𝑥 +𝑉2 +𝑆𝑅 += 𝛼2𝜋2 +4𝜎2 +� +�𝐸 +�𝑣2 +1 +� +. +(48) +Combining eqs. 47-48 with eq. 32 yields: +�𝐸2 𝜓 +�𝑣2 + �𝐸 +� +1+𝜓 + 𝜓 +�𝑣2 �𝑁 +� +−1+𝜓 �𝑁 = 0, +(49) +Solving �𝐸 using the quadratic formula and then plugging +the result into eq. 47 to solve for �𝐸𝐴 gives: +�𝐸𝐴 =�𝑣2+ +−1−𝜓 − 𝜓 +�𝑣2 �𝑁 + +√︂� +1+𝜓 + 𝜓 +�𝑣2 �𝑁 +�2 ++4 𝜓 +�𝑣2 +� +1−𝜓 �𝑁 +� +2 𝜓 +�𝑣2 +, +(50) +which may be written dimensionally as: +ECAPE𝐴 = +𝑉2 +𝑆𝑅 +2 ++ +−1−𝜓 − 2𝜓 +𝑉 2 +𝑆𝑅 +NCAPE +4 𝜓 +𝑉 2 +𝑆𝑅 ++ +√︄� +1+𝜓 + 2𝜓 +𝑉 2 +𝑆𝑅 +𝑁𝐶𝐴𝑃𝐸 +�2 ++8 𝜓 +𝑉 2 +𝑆𝑅 +(CAPE−𝜓NCAPE) +4 𝜓 +𝑉 2 +𝑆𝑅 +. +(51) +The solution for �𝐸𝐴 from eq. 51 (Fig. 6a) is similar to +that of �𝐸 from eq. 42 at small values of �𝑣, but diverges +notably from �𝐸 at large �𝑣, exceeding 1 (indicating that +ECAPEA surpasses CAPE). Similar behavior is evident in +the solutions for ECAPEA as a function of 𝑉𝑆𝑅 and CAPE +(Fig. 6b-d). Notably, ECAPEA is similar to ECAPE at +smaller values of 𝑉𝑆𝑅, but larger than ECAPEA at large +values of 𝑉𝑆𝑅, which is evident as a persistent downward +slant of ECAPEA as one moves from left-to-right on the fig- +ure. Again, we see that drastically different combinations +of 𝑉𝑆𝑅 and CAPE can yield the same value of ECAPEA. +For instance, an environment with NCAPE of 500 J kg-1, +1000 J kg-1 of CAPE, and a 𝑉𝑆𝑅 of 45 m s-1 will have an +ECAPEA of 2000 J kg-1. A starkly contrasting environ- +ment with NCAPE of 5000 J kg-1, 7000 J kg-1 of CAPE, +and a 𝑉𝑆𝑅 of 7 m s-1 will also have an ECAPEA of 2000 J +kg-1. +To illustrate the circumstances under which pressure ac- +celerations (as they have been formulated here) have the +greatest enhancement effect on updrafts, we examine the +quantity 𝐹 = +√︃ +ECAPE𝐴 +ECAPE − 1, which is equal to the ratio of +the fractional enhancement in 𝑤𝑚𝑎𝑥 due to pressure ac- +celerations. Fractional enhancement is quite small (< 0.1) +for most combinations of 𝑉𝑆𝑅 and CAPE. It only becomes +larger than 0.1 for smaller values of CAPE and/or larger +values of 𝑉𝑆𝑅. Physically, when CAPE is large and/or 𝑉𝑆𝑅 +is small, the kinetic energy generation from buoyancy dom- +inates the updraft kinetic energy budget. Whereas, when +CAPE is small and/or 𝑉𝑆𝑅 is large, the kinetic energy input +from the environmental wind becomes comparable to the +kinetic energy generation from buoyancy. Given this dis- +tribution of 𝐹, a potential explanation for why many past +studies have found that 𝑤𝑚𝑎𝑥 is primarily determined by +buoyancy is that the CAPE and 𝑉𝑆𝑅 in these simulations +fell within the region of the parameter space where 𝐹 is +small. In other words, the kinetic energy input into the +updraft via the background environmental flow is insignif- +icant compared to the kinetic energy generation via the +release of CAPE in most storm environments. +3. Evaluation of the formulas +a. Comparison of predicted 𝑤𝑚𝑎𝑥 with the output from +past simulations +We will compare the formula’s predictions to the vertical +velocities from simulations to evaluate the ECAPE and +ECAPE𝐴 formulas. +The simulations, which featured a +mix of supercells and multicellular clusters, originate from +four past studies: Coffer et al. (2022) (C23, 9 simulations), +Peters et al. (2023) (P23, 32 simulations), Peters et al. +(2020d) (P20, 48 simulations), and Peters et al. (2019) +(54 simulations). +All simulations used Cloud Model 1 +(CM1 Bryan and Fritsch 2002) and were initialized with +soundings that featured a variety of different wind and +thermodynamic profiles. Horizontal grid spacing was 100 +m in P23 and C23, and 250 m in P20, and P19. Vertical +grid spacing was 100 m or less in the troposphere in all +simulations. Additional details of the model configurations +are omitted here to save room, but are available in the +studies referenced in this paragraph. +We computed all subsequent quantities with the initial +model thermodynamic and wind profiles and storm mo- +tions in past simulations. Predictions of 𝑤𝑚𝑎𝑥 were derived +by taking the square root of half of the predicted CAPE +and ECAPE values. We compared the predicted values of +𝑤𝑚𝑎𝑥 to the median 𝑤𝑚𝑎𝑥 during the 1-3 hour time range +in the simulations, excluding tornadic periods in the P23 + +14 +Fig. 6. Same as Fig. 6, but showing � +𝐸𝐴 (panel a), and ECAPE𝐴 (panels b-d). +Fig. 7. 𝐹 (shading, nondimensional) as a function of 𝑉𝑆𝑅 (𝑥 axis, m s-1) and CAPE (𝑦 axis, J kg-1). Colored dots indicate the 𝑉𝑆𝑅 and CAPE +from the simulated storms analyzed in section 4. +and C23 simulations (see those studies for definitions of +“tornadic periods"). The parameter 𝑉𝑆𝑅 was computed by +subtracting the tracked motion vector of simulated updrafts +from the initial model profile, and averaging the resulting +storm-relative wind profile in the 0-1 km layer. Other layer +averages, including 0-500 m, 0-2 km, 0-3 km, and the +density weighted average from the surface to the EL gave +nearly identical results. +We will first see how well +√ +2CAPE, which is the tradi- +tional “thermodynamic speed limit”, predicts 𝑤𝑚𝑎𝑥 (Fig. +8a). +This parameter loosely captures the differences in +𝑤𝑚𝑎𝑥 among groups of simulations, but does not cap- +ture any of the variability in 𝑤𝑚𝑎𝑥 among simulations that + +15 +Fig. 8. All panels: predicted 𝑤𝑚𝑎𝑥 (𝑥 axis, m s-1) versus simulated 𝑤𝑚𝑎𝑥 (𝑦 axis, m s-1). Predictors are: the traditional “thermodynamic +speed limit" +√ +2CAPE (panel a), ECAPE with the fixed 𝜀 that minimized the RMSE (panel b), a multi-linear regression with 𝑉𝑆𝑅 and +√ +2CAPE as +predictors (panel c), ECAPE from P20 (panel d), ECAPE from the present study (panel e), and ECAPEA from the present study (panel f). Bias, +RMSE and 𝑅2 values are shown in the title of each plot. Colors correspond to the study where the simulations originated (see the legend in panel +e). +shared the same CAPE. Most 𝑤𝑚𝑎𝑥 were less than the tra- +ditional thermodynamic speed limit (i.e., below the 1-to-1 +line). However, the bulk of the P23 simulations and a few +of the P19 simulations exceeded this threshold, by up to 15 +m s-1. The 𝑉𝑆𝑅 and CAPE of these simulations puts them +in the portion of the parameter space where our theoretical +representation of pressure effects predicts that their 𝑤𝑚𝑎𝑥 +should exceed +√ +2CAPE (see the gray and red dots in Fig. +7). The coefficient of determination (𝑅2) of +√ +2CAPE with +simulated 𝑤𝑚𝑎𝑥 was 0.38, with a root-mean-square-error +(RMSE) of roughly 15 m s-1. +To see if we can do a better job of predicting 𝑤𝑚𝑎𝑥 with +ECAPE that uses a fixed entrainment rate, we found the +𝜀 that yielded the smallest RMSE between predictions by +eq. 24 and simulated 𝑤𝑚𝑎𝑥 (this value was 𝜀 = 2.25×10−5 +m-1). This prediction reduces the RMSE to 12.2 m s-1, +but does not improve the 𝑅2 much (Fig. 8b). Hence, with +no knowledge of how the variations in environmental wind +profiles affect entrainment, ECAPE with a fixed entrain- +ment rate only slightly improves predictions of the mean +𝑤𝑚𝑎𝑥 among groups of simulations, but does not capture +any of the variance in 𝑤𝑚𝑎𝑥 within a particular group. +We can do a better job of predicting 𝑤𝑚𝑎𝑥 by forming +a mult-linear regression with +√ +2CAPE and 𝑉𝑆𝑅 as predic- +tors, and 𝑤𝑚𝑎𝑥 as a predictand. This regression equation +takes the form 𝑤𝑚𝑎𝑥,𝑝𝑟𝑒𝑑 = 0.7823 +√ +2CAPE+1.503𝑉𝑆𝑅 − +13.3437. The predictions by this formula reduce RMSE +to 7.95 m s-1 and increase the 𝑅2 to 0.7 (Fig. 8c). This +formula also produces an improved subjective correspon- +dence between predicted and simulated 𝑤𝑚𝑎𝑥. +The ECAPE formula from P20, computed using all the +procedures and parameter values described in that study, +also better captures the variability in 𝑤𝑚𝑎𝑥 among simu- +lations with the same CAPE value than the +√ +2CAPE and +ECAPE with a fixed entrainment rate, with a 𝑅2 with 𝑤𝑚𝑎𝑥 +of 0.71. The RMSE of 13 m s-1, however, is inferior to that +of the linear regression and comparable to that of +√ +2CAPE +and ECAPE with a fixed entrainment rate. This large error +stems from a low bias in predictions from this formula, rel- +ative to the values in simulations, which is demonstrated +by the dots mostly falling to the left of the one-to-one line +in Fig. 8b). Recall that P20 used a 𝜀 ∼ 𝑅−1 scaling, and +the buoyancy formula from that study consequently over- +estimated the fractional reduction in undiluted buoyancy by + +16 +entrainment. Both of these factors may have contributed +to the formula’s bias. +To evaluate the ECAPE and ECAPEA derived in the +present study, we set 𝐿𝑚𝑖𝑥 = 120 m when evaluating the +ECAPE formulas derived in the present study against the +P23 and C23 simulations, and 𝐿𝑚𝑖𝑥 = 250 m when evalu- +ating against the P20, N20, and P19 simulations to account +for their coarser grid spacing. All other parameter values +were the same as those used to generate Figs. 5-6. The new +ECAPE formula improves correspondence (𝑅2 = 0.79), +reduces the low bias in prediction, and substantially de- +creases RMSE (8.2 m s-1) relative to the formula from P20 +and the linear regression. Dots in Fig. 8c fall close to the +1-1 line, suggesting that the 𝜀 ∼ 𝑅−2 scaling better reflects +the trends in entrainment-driven dilution in the simulations +than 𝜀 ∼ 𝑅−1. +The ECAPEA formula further improves correspondence +between predicted and simulated 𝑤𝑚𝑎𝑥 (𝑅2 = 0.82), de- +creases RMSE to 6.4 m s-1, and brings points closer to the +1-to-1 line. The most notable difference between ECAPEA +and ECAPE occurs with the P23 simulations, whose 𝑤𝑚𝑎𝑥 +substantially exceeded +√ +2CAPE (red dots above the 1-to-1 +line in Fig. 8a) and was under-predicted by the ECAPE +formulas from both P20 (red dots above the 1-to-1 line in +Fig. 8b) and the present study (red dots above the 1-to-1 +line in Fig. 8c). The ECAPEA brings the red dots much +closer to the 1-to-1 line, correctly reflecting that 𝑤𝑚𝑎𝑥 in +many of these simulations exceeded +√ +2CAPE. +The take home message is that the two formulas derived +in the present study are superior predictors of 𝑤𝑚𝑎𝑥 when +compared to CAPE and ECAPE with a fixed entrainment +rate. +They also perform better than a simple linear re- +gression that includes CAPE and 𝑉𝑆𝑅, suggesting that the +additional information contained in our formula about the +environmental thermodynamic profile via the NCAPE pa- +rameter is critical to accurately representing the effects of +entrainment on 𝑤𝑚𝑎𝑥. Finally, the new ECAPE formulas +correct a low bias in the older P20 formula. +b. Properties of ECAPE in severe weather proximity +soundings +Our final analysis examines the distribution of ECAPE𝐴 +within the Thompson et al. (2003) sounding dataset. Once +again, we use the 0-1 km mean 𝑉𝑆𝑅 computed with the ob- +served storm motion in our formulas, though we evaluate +other definitions of 𝑉𝑆𝑅 later in this sub-section. The dis- +tribution of ECAPE𝐴 for all nonsupercell severe weather +events is plotted against undiluted CAPE in Fig. 9a. Con- +tours of �𝐸𝐴 (the fraction of CAPE “realized") are also +shown for reference. There is substantial variability �𝐸𝐴, +with ECAPE𝐴 ≈ CAPE (�𝐸𝐴 ≈ 1) in some events, and +ECAPE𝐴 << CAPE (�𝐸𝐴 << 1) in others. Furthermore, +case-to-case variations in ECAPE𝐴 and CAPE only loosely +corresponded with one another, with 𝑅2 = 0.46 based on a +linear fit of these two quantities. In most events, particu- +larly those with significant CAPE (> 1000 𝐽/𝑘𝑔), ECAPE𝐴 +was less than CAPE suggesting that most nonsupercell +storms only realize a fraction of their available CAPE. +In contrast with nonsupercell events, there is a much +closer correspondence between ECAPE𝐴 and CAPE in su- +percell events, with 𝑅2 = 0.90 between these two variables +(Fig. 9b). Furthermore, �𝐸𝐴 > 0.5 for nearly every supercell +sounding, and this quantity was close to 1 in many cases, +and exceeded 1 in a handful of instances. This corroborates +the idea, proposed by Peters et al. (2019), that supercells re- +alize a larger percentage of their environmental CAPE than +nonsupercells. The primary reason for this difference is the +larger vertical wind shear, and consequently storm-relative +flow, in supercell environments relative to nonsupercell +environments. Hence, CAPE may be a better predictor of +storm-to-storm variations in updraft intensity in supercells +than it is in nonosupercells. However, there is still substan- +tial variability in the correspondence between ECAPE and +CAPE, particular for larger CAPE values, which suggests +that ECAPE provides added value over CAPE in supercell +environments. +To evaluate the sensitivity of ECAPE to how 𝑉𝑆𝑅 is cal- +culated, we re-computed ECAPE𝐴 with the 0-3 km mean +𝑉𝑆𝑅 with the observed storm motion, the density weighted +average of 𝑉𝑆𝑅 below the LFC with the observed storm +motion, the 0-1 km mean 𝑉𝑆𝑅 computed using the storm +motion estimate of Bunkers et al. (2000) which includes +components of storm motion driven by advection and prop- +agation, and the advective storm motion only, estimated as +half the 0-6 km bulk wind difference. Results with the 𝑉𝑆𝑅 +measures that use the observed storm motion yield nearly +identical results to one another in both nonsupercells (Fig. +9c) and supercells (Fig. 9d), with 𝑅2 ranging from 0.96 to +0.99. +In the case of supercells, the ECAPE𝐴 computed with +the observed storm motion corresponded well with the +ECAPE𝐴 computed using the Bunkers storm motion esti- +mate and half the bulk wind difference (Fig. 9d). However, +this correspondence was degraded slightly in nonsupercell +events, with the 𝑅2 ranging form 0.71 to 0.75 between +ECAPE𝐴 computed with the observed storm-motion, with +that computed using the bunkers estimate and bulk wind +difference. This likely reflects the fact that the motion of +nonsupercell storms is more often influenced by extraneous +factors like outflow and airmass boundaries, than in super- +cells. Hence, sounding-based estimates for storm motion +do not correspond with actual storm motions as well in +nonsupercell events as they do in supercell events. +In many contexts where this formula would be used, +such as in forecasting, the storm motion is unknown and +must be estimated. This analysis suggests that estimating +storm motion with the method of Bunkers et al. (2000) or +half the 0-6 km BWD are both viable choices. + +17 +Fig. 9. Top panels: scatter plots of ECAPE𝐴 (𝑥 axis, J kg-1) versus CAPE (𝑦 axis, J kg-1), computed with the Thompson et al. (2003) soundings. +Panel a: 351 nonsupercell events, and panel b: 834 supercell events. Contours of � +𝐸𝐴 are shown in red. Panels c-d: 𝑅2 between solutions for +ECAPE𝐴 computed using different definitions of 𝑉𝑆𝑅. A given cell shows the correlation coefficient between ECAPE𝐴 computed with the 𝑉𝑆𝑅 +definition on the 𝑥 axis, with that on the corresponding 𝑦 axis, with colors corresponding to the relative magnitudes. +4. Summary, conclusions, and discussion +In summary, we have derived a formula for ECAPE +that depends entirely on state variables available within +an atmospheric sounding. +This formula relies on three +concepts: a scaling between fractional entrainment and +updraft radius of 𝜀 ∼ 𝑅−2, the adiabatic conservation of +moist static energy, and a direct correspondence between +the cloud relative flow and the updraft radius. Finally, we +have accounted for the potential enhancement of updraft +kinetic energy via pressure accelerations. We recommend +using the following steps to compute this quantity in a +software routine: +1. Set the following constant values: 𝑐 𝑝 = 1005 J kg-1 +K-1, 𝐿𝑣,𝑟 = 2,501,000 J kg-1, 𝑔 = 9.81 m s-1, 𝜎 = 1.6, +𝛼 = 0.8, 𝑘2 = 0.18, 𝑃𝑟 = 1 +3, and 𝐿𝑚𝑖𝑥 = 120 m. +2. Compute CAPE, the 𝐿𝐹𝐶, and the 𝐸𝐿 for an undi- +luted parcel from an atmospheric profile using an ex- +isting software routine (e.g., SHARPy, Metpy). +3. Compute the following parameter: +𝜓 = 𝑘2𝛼2𝜋2𝐿𝑚𝑖𝑥 +𝑃𝑟𝜎2𝐻 +, +(52) +where 𝐻 is the equilibrium level. +4. Compute 𝑉𝑆𝑅 from an atmospheric profile. We rec- +ommend averaging 𝑉𝑆𝑅 in the 0-1 km layer, using +the method for estimating storm motion described by +Bunkers et al. (2000). +5. Evaluate the following formula, using a numerical +integration scheme. +�ℎ0(𝑧) = 1 +𝑧 +∫ +𝑧∗=𝑧 +𝑧∗=0 +�𝑐 𝑝𝑑𝑇0 + 𝐿𝑣,𝑟𝑞0 +𝑔𝑧∗� 𝑑𝑧∗, +(53) +This procedure only needs to be done once in a given +profile, and yields < ℎ0 > as a function of height. +6. Compute NCAPE, using the following formula: +NCAPE = − +∫ +𝑧=𝐸𝐿 +𝑧=𝐿𝐹𝐶 +𝑔 +𝑐 𝑝𝑑𝑇0 +� +�ℎ0 − ℎ∗ +0 +� +𝑑𝑧, +(54) + +18 +NCAPE is positive in most contexts though it may +become negative in environments with large free tro- +pospheric relative humidity. +7. Compute ECAPE𝐴, using the following formula: +ECAPE𝐴 = +𝑉2 +𝑆𝑅 +2 ++ +−1−𝜓 − 2𝜓 +𝑉 2 +𝑆𝑅 +NCAPE +4 𝜓 +𝑉 2 +𝑆𝑅 ++ +√︄� +1+𝜓 + 2𝜓 +𝑉 2 +𝑆𝑅 +𝑁𝐶𝐴𝑃𝐸 +�2 ++8 𝜓 +𝑉 2 +𝑆𝑅 +(CAPE−𝜓NCAPE) +4 𝜓 +𝑉 2 +𝑆𝑅 +. +(55) +In the case of a negative solution to this equation, set +the ECAPE𝐴 to 0. +Our results show that ECAPE provides a more accu- +rate prediction of updraft intensity than standard CAPE +when forecasting severe weather hazards that depend on +middle-to-upper tropospheric vertical velocities. Exam- +ples of these situations include forecasting heavy precipi- +tation, large hail, and intense cold pools and downdrafts. +Hence, it would benefit the forecasting community to dis- +play this quantity alongside standard CAPE on websites +that provide numerical weather prediction model output +graphics, such as the storm-prediction center Mesoanaly- +sis site. In addition, �𝐸𝐴, which is the fraction of CAPE +realized, is a powerful discriminator of supercellular from +nonsupercellular storm mode, with a True Skill Statistic +(TSS; e.g., section 2 in Peters et al. 2020d) of 0.76 in this +prediction. This is on par with the TSS for 0-1 km 𝑉𝑆𝑅, +which is 0.79 (these values are not statistically different). +The physical reason behind this discriminatory skill re- +lates to the conclusions of Peters et al. (2019), who showed +that supercells realize larger fractions of their CAPE than +nonsupercells (and hence have larger �𝐸𝐴). +A variety of research applications would also benefit +from the consideration of ECAPE, in addition to standard +CAPE. For instance, studies in past literature often contrast +storm dynamics in high-shear low-CAPE severe weather +events with events (e.g., Schneider and Dean 2008) occur- +ring in environments with higher CAPE (and sometimes +weaker shear). The premise behind this distinction is, be- +cause of the small updraft buoyancy in low-CAPE events, +the updrafts accelerations in these storms are dominated +by dynamic pressure accelerations rather than buoyancy +(Wade and Parker 2021). +However, it is possible that +because of the extreme shear in many low-CAPE severe +weather outbreaks, updrafts in these scenarios realize a +higher percentage of their CAPE than their counterparts in +high CAPE environments. Hence, ECAPE may more ac- +curately distinguish between storms with large and small +buoyancy than standard CAPE, and a reconsideration of +the analyses in these past studies with distinctions drawn +between high ECAPE and low ECAPE events may yield +additional insights into storm dynamics. +ECAPE may also yield novel insight into the influence +of climate change on thunderstorms. For instance, a subset +of studies that investigate the influence of climate change +on severe storm behavior use proxy analyses in global cli- +mate model (GCM) simulations, assessing the impacts of +global warming on parameters like CAPE and CIN. Future +changes to free tropospheric relative humidity, tempera- +ture, and vertical wind shear are also likely to influence +thunderstorms via the connection between these environ- +mental attributes and entrainment. Investigating changes +to the climatology of ECAPE in future climates is a con- +cise way of encapsulating these yet-to-be explored climate +change influences on storm entrainment, and consequently +storm intensity. Efforts to quantify the effects of climate +change among the authors of the present study are currently +underway. +Some of the intermediary formulas that express buoy- +ancy and ECAPE as an analytic function of fractional +entrainment may be useful in cumulus parameterization +schemes. +For instance, multi-plume schemes like the +scheme of Arakawa and Schubert (1974), the Relaxed +Arakawa-Schubert scheme Moorthi and Suarez (1992), the +EDMF𝑁 scheme Neggers (2015), and the MAP scheme +(Peters et al. 2020b) require the computation of diluted +buoyancy and ECAPE for each plume. In the traditional +approach for computing ECAPE, these schemes would ex- +ecute two numerical vertical integrations for each plume. +This procedure, however, is dramatically simplified by us- +ing eq. +24 in the present study, where only 3 vertical +integrations per grid cell are needed to obtain CAPE and +NCAPE, and then the ECAPE associated with each plume +is computed analytically. The MAP scheme from (Peters +et al. 2020b) was also formulated to use the formula from +P20 as part of its closure for convective mass flux. The +formula presented here is a more accurate alternative. +A potential caveat to using this parameter operationally +is that ECAPE𝐴 vanishes in the absence of 𝑉𝑆𝑅, whereas +we know that deep convection is possible in the absence of +substantial 𝑉𝑆𝑅. This discrepancy is likely a consequence +of the primary controls on updraft width shifting away from +vertical wind shear to other environmental factors when +shear is weak, such as the planetary boundary layer (PBL) +depth (e.g., Mulholland et al. 2021a) or the width scale +of terrain features (e.g., Nelson et al. 2021; Kirshbaum +2022). A potential way to circumvent this issue is to revert +to a standard ECAPE calculation (with a user-prescribed 𝜀) +in these weakly sheared environments, setting the updraft +radius to scale with the PBL depth or to a constant value +(e.g., 1500 m, as was done in Peters et al. 2020b). +Some may debate the semantics over whether the formu- +las derived are more appropriately described as predictive +equations for the maximum updraft vertical velocity, rather +than a modified CAPE that accounts for entrainment. Some + +19 +view CAPE as pertaining only to an isolated ascending +parcel with no explicit assumptions about updraft structure +and behavior. Hence, our inclusions of updraft dynamics +in our ECAPE calculation makes this calculation concep- +tually distinct from that of CAPE. However, we argue that +there are a variety of conceptual definitions of CAPE in past +literature, and that this quantity is often used in the fore- +casting community to predict how a given thermodynamic +environment may affect updraft vertical velocity. Because +of the familiarity of forecasters with CAPE, ECAPE (with +units of J kg-1) is a more relatable quantity to forecasters +than 𝑤𝑚𝑎𝑥. This is the primary reason why we have adver- +tised the quantity derived here as an ECAPE, rather than a +predictor of 𝑤𝑚𝑎𝑥. + +20 +Acknowledgments. +J. Peters’s efforts were supported +by National Science Foundation (NSF) grants AGS- +1928666, AGS-1841674, and the Department of Energy +Atmospheric System Research (DOE ASR) grants DE- +SC0000246356. +D. Chavas was supported by National +Science Foundation (NSF) grants 1648681 and 2209052. +H. 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American Meteorological Society, Boston, MA, +https://doi.org/10.1007/978-1-878220-63-9_5, URL https://doi.org/ +10.1007/978-1-878220-63-9_5. + diff --git a/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf b/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fae1fe179bdb1255c745b6bd3c1eea1180674f10 --- /dev/null +++ b/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0d63a348a71a8b75b4782713a96b1eefeee15c06bffe8727f109cb08076eb4af +size 677000 diff --git a/Q9FKT4oBgHgl3EQfiC5d/content/tmp_files/2301.11840v1.pdf.txt b/Q9FKT4oBgHgl3EQfiC5d/content/tmp_files/2301.11840v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..39e9b2c17da0b2a5d21b39aa3e0b450c89193d39 --- /dev/null +++ b/Q9FKT4oBgHgl3EQfiC5d/content/tmp_files/2301.11840v1.pdf.txt @@ -0,0 +1,354 @@ +Features of the Domain Boundaries +of a Highly Anisotropic (S = 1) Antiferromagnet +near the Transition to the Quantum Paramagnet Phase +V. V. Koneva, *, V. A. Ulitkoa, D. N. Yasinskayaa, Y. D. Panova, and A. S. Moskvina +aUral Federal University, Yekaterinburg, 620002 Russia +*e-mail: vitaliy.konev@urfu.ru +Abstract—It is shown that the structure of antiphase domain boundaries in the antiferromagnetic (AFM) +phase of a highly anisotropic magnet with S = 1 on a two-dimensional square lattice depends greatly on sin- +gle-ion anisotropy parameter D. Computer modeling on large square lattices illustrates the changes in the +boundary structure from the quantum paramagnet (QP) to the XY phase, including the intermediate QP–XY +phase at fairly small variations in positive D. +INTRODUCTION +In contrast to quantum magnets with S = 1/2 spin, +systems with S = 1 spin are characterized by more +complex Hamiltonian, single-ion anisotropy, biqua- +dratic intercentric interactions, and totally new phase +states of the quantum paramagnet (QP) type corre- +sponding to an easy-plane phase in the classical +approach. The interest in these systems is due to both +highly anisotropic magnets based on Ni2+ (S = 1) +(e.g., Y2BaNiO5 [YBNO], Ni(C2H8N2)2NO2(ClO4) +[NENP]) [1] and the so-called pseudo-spin systems of +the semi–hard core boson type with constraints on +filling lattice sites n = (0, 1, 2), or mixed valence ion +systems of the triplet type: Cu(1+, 2+, 3+) in cuprates +La(2 – x)SrxCuO4 and Bi(3+, 4+, 5+) in bismuthates [2, 3]. +In all cases, the phase diagrams of spin or pseudo-spin +systems with S = 1 is considerably richer than those of +similar systems with S = 1/2 quantum (pseudo)spin, +due primarily to the emergence in the Hamiltonian of +addends of the single-ion anisotropy and biquadratic +interaction types, plus ones of the quantum paramag- +net and spin-nematic phase types. +MODEL +Let bus consider a model cuprate that is a 2D sys- +tem of Cu centers in a CuO2 plane of cuprates that can +be in three different valence charge states: Cu(1+, 2+, 3+). +We associate this charge triplet with three states of S = 1 +pseudo-spin as Cu1+ → MS = −1, Cu2+ → MS = 0, +Cu3+ → MS = 1, and use the familiar ways of describ- +ing spin systems. The spin algebra of systems with S = 1 +(MS = 0, ±1) includes eight independent nontrivial +(three dipole and five quadrupole) functionals: Sz; +S± = ±(Sx ± iSy); + T± = {Sz, S±} = SzS± + S±Sz; and + Incremental/decremental functionals S± and T± +change the (pseudo)spin projection to ±1, but in differ- +ent ways: + = + = + and + = + = +1. Incremental/decremental function- +als + describe transitions + i.e., they gen- +erate on a site either a hole + or an electron +pair that is a composite local boson with kinematic +constraint + = 0, emphasizing its nature as a hard- +core boson. +Local (on-site) nondiagonal parameter XY of the +order of + which is actually a parameter of the local +superconducting order, is nonzero only when the site +hosts a quantum superposition of states + and +We write the effective Hamiltonian that commu- +tates with the z-component of the total spin + and thus maintains the system’s magne- +tization as the sum of potential and kinetic energies: +H = Hpot + Hkin: +(1) +In calculating the kinetic energy, we consider only the +contribution from double-ion biquadratic anisotropy +Hkin = –t + The first term in (1) +2; +z +S +2. +S± +0 +1 +S± ∓ +1 +0 +S± +± +1, +∓ +0 +1 +T± ∓ +1 +0 +T± +− ± +2 +S± +− +→ + +1 +1 ; +( +) +2 +S± +( +) +2 +S− +2 +S± +2 , +S± +1 +− +1 . ++ +1 +iz +i +n +S +N += + += ++ + + +2 +pot +. +iz +iz +jz +i +ij +H +D +S +J +S S +( +) +2 +2 +2 +2 . +i +j +j +i +ij S S +S S ++ +− ++ +− ++ + +1 + +FEATURES OF THE DOMAIN BOUNDARIES +(i.e., the single-ion anisotropy) describes the density– +density correlation effects on the sites, while the sec- +ond term describes inter-site interactions (correla- +tions) of the density–density type. Below, we consider +only the interactions between nearest neighbors with +positive (antiferromagnetic) signs of inter-center cor- +relation parameter J. +Depending on the relationship between the param- +eters of Hamiltonian (1) and magnetization (n), the +system ground state corresponds either to the homo- +geneous phase of the quantum paramagnet type with +Sz = + = 0, which is attained at high positive val- +ues of parameter D (a large D phase); or to the antifer- +romagnetic (AFM) phase along the z-axis; or to the XY +phase with a nonzero parameter on the order of +RESULTS AND DISCUSSION +We used an NVidia graphical processing unit for +the Monte Carlo modeling of the antiferromagnet +phase transition of highly anisotropic magnet S = 1 in +the two-sublattice approximation on a square lattice of +256 × 256 with periodic boundary conditions at +selected parameters t =1, J = 0.75, n = 0.04, which +ensured a ground state of the antiferromagnet ordering +type in a rather wide range of variations of single-ion +anisotropy parameter D. +At D = –5, a stripe domain structure formed +during rapid thermalization (annealing). At low tem- +peratures, a strongly pronounced filamentary XY +phase emerged at the center of the antiphase domain +boundaries of the AFM phase, which was character- +ized primarily by a nonzero module of the local +parameter of the order XY. Upon an increase in dou- +ble-ionic biquadratic anisotropy t, the domain bound- +ary gradually broadened and the volume of the +XY state grew up to the total displacement of the AFM +2 +z +S +2 . +S± +phase and the transition to the inhomogeneous +XY state. +It is interesting that both the AFM phase and the +XY structure of the domain boundary proved to be sta- +ble in relation to variations in local correlation param- +eter D over a wide range up to D ~ 1.0. Upon further +growth of local correlations, however, the domain +boundary structure reorganized radically. +The evolution of the antiphase domain boundary +upon an increase in parameter D is shown in Fig. 1. As +D grows gradually, the regular structure of the fila- +mentary XY phase on the edges of the antiphase +domain boundary is broken, while the QP phase +emerges and grows to completely displace the filamen- +tary XY phase at D ~ 1.2, accelerating the boundary +transition to QP. With further growth of local correla- +tions D > 1.5, the domain boundary broadens and +gradually displaces the AFM order. In other words, +the AFM → QP phase transition (the large D phase) +occurs with an increase in the local correlation param- +eter, due to expansion of the domain boundaries. +It is noteworthy that the QP phase nucleation on +the edges of the domain boundary occurs due to the +smaller difference between the energies of the QP and +XY phases there (Fig. 2). In other words, the emer- +gence of the QP phase on the edges is energetically +more advantageous than at the center. In Fig. 2, we +can see that the difference between the energies of +phases in the domain and at the center of the domain +boundary is much smaller when the QP phase emerges +at the center of the domain boundary (at D = 1.2) than +with the XY phase (D = −5). Upon the further growth +of D, the AFM phase becomes metastable in the +domains, and the QP phase becomes stable at the cen- +ter of the domain boundary. +The study of temperature effects shows that when +the temperature in the domain walls of the AFM phase +rises at D = 1.0, the system moves from the XY phase +Fig. 1. Average distribution across the domain boundary at local parameters on the order of AFM, XY, and QP phases marked by +solid, dashed-and-dotted, and dashed lines, respectively, on two sublattices A and B (the top and bottom parts of the figure, +respectively) at different values of parameter D: (a) −5.0, (b) 1.0, (c) 1.1, and (d) 1.2. The values along the horizontal axis are pre- +sented in terms of the lattice constant. +1 +0 +–1 +1 +0 +–1 +1 +0 +–1 +1 +0 +–1 +1 +0 +–1 +1 +0 +–1 +1 +0 +–1 +1 +0 +–1 +10 +20 +10 +20 +10 +20 +10 +20 +10 +20 +10 +20 +10 +20 +10 +20 +(а) +(b) +(c) +(d) +2 + +KONEV et al. +to the QP phase and then to a disordered paramagnetic +state. During subsequent cooling to very low tempera- +tures T = 0.0001, however, only the QP structure of the +domain boundaries is restored; i.e., a temperature hys- +teresis is observed in the structure of the boundaries. +CONCLUSIONS +We studied the effect single-ion anisotropy param- +eter D has on the structure of domain boundaries of +the antiferromagnetic phase. Using numerical Monte +Carlo modeling on large square lattices with rapid +annealing, we observed the formation of a stripe +domain structure, in whose antiphase domain bound- +aries a filamentary XY phase formed stably over a wide +interval of D variations up to positive D ~ 1. Upon fur- +ther growth of local correlations, however, the XY +phase was broken, and a filamentary QP phase formed +in the boundaries separating the domains with antifer- +romagnetic ordering. Our modeling of temperature +effects indicated there was a temperature hysteresis in +the structure of the boundaries. +FUNDING +This work was supported by Program 211 of the Govern- +ment of the Russian Federation, project no. 02.A03.21.0006; +and by the RF Ministry of Science and Higher Education, +project nos. 2277 and 5719. +REFERENCES +1. Rudowicz, C., Phys. B, 2014, vol. 436, p. 193. +2. Moskvin, A.S., J. Exp. Theor. Phys., 2015, vol. 121, +no. 3, p. 477. +3. Moskvin, A.S. and Panov, Yu.D., J. Supercond. Novel +Magn., 2018, vol. 31, no. 3, p. 677. +Fig. 2. Average distribution across the domain boundary for the local energy at different values of parameter D: (a) −5.0, (b) 1.0, +(c) 1.1, (d) 1.2. The values along the horizontal axis are presented in terms of the lattice constant; along the vertical axis, the energy +is shown in terms of parameter t. +20 +40 +20 +40 +20 +40 +20 +40 +0.5 +1 +0 +1 +0 +–0.5 +0 +–6 +–7 +(а) +(b) +(c) +(d) +3 + diff --git a/Q9FKT4oBgHgl3EQfiC5d/content/tmp_files/load_file.txt b/Q9FKT4oBgHgl3EQfiC5d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e4e7bb0162fe6ba16edca43d9609ff6bf64635d --- /dev/null +++ b/Q9FKT4oBgHgl3EQfiC5d/content/tmp_files/load_file.txt @@ -0,0 +1,135 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf,len=134 +page_content='Features of the Domain Boundaries of a Highly Anisotropic (S = 1) Antiferromagnet near the Transition to the Quantum Paramagnet Phase V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' V.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Moskvina aUral Federal University, Yekaterinburg, 620002 Russia e-mail: vitaliy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='konev@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='ru Abstract—It is shown that the structure of antiphase domain boundaries in the antiferromagnetic (AFM) phase of a highly anisotropic magnet with S = 1 on a two-dimensional square lattice depends greatly on sin- gle-ion anisotropy parameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Computer modeling on large square lattices illustrates the changes in the boundary structure from the quantum paramagnet (QP) to the XY phase, including the intermediate QP–XY phase at fairly small variations in positive D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' INTRODUCTION In contrast to quantum magnets with S = 1/2 spin, systems with S = 1 spin are characterized by more complex Hamiltonian, single-ion anisotropy, biqua- dratic intercentric interactions, and totally new phase states of the quantum paramagnet (QP) type corre- sponding to an easy-plane phase in the classical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' The interest in these systems is due to both highly anisotropic magnets based on Ni2+ (S = 1) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=', Y2BaNiO5 [YBNO], Ni(C2H8N2)2NO2(ClO4) [NENP]) [1] and the so-called pseudo-spin systems of the semi–hard core boson type with constraints on filling lattice sites n = (0, 1, 2), or mixed valence ion systems of the triplet type: Cu(1+, 2+, 3+) in cuprates La(2 – x)SrxCuO4 and Bi(3+, 4+, 5+) in bismuthates [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' In all cases, the phase diagrams of spin or pseudo-spin systems with S = 1 is considerably richer than those of similar systems with S = 1/2 quantum (pseudo)spin, due primarily to the emergence in the Hamiltonian of addends of the single-ion anisotropy and biquadratic interaction types, plus ones of the quantum paramag- net and spin-nematic phase types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' MODEL Let bus consider a model cuprate that is a 2D sys- tem of Cu centers in a CuO2 plane of cuprates that can be in three different valence charge states: Cu(1+, 2+, 3+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' We associate this charge triplet with three states of S = 1 pseudo-spin as Cu1+ → MS = −1, Cu2+ → MS = 0, Cu3+ → MS = 1, and use the familiar ways of describ- ing spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' The spin algebra of systems with S = 1 (MS = 0, ±1) includes eight independent nontrivial (three dipole and five quadrupole) functionals: Sz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' S± = ±(Sx ± iSy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' T± = {Sz, S±} = SzS± + S±Sz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' and Incremental/decremental functionals S± and T± change the (pseudo)spin projection to ±1, but in differ- ent ways: = = and = = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Incremental/decremental function- als describe transitions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=', they gen- erate on a site either a hole or an electron pair that is a composite local boson with kinematic constraint = 0, emphasizing its nature as a hard- core boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Local (on-site) nondiagonal parameter XY of the order of which is actually a parameter of the local superconducting order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' is nonzero only when the site hosts a quantum superposition of states and We write the effective Hamiltonian that commu- tates with the z-component of the total spin and thus maintains the system’s magne- tization as the sum of potential and kinetic energies: H = Hpot + Hkin: (1) In calculating the kinetic energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' we consider only the contribution from double-ion biquadratic anisotropy Hkin = –t The first term in (1) 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' z S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' S± 0 1 S± ∓ 1 0 S± ± 1, ∓ 0 1 T± ∓ 1 0 T± − ± 2 S± − → + 1 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' ( ) 2 S± ( ) 2 S− 2 S± 2 , S± 1 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' + 1 iz i n S N = \uf0e5 = + \uf0e5 \uf0e5 2 pot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' iz iz jz i ij H D S J S S ( ) 2 2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' i j j i ij S S S S + − + − + \uf0e5 1 FEATURES OF THE DOMAIN BOUNDARIES (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=', the single-ion anisotropy) describes the density– density correlation effects on the sites, while the sec- ond term describes inter-site interactions (correla- tions) of the density–density type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Below, we consider only the interactions between nearest neighbors with positive (antiferromagnetic) signs of inter-center cor- relation parameter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Depending on the relationship between the param- eters of Hamiltonian (1) and magnetization (n), the system ground state corresponds either to the homo- geneous phase of the quantum paramagnet type with \uf0e1Sz\uf0f1 = = 0, which is attained at high positive val- ues of parameter D (a large D phase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' or to the antifer- romagnetic (AFM) phase along the z-axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' or to the XY phase with a nonzero parameter on the order of RESULTS AND DISCUSSION We used an NVidia graphical processing unit for the Monte Carlo modeling of the antiferromagnet phase transition of highly anisotropic magnet S = 1 in the two-sublattice approximation on a square lattice of 256 × 256 with periodic boundary conditions at selected parameters t =1, J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='75, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='04, which ensured a ground state of the antiferromagnet ordering type in a rather wide range of variations of single-ion anisotropy parameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' At D = –5, a stripe domain structure formed during rapid thermalization (annealing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' At low tem- peratures, a strongly pronounced filamentary XY phase emerged at the center of the antiphase domain boundaries of the AFM phase, which was character- ized primarily by a nonzero module of the local parameter of the order XY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Upon an increase in dou- ble-ionic biquadratic anisotropy t, the domain bound- ary gradually broadened and the volume of the XY state grew up to the total displacement of the AFM 2 z S 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' S± phase and the transition to the inhomogeneous XY state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' It is interesting that both the AFM phase and the XY structure of the domain boundary proved to be sta- ble in relation to variations in local correlation param- eter D over a wide range up to D ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Upon further growth of local correlations, however, the domain boundary structure reorganized radically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' The evolution of the antiphase domain boundary upon an increase in parameter D is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' As D grows gradually, the regular structure of the fila- mentary XY phase on the edges of the antiphase domain boundary is broken, while the QP phase emerges and grows to completely displace the filamen- tary XY phase at D ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='2, accelerating the boundary transition to QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' With further growth of local correla- tions D > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='5, the domain boundary broadens and gradually displaces the AFM order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' In other words, the AFM → QP phase transition (the large D phase) occurs with an increase in the local correlation param- eter, due to expansion of the domain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' It is noteworthy that the QP phase nucleation on the edges of the domain boundary occurs due to the smaller difference between the energies of the QP and XY phases there (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' In other words, the emer- gence of the QP phase on the edges is energetically more advantageous than at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 2, we can see that the difference between the energies of phases in the domain and at the center of the domain boundary is much smaller when the QP phase emerges at the center of the domain boundary (at D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='2) than with the XY phase (D = −5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Upon the further growth of D, the AFM phase becomes metastable in the domains, and the QP phase becomes stable at the cen- ter of the domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' The study of temperature effects shows that when the temperature in the domain walls of the AFM phase rises at D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0, the system moves from the XY phase Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Average distribution across the domain boundary at local parameters on the order of AFM, XY, and QP phases marked by solid, dashed-and-dotted, and dashed lines, respectively, on two sublattices A and B (the top and bottom parts of the figure, respectively) at different values of parameter D: (a) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0, (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='1, and (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' The values along the horizontal axis are pre- sented in terms of the lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 1 0 –1 1 0 –1 1 0 –1 1 0 –1 1 0 –1 1 0 –1 1 0 –1 1 0 –1 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 (а) (b) (c) (d) 2 KONEV et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' to the QP phase and then to a disordered paramagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' During subsequent cooling to very low tempera- tures T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0001, however, only the QP structure of the domain boundaries is restored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=', a temperature hys- teresis is observed in the structure of the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' CONCLUSIONS We studied the effect single-ion anisotropy param- eter D has on the structure of domain boundaries of the antiferromagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Using numerical Monte Carlo modeling on large square lattices with rapid annealing, we observed the formation of a stripe domain structure, in whose antiphase domain bound- aries a filamentary XY phase formed stably over a wide interval of D variations up to positive D ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Upon fur- ther growth of local correlations, however, the XY phase was broken, and a filamentary QP phase formed in the boundaries separating the domains with antifer- romagnetic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Our modeling of temperature effects indicated there was a temperature hysteresis in the structure of the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' FUNDING This work was supported by Program 211 of the Govern- ment of the Russian Federation, project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='A03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' and by the RF Ministry of Science and Higher Education, project nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 2277 and 5719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Rudowicz, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' and Panov, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Novel Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=', 2018, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' Average distribution across the domain boundary for the local energy at different values of parameter D: (a) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='0, (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='1, (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' The values along the horizontal axis are presented in terms of the lattice constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' along the vertical axis, the energy is shown in terms of parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content=' 20 40 20 40 20 40 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='5 1 0 1 0 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} +page_content='5 0 –6 –7 (а) (b) (c) (d) 3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf'} diff --git a/QtE3T4oBgHgl3EQfDAk3/vector_store/index.pkl b/QtE3T4oBgHgl3EQfDAk3/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..bff53013a28be3d65b9035edeea9870a6fba8958 --- /dev/null +++ b/QtE3T4oBgHgl3EQfDAk3/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ce23b9f2e49055e36679a0b5967b1356f7b498af5edf5dd5ebef0d86c5bc9d21 +size 236949 diff --git a/U9E3T4oBgHgl3EQfawpi/content/tmp_files/2301.04509v1.pdf.txt b/U9E3T4oBgHgl3EQfawpi/content/tmp_files/2301.04509v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b037aaea424e5dd01ac954e9d7e3326939538913 --- /dev/null +++ b/U9E3T4oBgHgl3EQfawpi/content/tmp_files/2301.04509v1.pdf.txt @@ -0,0 +1,1043 @@ +Received 26 April 2016; +Revised 6 June 2016; +Accepted 6 June 2016 +DOI: xxx/xxxx +ORIGINAL ARTICLE +Alpha tensor and dynamo excitation in turbulent fluids with +anisotropic conductivity fluctuations +Oliver Gressel1,2 | Günther Rüdiger *1,3 | Detlef Elstner1 +1MHD & Turbulence Section, +Leibniz Institute for Astrophysics Potsdam, +An der Sternwarte 16, +14482 Potsdam, Germany +2Niels Bohr International Academy, +The Niels Bohr Institute, +Blegdamsvej 17, +DK-2100, Copenhagen Ø, Denmark +3Institute of Physics and Astronomy, +University of Potsdam, +Karl-Liebknecht-Str. 24-25, +14476 Potsdam, Germany +Correspondence +* Email: gruediger@aip.de +A mean-field theory of the electrodynamics of a turbulent fluid is formulated under +the assumption that the molecular electric conductivity is correlated with the tur- +bulent velocity fluctuation in the (radial) direction, 품. It is shown that for such +homogeneous fluids a strong turbulence-induced field advection anti-parallel to +품 arises almost independently of rotation. For rotating fluids, an extra 훼 effect +appears with the known symmetries and with the expected maximum at the poles. +Fast rotation, however, with Coriolis number exceeding unity suppresses this term. +Numerical simulations of forced turbulence using the NIRVANA code demonstrate +that the radial advection velocity, 훾, always dominates the 훼 term. We show finally +with simplified models that 훼2 dynamos are strongly influenced by the radial pump- +ing: for 훾 < 훼 the solutions become oscillatory, while for 훾 > 훼 they become highly +exotic if they exist at all. In conclusion, dynamo models for slow and fast solid- +body rotation on the basis of finite conductivity–velocity correlations are unlikely to +work, at least for 훼2훺 dynamos without strong shear. +KEYWORDS: +astrophysical plasma – dynamo theory +1 +INTRODUCTION +If apart from the velocity and magnetic field, by any reason +also the electric conductivity in a turbulent fluid fluctuates +around a certain value then also the local magnetic diffusiv- +ity fluctuates around its average. Krause & Roberts (1973) +started to consider the consequences of this constellation with +the result that the effective decay time of a large-scale nonuni- +form magnetic field is changed by reducing the effective eddy +diffusivity of the turbulence field. +Moreover, in convection-driven turbulent fields the always +existing temperature fluctuations should produce magnetic +resistivity fluctuations which are correlated with one of the +velocity components, e.g., the vertical one. In this case, +even a turbulent diffusivity-flux vector ⟨휂′풖′⟩ —with 휂 = +1∕휇0휎 denoting the magnetic resistivity and 풖′ the velocity +fluctuations— occurs. This, in connection with the magnetic +background field or electric current, may form new terms +in the mean-field induction equation. Pétrélis, Alexakis, & +Gissinger (2016) suggested a new sort of 훼 effect arising in +such systems. +They derived an expression for the diffusivity-current cor- +relation, in which the diffusivity-flux vector, multiplied with +the mean magnetic field, ̄푩, appears so that a new 훼 effect +could be possible in spite of the assumed homogeneity of the +turbulence field. However, there are two possibilities for the +relation between the electromotive force and the mean mag- +netic field: the latter can be i) parallel to the electromotive +force or ii) perpendicular to the electromotive force. Only in +the first case, one formally speaks of an 훼 effect, which may +lead to self-excitation of large-scale magnetic fields, while in +the second case the expression describes a turbulent diamag- +netism (also called “topological pumping”) which is known +arXiv:2301.04509v1 [physics.flu-dyn] 11 Jan 2023 + +2 +Gressel, Rüdiger & Elstner +to hamper dynamo instability. If the correlation ⟨휂′풖′⟩ exclu- +sively defines a preferred direction 품 the resulting turbulent +electromotive force is perpendicular to the mean magnetic +field and an alpha-effect is not obtained. +Later on, quasi-linear SOCA calculations applicable to +rotating forced turbulence and/or magneto-convection indeed +confirmed the existence of an 훼 effect in the presence of global +rotation. Without rotation, the conductivity fluctuations lead to +a reduction of the eddy diffusivity and —if correlated with one +of the velocity components— to a new but rather strong dia- +magnetic pumping effect (Rüdiger, Küker, & Käpylä 2020). In +that work, rotating magneto-convection was numerically used +to derive the radial turbulent electric current flux ⟨푢′ +푟curl푩′⟩ — +where 푟 is the radial coordinate— which serves as a proxy of +the turbulent diffusivity-current vector ⟨휂′curl푩′⟩ if 휂′ and 푢′ +푟 +are correlated or anti-correlated. The flux vector always exists +for rotating convection under the influence of an azimuthal +magnetic background field. The result is a well-defined dia- +magnetic pumping and, with rotation, an 훼 effect which is +anti-symmetric with respect to the equator. +However, convection only exists if the fluid is stratified +in the radial direction, 품. The main difference caused by +the fluctuating-conductivity concept is the occurrence of an +훼 effect in fully uniform fluids in which an anisotropy exists +rather than any form of stratification. This makes the idea a +promising one for a dynamo theory of planetary magnetism. +In the present paper, therefore, the existence of the 훼 effect +in absolutely homogeneous fluids is shown by numerical sim- +ulations of forced rotating turbulence. We shall demonstrate +that the 훼 effect indeed occurs, if the global rotation is not too +slow or too fast but that it is, however, always accompanied +by a dominating diamagnetic pumping term, 훾. Even without +rotation (and only slightly suppressed in its presence) a strong +radial advection term occurs by which the horizontal field (i.e. +perpendicular to 품) is lifted to either of the radial boundary +layers, depending on the sign of the effect. +We note that a large-scale 훼2 dynamo can in principle oper- +ate for very weak 훼 effect if only the region is big enough, or +—with other words— if it hosts a sufficiently large number +of eddies. In our final Section, the consequences of this puz- +zling situation are shown by the presentation of a sequence of +mean-field 훼2 dynamo models with stronger and stronger mag- +netic pumping term (i.e. turbulence-induced diamagnetism). +We shall show that such dynamos can only operate as long as +the 훼 term (in form of a pattern velocity) exceeds the pump- +ing velocity. This condition is unfortunately not met – at least, +according to the results of the derived electrodynamics, which +is based on the correlations with conductivity fluctuations. +2 +THE EQUATIONS +The basic equation of the problem is the induction equation +휕푩 +휕푡 = curl +( +풖 × 푩 − 휂 curl푩 +) +, +(1) +with the continuity condition div 푩 += 0. Moreover, we +assume div 풖 = 0 as the condition for an incompressible fluid +for the analytic derivations, while for the numerical experi- +ments, this constraint is relaxed. Here, 풖 is the fluid velocity, 푩 +is the magnetic field vector and 휂 the (molecular) magnetic dif- +fusivity. We consider a turbulent fluid with 풖 = ̄풖+풖′ and with +a fluctuating magnetic diffusivity 휂 = ̄휂 + 휂′. For the expec- +tation values of the perturbations we shall use the notations +푢rms = ⟨풖′2⟩1∕2 and 휂rms = ⟨휂′2⟩1∕2. Large-scale observables +(i.e., mean values) are marked with overbars, while brack- +ets are used for the correlations of fluctuations. Low or high +values of the magnetic Reynolds number +Rm = 푢rms퓁∕̄휂 +(2) +(for Strouhal number ≃ 1, and with 퓁 the correlation length) +distinguish between the regimes of low / high conductivity. +Within the realm of the electrodynamics with finite fluctua- +tions, the high-conductivity limit ̄휂 →0 may not be allowed. +If the fluctuations 풖′ and 휂′ exist and are correlated, then the +turbulence-originated diffusivity flux +푼 = ⟨휂′풖′⟩ +(3) +forms a vector, which is polar by definition. The existence +of the radial component of this vector is obvious for thermal +convection, where both the radial velocity and the electric con- +ductivity are due to temperature fluctuations. The correlation +(3) can be understood as transport of magnetic diffusivity in a +certain direction. If, e.g., the correlation between 휂′ and 푢′ +푟 is +positive then resistivity is transported upwards – balanced by a +downward radial velocity ∇(−휂) which “pumps” the horizon- +tal field downwards in the direction where the magnetic decay +is maximum (the “diamagnetic effect” of turbulent origin). +Also the magnetic field will fluctuate, hence 푩 = ̄푩+푩′. The +magnetic fluctuation 푩′ fulfils a nonlinear induction equation +which follows from (1). The turbulence-originated electromo- +tive force  = ⟨풖′ ×푩′⟩ and the diffusivity-current correlation + = −⟨휂′curl푩′⟩ enter the induction equation for large-scale +magnetic field via +휕 ̄푩 +휕푡 = curl +( + +  − ̄휂 curl ̄푩 +) +. +(4) +Under the assumption that the large-scale field, ̄푩, varies suf- +ficiently slowly in space and time, the electromotive force can +be written as + = 훼◦ ̄푩 − 휂tcurl ̄푩 , +(5) +where the tensor 훼 and the coefficient 휂t represent the 훼 effect +and the turbulent magnetic diffusivity (Krause & Rädler + +Gressel, Rüdiger & Elstner +3 +1980), respectively, and where ‘◦’ denotes a tensor multi- +plication. The tensorial structure of 휂t under the presence +of magnetic field and rotation has been discussed later by +Kitchatinov, Pipin, & Rüdiger (1994). As in Rüdiger et al. +(2020), the spectral vector of the correlation (3) may be +written as +̂푈푖 = 푢1(푘, 휔) +( +푔푖 − (품⋅풌) 푘푖 +푘2 +) +. +(6) +The vector 품 gives the unit vector of the direction in which the +correlation between velocity and diffusivity is non-vanishing. +The expression (6) must be odd in 품 and its real part must +be even in the wave number 풌. The quantity 푢1 reflects the +correlation of the velocity component 품⋅풖′ with 휂′ where 휔 +is the Fourier frequency of the spectrum. As it should, the +transformation 품 → −품 only changes the sign of 푼. +3 +THE DIFFUSIVITY-CURRENT +CORRELATION +It has been shown earlier that a relation + = −훾 품 × ̄푩 +(7) +between the the diffusivity-current correlation,  , and the +large-scale magnetic field, ̄푩, results with +훾 = 1 +3 ∫∫ +̄휂푘4푢1 +휔2 + ̄휂2푘4 d풌 d휔 , +(8) +representing a turbulent advection of the magnetic background +field where 풖adv = −훾품 is the advection velocity (Rüdiger et +al. 2020). We find a coefficient 훾 of the same sign as the dif- +fusivity flux (3). For positive 푢1 (i.e., for positive correlation +of 휂′ and 푢′ +푟), the advection velocity, 풖adv, points downward +if 품 is the radial unit vector. Anti-correlated 휂′ and 푢′ +푟 lead +to an upward turbulent transport of the mean magnetic field. +This means that the field is always attracted by the islands of +lower resistivity – or, equivalently, of higher electric conduc- +tivity. As a consequence, the large-scale magnetic field favours +the direction towards longer diffusive decay times. The advec- +tion velocity is opposite to the diffusivity flux (3). The integral +expression for 훾 of Eq. (8) scales linearly with Rm until it +saturates for large magnetic Reynolds numbers. +Let ̂푉 be the spectral function of the two-point autocor- +relation function 푉 (흃, 휏) = ⟨휂′(풙, 푡) 휂′(풙 + 흃, 푡 + 휏)⟩ of the +diffusivity fluctuations. For the diffusivity-current correlation + the term with ̂푉 leads to + = ⋯ + 2 +3 ∫∫ +푘2 ̂푉 +−i휔 + ̄휂푘2 d풌 d휔 curl ̄푩 , +(9) +which provides an extra contribution to the magnetic field +dissipation. The question is whether this term reduces or +enhances the eddy diffusivity 휂t representing turbulence with- +out 휂-fluctuations. The small-scale diffusivity fluctuations +obviously lead to a reduction of the large-scale eddy diffusiv- +ity 휂t which, however, is only weak as it runs with the small +value (휂rms∕̄휂) in second order (Krause & Roberts 1973; Rüdi- +ger et al. 2020). The actual value of the turbulence dissipation +will not have relevance for the results of the present paper. +Our assumed background turbulence is homogeneous but +anisotropic, where the anisotropy is only implicit. If the turbu- +lence rotates, an additional pseudo-scalar 품 ⋅ 휴 appears with +which a relation + = −훾 품 × ̄푩 − +− 훼1 +[ (품⋅ ̄푩) 휴 + (품⋅휴) ̄푩 ] − 훼2( ̄푩⋅휴) 품 +(10) +can be formulated – with yet unknown coefficients 훼1 and +훼2 for the diffusivity-current correlation,  , in presence of a +large-scale magnetic field and rotation. For the above expres- +sion, 훾 is again given by Eq. (8). Relation (10) formally +describes the existence of an 훼 tensor which connects the +correlation  with the large-scale magnetic field ̄푩. This con- +nection exists despite the turbulence model being assumed as +strictly homogeneous (so that the standard 훼 tensor cannot +appear). The 훼 effect according to (10) is highly anisotropic, +the middle term with the coefficient 훼1 provides the rotation- +induced standard 훼 expression. While the diamagnetic term +with 훾 also exists for 훺 = 0, the 훼 terms need global rotation. +We shall show below that, independently of the sign of the cor- +relations ⟨휂′푢′ +푟⟩, the values of 훼1 and 훾 are always of opposite +sign. +The dimensionless ratio +̂훾 = +훾 +훼1훺 +(11) +of the pumping velocity 훾 and the rotation-induced 훼 effect +indicates the ratio of anti-symmetric and symmetric elements +in the complete 훼 tensor. Simulating electromotive forces +for models of rotating magnetoconvection, Ossendrijver, Stix, +& Brandenburg (2001); Ossendrijver, Stix, Brandenburg, & +Rüdiger (2002) found ̂훾 +≃ 1 where both 훼 and 훾 were +about 10% of the rms value of the convective velocity. Also +Käpylä, Korpi, & Brandenburg (2009) reached typical val- +ues of order unity in their numerical models of turbulent +magnetoconvection. Additionally, with their extensive numer- +ical simulations, Gressel, Ziegler, Elstner, & Rüdiger (2008) +derived ̂훾 = 푂(1) for interstellar turbulence driven by col- +lective supernova explosions. All these examples summarise +the results of 훼 effect calculations from the relation between +the electromotive force  and the mean magnetic field ̄푩, +which only appears if the turbulence is nonuniform. On the +other hand, we shall demonstrate in the following that for +homogeneous models with fluctuating conductivities, the cor- +responding ratio (11) reaches values even exceeding unity – +with severe consequences for associated dynamo models. + +4 +Gressel, Rüdiger & Elstner +4 +NUMERICAL METHODS +To probe the theoretical predictions we run artificially forced, +fully nonlinear numerical simulations with the NIRVANA +MHD code (Ziegler 2004), which solves the equations of com- +pressible magnetohydrodynamics by means of a second-order +Godunov approach. In the simulations, the fluctuating compo- +nent of the magnetic diffusivity is prescribed by 휂′ = 푐푢푢푧, +where the coefficient 푐푢 is used to control the strength of the +correlation. We furthermore use 휂rms = 푐푢푢푧,rms to quantify the +amplitude of the fluctuating part of the magnetic diffusivity. +The simulation domain is a fully periodic cube with volume +퐿3. The units of length and time are [푥] = 푘−1 +1 , [푡] = (푐s푘1)−1 +where 푘1 is the wave number corresponding to the system size +and 푐s is the constant speed of sound. The simulations employ +standard non-helical forcing according to eqn. (7) of Hau- +gen, Brandenburg, & Dobler (2004) and are characterised by +the magnetic Reynolds number (2) with 푢rms volume averaged +and 퓁 = (푘f)−1. The flows under consideration are weakly +compressible with Mach number Ma = 푢rms∕푐s ≈ 0.1. All +simulations have 푘f ≃ 4.5 (using isotropically sampled dis- +crete wave vectors obeying 4 ≤ 푘f ≤ 5) and employ a grid +resolution of 803. In code units, the molecular diffusivity is +fixed at ̄휂 = 0.02. +5 +THE TURBULENT FLUX OF ELECTRIC +CURRENT +Consider a homogeneous and isotropic turbulence that is influ- +enced by uniform magnetic fields and global rotation. Let us +write its correlation tensor, ⟨푢′ +푖 curl푗푩′⟩, as +⟨푢′ +푖 curl푗푩′⟩ = += 휅′휖푗푖푘 ̄퐵푘 + 휅1훺푖 ̄퐵푗 + 휅2훺푗 ̄퐵푖 + 휅3(휴 ⋅ ̄푩)훿푖푗 . (12) +The tensor is not a pseudo-tensor and there is no reason that +the dimensionless coefficients 휅 identically vanish. It does not +play a known role in the mean-field electrodynamics but it +is exploited here as a proxy of the desired diffusivity-current +correlation. The correlation vector ⟨푢′ +푟 curl푩′⟩ describes an +upward or downward radial flux of electric current in a rotating +magnetised turbulence which we shall use below to estimate +the diffusion-current correlation  . We note that for 훺 = 0 it +is ⟨(품 ⋅ 풖′) curl푩′⟩ = 휅′품 × ̄푩 for all directions 품. With 품 as +the radial direction, one finds +⟨푢′ +푟curl휃푩′⟩ = −⟨푢′ +휃curl푟푩′⟩ = −휅′ ̄퐵휙 , +(13) +if the magnetic background field only has an azimuthal com- +ponent. Based on SOCA calculations, the coefficient 휅′ is +휅′ = 1 +15 +∞ +∫ +0 +∞ +∫ +0 +휂푘4퐸(푘, 휔) +휔2 + 휂2푘4 d푘 d휔 , +(14) +with the positive spectral function 퐸 of the turbulence inten- +sity, +푢2 +rms = +∞ +∫ +0 +∞ +∫ +0 +퐸(푘, 휔) d푘 d휔 . +(15) +As the spectrum 퐸(푘, 휔) is positive-definite, the tensor coeffi- +cient 휅′ is positive-definite, too. +Figure 1 gives a numerical representation of the complete +tensor (12) in Cartesian coordinates (푟, 휃, 휙) → (푥, 푦, 푧) where +the rotation vector is 휴 = 훺0(cos 휃, − sin 휃, 0) and the mag- +netic field ̄푩 = (0, 0, 퐵0). The details of the simulations were +given in the previous Section. Obviously, the 휅3 coefficient +in (12) cannot be determined for this geometry as always +휴 ⟂ 푩. It is clear from the uppermost and the lowermost +curves in the left and the right panel that after (13) the sim- +ulation gives 휅′ > 0 in accordance to the result (14) of the +quasi-linear theory. Only the 푥푦-component is anti-symmetric +in its indices but the cross correlations 푥푧 and 푦푧 are sym- +metric. The diagonal components 푥푥, 푦푦 and 푧푧 vanish (not +shown) in accordance to the relation (12). +For the remaining off-diagonal tensor components, one +finds 휅1 = 휅2 = 휅 with 휅 < 0 as +⟨푢′ +푟curl휙푩′⟩ = ⟨푢′ +휙curl푟푩′⟩ = 휅훺 ̄퐵0 cos 휃 < 0 +(16) +and +⟨푢′ +휃curl휙푩′⟩ = ⟨푢′ +휙curl휃푩′⟩ = −휅훺 ̄퐵0 sin 휃 > 0 , +(17) +hence for rotating and magnetised (but otherwise isotropic) +turbulence, the tensor expression (12) becomes +⟨푢′ +푖 curl푗푩′⟩ = 휅′휖푗푖푘 ̄퐵푘 + ++ 휅(훺푖 ̄퐵푗 + 훺푗 ̄퐵푖) + 휅3(휴 ⋅ ̄푩)훿푖푗 . +(18) +In a rotating but otherwise isotropic turbulence with an +azimuthal background field, the meridional flow fluctuations +will always be correlated with the azimuthal electric-current +fluctuations. We note that the simulations show that the anti- +symmetric (푥푦)-component of the tensor is always much +larger than the symmetric (푥푧)-component – which, in fact, +will have important consequences. +Replace now in the relations (13) and (16) 푢′ +푟 by 휂′ and the +existence of correlations such as ⟨휂′curl휃푩′⟩ and ⟨휂′curl휙푩′⟩ +becomes obvious in (rotating) homogeneous turbulence fields +magnetised with an azimuthal background field. Just this find- +ing is formulated by Eq. (10). For positive correlation of the +휂-fluctuation and the radial velocities (i.e., positive 푈푟), the +훼1 in (10) becomes negative and for negative correlations it +becomes positive. Note the negative sign in the definitions. In +the same relation, the 훾 results as positive – hence the pumping + +Gressel, Rüdiger & Elstner +5 +FIGURE 1 The off-diagonal components (expectation value plus temporal variations) of the turbulence-induced electric- +current flux tensor ⟨푢′ +푖curl푗푩′⟩ normalised with 푢rms푘f퐵0 for various co-latitudes. The plot reflects the symmetry of the tensor +except the 푥푦-component which is anti-symmetric in accordance with Eq. (18). Rm = 100, 퐵0 = 2 × 10−8, Pm = 1, 훺 = 1. +is downward (i.e., opposite to 푈푟). We always obtain 훾훼1 ≤ 0 +for both signs of 푈푟. +Another basic finding is that the term with 휅′ always +exceeds those with 휅, which – in other words – means that, for +rotating turbulence, the pumping term (a velocity) will always +be larger than the 훼 term (also a velocity). As a consequence, +in rotating conducting fluids, the diamagnetic effect may by +far exceed the inducting action of the 훼 terms. The remainder +of this paper will confirm this suggestion and will show that +a dominating turbulent pumping precludes dynamo instability +of the 훼2-type, that is, in the absence of large-scale shear. +Figure 2 +numerically shows the influence of the mag- +netic Prandtl number on the off-diagonal components of tensor +(18). The values are taken for mid-latitudes. The Pm varies +by more than one order of magnitude. The numerical values +basically grow for growing Prandtl number. Nevertheless, the +ratio of the negative quantities ⟨푢′ +푥curl푦푩′⟩ and ⟨푢′ +푥curl푧푩′⟩ +remains numerically always much larger than one, also for the +important case of Pm < 1. +The following numerical simulations in a Cartesian box +with the vertical (radial) vector 품 = (1, 0, 0) have been done +with a negative correlation between diffusivity fluctuation and +vertical velocity, hence 푈푥 < 0. Again, the applied magnetic +field is azimuthally directed, ̄푩 = (0, 0, 퐵0). We find +훾 = − +⟨휂′curl푦푩′⟩ +퐵0 +, +훼1훺 = ⟨휂′curl푧푩′⟩ +퐵0 +. +(19) +Figure 3 displays the three components of the diffusivity- +current vector as function of the co-latitude 휃. As it should, its +radial component vanishes (left panel). It is also understand- +able that the advection term ⟨휂′curl푦푩′⟩ is positive and does +hardly depend on the latitude. According to the first relation +in (19), 훾 < 0 – so that the advection velocity 풖adv is directed +upwards (i.e., opposite to 푈푥). +Contrary to this, the 푧-component of the correlation vector +vanishes at the equator – as it is expected for a rotation- +induced 훼-term. Its maximum is obtained at the poles. Accord- +ing to the second definition (19), one finds a positive 훼1. Note +that the negative sign of the product 훾훼1 is independent of the +sign of the correlation of 휂′ and 푢′ +푥. +The simulated components of the correlation vector +⟨휂′curl푩⟩ for fixed rotation rate have been given in Fig. 3 . +For a characteristic value 휂rms∕̄휂 = 0.1 of the diffusivity fluc- +tuations, the rotation frequency is varied in Fig. 4 +to obtain +the characteristic numbers at the northern pole. Obviously, the +maximal correlation appears for rotation 훺 ≃ 1 and will be +suppressed by faster rotation. +For the ratio (11) we generally obtain a value of about five. +The normalised 훼 effect is +퐶훼 = 훼1훺퐿 +̄휂 + 휂t +≃ 3훼1훺 +푢rms +퐿 +퓁 , +(20) +with 퐿 as the box length in code units. The characteristic +turnover time of the turbulence is 휏corr ≃ 퓁∕푢rms ≃ 2 in the +simulation (also in code units), where 푢rms ≃ 0.11 is set by +the amplitude of the forcing. It is 휂t∕̄휂 ≃ 0.3푢2 +rms휏corr∕̄휂 ≃ 0.3. +According to Fig. 3 +and Eq. (19), we have 훼1훺∕푢rms ≃ +5 ⋅ 10−3 so that +퐶훼 ≃ 1.5 ⋅ 10−2 퐿 +퓁 . +(21) + +×10-1 +×10-1 +×10-1 +4 +2 +0 +-2 +-4 + + + + + +-6 +0°15° 30° 45° 60° 75° 90° +0°15° 30° 45° 60° 75° 90° +0°15° 30° 45° 60° 75° 90 +co-latitude e +co-latitude e +co-latitude e6 +Gressel, Rüdiger & Elstner +FIGURE 2 Similar to Fig. 1 +but for 휃 = 45◦ and for various magnetic Prandtl numbers. The blue line in the middle panel +(⟨푢′ +푥curl푦푩′⟩, leading to the advection term) and the orange line in the right panel (⟨푢′ +푥curl푧푩′⟩, leading to the 훼 effect) are of +particular relevance. The ratio ⟨푢′ +푥curl푦푩′⟩∕⟨푢′ +푥curl푧푩′⟩ for all Pm always exceeds unity. Rm = 11, 훺 = 1. +FIGURE 3 The three components of the diffusivity-current vector ⟨휂′curl푩′⟩∕(푢rms퐵0) versus co-latitude. Rm = 11. 휂rms∕̄휂 = +0.1, 퐵0 = 2 × 10−8, Pm = 1, 훺 = 1. +The ratio 퐿∕퓁 gives the number of cells along the vertical +direction, which obviously must exceed 70 to reach 퐶훼 of +order unity. This is one of the arguments that it would not be +easy to simulate such a dynamo in a box. +The dependencies of the diffusivity-current vector compo- +nents on the rotation rate 훺0 are shown in Fig. 4 , where for +훺0 = 1, the rotation period is 2휋. As usual, the Strouhal num- +ber St = 푢rms휏corr∕퓁 results of order unity. We also note that +훺0 = 1 describes a rapid rotation with a Coriolis number of +2휏corr훺 ≃ 4.4 beyond which the 훼 effect is strongly quenched +by the rotation (Fig. 4 , right panel). The maximum correla- +tion exists for 훺 = 1; one cannot increase this value by faster +rotation. For 훺 = 1 it is ̂훾 ≃ 5, and this ratio even grows +for slower and/or faster rotation. A weak rotational quenching +can also be observed in the middle panel, where the advection +term is reduced (only) by a factor of three when 훺0 grows by +two orders of magnitude. + +×10-1 +x10-1 +x10-1 +2 +0 +-2 +-4 + + + +-6 + + + +2 +0.25 +0.5 +1 +4 +0.25 +0.5 +1 +2 +4 +0.25 +0.5 +1 +2 +4 +magn. Prandtl number +magn. Prandtl number +magn. Prandtl number×10-2 +×10-2 +×10-2 +3 +2 +1 +0 + + + +0°15° 30° 45°60°75° 90° +0°15° 30° 45° 60° 75° 90' +0°15° 30° 45° 60°75° 90 +co-latitude A +co-latitude e +co-latitude eGressel, Rüdiger & Elstner +7 +FIGURE 4 Similar to Fig. 3 but at the northern pole and for increasing rotation rate 훺. +FIGURE 5 The three components of the vector ⟨휂′curl푩′⟩∕(푢rms퐵0) for non-rotating turbulence. In accordance with Eq. (10) +only the 푦-component (representing the topological pumping) remains finite. Rm = 11, Pm = 1, 훺 = 0. +Figure 5 +refers to non-rotating turbulence with growing +ratio of the diffusivity fluctuations, 휂rms∕̄휂. As expected, the +curve in the middle panel linearly runs with the normalised +diffusivity fluctuation in accordance with the 훾 expression (8), +and it vanishes for 휂′ → 0. For non-rotating turbulence, of +course, the two remaining components (including the 훼 effect) +are identical zero – as shown in the left and the right panel of +Fig. 5 . As it should, the advection term plotted in the middle +panel also exists for non-rotating turbulent fluids. We still have +to find out how the calculated large values of ̂훾 influence the +operation of a global dynamo. +6 +KINEMATIC 훂ퟐ DYNAMO MODELS +WITH FIELD ADVECTION +It has been shown that for rotating turbulence, the above for- +mulated 훼 effect is always accompanied by a pumping effect +in the direction of the component of the flow vector which +is correlated with conductivity fluctuations. For all rotation +rates, the ratio ̂훾 exceeds unity. We now turn our inquiry to +the influence of the turbulent field advection on the operation +of an 훼2 dynamo. In earlier papers, we already found for disk + +×10-2 +×10-2 +×10-2 +3 +2 +1 +0 + + +-1 +0.1 +1 +10 +0.1 +1 +10 +0.1 +1 +10 +angular velocity Qo +angular velocity 20 +angular velocity Qo×10-2 +×10-2 +x10-2 +3 +2 +1 +0 +n'curlxB'> +I.B'> +-1 +0.01 +0.1 +1 +0.01 +0.1 +1 +0.01 +0.1 +1 +n'/n (nominal) +n'/n (nominal) +n'/n (nominal)8 +Gressel, Rüdiger & Elstner +dynamo models that a too strong field advection suppresses +the field excitation even under the presence of differential +rotation (Schultz, Elstner, & Rüdiger 1994). +The geometrically simplest model is posed by uniform +quantities 훼 and 훾 existing in a gap between two parallel +plates embedded in vacuum. The vertical distance between the +boundaries is 퐻. The eddy diffusivity 휂0 between the plates +is assumed as a free parameter, whose actual value is not +important for the result. All quantities are assumed as uniform +in the two horizontal directions 푦 and 푧. Then the condition +div 푩 = 0 requires that the vertical field 퐵푥 does not depend +on 푥 hence 퐵푥 = 0 without lost of generality. +The equations for this kinematic 1D slab model may be +formulated with the normalised quantities +퐶훼 = 훼퐻 +휂0 +, +퐶훾 = 훾퐻 +휂0 +(22) +(let 훺 = 1 for simplicity) so that +i휔퐵푦 − +d2퐵푦 +d푥2 = −퐶훾 +d퐵푦 +d푥 − 퐶훼 +d퐵푧 +d푥 +(23) +and +i휔퐵푧 − d2퐵푧 +d푥2 = −퐶훾 +d퐵푧 +d푥 + 퐶훼 +d퐵푦 +d푥 , +(24) +— see Moss, Shukurov, & Sokoloff (1999); Parker (1979); +Rüdiger & Kitchatinov (2006). The real part of the complex +frequency 휔 determines the oscillation frequency (in units of +the diffusion rate) of a possible dynamo wave along the verti- +cal direction, while the growth rate of the dynamo is given by +the negative value of its imaginary part. We are mainly inter- +ested to know the critical 퐶훼,0 for neutral instability, ℑ(휔) = 0. +Let us define the ratio +̂훾0 = +퐶훾 +퐶훼,0 +(25) +as describing the influence of the pumping effect on the +excitation of kinematic 훼2 dynamos. +The vacuum boundary conditions 퐵푦(0) = 퐵푧(0) = 퐵푦(1) = +퐵푧(1) = 0 are applied. For 훾 = 0 the lowest nontrivial eigen- +value of a stationary solution is 퐶훼,0 = 2휋. The solutions do +not depend on the sign of 퐶훾 as they do also not depend on the +sign of 퐶훼. +The upper panel of Fig. 6 +gives the dynamo’s growth +rates for three values of 퐶훾 as function of 퐶훼. As usual, for +sub-critical (super-critical) 훼 the modes are decaying (grow- +ing), and we find that the 퐶훼,0 for neutral instability grows +with growing 퐶훾. All the critical dynamo solutions for non- +vanishing 훾 are oscillating. The lower panel of this figure +demonstrates that for these eigensolutions the ratio ̂훾0 does +never exceed unity. The 1D 훼2 dynamo, therefore, has no neu- +tral dynamo solution for 퐶훾 > 퐶훼. A too strong radial advec- +tion effect is not compatible with the operation of 훼2 dynamos. +The reason for the suppression of the dynamo instability by +FIGURE 6 Upper panel: Growth rates multiplied with the +diffusion time vs. 퐶훼 for three plane dynamos with 퐶훾 = +8, 16, 24. All solutions describe waves travelling in vertical +direction. Lower panel: The dimensionless ratio ̂훾0 versus 퐶훾 +for neutral excitation. Kinematic dynamos only exist as long +̂훾 ≤ 1, the pumping term 퐶훾 suppresses the dynamo action. +dominating radial advection is that the field components per- +pendicular to the advection vector are concentrated inwards +(or outwards) so that the dynamo domain is reduced and the +critical 퐶훼 must grow. This destructive action proves to be +even more drastic for 훼2 dynamos than for those of 훼훺-type +(Brandenburg, Moss, & Tuominen 1992; Moss et al. 1999). +Results for a very special spherical model with 훼 effect and +pumping term are plotted in Fig. 16.10 in Krause & Rädler +(1980). The 훼 effect only exists in an outer hemisphere while +the diamagnetic pumping only exists in its inner part. Simi- +lar to the above slab model, for growing 퐶훾 also the critical +퐶훼,0 grows linearly so that the ̂훾0 never exceeds unity. The +mode with the lowest dynamo number is a nonaxisymmetric +quadrupole with an azimuthally drifting magnetic field. +Because of its relevance for the concept of conductivity +fluctuations, we have designed a simple shell-type dynamo +model with an outer turbulence domain filled with 훼 effect + +Gressel, Rüdiger & Elstner +9 +independent of the radius, and with uniform radial 훾. The 훼 +term is anti-symmetric with respect to the equator. The def- +initions (22) have been used with the replacement 퐻 → 퐷 +with 퐷 = (1 − 푟in)푅 and 푅 the radius of the sphere. The inner +boundary is a perfect-conducting one while the outer boundary +mimics vacuum, so that the Poynting flux is zero. To illus- +trate the performance of the advection term, examples for the +excited magnetic fields are plotted in Fig. 7 for a turbulence +with outward pumping (top) and inward pumping (bottom). +The inner part (or the outer part, in dependence on the sign +of 훾) of the shell are field-free. Eigensolutions with dipolar +symmetry have the same eigenvalue as those with quadrupolar +symmetry. The sign of 퐶훾 differs in both models but with- +out consequences for the excitation condition. For both cases +|̂훾| = 0.8 is prescribed. The radial advection produces nonax- +isymmetric solutions drifting in the azimuthal direction. For +훾 = 0 the critical eigenvalue for neutral excitation is 퐶훼,0 ≃ 5, +independent of the value of 푟in (see Fig. 7 , middle panel). +For increasing ̂훾, the horizontal field will be more and more +concentrated at the inner or the outer boundary (in dependence +on the sign of 훾) while the bulk of the shell becomes field-free. +The values of 퐶훼 necessary for dynamo excitation grow to +unrealistic high values (Fig. 8 ). A fluid with values of ̂훾 > 1 +and without shear cannot maintain large-scale fields via the +훼2 mechanism. For the above calculated high value of ̂훾 ≃ 5, +therefore, kinematic 훼2 dynamos are not possible. With other +words, the dynamo only works for 퐶훼 >∼ Max(5, 퐶훾). In case +that 훼 ≃ ̂훼훺 (which is true for slow rotation), the dynamo +only operates as long as the rotation rate exceeds the critical +value of 훺 ≃ 훾∕̂훼. The dynamo decays for 훺 < 훺1 where +훺1 denotes the rotation rate where ̂훾 = 1. The above men- +tioned simulations for solar magneto-convection suggest that +indeed 훺1 ≃ 훺⊙. Figure 8 +also contains eigenvalues for an +훼2훺 dynamo with the rather flat rotation law 훺 = 훺0∕푟0.3. +For the normalised rotation rate 퐶훺 = 훺0퐷∕휂0 the value +퐶훺 = 460 is used. One only finds small deviations from the +curves for the 훼2 dynamo with 퐶훺 = 0. For weak field advec- +tion the solutions with the lowest 훼0 are axisymmetric and +oscillating while for stronger pumping the nonaxisymmetric +modes prevail which are drifting in the azimuthal direction. +We note that we only considered the kinematic approxima- +tion where any nonlinear feedback of the induced fields onto +the turbulence is ignored. In any case, if dynamos ever existed +for large values of ̂훾, they must be rather exotic. +7 +CONCLUSIONS +If an anisotropy in a conducting turbulent fluid is defined as +one that is (only) in the direction of the conductivity fluc- +tuations, and the velocity fluctuation is correlated, then a +FIGURE 7 Influence of the advection term on 훼2 dynamos. +The nonaxisymmetric dipolar mode A1 (top) and the quadru- +polar mode S1 (bottom) for |̂훾0| = 0.8 are excited by the +same value of 퐶훼,0. The kinematic axisymmetric 훼2 dynamo +for 훾 = 0 (middle) is shown for reference. The bottom of the +turbulence domain is at a 푟 = 0.5, with a perfect-conducting +boundary. The models are embedded in vacuum. + +1.0 +0.5 +60. +0.0 +50 +20 +AD +E +:70: +.30 +60 +60 +50 +.20 +50 +40 +.1.0 +40 +-20 +20. +.1.9 +9 +10 +0.5 +20 +1.0 +0.0 +0.2 +D.4 +0.6 +0.8 +1.01.0 +0.5 +60. +0.0F +.50 +20 +E +:70: +60 +60 +50 +.20 +50 +40 +.10 +40 +-20 +20. +30 +.1.9 +9 +10 +0.5 +-.1.0 +20 +0.0 +0.2 +D.4 +0.6 +0.8 +1.01.0 +0.5 +.70 +60. +0.0 +.50 +20 +.40 +B +:70: +60 +.30. +60 +50 +.20 +50 +40 +.10 +40 +-20 +20. +9 +10. +0.5 +20 +G.2 +D.4 +0.6 +0.8 +1.010 +Gressel, Rüdiger & Elstner +0 +50 +100 +150 +200 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ˆγ +Cγ +FIGURE 8 The values ̂훾0 critical for excitation versus 퐶훾 of +spherical shell dynamo models. The nonaxisymmetric (dashed +lines) solutions possess (slightly) smaller 퐶훼 than the axisym- +metric solutions (solid line). Dynamo solutions for 훾 > 훼0 +do not exist. 퐶훺 = 0 (dark), 퐶훺 = 460 (red). The smallest +eigenvalue is 퐶훼,0 = 5 for 훾 = 0. 푟in = 0.5. +turbulent field-advection exists in this direction. It lifts large- +scale fields oriented perpendicular to this direction downward +or upward, depending on the sign of the correlation. +Our simulations provide the amplitude of this advection +term in units of the turbulence velocity. They are on the order +of about 10% of the normalised resistivity fluctuation 휂rms∕̄휂, +while the 훼 effect is generally smaller. Its amplitude grows +for growing rotation rate until 훺 ≃ 1 – declining, however, +for faster rotation. On the other hand, the advection term 훾 +is numerically almost uninfluenced by the rotation, in accor- +dance with general expectations. As we have also shown that +the Pm-dependence of the results is only weak, one can be +sure that in rotating fluids with velocity-correlated conductiv- +ity fluctuations, the resulting pumping term 훾 always exceeds +the alpha term velocity 훼1훺. +As demonstrated in Section 6, this constellation has severe +consequences for associated dynamo models. There we have +considered two dynamo models with different geometries. +First, a simplified slab dynamo model with two insulat- +ing plates and with a uniform 훼 effect, including a vertical +turbulence-induced field advection. This model only yields +solutions with neutral stability if the 훼 velocity exceeds the +advection velocity. The solution for 훾 = 0 is stationary while +otherwise it forms a vertical dynamo wave. For 훾 +≥ 훼, +dynamo solutions no longer exist. The results are very simi- +lar for spherical shell dynamos. For growing advection effect +the most unstable modes become oscillatory but always the +dynamos need 퐶훼 > 퐶훾, i.e. the ratio ̂훾 never exceeds unity. +Pure 훼2 dynamos on the basis of resistivity fluctuations can +thus not work. The same holds for shell dynamos with rather +flat rotation laws while the behaviour of 훼훺 dynamos with +large shear is still unknown for the case of strong pumping. +ACKNOWLEDGEMENTS +OG thanks Petri Käpylä for a helpful correspondence. This +work used the NIRVANA code version 3.3, developed by Udo +Ziegler at the Leibniz-Institut für Astrophysik Potsdam (AIP). +All direct computations were performed on the Steno node +at the Danish Center for Supercomputing (DCSC). +REFERENCES +Brandenburg, A., Moss, D., & Tuominen, I. 1992, Turbulent Pump- +ing in the Solar Dynamo. +K. L. Harvey (Ed.), The Solar Cycle +Vol. 27, p. 536. +Gressel, O., Ziegler, U., Elstner, D., & Rüdiger, G. +2008, July, +Astronomische Nachrichten, 329, 619. doi: +Haugen, N. E., Brandenburg, A., & Dobler, W. 2004, July, Physical +Review E, 70(1), 016308. doi: +Käpylä, P. J., Korpi, M. J., & Brandenburg, A. 2009, June, Astron- +omy & Astrophysics, 500, 633-646. doi: +Kitchatinov, L. L., Pipin, V. V., & Rüdiger, G. +1994, February, +Astronomische Nachrichten, 315, 157-170. +Krause, F., & Rädler, K. H. 1980, Mean-field magnetohydrodynam- +ics and dynamo theory. +Krause, F., & Roberts, P. H. 1973, May, The Astrophysical Journal, +181, 977-992. doi: +Moss, D., Shukurov, A., & Sokoloff, D. 1999, March, Astronomy & +Astrophysics, 343, 120-131. +Ossendrijver, M., Stix, M., & Brandenburg, A. 2001, September, +Astronomy & Astrophysics, 376, 713-726. doi: +Ossendrijver, M., Stix, M., Brandenburg, A., & Rüdiger, G. 2002, +November, Astronomy & Astrophysics, 394, 735-745. doi: +Parker, E. N. 1979, Cosmical magnetic fields. Their origin and their +activity. +Pétrélis, F., Alexakis, A., & Gissinger, C. +2016, April, +Physical +Review Letters, 116(16), 161102. doi: +Rüdiger, G., & Kitchatinov, L. L. +2006, May, +Astronomische +Nachrichten, 327(4), 298. doi: +Rüdiger, G., Küker, M., & Käpylä, P. J. 2020, June, Journal of +Plasma Physics, 86(3), 905860318. doi: +Schultz, M., Elstner, D., & Rüdiger, G. 1994, June, +Astronomy & +Astrophysics, 286, 72-79. +Ziegler, U. +2004, March, +Computer Physics Communications, +157(3), 207-216. doi: +How cite this article: O. Gressel G. Rüdiger, and D. Elstner +(2021), Alpha tensor and dynamo excitation in turbulent flu- +ids with anisotropic conductivity fluctuations, Astronomical Notes, +2021;00:x–y. +How cite this article: O. Gressel G. Rüdiger, and D. Elst- +ner (2021), Alpha tensor and dynamo excitation in turbulent +fluids with anisotropic conductivity fluctuations, Astronomical +Notes, 2021;00:x–y. + diff --git a/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf b/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c7394cd82d2442caf0999b5a7a32bd5c3fb3eb57 --- /dev/null +++ b/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:76e0a9424c652bf0ca8cced6314d4443210fbe8339788caa048ffa81b3ba741a +size 574482 diff --git a/VNAyT4oBgHgl3EQf8vqZ/content/tmp_files/2301.00862v1.pdf.txt b/VNAyT4oBgHgl3EQf8vqZ/content/tmp_files/2301.00862v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b05efca6e63b9f8a1d9a04423ba43771edfebf1c --- /dev/null +++ b/VNAyT4oBgHgl3EQf8vqZ/content/tmp_files/2301.00862v1.pdf.txt @@ -0,0 +1,2963 @@ +arXiv:2301.00862v1 [math.AG] 2 Jan 2023 +Catalan varieties∗ +Syu Kato† +January 4, 2023 +Abstract +We construct two smooth projective algebraic varieties XΨ and X♯ +Ψ +that compactify an equivariant vector subbundle of the cotangent bundle +of the flag variety of GL(n) (corresponding to a root ideal Ψ). The variety +XΨ carries natural class of line bundles whose spaces of global sections +realize the Catalan functions defined in Chen-Haiman [thesis, UCBerke- +ley] and studied in Blasiak-Morse-Pun-Summers [J. Amer. Math. Soc. +(2019)]. We prove the vanishing conjectures of Chen-Haiman and Blasiak- +Morse-Pun [arXiv:2007.04952] (in their full generalities), as well as the +monotonicity conjectures of Shimozono-Weyman [Electronic J. Combin. +(2000)] using the geometry of X♯ +Ψ and XΨ. +Introduction +In search of better understanding of the internal structure of Macdonald poly- +nomials ([34]) after Haiman’s solution ([18]) of the Macdonald positivity con- +jecture, LaPointe-Lascoux-Morse [33] proposed the notion of k-Schur functions. +They are shown to represent Schubert classes of affine Grassmannians ([31]), +and hence has a rˆole in quantum cohomology of flag variety X of G = GL(n, C) +([39, 32]). +However, we still do not fully understand its precise relationship +with the Macdonald polynomials (without specializations) and computations in +quantum cohomology. +Chen-Haiman [11] made remarkable conjectures on the internal structure of +k-Schur functions and its generalizations, sometimes called the Catalan func- +tions, by offering their geometric interpretation using certain vector bundles on +X. Their conjectures include the conjectures posed by Broer [7, 3.16] (for type +A) and Shimozono-Weyman [40] as their particular cases. Although the numer- +ical portion of their conjectures are established by Blasiak-Morse-Pun-Summers +[4, 3], we still do not know some cohomology vanishing conjectures, that is fur- +ther refined in [3]. Since these conjectures are central in the geometric picture in +[11] and also in the logic in the monotonicity conjectures in [40, §2.10], we might +say they are the final missing pieces to establish the framework anticipated by +many people for more than 30 years. +In this paper, we define and study two compactifications XΨ and X♯ +Ψ of a +G-equivariant vector subbundle T ∗ +ΨX ⊂ T ∗X employed by Chen-Haiman [11] +∗MSC2010: 14N15,20G44 +†Department of Mathematics, Kyoto University, Oiwake Kita-Shirakawa Sakyo Kyoto 606- +8502 JAPAN E-mail:syuchan@math.kyoto-u.ac.jp +1 + +indexed by a Dyck path Ψ of size n. Let a(Ψ) denote the area statistic of the +Dyck path. +Let Pn denote the set of partitions of length ≤ n. +The set of +irreducible polynomial representations of G is parametrized by Pn as Pn ∋ λ �→ +Vλ (up to isomorphisms). The character of Vλ is the Schur polynomial sλ. By +recording the character of C× by q•, we have the notion of the graded character +gch V of a rational (G × C×)-module V . +Theorem A ( .= Theorems 3.14 and 4.1 and 4.5 + Corollary 3.17). The ex- +ists two varieties XΨ and X♯ +Ψ equipped with (G × C×)-actions and (G × C×)- +equivariant morphisms +XΨ +ηΨ +←− X♯ +Ψ +πΨ +−→ X +with the following properties: +1. The varieties XΨ and X♯ +Ψ are smooth of dimension 2 dim X − a(Ψ); +2. We have an embedding T ∗ +ΨX ⊂ X♯ +Ψ, and the restriction of ηΨ to the image +of T ∗ +ΨX is an isomorphism; +3. For each λ ∈ Pn, we have (G × Gm)-equivariant line bundles OXΨ(λ) and +OX♯ +Ψ(λ) := η∗ +ΨOXΨ(λ) such that +H>0(XΨ, OXΨ(λ)) = 0 = H>0(X♯ +Ψ, OX♯ +Ψ(λ)) +and +� +HΨ(λ) +� +q�→q−1 = gch H0(XΨ, OXΨ(λ))∗ = gch H0(X♯ +Ψ, OX♯ +Ψ(λ))∗, +where HΨ(λ) is the Catalan polynomial ([4, (2.2)]); +4. There is a (G × C×)-equivariant effective Cartier divisor ∂ supported on +X♯ +Ψ \ T ∗ +ΨX such that we have +H>0(X♯ +Ψ, OX♯ +Ψ(λ + m∂)) = 0 +λ ∈ Pn, m ∈ Z≥0. +In particular, we have +H>0(T ∗ +ΨX, OT ∗ +ΨX(λ)) = +� +m≥0 +H>0(XΨ, OX♯ +Ψ(λ + m∂)) = 0 +λ ∈ Pn. +Theorem A 4) resolves [11, Conjecture 5.4.3 2)]. In conjunction with [3, +Theorem 2.18], this establishes [11, Conjecture 5.4.3] in its full generality. As +[11, Conjecture 5.4.3] include conjectures by Broer [7, 3.16] (for type A, in +the form that the desired supports are always empty) and Shimozono-Weyman +[40, §2.4, §2.10], our consideration resolves (the geometric portions of) these +conjectures as well. As a corollary of Theorem A, we find: +Corollary B ( .= Corollary 3.22). There exists an action of GL(n, C[[z]]) ⋊ Gm +on XΨ such that +H0(T ∗ +ΨX, OT ∗ +ΨX(λ))∨ −→ +→ H0(XΨ, OXΨ(λ))∨ +is a quotient as (a graded) cyclic representations of gl(n, C) ⊗ C[[z]]. +2 + +In case T ∗ +ΨX ∼= T ∗X, Corollary B tells us the exact relation between two +realizations of Hall-Littlewood functions via the geometry of X ([17]) and via +representation theory of gl(n, C[[z]]) ([35, 9]), in conjunction with the existing +relations ([10, 23, 14]). +Let B ⊂ G be the subgroup consisting of upper triangular matrices. The set +of B-orbits of X is parametrized by the symmetric group Sn. For w ∈ Sn, let +us denote by X(w) the closure of the corresponding B-orbit (a Schubert variety +of X). We set X♯ +Ψ(w) := π−1 +Ψ (X(w)) and +T ∗ +ΨX(w) := T ∗ +ΨX ∩ X♯ +Ψ(w) ⊂ X♯ +Ψ. +This is the restriction of the vector bundle T ∗ +ΨX to X(w), together with its +compactification. By a similar logic as in Theorem A 4), we deduce: +Theorem C ( .= Theorem 4.5). For each λ ∈ Pn and w ∈ Sn, we have +H>0(T ∗ +ΨX(w), OT ∗ +ΨX(w)(λ)) = 0, +where OT ∗ +ΨX(w)(λ) is the line bundle on T ∗ +ΨX(w) obtained as the restriction of +OXΨ(λ). +Theorem C resolves [3, Conjecture 3.4 (ii)], that generalizes many previous +results and conjectures (see Remark 4.6). As a bonus of our consideration, we +have: +Corollary D ( .= Corollary 5.4). Let Ψ′ ⊂ Ψ be an inclusion of Dyck paths that +yields T ∗ +Ψ′X ⊂ T ∗ +ΨX. For each λ ∈ Pn, the restriction map +H0(T ∗ +ΨX, OT ∗ +ΨX(λ)) −→ H0(T ∗ +Ψ′X, OT ∗ +Ψ′X(λ)) +is surjective. +Corollary D resolves conjectures in [40, §2.10] as explained in Remark 5.7. +Our constructions of the varieties XΨ and X♯ +Ψ are explicit constructions of +their coordinate rings, as well as their blowing-ups along explicit loci. +The +main technical portion of the vanishing results is to afford suitable Frobenius +D-splittings on the positive characteristic analogues of XΨ and X♯ +Ψ. In order +to carry this out, we tweak standard results in compatible Frobenius splittings +enforcing [5]. +The organization of this paper is as follows: We fix notations and record basic +results §1. We present some basic materials and our construction of Frobenius +splittings in §2 by assuming the characteristic is positive. In §3, we work over an +arbitrary algebraically closed field and construct our variety XΨ, and establish +a part of Theorem A and Corollary B. In §4, we construct our variety X♯ +Ψ, and +establish the remaining part of Theorem A, as well as Theorem C. We discuss +consequences of our results including Corollary D in §5. +The last section is +devoted to example calculations. +So far, we have introduced algebraic varieties that are natural geometric +counterparts of the Catalan functions. +An obvious question is how to place +them in the context of topological field theories and geometric realizations of +Macdonald polynomials arising from G = GL(n). We hope to answer these +questions in the sequel. +3 + +1 +Preliminaries +We work over an algebraically closed field k unless specified otherwise. A variety +means a separated (integral) normal scheme of finite type over k. For a scheme +X over a ring S and a ring map S → R, we denote by X(R) the set of R-valued +points of X. +For a k-vector space V , let S•V = � +i≥0 SiV denote its symmetric power +ring. Let L be an abelian free monoid. A L-graded k-vector space V is a k- +vector space V equipped with a direct sum decomposition V = � +a∈L Va such +that dimk Va < ∞ for each a ∈ L. For a L-graded vector space V = � +a∈L Va, +we set +V ∨ := +� +a∈L +V ∗ +a . +A L-graded ring R is an unital k-algebra that is a L-graded k-vector space such +that k1 = R0 and Ra · Ra′ ⊂ Ra+a′ (a, a′ ∈ L). +1.1 +Algebraic Groups +We fix an integer n > 0 and consider the algebraic group +G = GL(n) ⊂ Mn(k) = Mn ∼= An2 +k +over k. Here we omit k when the meaning is clear from the context. We also +define an algebraic group G = GL(n, k[[z]]) over k. We also set +G((z)) := GL(n, k((z))) +and regard it as a (topological) group. Let Eij ∈ Mn(k) (1 ≤ i, j ≤ n) be the +matrix unit. Let T ⊂ G be the diagonal torus and let B ⊂ G (resp. B− ⊂ G) +be the upper (resp. lower) triangular part of G. The group N := [B, B] ⊂ B is +the group of upper unitriangular matrices. We have the evaluation map +ev0 : G −→ G +z �→ 0. +We set B := ev−1 +0 (B). For each 1 ≤ i < n, we set Pi ⊂ G (resp. Pi ⊂ G) as the +(algebraic or pro-algebraic) subgroup generated by B (resp. I) and Id + kEi+1,i +inside G (resp. G). We set P0 as the (pro-algebraic) group generated by I +and Id + kz−1E1,n inside G((z)). Note that we have the extra loop rotation +Gm-action on each of I ⊂ Pi and G (that we denote by Grot +m ). We denote by +�I, �Pi, and �G the semi-direct products of I, Pi, and G with Grot +m , respectively. +In addition, the group G((z)) admits a central extension by k×, that induces a +(trivial) central extension �Pi (0 ≤ i < n) of �Pi by Gm (that we denote by Gc +m). +In particular, we have extended tori +�T := T × Grot +m × {1} ⊂ T × Grot +m × Gc +m =: �T +such that �B := �B× Gc +m contains �T such that �B∩ �T = �T and �Pi ∩ �Pj = �B when +i ̸= j. For each 0 ≤ i < n, we define a map of algebraic groups +ui : Ga(k) = k ∋ x �→ +� +1 + xEi,i+1 +(i ̸= 0) +1 + xzEn,1 +(i = 0) ∈ �B(k). +4 + +We also have +�G((z)) := Grot +m (k) ⋉ G((z)) ⋉ Gc +m(k) +as a group. Let �B− ⊂ �G((z)) be the subgroup generated by �T, the lower trian- +gular part of G, and Id+kz−1E1,n. Let �G− ⊂ �G((z)) be the subgroup generated +by �B− and G. We warn that the groups �G((z)), �B−, and �G− are not algebraic. +For 1 ≤ i ≤ n, we have a(n algebraic) character ǫi : T → Gm that extracts +the i-th (diagonal) entry of T (k). We set P := �n +i=1 Zǫi and consider its subsets +P := +� +i=1 +Z≥0ǫi, +P+ := { +n +� +i=1 +miǫi ∈ P | m1 ≥ m2 ≥ · · · ≥ mn}. +For λ = �n +i=1 λiǫi ∈ P, we set |λ| := �n +i=1 λi ∈ Z. The permutation of indices +define Sn-actions on P and P. We set P+ := (P+ ∩ P) and identify it with the +set of partitions with its length at most n. The semi-group P+ is generated by +̟i := ǫ1 + · · · + ǫi +1 ≤ i ≤ n. +For λ ∈ P+, we may write λ ≫ 0 whenever all the expansion coefficients of +̟1, . . . , ̟n are sufficiently large. We may regard ̟i as a character of �T through +the projection to T . We set ρ := �n +i=1(n + 1 − i)̟i. Let ℘ and δ denote the +degree one character of Gc +m and Grot +m extended to �T trivially, respectively. We +define another (non-standard) lift of ̟i to �T as: +Λi := ̟i + ℘ +(1 ≤ i < n), +̟n + ℘ +(i = 0). +We set Iaf := {0, 1, . . ., (n − 1)}. We frequently identify 0 with n in the se- +quel, and hence {̟i}i is indexed by Iaf. Note that {̟i}i∈Iaf and {Λi}i∈Iaf +corresponds to each other by restriction. We set +Paf := +� n +� +i=1 +Z̟i +� +⊕ Z℘ ⊕ Zδ +and +P+ +af := ( +� +i∈Iaf +Z≥0Λi) + Z̟n + Zδ ⊂ Paf. +The set Paf is the character group of �T . +The set of positive roots ∆+ of G is ∆+ := {ǫi − ǫj}1≤i0, we find w ∈ Waf such that +λ + kΛ0 = wΛ ∈ P+ +af. +(1.3) +We set +D(k) +λ +:= Dw(kΛ) ≡ Lw(Λ) ⊂ L(Λ) +and call it the Demazure module (of level k). They are finite-dimensional �B- +modules, and does not depend on the choice of w in (1.3). +Definition 1.12. A finite-dimensional �B-module M is said to be D(k)-filtered +if it admits a finite filtration whose associated graded is the direct sum of De- +mazure modules of level k. +Theorem 1.13 (Joseph [22], see also [37, 26]). For each λ ∈ P and k ∈ Z>0, it +holds: +1. For each i ∈ Iaf, the module D(k) +λ +⊗ kΛi is D(k+1)-filtered; +2. For a D(k)-filtered module M and i ∈ Iaf, we have L<0Di(M) = 0 and +Di(M) is D(k)-filtered. +✷ +Proposition 1.14 (Joseph, see also [26] Lemma 4.1). For each λ ∈ P+ and +k ∈ Z>0, we have +gch D(k) +λ +∈ Z[q−1][X1, . . . , Xn], +where we set Xi = eǫi for 1 ≤ i ≤ n. +Proof. By the comparison of defining equation of D(k) +λ +([21, 3.4 Theorem], see +[15, Theorem 1] or [26, Proof of Lemma 4.1] for explicit equations), we find +that D(k) +λ +is a quotient of D(1) +λ . +Then, Dw0(D(1) +λ ) is the local Weyl module +whose highest weight is a monomial in X1, . . . , Xn as a character. Therefore, +the identification of local Weyl modules with the Garsia-Procesi modules ([16], +see also [25, 27]) through the Schur-Weyl functor ([14]) yields that +gch Dw0(D(1) +λ ) ∈ Z[q−1][X1, . . . , Xn]Sn. +Taking Theorem 1.11 2) into account, we deduce +gch Dw0(D(1) +λ ) ≥ gch D(1) +λ . +Therefore, we conclude the assertion. +Corollary 1.15. Let k ∈ Z>0 and w ∈ Waf. For a D(k)-filtered module M and +m ∈ Z≥0, we have L<0Dw(kmΛi ⊗ M) = 0, and the �B-module Dw(kmΛi ⊗ M) +is D(m+k)-filtered. +Proof. Apply Theorem 1.13 1) and 2) repeatedly. +10 + +1.6 +Lifting results +For a finite type scheme X over Z, the following is well-known: +Theorem 1.16 (see e.g. [5] §1.6). There exists a finite collection of primes +S ⊂ Z such that X ⊗Z Z[S−1] is flat over Z[S−1]. If a line bundle L on X +satisfies +H>0(X ⊗Z Fp, L ⊗OX OX⊗ZFp) = 0 +for a prime p ̸∈ S, then we have +H>0(X ⊗Z Q, L ⊗OX OX⊗ZQ) = 0. +2 +Frobenius splittings +Keep the setting of the previous section. +We additionally suppose char k = p > 0 throughout this section. Let L be +a free abelian monoid. For an L-graded commutative k-algebra R, we have a +k-algebra map +R = +� +a∈L +Ra +Fr +−→ +� +a∈L +Rpa ⊂ +� +a∈L +Ra = R +(2.1) +defined by the p-th power map on R (with its effect on R0 = k normalized to +be an identity by the action of AutFp(k)). This algebra map equips R in the +RHS of (2.11) with a R-module structure that we refer as R(1). +We set +ProjL R := +� +Spec R \ +� +0̸=a∈L +AnnR Ra +� +/(Gm)rank L, +(2.2) +where AnnR Ra := {f : R → R/I | I ∈ Spec R, f(Ra) = 0}. +Definition 2.1 (Frobenius splittings of a ring). Let L be a free abelian monoid. +A Frobenius splitting of a L-graded commutative k-algebra R is a R-module +map φ : R(1) → R such that φ ◦ Fr is an identity. +A L-graded ideal I ⊂ R, the ideal I is said to split compatibly if and only +if φ(I(1)) ⊂ I, where I(1) is the image of I under the isomorphism R ∼= R(1) of +k-vector spaces. +Assume that R admits a rational �T-action that preserves each Ra (a ∈ L). +A Frobenius splitting φ of R is said to be �T-invariant if φ(t · •) = tp · φ(•) for +each t ∈ �T(k). +Let S ⊂ Iaf. In case the algebra R admits an action of (L-degreewise) �B(S)- +action, then we say that φ is �B(S)-canonical if and only if φ is �T-invariant and +ui(−ap)φ(ui(a)f) +i ∈ S +(2.3) +is a polynomial of degree < p on a ∈ k for each f ∈ R. A �B-canonical Frobenius +splitting φ on R is B-canonical if Grot +m and Gc +m acts on R trivially. +Let X be a k-scheme. We have the (k-linear) Frobenius morphism Fr : X −→ +X that sends the local section of OX to its p-th power (up to rearrangement of +scalars). This induces a morphism +OX −→ Fr∗Fr∗OX. +11 + +Recall that a semi-ample line bundle L on a k-scheme X is a line bundle such +that L⊗m is base-point-free for m ≫ 0. +Definition 2.2 (Frobenius splittings of a variety). Let X be a k-scheme. The +Frobenius splitting φ : Fr∗Fr∗OX −→ OX is a morphism of quasi-coherent +sheaves such that the composition +OX −→ Fr∗Fr∗OX +φ +−→ OX +is the identity. A closed subscheme Y ⊂ X is called compatibly Frobenius split +(with respect to φ) if and only if we have φ(IY) ⊂ IY, where IY is the ideal +sheaf that defines Y. +Suppose that X admits a Frobenius splitting φ. For a surjective morphism +f : X −→ Y of k-schemes, we have an induced morphism +f∗(φ) : Fr∗Fr∗f∗OX −→ f∗OX +coming from the base change map +Fr∗Fr∗f∗OX −→ Fr∗f∗Fr∗OX ֒→ f∗Fr∗Fr∗OX. +Lemma 2.3. Let L be a free abelian monoid. If a L-graded k-algebra R admits +a �B-canonical Frobenius splitting φ, then so is the k-algebra +� +a∈L +Ra ⊗ kυ(a), +where υ : L → Paf is a monoid map. +Proof. It is enough to notice that (2.3) imposes no restriction on �T-weights. +Lemma 2.4 (Brion-Kumar [5] 1.1.8 Lemma). Let X be a variety equipped with +a Frobenius splitting φ. Let f : X −→ Y be a morphism of schemes such that +the natural map OY → f∗OX is an isomorphism. Let Z be a closed subscheme +of X and let W be the scheme-theoretic image of Z in Y. Identifying f∗OX with +OY, we have: +1. IW = f∗IZ; +2. If X admits a Frobenius splitting φX, then so is Y. If, in addition, Z is +compatibly split with φX, then W is compatibly split with f∗(φX). +✷ +Corollary 2.5. Keep the setting of Lemma 2.4. Assume that X and Y are pro- +jective varieties admitting �B-action, f is �B-equivariant, and φX is �B-canonical +in addition. We have: +1. The Frobenius splitting φY = f∗φX is �B-canonical; +2. In addition, if a �T-stable subvariety W ⊂ Y is compatibly split with Y +under φY, then the scheme-theoretic preimage f −1(W) ⊂ X is compatibly +split under φX. +12 + +Proof. For an ample line bundle L on Y, we have an inclusion f ∗L ⊂ L′ into +an ample line bundle on X. By [41], we can rearrange L and L′ if necessary to +assume that the both of L and L′ are �B-equivariant. It follows that L′ ⊗OX f ∗L +is a �B-equivariant line bundle on X and its space of global sections is a non- +zero finite-dimensional �B-module. Thus, we can choose 0 ̸= s ∈ Γ(X, L′ ⊗OX +f ∗L), that is a �B-eigensection. Renormalizing the �B-action on L′ by tensoring +with a �T-character if necessary, we can additionally assume that s is �B-fixed. +The section s gives a non-zero �B-equivariant morphism f ∗L −→ L′, that is +necessarily an inclusion (as X is integral). It induces a �B-equivariant embedding +of the section rings +� +m≥0 +Γ(Y, L⊗m) = Γ(X, (f ∗L)⊗m) ⊂ +� +m≥0 +Γ(X, (L′)⊗m), +(2.4) +that is preserved by φX by Lemma 2.4. This transplants the condition (2.3) +from the LHS to the RHS of (2.4), that is the first assertion. +For the second assertion, it suffices to notice that we can employ the pull- +backs of the �T-invariant sections of L that defines W ⊂ Y to define its scheme- +theoretic preimages in (2.4). +Theorem 2.6 ([1], cf. [5] 4.1.15 Theorem and 4.1.17 Proposition). For each w ∈ +Sn, the B-canonical Frobenius splitting of X(w) is unique (up to a character +twist in Lemma 2.3). In addition, this Frobenius splitting is compatible with the +embedding X(v) ⊂ X(w) for each w ≥ v ∈ Sn. +✷ +Proposition 2.7 (Brion-Kumar [5] 4.1.17 Proposition). Let X be a k-scheme +equipped with an action of �B, i ∈ Iaf and let φ denote a �B-canonical Frobenius +splitting on X. Then, there exists a �B-canonical splitting �φ on �Pi ×�B X that +restricts to φ via the inclusion +X ∼= �B × +�B X ⊂ �Pi × +�B X. +Corollary 2.8. Keep the setting of Proposition 2.7. We have a �B-equivariant +affine fibration +(�Pi × +�B X) \ X ∼= A1 × X −→ X +that admits a �B-canonical Frobenius splitting �φ obtained from the standard one +of A1 and φ by taking the product. +Proof. Let ˙s denote a lift of si ∈ Waf to �Pi. Then, our fibration is obtained by +applying ˙s to the action map +�Pi × +�B X ⊃ (Im ui) × X −→ X. +Here we twist the �B-action by ˙s, and hence our X in the middle term should +be regarded to admit an action of (�B ∩ Ad( ˙s)�B). Now we have a desired �B- +canonical Frobenius splitting of the LHS by Proposition 2.7 and [5, Exercise +4.1E 1)]. +13 + +Corollary 2.9. Let L be a free abelian monoid. Assume that a L-graded k- +algebra R admits a �B-canonical Frobenius splitting. For each i ∈ Iaf, the space +of global sections +� +a∈L +D† +i(Ra) = +� +a∈L +H0(�Pi/�B, Ei(Ra)), +admits a �B-canonical Frobenius splitting. +Proof. If we set X := Spec R, then the set of �B-canonical Frobenius splitting +of X injects into the set of �B-canonical Frobenius splitting of �Pi ×�B X that +is compatible with the closed subvariety X by Proposition 2.7. From this and +Lemma 2.4 2), we conclude the result. +Proposition 2.10. Let V0 be a d-dimensional Fp-vector space and let V ′ +0 ⊂ V0 +be its proper subspace. Let V and V ′ be the scalar extensions of V0 and V ′ +0 from +Fp to k, respectively. Then, the variety +P(V/V ′) +η +←− B(V, V ′) +π +−→ P(V ) +obtained by blowing up P(V ′) ⊂ P(V ) admits a Frobenius splitting φ with the +following properties: +1. There exist semi-ample line bundles OB(V,V ′)(1, 0) and OB(V,V ′)(0, 1) on +B(V, V ′) such that +R•η∗OB(V,V ′)(0, 1) ∼= OP(V/V ′)(1) +and +R•π∗OB(V,V ′)(1, 0) ∼= OP(V )(1); +2. π∗(φ) and η∗(φ) define Frobenius splittings of P(V ) and P(V/V ′), respec- +tively; +3. The Frobenius splitting φ is uniquely determined as the �T-invariant split- +ting if �T acts on V and V ′, and we have +V ∼= +d +� +j=1 +χi +such that +χi ̸∼= χj +if +i ̸= j; +4. Let S ⊊ Iaf. Fix a morphism of algebraic groups +�B −→ GL(|S|, k) +whose image is the upper-triangular part of GL(|S|, k). If V is a vector +representation of GL(|S|, k) and V ′ is a �T-stable subspace of V , then we +can rearrange φ if necessary to assume +• π∗φ is the unique �B(S)-canonical splitting of P(V ); +• In case �B(S′) preserves V ′ for S′ ⊂ S, then φ is a �B(S′)-canonical +splitting of B(V, V ′). +14 + +Proof. Let us find a Fp-basis +e1, e2, . . . , ec, ec+1, . . . , ed ∈ V0 +(2.5) +such that e1, . . . , ec spans V ′ +0. Consider the dual vectors ξ1, . . . , ξd ∈ V ∗ such +that ξi(ej) = δij. For r1, r2 ≥ 0, we define +R(r1,r2) := SpanFp{ξa1 +1 · · · ξad +d +| +d +� +j=1 +aj = r2 + r1, +d +� +j=c+1 +aj ≥ r2} ⊂ k[ξ1, . . . , ξd]. +We set L := Z2 +≥0 and form a L-graded k-algebras +R := +� +(r1,r2)∈L +R(r1,r2) ⊂ +� +r2≥0 +k[ξ1, . . . , ξd]e(r2) = k[ξ1, . . . , ξd, e(1)], +where the multiplication of both rings are coming from k[ξ1, . . . , ξd] with addi- +tional requirements +R(r1,r2)·R(r′ +1,r′ +2) ⊂ R(r1+r′ +1,r2+r′ +2) +and +e(r2)·e(r′ +2) = e(r2 +r′ +2) +(r1, r2) ∈ L. +These rings are integral and integrally closed by inspection. +We find B(V, V ′) ∼= ProjL R, and it admits line bundles OB(V,V ′)(r1, r2) such +that +Γ(B(V, V ′), OB(V,V ′)(r)) = Rr +r ∈ L. +(2.6) +Here, the ring k[ξ1, . . . , ξd, e(1)] admits a Frobenius splitting φ extending +ξa1 +1 · · · ξad +d e(r) �→ +� +ξa1/p +1 +· · · ξad/p +d +e(r/p) +(p|gcm(a1, . . . , ad, r)) +0 +(else) +. +(2.7) +The L-graded subring R inherits this Frobenius splitting as +d +� +j=c+1 +aj ≥ r2 +implies +d +� +j=c+1 +aj +p ≥ r2 +p . +The ring inclusions +� +r1≥0 +R(r1,0) ⊂ R ⊃ +� +r2≥0 +R(0,r2) +induce morphisms +P(V/V ′) +η +←− B(V, V ′) +π +−→ P(V ) +such that +R•η∗OB(V,V ′)(0, r2) ∼= OP(V/V ′)(r2) +and +R•π∗OB(V,V ′)(r1, 0) ∼= OP(V )(r1) +(2.8) +since η is a Pd−c−1-bundle and π is the blow-up along an ideal sheaf that defines +P(V ′) ⊂ P(V ). This is the first assertion. +In particular, the line bundle OB(V,V ′)(r) (r ∈ L) is itself base-point-free. +Thus, OB(V,V ′)(r) (r ∈ L) is semi-ample. The section rings of OB(V,V ′)(1, 0) and +OB(V,V ′)(0, 1) are polynomials rings, and hence it acquires a Frobenius splitting +from φ by (2.7). +15 + +As our construction has no ambiguity when we fix the �T-stable basis (2.5), +we find the third assertion. +The vector representation of GL(|S|, k) has multiplicity-free �T-weight spaces. +Hence, the �T-eigenbasis of V is unique (up to scalar and reordering). If (2.5) is +the unique �T-eigenbasis, then π∗φ gives rise to a �T-invariant Frobenius splitting +on P(V ). Such a Frobenius splitting is unique, and the Gm-action on V through +ui is the restriction of the direct sum of a vector representation of SL(2) and a +trivial representation to Gm. Hence, π∗φ is �B(S)-canonical. +In particular, the ring k[ξ1, . . . , ξd, e(1)] admits a �B(S)-canonical splitting, +that restricts to a �B(S′)-canonical splitting. As �B(S′) preserves R, we conclude +that φ must be �B(S′)-canonical. This is the fourth assertion. +These complete the proof. +Theorem 2.11. Let X be an integral projective k-scheme admitting �T-action, +equipped with a �T-invariant Frobenius splitting φ. Assume that we have a sur- +jective �T-equivariant morphism +f : X −→ P(V ), +where V is a (d + 1)-dimensional �T-representation over k. Suppose that V is +the direct sum of �T-characters without multiplicities, and φ0 is the �T-invariant +Frobenius splitting of P(V ) from Proposition 2.10. +If f∗OX = OP(V ) and f∗φ = φ0, and we have a �T-stable affine open sub- +set Ad+e ⊂ X (d, e ∈ Z≥0) such that f induces a �T-equivariant projection +Ad+e → Ad and φ restricts to an external tensor product of �T-invariant Frobe- +nius splittings of Ad and Ae, then the fiber product +X +π +←− X ×P(V ) B(V, V ′) +η +−→ B(V, V ′) +admits a Frobenius splitting for each �T-stable subspace V ′ ⊊ V that recovers φ +and φ0 by the pushforwards. +Proof. Consider the section rings +S := +� +r2≥0 +Γ(P(V ), OP(V )(r2)) ⊂ +� +r2,r3≥0 +Γ(B(V, V ′), OB(V,V ′)(r2, r3)) =: S+. +In view of the proof of Proposition 2.10, the localization with respect to the +�T-eigenfunctions yield +k[ξ1, . . . , ξd+1] = S ⊂ S+ ⊂ k[ξ1, . . . , ξd+1, e]. +(2.9) +We denote the most RHS of (2.9) by S♯. The Frobenius splitting �φ1 of S♯ given +by sending a monomial to its p-th root (see (2.7)) induces a Frobenius splitting +φ1 of S+, that recovers the Frobenius splitting φ0 on S by restriction. +We set L := Z2 +≥0, and form a multi-section ring +R := +� +(r1,r2)∈L +Γ(X, L⊗r1 ⊗ f ∗OP(V )(r2)) ⊃ +� +r2≥0 +Γ(P(V ), OP(V )(r2)) = S. +by the adjunction isomorphism +f∗f ∗OP(V )(r2) ∼= f∗OX ⊗OP(V ) OP(V )(r2) ∼= OP(V )(r2), +16 + +where L denotes a �T-linearized ample line bundle of X (that exists by [41]). +By inverting a �T-invariant section of Γ(X, L⊗r1) (r1 ≫ 0) that defines the �T- +invariant affine open set Ad+e, we obtain a ring R ⊂ Rloc that carries a Frobenius +splitting �φ′ +loc that restricts to the Frobenius splitting φ of the section ring R. +Here the complement of f(Ad+e) = Ad is a prime �T-invariant divisor. We can +assume that this divisor is defined by ξ1 without the loss of generality. We set +Sloc := S[ξ−1 +1 ], S+ +loc := S+[ξ−1 +1 ], and S♯ +loc := S♯[ξ−1 +1 ]. We have Sloc ⊂ Rloc. We +have algebra inclusions +R ⊗S S+ ⊂ R ⊗S S♯ ⊂ Rloc ⊗Sloc S♯ +loc +(2.10) +In view of the fact that the Frobenius splitting �φ1 is to extract the p-th power +parts, we find that the Frobenius splitting �φ′ +loc extends to a Frobenius split- +ting �φloc on (Rloc ⊗Sloc S♯ +loc) that recovers all previous Frobenius splittings by +restrictions. +By construction, (Rloc ⊗Sloc S♯ +loc) is a (Laurent) polynomial ring that is free +over S♯ +loc. By assumption, �φloc is the external tensor product of the Frobenius +splitting of three Laurent polynomial rings, one is given by Sloc, and another is +its complementary tensor factor in Rloc, and the other is k[e]. It follows that +�φloc respects the grading coming from the degrees of ξ1, . . . , ξd. Since we have +�φloc(R) ⊂ R, +�φloc(S+) ⊂ S+, +and +�φloc(S) ⊂ S, +we conclude that �φloc restricts to a Frobenius splitting of R ⊗S S+. +Here, �φloc respects the Z3 +≥0-graded structure of Rloc ⊗S S+ induced from +R, S, and S+ by construction. +Hence, we find that �φloc is compatible with +the graded ideals of (R ⊗S S+) corresponding to every proper ideal of the free +abelian monoid Z3 +≥0. +Therefore, we conclude that �φloc descends to a Frobenius splitting of the +section ring of +X ×P(V ) B(V, V ′) = ProjZ3 +≥0(R ⊗S S+). +As our Frobenius splitting restricts to the original Frobenius splittings φ, φ0 of +R, S ⊂ (R ⊗S S+), we conclude the last part of the assertion. +Definition 2.12 (Frobenius D-splitting). Let X be a variety over k equipped +with a Frobenius splitting φX. Let D be a Cartier divisor of X. We say φX a +D-splitting (with respect to the divisor D) if we have a morphism OX ֒→ OX(D) +that induces +Fr∗OX −→ Fr∗(OX(D)) −→ OX +such that the composition map is φX. A Frobenius D-splitting φX is compatible +with a subvariety Y if supp D does not contain an irreducible component of Y +and φX induces a D-splitting of Y with respect to the divisor (D ∩ Y). +Remark 2.13. 1) If we have a Frobenius sD-splitting for a Cartier divisor D and +a positive integer s, then it is also Frobenius D-splitting ([5, 1.4.2 Remark]); 2) +Let V be a k-vector space with a basis x1, . . . , xn. We set Hi := {xi = 0} for +1 ≤ i ≤ n. Then, a Frobenius Hi-splitting of P(V ∗) is given as +Γ(P(V ∗), OP(V ∗)(pm)) +xi +−→ Γ(P(V ∗), OP(V ∗)(pm + 1)) +φ→ Γ(P(V ∗), OP(V ∗)(m)), +17 + +where m ≥ 0 is an integer, and the map φ sends xif p �→ f for a monomial +f ∈ Γ(P(V ∗), OP(V ∗)(m)), and annihilates a monomial in Γ(P(V ∗), OP(V ∗)(pm+ +1)) that is not this type. +The corresponding original Frobenius splitting is +independent of the choice of i. +Theorem 2.14. Let X be a G-variety over k equipped with a G-equivariant +closed embedding +ψ : X −→ P(V1) × P(V2), +where V1 and V2 are rational G-modules. Assume that +1. pr1 ◦ ψ(X) = G/P for a parabolic subgroup B ⊂ P ⊂ G; +2. pr1 induces a smooth fibration τ : X → G/P; +3. one of its fiber +F := pr−1 +1 (P/P) ∩ ψ(X) ⊂ P(V2) +admits a Frobenius D-splitting with respect to the B-stable divisor +D2 := {v∗ = 0} +for a B−-eigenvector v∗ ∈ V ∗ +2 ; +4. the partial flag manifold G/P admits a Frobenius D-splitting with respect +to an ample divisor H on G/P. +Then, the variety X admits a Frobenius D-splitting with respect to +τ −1(H) + G.D2, +where G.D2 is the G-translate of D2. +Remark 2.15. It is well-known that all partial flag manifolds admit D-splittings +with respect to some ample divisor H ([5, 2.2.5 Theorem]). +Proof of Theorem 2.14. Note that we have +Fr = Fr1 ◦ Fr2 = Fr2 ◦ Fr1, +(2.11) +where Fr is the Frobenius morphism of X, Fr1 is the Frobenius morphism on +P(V1), and Fr2 is the Frobenius morphism of P(V2). We have a unipotent sub- +group U − ⊂ B− such that +U − × P/P → (U −P/P) ⊂ G/P +defines a Zariski open subset. Then, the Frobenius D-splitting of F induces a +relative D-splitting +(Fr2)∗OU−×F −→(Fr2)∗OU−×F (G.D2) −→ OU−×F +(2.12) +induced by the multiplication of the section (Fr2)∗v∗ pulled back to X. Here, +the first map in (2.12) is the restriction of the map defined over all of X and +yields +(Fr2)∗OX −→ (Fr2)∗OX(G.D2) −→ OX(G.D2). +(2.13) +In view of (2.12), the image of the composition map in (2.13) must land on a +OX-submodule I ⊂ OX(G.D2) that coincides with OX on U − × F. Since X is +18 + +normal, every local section of I regarded as a rational section of OX must have +a pole along +supp (G.D2) ∩ τ−1(G/P \ (U −P/P)), +that has codimension two in X. This is impossible, and hence we have I = OX. +Therefore, we obtain +(Fr2)∗OX −→ (Fr2)∗OX(G.D2) −→ OX. +(2.14) +By pulling back the Frobenius D-splitting +(Fr1)∗OG/P −→ (Fr1)∗OG/P (H) −→ OG/P +induced by a section s of an ample line bundle of G/P, we deduce +(Fr1)∗OX −→ (Fr1)∗OX(τ −1(H)) −→ OX. +(2.15) +Combining these two maps, we obtain +(Fr)∗OX −→ (Fr)∗OX(τ −1(H) + G.D2) −→ OX +induced by the external tensor product of the sections s and (Fr2)∗v∗ by (2.11) +as required. +Corollary 2.16. Keep the setting of Theorem 2.14. Let Y ⊂ G/P be a closed +subvariety that is compatibly split with the Frobenius D-splitting of G/P with +respect to H, and supp H does not contain Y. +Then, the variety τ −1(Y) is +D-split with respect to (τ −1(H) + G.D2) ∩ Y. +Proof. Note that τ(G.p) = G/P for each p ∈ supp D2, and hence (G.D2 ∩ +τ −1(Y)) is a divisor. We have a Frobenius D-splitting +Fr∗OY → Fr∗OY(H) → OY +by [5, 1.4.6 Lemma]. We combine the pullback of this under τ instead of (2.15) +and (2.14) to deduce the result by the locally trivial fibration structure of the +map τ (that separates out the local effect of two Frobenius splittings Fr1 and +Fr2). +Theorem 2.17 ([5] 1.4.8 Theorem). Let X be a proper scheme over k; let L +be a semi-ample invertible sheaf on X, and let D be an ample effective Cartier +divisor on X. +1. If X is D-split, then Hi(X, L) = 0 for all i ≥ 1; +2. If a closed subscheme Y is compatibly D-split, then the restriction map +H0(X, L) −→ H0(Y, L) is surjective, and Hi(Y, L) = 0 for all i ≥ 1. +Theorem 2.18 ([5] 2.2.3 Theorem). Let i be a sequence in Iaf. Then, X(i) has +a Frobenius D-splitting φ relative to an ample Cartier divisor. Moreover, it is +compatible with X(i′) for any subsequence i′ of i. +Theorem 2.19 ([5] 2.3.2 Theorem). For each w ∈ Sn, we have an ample B−- +stable divisor H corresponding to OX(ρ) on X that gives a Frobenius D-splitting +of X, compatible with X(w) for each w ∈ Sn. +✷ +19 + +3 +Construction of the variety XΨ +We employ the setting of §1. For each λ ∈ P+, we define +m1(λ) := λ1 − λ2, m2(λ) := λ2 − λ3, . . . , mn−1(λ) := λn−1 − λn, mn(λ) := λn. +For each 1 ≤ i < n, 1 ≤ e ≤ n, and m ∈ Z≥0, we define the composition functors +as: +Ci,e := (Di−1 ◦ Di−2 ◦ · · · ◦ De) +Ci,e(λ)(•) := (Di−1 ◦ Di−2 ◦ · · · ◦ De) (kme(λ)Λe ⊗ •), +where the indices of D are understood to be modulo n, and the number of D’s +in each definition is (i+n−e) in total when n ≥ e ≥ i, and (i−e) in total when +0 < e < i. +For a root ideal Ψ and 1 ≤ j ≤ ℓ(Ψ), we define +CΨ +j (λ, •) := +� +Cij(Ψ),ej(Ψ)(λ) ◦ Cij(Ψ),ej(Ψ)+1(λ) ◦ · · · ◦ Cij(Ψ),e(j+1)(Ψ)−1(λ) +� +(•). +We also set +λ(Ψ) := +d(Ψ)1 +� +j=1 +mj(λ)Λj. +Using these, we define +N Ψ +w (λ) := Dw +� +kλ(Ψ) ⊗ (CΨ +1 (λ) ◦ CΨ +2 (λ) ◦ · · · CΨ +ℓ(Ψ)(λ))(k) +� +, +and +M Ψ +w (λ) := Dw +� +km1(λ)Λ1 ⊗ (C1,e1(Ψ)(km2(λ)Λ2 ⊗ C2,e2(Ψ)(km3(λ)Λ3⊗ +C3,e3(Ψ)(· · · (kmn−1(λ)Λn−1 ⊗ Cn−1,en−1(Ψ)(km0(λ)Λ0) · · · ) +� +(3.1) +for each w ∈ Sn. +Lemma 3.1. Let 1 ≤ i < e ≤ n. If Dj(M) ∼= M for each 0 < j < i, then we +have Dj(Ci,e(M)) ∼= Ci,e(M) for each 0 ≤ j < i. +Proof. Let v′ and v be the longest elements in +⟨si−1, si−2, . . . , s1⟩ ⊂ ⟨si−1, si−2, . . . , s0⟩ ⊂ �Sn, +respectively, i.e. v′ is the longest element in Si and v is the longest element of +Si+1. We have +Ci,e ◦ Dv′ ∼= (Di−1 ◦ · · · ◦ D1 ◦ D0) ◦ (Dn−1 ◦ · · · ◦ De) ◦ Dv′ ∼= Dv ◦ Ci,e +by inspection using Theorem 1.10 2) and 4). Here we have sjv < v for 0 ≤ j < i. +Thus, we deduce Dj ◦ Dv ∼= Dv by Theorem 1.10 4) and 2). This yields the +result. +Corollary 3.2. Let 1 ≤ i < e ≤ e′ ≤ n. If Dj(M) ∼= M for each 0 ≤ j < i and +each e′ < j < n, then we have Dj(Ci,e(M)) ∼= Ci,e(M) for each 0 ≤ j < i and +e′ ≤ j < n. +20 + +Proof. Taking into account the fact that the affine Dynkin diagram of type A(1) +n−1 +is invariant under the cyclic rotation, we simply add (n − e′) to all the indices +(modulo n) to deduce the result from Lemma 3.1. +Lemma 3.3. Let 1 ≤ i < e ≤ n. For each e ≤ j < n or 0 ≤ j < i − 1, we have +Dj ◦ Ci,e ∼= Ci,e ◦ Dj+1. +Proof. By the isomorphism of functors +Dj ◦ Dj+1 ◦ Dj ∼= Dj+1 ◦ Dj ◦ Dj+1, +that is a special case of Theorem 1.10 4), the assertion reduces to the fact that +Di−1, . . . , Dj+2 commutes with Dj, and Dj−1, . . . , De commutes with Dj+1. +Corollary 3.4. Let 1 ≤ i < e < n. For each e ≤ e′ < n or 0 ≤ e′ < i − 1, we +have +Ci−1,e′ ◦ Ci,e ∼= Ci,e ◦ Ci,e′+1. +Proof. Apply Lemma 3.3 to Ci−1,e′ ◦ Ci,e ≡ Di−2 ◦ · · · ◦ De′ ◦ Ci,e repeatedly to +deduce +Di−2 ◦ · · · ◦ De′ ◦ Ci,e ∼= Ci,e ◦ Di−1 ◦ · · · ◦ De′+1, +that is equivalent to the assertion. +Theorem 3.5 (Blasiak-Morse-Pun [3]). Let Ψ ⊂ ∆+ be a root ideal. Assume +that w ∈ Sn is Ψ-tame. Then, we have +H(Ψ; λ; w) = +� +gch M Ψ +w (λ) +� +q�→q−1 +λ ∈ P+, +where H(Ψ; λ; w) is borrowed from [3, (2.2)]. +Remark 3.6. The grading shift in the convention of [3] arises from the effect +of the extended Dynkin diagram automorphism, and hence is irrelevant in our +current formulation. +Proposition 3.7. Let Ψ ⊂ ∆+ is a root ideal. Assume that w ∈ Sn is Ψ-tame. +Then, we have N Ψ +w (λ) ∼= M Ψ +w (λ) for each λ ∈ P+. +Example 3.8. Let us exhibit the contents of the proof of Proposition 3.7 in +Example 1.3. We need to transform +C2,3 ◦ C2,4 ◦ C2,5 ◦ C3,6 = (D1D0D5D4D3)(D1D0D5D4)(D1D0D5)(D2D1D0) +into +(D0D5D4D3)(D1D0D5D4D3)(D2D1D0)(D3D2D1)(D4D3D2D1) +(3.2) +by applying a character kΛ0 from the RHS, and let D3, D4, D5 act freely from +the LHS. Note that (3.2) can be transformed into +(D0D5D4D3)(D1D0D5D4D3)(D2D1D0) +(3.3) +as Di(kΛ0) = kΛ0 for i ̸= 0. Here, we have +Di(D1D0D5D4D3) = (D1D0D5D4D3)Di+1 +i = 3, 4, 5, 0. +21 + +This transforms (3.3) into +(D1D0D5D4D3)(D1D0D5D4)(D2D1D0) +(3.4) +For i = 3, 4, 5, we have +Di(D1D0D5D4D3)(D1D0D5D4) = (D1D0D5D4D3)(D1D0D5D4)Di+2. +Hence, the left actions of D3, D4, D5 allow us to rewrite (3.4) into +(D1D0D5D4D3)(D1D0D5D4)(D1D0D5)(D2D1D0) = C2,3 ◦ C2,4 ◦ C2,5 ◦ C3,6. +Proof of Proposition 3.7. By Theorem 1.10, we find an isomorphism L•Di(kΛj ⊗ +•) ∼= kΛj ⊗ L•Di(•) for distinct i, j ∈ Iaf. In addition, we have L•Di(k) ∼= k. +Using these two facts repeatedly, we shift the character twists to the left and +discards Di’s with trivial effects repeatedly to obtain +M Ψ +w (λ) ∼= Dw +� +kµ1 ⊗ (C1,e1(Ψ)(kµ2 ⊗ C2,e2(Ψ)(· · · (kµr−1 ⊗ Cr,er(Ψ)(kµr) · · · ) +� +, +(3.5) +where r = max{1 ≤ s < n | s + d(Ψ)s ≤ n} = iℓ(Ψ)(Ψ), and +µi = +ei+1(Ψ)−1 +� +j=ei(Ψ) +mj(λ)Λj. +In particular, µi ̸= 0 (i ≥ 1) only if i ∈ I(Ψ). For 0 ≤ i ≤ ℓ(Ψ), the value ek(Ψ) +is constant for ii(Ψ) < k ≤ ii+1(Ψ). +In the below (during this proof), we drop Ψ from the notation of numbers +presented by the typesetting fonts (i.e. i and e). We discard kµi with µi ≡ 0 +in (3.5). Then, we inductively transform the sequence of terms +kµi(j−1) ⊗ (Ci(j−1)+1,ej ◦ Ci(j−1)+2,ej ◦ · · · ◦ Cij,ej)(kµij ⊗ •), +(3.6) +that is a part of (3.5), into +kµi(j−1) ⊗ (Cij,ej ◦ Cij,ej+1 ◦ · · · ◦ Cij,ij−1)(kµij ⊗ •) +(3.7) +for each 1 ≤ j < ℓ(Ψ) provided if we can freely apply +Di(j−1)−1, Di(j−1)−2, . . . , Dej +(3.8) +to (3.6) without modifying the total output (3.5). For the initial case j = 1, the +functors in (3.8) arises from Dw since we have wsi < w for e1(Ψ) ≤ i < n, that +implies Dw ∼= Dw ◦ Di for e1(Ψ) ≤ i < n (and we have i0 = 0 by convention). +Note that each of the terms in (3.8) commute with kµi(j−1) since we have +i(j−1) − 1 < i(j−1) + di(j−1)(Ψ) = e(j−1) < ej, +by Lemma 1.5. +In particular, we can add each of (3.8) freely in front of +Ci(j−1)+1,ej in (3.6). Applying Lemma 3.3 repeatedly, this is the same as adding +each of +Dij−1, Dij−2, . . . , Dej+ij−i(j−1). +(3.9) +22 + +freely just after Cij,ej in (3.6) without modifying the output. In particular, we +can freely insert +Cij,ej+ij−i(j−1), . . . , Cij,ij−1 +(3.10) +just after Cij,ej in addition to (3.9). +By using Corollary 3.4 repeatedly, we have +Ci(j−1)+1,ej ◦ Ci(j−1)+2,ej ◦ · · · ◦ Cij,ej ∼= Cij,ej ◦ Cij,ej+1 ◦ · · · ◦ Cij,ej+ij−i(j−1)+1. +In conjunction with (3.10), we obtain (3.7). Here the product of C’s in (3.7) +gives a reduced expression of the longest element of +� +sij−1, . . . , s0, . . . , sej +� +⊂ �Sn. +(3.11) +Thus, we can add each of +Dij−1, Dij−2, . . . , De(j+1) +just after Cij,ij−1 in (3.7) without modifying the output. This proceeds the +induction on j. Hence, we can replace every (3.6) in (3.5) into (3.7) inductively. +The terms +Cij,e(j+1), Cij,e(j+1)+1, · · · , Cij,ij−1 +(3.12) +in (3.5) commutes with kµij , and can be moved to the component (3.7) for j +replaced with (j + 1). Then, each of (3.12) is the composition of Demazure +functors corresponding to simple reflection of (3.11) for j replaced with (j + 1) +(as ij < i(j+1)). Thus, we can delete them by using Theorem 1.10 4) and 2) +successively. +This procedure further replaces (3.6) in (3.5) with +kµi(j−1) ⊗ (Cij,ej ◦ Cij,ej+1 ◦ · · · ◦ Cij,e(j+1)−1)(kµij ⊗ •). +This is precisely the definition of CΨ +j (λ). +Therefore, we conclude the result. +Proposition 3.9. Let Ψ ⊂ ∆+ be a root ideal, w ∈ Sn, and λ ∈ P+. +1. The following vanishing of the total homology complex holds: +L<0Dw +� +kλ(Ψ) ⊗ (CΨ +1 (λ) ◦ CΨ +2 (λ) ◦ · · · CΨ +ℓ(Ψ)(λ))(k) +� += 0; +2. The module N Ψ +w (λ) has simple socle kλ as �B-modules. +Proof. The first assertion is the Leray spectral sequence applied to a repeated +application of Corollary 1.15. For the second assertion, we first see +M ⊂ Di(M) +i ∈ Iaf +(3.13) +for a D(k)-filtered module M for each k > 0. This holds for a Demazure module +of a fixed level k > 0 by Theorem 1.11 2), and the vanishing of L<0Di there +yields the general case by the exactness of the functor Di applied to D(k)-filtered +modules. +23 + +In (3.13), the module Di(M) is cocycic to M, i.e. every 0 ̸= v ∈ Di(M) +satisfies +Span +��Bv +� +∩ M ̸= {0}. +(3.14) +In addition, cocylicity of a �B-module is the same when we restrict to [�B, �B]- +module as the �T-action on N Ψ +w (λ) is semi-simple. Thus, we apply (3.13) and +(3.14) repeatedly to the construction of N Ψ +w (λ) using Corollary 1.15. As the our +construction begin from a one-dimensional �B-module that is twisted into kλ in +N Ψ +w (λ), we conclude the second assertion. +Corollary 3.10. Let Ψ ⊂ ∆+ be a root ideal, and let w ∈ Sn. Then, the +P+-graded vector space +� +λ∈P+ +N Ψ +w (λ)∗ +(3.15) +acquires the structure of P+-graded commutative k-algebra with �B-action. More- +over, the ring (3.15) is integral and generated by degree ̟i-parts for i ∈ Iaf. +Proof. Since the character twists of the modules N Ψ +w (λ) are linear with respect +to the monoid structure of P+, we apply Lemma 1.9 repeatedly to deduce the +first assertion from the definition of N Ψ +w (λ)’s. +Note that (3.15) is a section ring of a scheme X(i) for the sequence i of +elements of Iaf obtained by reading the definition of Nw(λ) from the left. Since +X(i) is integral, so its the ring (3.15). For λ, µ ∈ P+, we have the dual multi- +plication map +mλ,µ : N Ψ +w (λ + µ) −→ N Ψ +w (λ) ⊗ N Ψ +w (µ) +λ, µ ∈ P+, +that is a �B-module map. For each γ ∈ P+, the inclusion kγ ⊂ N Ψ +w (γ) is the +socle, and hence γ ∈ P+ ⊂ Paf is the largest �T -weight of N Ψ +w (λ) with respect to +≺. By the comparison of �T-weights, we see that the socle kλ+µ ⊂ N Ψ +w (λ + µ) +must be sent to +kλ ⊗ kµ ⊂ N Ψ +w (λ) ⊗ N Ψ +w (µ). +It follows that mλ,µ is injective, and hence the ring (3.15) is generated by its +primitive P+-degrees. This yields the second assertion. +For a root ideal Ψ ⊂ ∆+ and w ∈ Sn, we define a k-scheme +XΨ(w) := ProjP+ +� +λ∈P+ +N Ψ +w (λ)∗ +following (2.2) equipped with an action of �B. +Lemma 3.11. For a root ideal Ψ ⊂ ∆+ and w ∈ Sn, the scheme XΨ(w) is +integral and normal. +Proof. The homogeneous coordinate ring of XΨ(w) is obtained as a repeated +application of a character twist, inflation to P1 from a point by the SL(2)-action, +and taking global sections. Since each of the operation preserves the integrality +and normality, we conclude the result. +This defines a �B-equivariant line bundle OXΨ(w)(λ) on XΨ(w) for each λ ∈ P, +extended to λ ∈ P by taking the duals and tensor products. +24 + +Corollary 3.12. Let Ψ ⊂ ∆+ be a root ideal, and let w ∈ Sn. Then, we have +a �B-equivariant closed embedding +XΨ(w) ֒→ +� +j∈Iaf +P(N Ψ +w (̟j)) ֒→ +� +j∈Iaf +P(L(Λj)). +(3.16) +In particular, XΨ(w) is a variety. +Proof. Since Di(k) = k for i ∈ Iaf, we can omit unnecessary Demazure functors +in the definition of N Ψ +w (̟j) to find that N Ψ +w (̟j) is a Demazure module of +L(Λj). This yields the second embedding. In view of Corollary 3.10, collecting +the primitive degree part of (3.15) yields the first embedding. These imply the +first assertion. The second assertion follows from the first assertion and Lemma +3.11. +Lemma 3.13. Let Ψ ⊂ ∆+ be a root ideal, and let w ∈ Sn. We have +N Ψ +w (λ) ∼= N Ψ +wsi(λ) +λ ∈ P+, e1(Ψ) ≤ i < n. +Proof. It suffices to prove N Ψ +e (λ) ∼= N Ψ +si(λ) for each λ ∈ P+ and e1(Ψ) ≤ i < n +as the general case is obtained by the postcomposition of Dw. By Theorem +1.10 3) and Lemma 3.1, we find that CΨ +j (λ) in the definition of N Ψ +e (λ) can be +replaced with +� +CΨ +j (λ) ◦ Cij(Ψ),e(j+1)(Ψ) ◦ · · · Cij(Ψ),ij(Ψ)−1 +� +(•) +(3.17) +by using an induction on 1 ≤ j ≤ ℓ(Ψ) from the initial case j = ℓ(Ψ) using the +monotonicity of ij(Ψ) and ej(Ψ) as a function on j. We can further replace +(3.17) with +Dwj(kµj ⊗ •) +and +µj = +ej+1(Ψ)−1 +� +s=ej(Ψ) +ms(λ)Λs +without modifying the output, where +wj ∈ +� +sij(Ψ)−1, . . . , s0, . . . , sej(Ψ) +� += Sij+n−ej(Ψ)+1 ⊂ �Sn +is the longest element, using the commutativity of +kms(λ)Λs +(ej(Ψ) ≤ s < e(j+1)(Ψ)) +and +Cij(Ψ),t +(s < t ≤ n, 0 ≤ t < ij(Ψ)). +The final case j = 1 implies that +N Ψ +e (λ) ∼= Dw1(N Ψ +e (λ)) ∼= (Di ◦ Dw1)(N Ψ +e (λ)) ∼= Di(N Ψ +e (λ)) +for i = i1(Ψ) − 1, . . . , 0, . . . , e1(Ψ) as required. +Theorem 3.14. Let Ψ ⊂ ∆+ be a root ideal, and let wΨ +0 be the longest element +in +� +se1(Ψ), se1(Ψ)+1, . . . , s(n−1) +� +⊂ Sn. +The variety XΨ(wΨ +0 ) is smooth, and we have +dim XΨ(wΨ +0 ) = ℓ(wΨ +0 ) + |Ψ|. +(3.18) +25 + +For each e1(Ψ) ≤ k ≤ n, we have a �B-equivariant morphism +θk +Ψ : XΨ(wΨ +0 ) −→ +k +� +j=e1(Ψ) +P(L(Λj)), +(3.19) +that defines a smooth fibration over its image. +Proof. We drop w = wΨ +0 from the notation during this proof for the sake of +simplicity. Note that N Ψ +wΨ +0 (λ) ≡ N Ψ(λ) ∼= N Ψ +e (λ) by Lemma 3.13. Since the �T- +character twist does not change the ring structure of (3.15), we conclude that the +P(L(Λj))-component of the image of (3.16) is kvΛj ⊂ L(Λj) for 1 ≤ j ≤ d1(Ψ). +Thus, we obtain the �B-equivariant embedding (3.19) for k = n. +Note that +OXΨ(̟i) (i ∈ Iaf) is the pullback of O(1) from P(L(Λi)) in (3.16) by inspection. +We construct a sequence of varieties Y (r) (r ≥ 0) from Y (0) := [vΛ0] ∈ +P(L(Λ0)) by the following rule: +• We read the definition of N Ψ +e (λ) from the right, and count the appearance +of the Demazure functor Dj (j ∈ Iaf) and the character twist by kmjΛj: +1. If the (r + 1)-th operation is the character twist by kmjΛj, then we +take product with P(L(Λj)) by adding the new factor [vΛj ] to obtain +an isomorphic variety with bigger ambient space; +2. If the (r + 1)-th operation is the application of Dj (j ∈ Iaf), then we +set Y (r+1) := �PjY (r) in the ambient space. +Since �Pi ×�BY (r) is integral and proper, we find that Y (r+1) is in fact an integral +(closed) k-subscheme of the RHS of (3.19) for k = n. +As the definition of N Ψ +e (λ) consists of only finite steps, we obtain a closed +k-subscheme Y = � +r≥0 Y (r) of the RHS of (3.19). Here, the character twist +by Λi appears only once for each i ∈ Iaf, and we apply additional Demazure +functor only if e1(Ψ) ≤ i ≤ n. Therefore, we find a closed embedding +Y ֒→ +� +e1(Ψ)≤i≤n +P(L(Λi)). +Note that the application of each P0 enlarges the Gm-grading of a finite- +dimensional �B-module by finite amplitude. Hence, we deduce that Y is of finite +type. Thus, we have a line bundle OY (ǫi) for each 1 ≤ i ≤ n as the pullback of +O(1) from P(L(Λi)), and hence OY (λ) for λ ∈ P by their tensor product. +By examining the definition of the Demazure functors and the construction +of N Ψ(λ) (together with Corollary 3.10), we conclude that the both of XΨ and +Y are closed subschemes of a common scheme with the same set of closed points +and the both are integral and are of finite types. Therefore, we conclude that +XΨ ∼= Y as integral k-varieties equipped with �B-actions. +In the process of the construction of Y , we have a (consecutive sub)sequence +(of the definition of N Ψ(λ)) for each 1 ≤ j ≤ ℓ(Ψ): +Cij(Ψ),k(λ)(•) = (Dij(Ψ)−1 · · · D0◦· · ·◦Dk)(kmkΛk ⊗•) +ej(Ψ) ≤ k < ej+1(Ψ). +(3.20) +Here, we have ij(Ψ) < ej(Ψ) ≤ k. Thus, the Dynkin subdiagram constructed +by gathering indices in (3.20) yields the algebraic group SL(ij(Ψ)+n−k+1, k). +26 + +Therefore, the application of the corresponding �Pi’s to [vΛk] ∈ P(L(Λk)) yields +Pij(Ψ)+n−k. In particular, we have +dim XΨ ≥ +ℓ(Ψ) +� +j=1 +ej+1(Ψ)−1 +� +k=ej(Ψ) +(ij(Ψ) + n − k) +(3.21) +by counting the dimension of the base direction for each 1 ≤ j ≤ ℓ(Ψ). As the +RHS of (3.21) is precisely the number of Demazure functors appearing in the +construction of Y , and each application of �Pi for a �B-stable variety increases +the dimension at most one, we deduce +dim XΨ ≤ +ℓ(Ψ) +� +j=1 +ej+1(Ψ)−1 +� +k=ej(Ψ) +(ij(Ψ) + n − k). +(3.22) +The comparison of (3.21) and (3.22) implies +dim XΨ = +ℓ(Ψ) +� +j=1 +ej+1(Ψ)−1 +� +k=ej(Ψ) +(ij(Ψ) + n − k). +(3.23) +We have +ℓ(wΨ +0 ) = (n − e1(Ψ))(n − d1(Ψ)) +2 += +n +� +k=e1(Ψ) +(n − k) = +ℓ(Ψ) +� +j=1 +ej+1(Ψ)−1 +� +k=ej(Ψ) +(n − k) +(3.24) +and +|Ψ| = +n +� +i=1 +(n + 1 − ei(Ψ)) = +n +� +i=1 +(ei+1(Ψ) − ei(Ψ))i += +ℓ(Ψ) +� +j=1 +(ej+1(Ψ) − ej(Ψ))ij(Ψ) = +ℓ(Ψ) +� +j=1 +ej+1(Ψ)−1 +� +k=ej(Ψ) +ij(Ψ), +(3.25) +where the second equality comes from changing the row-wise counting to the +column-wise counting. +The comparison of (3.23), (3.24), and (3.25) implies +(3.18). +For each e1(Ψ) ≤ k ≤ n, we set Y (k) to be Y (r) such that the r-th step (the +last step) is the character twist by kmk−1Λk−1. Consider the condition +(⋆)k Y (k) ⊂ �n +s=k P(L(Λs)) is an irreducible and reduced subscheme stable +under the action of �Pi for k ≤ i ≤ n and 1 ≤ i < ij(Ψ) when ej(Ψ) ≤ +k < ej+1(Ψ). +We have Y (n+1) = Y (0) = pt ⊂ P(L(Λ0)), that is in fact invariant with respect +to G, and hence (⋆)n+1 holds. We have +Y (k) ∼= �Pij(Ψ)−1 �Pij(Ψ)−2 · · · �P0 · · · �PkY (k + 1) ⊂ +n +� +j=k +P(L(Λj)), +27 + +where j is borrowed from (⋆)k. By induction hypothesis, we find that Y (k + 1) +is stable under an individual applications of �Pij(Ψ)−1, . . . , �Pk+1. It follows that +we have a diagram +Y (k) +πk +←− G(k) ×P (k) Y (k + 1) −→ G(k)/P(k) ∼= Pij(Ψ)+n−k, +(3.26) +where πk is the action map, G(k) := GL(ij(Ψ) + n − k + 1, k) is the algebraic +group generated by Im ǫ1 and Ps for k ≤ s ≤ n and 1 ≤ s < ij(Ψ), and +P(k) is the algebraic subgroup of G(k) generated by Ps for k < s ≤ n and +1 ≤ s < ij(Ψ). In view of the discussion after (3.20), we find that πk is an +isomorphism of topological spaces. As the both of Y (k) and G(k)×P (k) Y (k+1) +define reduced subschemes of �n +s=k P(L(Λs)) (of finite types) with identical +closed points (as Y (k +1) is reduced), we conclude that πk is an isomorphism of +varieties. Therefore, Y (k) satisfies (⋆)k if Y (k + 1) satisfies (⋆)(k+1), and hence +(⋆) holds by the downward induction from k = (n + 1). +From these, we find that the induced map from XΨ to Im θk +Ψ is a fibration +whose fiber is isomorphic to Y (k + 1). As the both of the base and fiber are +successive projective space fibrations, we conclude that that θk +Ψ induces a smooth +fibration. In particular, XΨ is a smooth variety. +These complete the proof. +Corollary 3.15. Keep the setting of Theorem 3.14. Let Y (k) be the fiber of θk +Ψ +in (3.19) for ej(Ψ) ≤ k < ej+1(Ψ), where 1 ≤ j ≤ ℓ(Ψ). Set r := ij(Ψ) + n − +k + 1. We have: +1. There exist an embedding +GL(r)(k) ⊂ �G((z)) +such that ( �T (k) ∩ GL(r)(k)) gives rise to a maximal torus of GL(r), and +we have +GL(r)(k) ⊃ {sij(Ψ)−1, . . . , s1, s0, sn−1, . . . , sk} ⊂ Waf. +It gives rise to an action of GL(r) on Xk +Ψ through (3.19) that makes Xk +Ψ +into a GL(r)-equivariant Y (k + 1)-fibration over Pr−1; +2. The intersections +GL(r) ∩ �G, +and +GL(r)(k) ∩ �G− +define parabolic subgroups of GL(r) with their common Levi subgroup +GL(ij(Ψ)) × GL(n − k + 1) +that contains {si−1, . . . , s1, sn−1, . . . , sk}. +Proof. We borrow the convention of the proof of Theorem 3.14. The first as- +sertion paraphrases of (3.26). The second assertion follows as Im u0 is the only +one-parameter subgroup (corresponding to a simple positive root) of GL(r) that +has non-zero Gm-degree. +28 + +Corollary 3.16. Let Ψ ⊂ ∆+ be a root ideal, and let w ∈ Sn is Ψ-tame. We +have the following �B-equivariant closed embedding +XΨ(w) ֒→ +n +� +i=1 +P(L(Λi)). +(3.27) +The scheme XΨ(w) is a variety, and we have +dim XΨ(w) = ℓ(w) + |Ψ|. +(3.28) +The variety XΨ(w) is smooth if and only if X(w) is smooth. +Proof. Since w is Ψ-tame, we have w = vwΨ +0 for some v ∈ Sn such that ℓ(w) = +ℓ(v) + ℓ(wΨ +0 ). Let v = si1si2 · · · siℓ be a reduced expression of v. Recall that +each of the direct product component of the RHS of the closed embedding +XΨ(wΨ +0 ) ֒→ +n +� +i=e1(Ψ) +P(L(Λi)) +induces a line bundle corresponding to {̟i}n +i=e1(Ψ) by the pullback of O(1), and +their global sections gives rise to the primitive degree components of (3.15), that +generates the homogeneous coordinate ring of XΨ(w). In addition, applying D† +i +to the homogeneous coordinate ring of XΨ(w) yields that of XΨ(siw) when +siw > w. From these, we deduce that +XΨ(w) = �Pi1 �Pi2 · · · �PiℓXΨ(wΨ +0 ) ⊂ +n +� +i=1 +P(L(Λi)). +(3.29) +In particular, we have XΨ(siw) = �PiXΨ(w) when siw > w for i ∈ I. Moreover, +XΨ(w) must be irreducible. Therefore, we have +dim XΨ(siw) ≤ dim XΨ(w) + 1 +if +siw > w. +On the other hand, we have +dim �Pi1 �Pi2 · · · �Piℓ{[vi]}d1(Ψ) +i=1 += ℓ, +where we consider the actions in �d1(Ψ) +i=1 +P(L(Λi)). By the comparison of these +two, we conclude the dimension formula (3.28). Since each �Pi (e1(Ψ) ≤ i < n) +preserves XΨ(wΦ +0 ) by construction, the parabolic subgroup Q ⊂ G generated +by the Gm-fixed parts of �Pi (e1(Ψ) ≤ i < n) preserves XΨ(wΨ +0 ). Therefore, we +find an isomorphism +G ×Q � +{[vi]}d1(Ψ) +i=1 +× XΨ(wΨ +0 ) +� +≡ G ×Q XΨ(wΨ +0 ) ∼= XΨ(w0) ⊂ +n +� +j=1 +P(L(Λj)), +(3.30) +that projects onto G/Q through the projection +η : +n +� +j=1 +P(L(Λj)) −→ +d1(Ψ) +� +j=1 +P(L(Λj)). +29 + +By (3.29), we find that XΨ(w) projects onto BwQ/Q. Hence, we have +XΨ(w) ⊂ η−1(BwQ/Q) ∩ XΨ(w0) +by XΨ(w) ⊂ XΨ(w0). +Now the irreducibility of XΨ(w0) and the dimension +formula (3.28) forces XΨ(w) = η−1(BwQ/Q). +In particular, XΨ(w) is the +restriction of a smooth fibration to BwQ/Q ⊂ G/Q. Therefore, XΨ(w) is a +variety (as BwQ/Q ⊂ G/Q is a variety), and it is smooth if and only if BwQ/Q +is smooth. +These complete the proof. +Corollary 3.17. Let Ψ ⊂ ∆+ be a root ideal, and let w ∈ W be Ψ-tame. For +each λ ∈ P+, we have: +1. H>0(XΨ(w), OXΨ(w)(λ)) = 0; +2. H0(XΨ(w), OXΨ(w)(λ)))∗ ∼= N Ψ +w (λ) as �B-modules. +In particular, this +module admits a D(λ1)-filtration, and hence admits an excellent filtration +when regarded as a B-module in the sense of van der Kallen [42]. +Proof. We replace w with w(wΨ +0 )−1 by Lemma 3.13 to achieve +ℓ(wwΨ +0 ) = ℓ(w) + ℓ(wΨ +0 ) +with keeping N Ψ +w (λ) (λ ∈ P+) as the original. Let i be the sequences in Iaf +extracted from the definition of N Ψ +w (λ) (λ ∈ P+) by fixing a reduced expression +of w. In particular, the length ℓ of i is dim XΨ(w) by the definition of N Ψ +w (λ), +Lemma 3.13, and (3.23). +By construction (in the proof of Theorem 3.14), we obtain a �B-equivariant +surjection +π : X(i) −→ XΨ(w) +of varieties. We have π∗OX(i) = OXΨ(w) by Proposition 3.9. +We assume R>0π∗OX(i) ̸= 0 to deduce contradiction. We have +Hk′(XΨ(w), (Rkπ∗OX(i)) ⊗OXΨ(w) OXΨ(w)(λ)) = 0 +λ ≫ 0 +for each k′ > 0 by the Serre vanishing theorem. By the degeneration of the +Leray spectral sequence, we find that +H0(XΨ(w), (Rkπ∗OX(i)) ⊗OXΨ(w) OXΨ(w)(λ)) ̸= 0 +λ ≫ 0 +yields +Hk(X(i), π∗OXΨ(w)(λ)) ̸= 0 +λ ≫ 0, +that contradicts with Proposition 3.9. Therefore, we have necessarily +R>0π∗OX(i) = 0. +Again by Proposition 3.9, we find +Hk(XΨ(w), OXΨ(w)(λ))∗ ∼= +� +N Ψ +w (λ) +(k = 0) +0 +(k > 0) +λ ∈ P+. +30 + +This proves the first assertion. The �B-module N Ψ +w (λ) admits a D(λ1)-filtration +by applying Corollary 1.15 repeatedly to the definition of N Ψ +w (λ). Taking ac- +count into the fact that D(k) +λ +admits a D(k+1)-filtration for each k > 0 and D(k′) +µ +(µ ∈ P) is a Demazure module of G for k′ ≫ 0 (see [22, 37, 26]), we conclude +the second assertion. +Corollary 3.18. Let Ψ ⊂ ∆+ be a root ideal and w ∈ Sn is tame. For each +λ ∈ P+, we have +gch H0(XΨ(w), OXΨ(w)(λ)))∗ = +� +H(Ψ; λ; w) +� +q�→q−1. +Proof. Combine Corollary 3.17 with Theorem 3.5 using Proposition 3.7. +Corollary 3.19. Keep the setting of Corollary 3.10. If char k = p > 0, then +the P+-graded k-algebra +� +λ∈P+ +N Ψ +w (λ)∗ +admits a �B-canonical Frobenius splitting. +Proof. Apply Lemma 2.3 and Corollary 2.9 repeatedly from the case R = +� +a∈L k0 in accordance with the construction of the ring (3.15). +For each i ∈ I, we have an embedding V (̟i) ⊂ L(Λi) of �G-modules, that +can be also understood to be the Gm-fixed part of L(Λi). We also have a �G- +module embedding k ∼= V (0) ⊂ L(Λ0). These induce a �G-equivariant closed +embedding +� +i∈I +P(V (̟i)) ֒→ +� +i∈Iaf +P(L(Λi)). +Lemma 3.20. For a root ideal Ψ ⊂ ∆+ and w ∈ Sn that is Ψ-tame, the +intersection +XΨ(w) ∩ +n−1 +� +j=e1(Ψ) +P(V (̟i)) +is isomorphic to X(w). +Proof. Thanks to the construction of XΨ(wΨ +0 ) in the proof of Theorem 3.14, we +find that the image of the composition map +fi : XΨ(wΨ +0 ) ֒→ +n +� +i=1 +P(L(Λi)) −→ P(L(Λi)) +1 ≤ i ≤ n +satisfies (Im fi ∩ P(V (̟i))) = [v̟i] for 1 ≤ i ≤ d1(Ψ). +We prove: +(♠)k GL(n − d1(Ψ)){[v̟j]}k +j=e1(Ψ) = Im θk +Ψ ∩ �k +j=e1(Ψ) P(V (̟j)); +for k ≥ e1(Ψ), where GL(n − d1(Ψ)) ⊂ G is borrowed from Corollary 3.15 2). +The assertion (♠)e1(Ψ) follows as the image is the projective space homogeneous +under the action of GL(i1(Ψ) + n − d1(Ψ)), and its Gm-attracting fixed points +are homogeneous under the action of GL(n − d1(Ψ)) by Corollary 3.15. Then, +31 + +we examine the fiber structure offered by Corollary 3.15 repeatedly to deduce +that (♠)k+1 holds when (♠)k holds. +Therefore, the induction proceeds and (♠)k holds for each k ≥ e1(Ψ). In +particular, we have +XΨ(wΨ +0 ) ∩ +� +i∈I +P(V (̟i)) = X(wΨ +0 ). +We have XΨ(siw) = (�Pi ∩ G)XΨ(w) when w ∈ Sn is Ψ-tame and siw > w by +(3.29). Since (�Pi ∩ G) preserves � +i∈I P(V (̟i)), we conclude +XΨ(siw) ∩ +� +i∈I +P(V (̟i)) = (�Pi ∩ G)(XΨ(w) ∩ +� +i∈I +P(V (̟i))) +in this case. Therefore, we conclude the assertion by induction on w. +Theorem 3.21. For a root ideal Ψ ⊂ ∆+, the Gm-attracting set of X = +X(w0) ⊂ XΨ(w0) is open dense, and is isomorphic to T ∗ +ΨX. +Proof. Since XΨ(w0) is a connected smooth variety and X ⊂ XΨ(w0) is a +connected component of its Gm-fixed part, we find that the attracting locus +˚ +XΨ ⊂ XΨ(w0) is defined as the intersection of the product of attracting loci of +the spaces P(V (̟i)) ⊂ P(L(Λi)) (i ∈ Iaf) and XΨ(w0) through the embedding +(3.27). In particular, ˚ +XΨ(w0) is a Zariski open subset of XΨ(w0). +By Bia�lnyki-Birula’s theorem [2], we find that ˚ +XΨ is an affine bundle over +X, that admits an action of (G × Gm). By X ∼= G/B, we take a base point +p = X(e) = B/B. We have a direct sum decomposition +TpXΨ(w0) ∼= TpX ⊕ E, +(3.31) +where TpX admits trivial Gm-action and E has strictly negative Gm-degree. +Note that each direct summand of (3.31) is B-stable. In view of the fiber bundle +structure of XΨ(w0), the �T-character of E is calculated from the tangent space +the projective spaces +P(GL(ij(Ψ)+n−k+1)vΛk) ⊂ P(L(Λk)) +ej(Ψ) ≤ k < ej+1(Ψ), 1 ≤ j ≤ ℓ(Ψ), +borrowed from Corollary 3.15. In particular, we find +T[vΛk]P(GL(ij(Ψ) + n − k + 1)vΛk) ∼= +� � +k 0, the variety � +XΨ admits a Frobenius splitting that in- +duces the Frobenius splittings of XΨ constructed in Corollary 3.19, and the +B-canonical Frobenius splitting of X in Theorem 2.6 by the push-forward. +33 + +Proof of Theorem 4.1. During this proof, we drop the root ideal Ψ from the +notation of ej(Ψ), ej(Ψ), dj(Ψ), and ij(Ψ) (j ∈ Z) for the sake of simplicity +(with keeping them on wΨ +0 ). +For 1 ≤ k < n, let Qk ⊂ G denote the parabolic subgroup that is generated +by B and Id + kEi+1,i for k ≤ i < n. +We borrow the notation from Corollary 3.15. We set +G(k) := GL(ij + n − k + 1) +and define six algebraic subgroups +L[k], Q[k], Q[k], Q(k), G[k], G[k] ⊂ G(k) +for each e1 ≤ k ≤ n as follows: We first find a unique 1 ≤ j ≤ ℓ(Ψ) such that +ej ≤ k < ej+1. We set G[k] = GL(n − k + 1), that is the second factor of +the Levi subgroup in Corollary 3.15. We set G[k] = GL(ij + 1), that contains +the first factor of the Levi subgroup in Corollary 3.15, as well as Im ǫk, and +it commutes with Im ǫr for k < r ≤ n. We also set L[k] := GL(ij) coming +from the first factor of the Levi subgroup in Corollary 3.15. We set Q(k) := +ZG(k)(ǫk)(�B ∩ G(k)), +Q[k] := ZG[k](ǫk)(�B ∩ G[k]) ⊂ G[k], +and +Q[k] := L[k](�B ∩ G[k]) ⊂ G[k]. +We have (Q[k] ∩ Q[k]) = Gm, G[k]/Q[k] ∼= Pij, G[k]/Q[k] ∼= Pn−k, +(Gm × L[k]) = (Q[k] ∩ G), +G[k]B = Qk, +G[k] ⊃ G[k+1], +and +L[k] ⊂ L[k+1] +for each 1 ≤ k < n by inspection. +We have Q[k], Q[k] ⊂ Q(k) that induce +embeddings +G[k]/Q[k] ֒→ G(k)/Q(k) ←֓ G[k]/Q[k], +(4.2) +where each space is of the form P• and +(G[k]/Q[k]) ∩ (G[k]/Q[k]) = {[vΛk]} ∈ G(k){[vΛk]} ⊂ P(L(Λk)) +by Corollary 3.15. +Let Y (k) denote the fiber of the map θk +Ψ presented at (3.16) along the point +{[vΛj]}k +j=1 for 1 ≤ k < n (notation borrowed from the proof of Theorem 3.14). +In view of (3.30), we have a surjective map +XΨ(w0) −→ G/Qe1 ⊂ +d1 +� +j=1 +P(V (̟j)), +whose fiber is XΨ(wΨ +0 ). +For each (d1 + 1) = e1 ≤ k ≤ n, we construct a variety X[k] +Ψ equipped with +a commutative diagram +X[k−1] +Ψ +X[k] +Ψ +ηk +� +πk � G/Qk� � +� �k−1 +j=1 P(V (̟j)) +Z[k−1] +�� +ık−1 +� +Z[k] +η′ +k +�� +�� +ık +� +(4.3) +34 + +such that (πk, ık) forms a G-equivariant fibration, and +Z[k] := G[e1]×Q[e1] � +G[e1+1]×Q[e1+1] � +· · ·×Q[k−1] � +G[k]×Q[k] Y (k+1) +�� +· · · +� +(4.4) +is the fiber bundle induced from the Cartesian diagram +G[k] ×Q[k] Y 0(k) +� +� +B(V (k), V −(k)) +� +� P(V (k)/V −(k)) = G[k]/Q[k] +Y (k) +� G(k)/Q(k) = P(V (k))� � +� P(L(Λk)) +, +(4.5) +where V (k) := G(k)vΛk ⊂ L(Λk) is a vector subspace equipped with a linear +�T-action inherited from L(Λk), and V −(k) denote the (strictly) negative Grot +m - +degree part of V (k). Here we understand that Z[d1] = Y (d1) = XΨ(wΨ +0 ). Since +the fiber of B(V (k), V −(k)) is G[k]/Q[k], we conclude that +Y 0(k) ∼= G[k] ×Q[k] Y (k + 1). +Here, (4.5) is precisely the construction in Proposition 2.10 applied to the +map associated to the sections of the semi-ample line bundle OY (k)(ǫk). +We examine whether (4.5) induces (4.4) for k provided if we have (4.4) for +(k − 1). In view of the fiber bundle structure of XΨ, we can reorganize the +embedding +XΨ(wΨ +0 ) ֒→ +n +� +k=e1 +P(L(Λi)) +(4.6) +(borrowed from Theorem 3.14) into +XΨ(wΨ +0 ) ֒→ P(L(Λe1))× +n +� +k=e1+1 +G(e1)×Q(e1) � +· · · G(k−1)×Q(k−1) P(L(Λk) +� +· · · +� +(4.7) +through the map obtained by replacing each G(k) and Q(k) with G(k) := G(k)�B +and Q(k) := Q(k)�B (e1 ≤ k ≤ n) and taking product +G(e1) ×Q(e1) � +· · · G(k − 1) ×Q(k−1) P(L(Λk) +� +· · · +� +−→ P(L(Λk)) +e1 < k ≤ n. +(4.8) +Taking into account the fiber space structure of XΨ(wΨ +0 ), we can replace P(L(Λk)) +in (4.7) with V (k) since it is stable under the action of G(k) (and P(j−1) ⊂ G(j) +for each e1(Ψ) < j ≤ n). We restrict this to Z[k−1] by assuming (4.4), and obtain +an embedding +Z[k−1] ֒→ G[e1]×Q[e1]� +· · ·×Q[k−1] +n +� +j=k +G(k)×Q(k)· · · +� +G(j−1)×Q(j−1)P(V (j)) +� +· · · +� +. +(4.9) +Consider the unipotent part U [m] ⊂ Q[m] (e1 ≤ m < k) that is stable by the +Levi part (Gm × L[m]). We examine the U [m]-action on P(V −(j)) ⊂ P(L(Λj)) +(k ≤ j ≤ n). Here the set of �T-weights of U [m] is contained in +{−ǫt + ǫm + δ | 1 ≤ t < m}, +(4.10) +35 + +while the set of �T-weights of V −(j) is contained in +{Λj + ǫs − ǫj − δ | 1 ≤ s < j}. +(4.11) +The sum of (4.10) and (4.11) yields +Λj + ǫa + ǫb − ǫc − ǫj +1 ≤ a, b, c < j, +whose restriction to T is not a weight of V (̟j), that has shape +ǫi1 + ǫi2 + · · · + ǫij +i1 < i2 < · · · < ij. +It follows that the action of U [m], as well as its twist by L[j−1], on P(V (j)) +is trivial for m < j. Therefore, replacing Y (k) with G[k] ×Q[k] Y 0(k) in (4.5) +defines a k-scheme +G[e1] ×Q[e1] � +· · · ×Q[k−1] � +G[k] ×Q[k] Y 0(k) +� +· · · +� +(4.12) +that maps to Z[k−1]. Since the actions of U [e1], . . . , U [k−1] on Y 0(k) are trivial, +we conclude that +(4.12) ∼= G[k] ×Q[k] +� +G[e1] ×Q[e1] � +· · · ×Q[k−1] Y 0(k) +� +∼= G[k] ×Q[k] +� +G[e1] ×Q[e1] � +· · · ×Q[k−1] (G[k] ×Q[k] Y (k + 1)) +�� +. +In particular, we have a fiber bundle structure on (4.12) with its fiber iso- +morphic to Z[k] and its base isomorphic to G[k]/Q[k]. From this description, +we conclude that Z[k] is also a variety. Therefore, induction proceeds and we +obtain a variety Z[k] for e1 ≤ k ≤ n. +Now we inflate Z[n] from Qd1 to G to obtain a desired variety X♯ +Ψ equipped +with the maps +XΨ +ηΨ +←− X♯ +Ψ +πΨ +−→ X, +since X♯ +Ψ is a successive blow-up of XΨ described in (4.5), and X is a successive +G[k]/Q[k]-fibrations (e1 ≤ k ≤ n) over G/Qd1. By construction, we have a closed +embedding +X♯ +Ψ ֒→ +�� +i∈I +P(V (̟i)) +� +× +� � +i∈Iaf +P(L(Λi)) +� +, +that establishes the first assertion (and particularly the projectivity of X♯ +Ψ). +Our blow-ups are (G × Gm)-equivariant as being the inflation of (B × Gm)- +equivariant blow-ups. In addition, the blow-up center is understood to be the +strictly negative Gm-degree parts of P(L(Λk)) for e1 ≤ k ≤ n (up to G[k]- +actions). +Therefore, the exceptional locus of ηΨ does not contain the Gm- +attracting sets of X ⊂ XΨ(w0). These yields the second assertion. +Being a (successive) blow-up, we have +R•(ηΨ)∗OX♯ +Ψ +∼= OXΨ. +Since πΨ is a fibration over X whose fiber is connected and projective by (4.4) +and X♯ +Ψ is a variety, we conclude that +R•(πΨ)∗OX♯ +Ψ +∼= OX. +36 + +These complete the proof of the third assertion. +To see the Frobenius splitting of X♯ +Ψ(w0), it suffices to find a �B-canonical +Frobenius splitting of each of Z[k] for d1 ≤ k ≤ n. The case of Z[d1] = XΨ(wΨ +0 ) is +Corollary 3.19. Assume that Z[k] inherits a Frobenius splitting from its ambient +space Z[k−1]. Then, the resulting Frobenius splitting must be �B-canonical. Since +the image of the morphism +F(j; k) := G[j] ×Q[j] � +G[j+1] ×Q[j+1] � +· · ·×Q[k−1] Y (k) +� +· · · +� +−→ P(L(Λj)) (4.13) +lands on P(V −(j) ⊕ kvΛj) for each e1 ≤ j ≤ k, if the Frobenius splitting φj,k of +the LHS of (4.13) is �T-invariant and its push-forward yields the �T-invariant +Frobenius splitting of P(L(Λj)) (cf. +Proposition 2.10), then the subvariety +F(j +1; k) is compatible with φj,k by Corollary 2.5. In particular, the construc- +tion of the Frobenius splitting of Z[k] reduces to that of (4.12) by (a repeated +application of) Proposition 2.7. Let i be the sequence of simple reflections used +to construct Y (k) through the applications of �P’s (in the proof of Theorem +3.14). It consists of two subsequences i1 and i2 such that i1 gives the action +of G(k), and i2 gives Y (k + 1). +Then, we have a �B-equivarinant birational +projective morphism +X(i) −→ Y (k), +that induces Y (k) a �B-canonical Frobenius splitting from X(i). In view of the +P•-fibration structure of Y (k) provided in Corollary 3.15, we find a �B-stable +affine open subset U ⊂ X(i) that maps isomorphically to an affine open subset +of Y (k), that we refer as U. The space U inherits a �B-canonical splitting from +X(i), and it is compatible with the �B-canonical Frobenius splitting of X(i2) +through its inflation to X(i) (obtained by using Corollary 2.9). Thus, we apply +Theorem 2.11 to equip G[k] ×Q[k] Y 0(k) in (4.5) with a �B-canonical splitting +compatible with Y 0(k). Therefore, we conclude that each Y (k) (d1 ≤ k ≤ n) +admits a �B-canonical splitting that is compatible with Y 0(k). +Inducing the �B-canonical Frobenius splitting of Z[k], we find a �B-canonical +Frobenius splitting φ♯ of X♯ +Ψ. Being a lift of XΨ, the push-forward of φ♯ re- +covers the �B-canonical Frobenius splitting of XΨ offered in Corollary 3.19. By +construction, (πΨ)∗φ♯ yields a �B-canonical splitting of X that is unique by The- +orem 2.6. These completes the proof of the third item, and hence the proof of +Theorem 4.1 is complete. +Lemma 4.2. Let Ψ be a root ideal. +We have an embedding π−1 +Ψ (X(e)) ⊂ +XΨ(wΨ +0 ) of varieties. +Proof. Since the construction of X♯ +Ψ is via the blow-up along strictly negative +Gm-degree part, we find that fixing the image under πΨ restricts our blow- +ups along smooth hyperplanes. +This does nothing, and hence we find that +π−1 +Ψ (X(e)) ⊂ XΨ. By (3.30), it is contained in XΨ(wΨ +0 ) as required. +Corollary 4.3. Keep the setting of Theorem 4.1, and assume that char k = +p > 0. Then, the preimage π−1 +Ψ (X(e)) admits a Frobenius splitting compatible +with the boundary of n(Ψ) ⊂ π−1 +Ψ (X(e)). In addition, the boundary of n(Ψ) ⊂ +π−1 +Ψ (X(e)) supports an ample Cartier divisor ∂Ψ that splits compatibly. +37 + +Proof. We apply Corollary 2.5 to πΨ to obtain a �B-canonical Frobenius splitting +of π−1 +Ψ (X(e)). We fix 0 ≪ λ ∈ P+. +The line bundle η∗OXΨ(λ) restricts to a very ample line bundle on π−1 +Ψ (X(e)) +by Lemma 4.2. Here +n(Ψ) = (π−1 +Ψ (X(e)) ∩ T ∗ +ΨX) +is a �T-invariant Zariski open subset with a unique fixed point. Hence, we have +a �T-eigensection s ∈ Γ(XΨ, OXΨ(λ)) that is not zero at 0 ∈ n(Ψ). Since the +support U of s contains a �T-invariant open neighbourhood of 0, we find that +n(Ψ) ⊂ U. If n(Ψ) ⊊ U, then we have +k[U] ⊂ k[n(Ψ)]. +It follows that a maximal ideal of k[U] is contained in a maximal ideal of k[n(Ψ)], +that contradicts with n(Ψ) ⊊ U. Therefore, we have necessarily +n(Ψ) = U. +Therefore, the �T-invariant divisor defined by s yields an effective Cartier divisor +that is fully supported on (π−1 +Ψ (X(e)) ∩ T ∗ +ΨX). In view of the fact that all the +T -weights of n(Ψ) belongs to ∆+ that spans a cone in P ⊗Z R, we deduce that +our section s must be unique (up to scalar) after the restriction to π−1 +Ψ (X(e)). +It follows that the ring map +k[U] = +� +m≥0 +1 +sm Γ(π−1 +Ψ (X(e)), Oπ−1 +Ψ (X(e))(mλ)) ←− +� +m≥0 +Γ(Oπ−1 +Ψ (X(e))(mλ)) +(4.14) +intertwines our Frobenius splitting. In particular, the LHS of (4.14) induces a +filtration of the RHS that is compatible with the Frobenius splitting. Hence, +the divisor on π−1 +Ψ (X(e)) afforded by s is compatibly split with our �B-canonical +splitting as required. +For each w ∈ Sn, we set +∂Ψ(w) := (G ×B ∂Ψ) ∩ π−1 +Ψ (X(w)) ⊂ π−1(X(w)) =: X♯ +Ψ(w), +where we renormalize the �T-action on ∂Ψ so that the �T-invariant section of +H0(X♯ +Ψ(e), OX♯ +Ψ(e)(∂Ψ)) that defines nΨ have �T-weight zero. +Lemma 4.4. For a root ideal Ψ ⊂ ∆+ and w ∈ Sn that is Ψ-tame, we have a +(G × Gm)-equivariant birational projective morphism +X♯ +Ψ(w) −→ XΨ(w). +Proof. Since T ∗ +ΨX ⊂ X♯ +Ψ is an open dense G-equivariant embedding, we know +that +π−1 +Ψ (X(w)) ∩ T ∗ +ΨX = π−1 +Ψ (X(w)). +Here ηΨ is proper, and hence +ηΨ(π−1 +Ψ (X(w)) ∩ T ∗ +ΨX) ⊂ ηΨ(π−1 +Ψ (X(w)) ∩ T ∗ +ΨX) = ηΨ(π−1 +Ψ (X(w))). +This particularly implies that X♯ +Ψ(w) and XΨ(w) shares a Zariski open sub- +set, and hence the map is birational. It is projective as ηΨ is projective by +construction. +38 + +We set +OX♯ +Ψ(w)(λ) := η∗ +ΨOXΨ(w)(λ) +w ∈ Sn, λ ∈ P. +By Corollary 3.24 and Theorem 4.1, we find that +j∗ +ΨOX♯ +Ψ(w)(λ) ∼= j∗ +Ψπ∗ +ΨOX(w)(λ) +w ∈ Sn, λ ∈ P, +where jΨ : T ∗ +ΨX ⊂ X♯ +Ψ is the natural embedding. +Theorem 4.5 (Conjecture 3.4 (ii) in [3]). For a root ideal Ψ ⊂ ∆+ and w ∈ Sn, +we have +H>0(X♯ +Ψ(w), OX♯ +Ψ(w)(λ + m∂Ψ(w))) = 0 +λ ∈ P+, m ∈ Z≥0. +In particular, we have +H>0(T ∗ +ΨX(w), π∗ +ΨOX(w)(λ)) = 0 +λ ∈ P+. +Proof. The line bundle OX♯ +Ψ(e)(λ) (λ ∈ P+) is semi-ample by construction. The +relative ample divisor ∂Ψ(e) contains a canonical �T-eigensection +θ ∈ H0(X♯ +Ψ(e), OX♯ +Ψ(e)(∂Ψ(e))) +that is the cocyclic vector with respect to the B-action. By Lemma 4.2, we can +assume that ∂Ψ(e) is the restriction of OXΨ(wΨ +0 )(Ω), where we set +Ω := +n +� +j=e1(Ψ) +̟j, +and θ is the image of the cocyclic �B-eigenvector through the restriction map +H0(XΨ(w0), OXΨ(w0)(Ω)) −→ H0(X♯ +Ψ(e), OXΨ(wΨ +0 )(Ω)|X♯ +Ψ(e)). +Our construction of X♯ +Ψ(e) is by a successive construction of projective bundles, +and each of ̟i corresponds to degree a one line bundle on a projective space. +More precisely, each fiber structure is obtained as the G[k]-equivariant fibration +over the projective space G[k]/Q[k] (d1(Ψ) ≤ k ≤ n). In view of the fact that +our Frobenius splitting is �T-invariant, our Frobenius splitting must be identical +to the one obtained from a successive application of Theorem 2.14, together +with Remark 2.13. In particular, X♯ +Ψ(e) is Frobenius D-split with respect to the +B-stable ample effective Cartier divisor ∂Ψ(e). +In view of the fact that X admits a Frobenius D-splitting with respect to +an ample B−-stable section of OX(ρ) (Theorem 2.19), we apply Theorem 2.14 +once again to conclude that X♯ +Ψ(w0) is D-split for an ample effective Cartier +divisor corresponding to the line bundle +η∗ +ΨOXΨ(w0)(Ω) ⊗OX♯ +Ψ(w0) π∗ +ΨOX(ρ). +As our Frobenius D-splitting of X is compatible with the Schubert varieties, +we apply Corollary 2.16 to deduce that X♯ +Ψ(w) is D-split for an ample effective +Cartier divisor corresponding to the line bundle +η∗ +ΨOXΨ(w0)(Ω) ⊗OX♯ +Ψ(w0) π∗ +ΨOX(ρ) ⊗OX♯ +Ψ(w0) OX♯ +Ψ(w). +39 + +Now, we apply Theorem 2.17 to OX♯ +Ψ(w)(λ + m∂Ψ(w)) to obtain the first +assertion when char k > 0. Since our construction of the varieties are flat over +Z, we apply Theorem 1.16 to transplant into the case of char k = 0 as well. For +the second assertion, note that the embedding T ∗ +ΨX(w) ⊂ X♯(w) is affine as the +embedding n(Ψ) ⊂ π−1(X(e)) is an affine open subset of a smooth projective +variety whose complement supports an ample Cartier divisor. We have +H>0(T ∗ +ΨX(w), π∗ +ΨOX(w)(λ)) = lim +−→ +m +H>0(X♯ +Ψ(w), OX♯ +Ψ(w)(λ + m∂Ψ(w))) = 0 +by the commutation of the cohomology with direct limits ([19, III, Proposition +2.9]). +Remark 4.6. Theorem 4.5 is conjectured in [3, Conjecture 3.4 (ii)] as a unifica- +tion of many previous results in the literatures: Theorem 4.5 for Ψ = ∅ is a part +of the Demazure character formula ([13, 1, 21]), and Theorem 4.5 for Ψ = ∆+ +and w = w0 is the Broer’s vanishing theorem [6, 29]. For relatively large λ, +Theorem 4.5 for w = w0 follows from Panyshev’s vanishing theorem [38], as +explained in [3, §3]. +Corollary 4.7. Keep the setting of Theorem 4.5. For each w′ ∈ Sn such that +X(w′) ⊂ X(w), the natural restriction maps +H0(X♯ +Ψ(w), OX♯ +Ψ(w)(λ)) −→ H0(X♯ +Ψ(w), OX♯ +Ψ(w′)(λ)) +and +H0(T ∗ +ΨX(w), π∗ +ΨOX(w)(λ)) −→ H0(T ∗ +ΨX(w′), π∗ +ΨOX(w′)(λ)) +λ ∈ P+ +are surjective. +Proof. For the first assertion, we apply Theorem 2.17 by the compatible Frobe- +nius D-splittings offered in the proof of Theorem 4.5. +The second assertion +follows by the commutative diagram +H0(T ∗ +ΨX(w), π∗ +ΨOX(w)(λ)) +� H0(T ∗ +ΨX(w′), π∗ +ΨOX(w)(λ)) +H0(X♯ +Ψ(w), OX♯ +Ψ(w)(λ + m∂Ψ(w′))) +�� +� +� � H0(X♯ +Ψ(w′), OXΨ(w′)(λ + m∂Ψ(w′))) +�� +� +for each λ ∈ P+ and m ≥ 0 combined with the isomorphism +H0(T ∗ +ΨX(w), π∗ +ΨOX(w)(λ)) = lim +−→ +m +H0(X♯ +Ψ(w), OX♯ +Ψ(w)(λ + m∂Ψ(w))) +obtained by the commutation of the global section functors and direct limits. +5 +Consequences +Keep the setting of the previous section. +Proposition 5.1. Let Ψ′ ⊂ Ψ ⊂ ∆+ be root ideals, and let w′, w ∈ Sn be Ψ- +tame elements such that X(w′) ⊂ X(w). Then, we have an inclusion X′ +Ψ(w′) ⊂ +XΨ(w) that induces a surjection +H0(XΨ(w), OXΨ(w)(λ)) −→ +→ H0(XΨ′(w′), OXΨ(w′)(λ)) +λ ∈ P+. +40 + +Proof. Note that a Ψ-tame element is automatically Ψ′-tame, and hence w′ is +Ψ′-tame. Recall that the homogeneous coordinate ring of XΨ(w) is written as +� +λ∈P+ +(M Ψ +w (λ))∗ ≡ +� +λ∈P+ +(N Ψ +w (λ))∗ +by Proposition 3.7. By interpreting (3.1) as successive applications of Demazure +functors together with the character twists, Corollary 2.9 implies that we have +a sequence i of elements of Iaf of length ℓ such that +(N Ψ +w (λ))∗ = H0(X(i), Lλ), +where X(i) is defined in (1.2), and {Lλ}λ∈P+ is a collection of �B-equivariant +line bundles on X(i) such that +Lλ ⊗ Lµ ∼= Lλ+µ +λ, µ ∈ P+. +Examining the sequence offered in (3.1), we have its subsequence i′ that realizes +N Ψ′ +w′ (λ). By Theorem 2.18 and Theorem 2.17, we have a surjection +N Ψ +w (λ)∗ = H0(X(i), Lλ) −→ +→ H0(X(i′), Lλ) = N Ψ′ +w′ (λ)∗. +This implies that the homogeneous coordinate ring of XΨ′(w′) is a quotient of +that of XΨ(w). In view of Corollary 3.17, we conclude the desired surjection. +Theorem 5.2. Let Ψ ⊂ ∆+ be a root ideal. Then, the line bundle OXΨ(w0)(̟0) +corresponds to an effective Cartier divisor D such that +supp D = XΨ(w0) \ T ∗ +ΨX. +Moreover, we have +H0(XΨ(w), OXΨ(w)(λ)) = +� +m≥0 +H0(XΨ(w), OXΨ(w)(λ+m̟n))⊗km̟n +λ ∈ P+ +whenever w is Ψ-tame. +Remark 5.3. As the both of π−1 +Ψ (X(e)) and XΨ(wΨ +0 ) are successive projective +space bundles, we can apply Theorem 2.14 directly to XΨ(wΨ +0 ) to obtain the +higher cohomology vanishing in Theorem 5.2 (that follows from Theorem 4.5 +and Theorem 4.1). +Proof of Theorem 5.2. Let us fix D as the divisor realized by imposing zero in +the coefficient of vΛ0 through the map +XΨ(w0) −→ P(L(Λ0)). +Since D is �T-stable, we have +supp D ∩ T ∗ +ΨX = ∅ +by the description of XΨ(w0) near the �T-fixed point {[vΛi]}i∈Iaf in Theorem +3.21. In the construction of the variety XΨ(wΨ +0 ) in Theorem 3.14 and Corollary +3.15, we can realize β = ǫk′ − ǫk ∈ ∆+ as +β − δ = sk′−1 · · · s1s0sn−1 · · · skΛk − Λk. +41 + +By additionally multiplying Sn by the G-invariance of XΨ(w0) and vΛ0, we can +apply the one-parameter subgroup whose weight is β − δ as +sβ−δ = (s1 · · · sn−1)(sk′−1 · · · s1)s0(sn−1 · · · sk)(s1 · · · sk′−1). +In particular, we find that +P(kvΛ0 ⊕ n(Ψ)[z−1]vΛ0) +belongs to the image of XΨ(w0) in P(L(Λ0)). Therefore, the closure of a fiber +of T ∗ +ΨX at B/B ∈ X has supp D as its boundary. Taking the G-action into +account, we conclude that +supp D = XΨ(w0) \ T ∗ +ΨX, +that is the first assertion. From this, we conclude that +H0(T ∗ +ΨX(w), π∗ +ΨOX(w)(λ)) = lim +−→ +m +H0(XΨ(w), OXΨ(w)(λ + m̟n)) ⊗ km̟n +by the commutation of the global section functor with direct limits. This is the +second assertion. +Corollary 5.4. Let Ψ′ ⊂ Ψ ⊂ ∆+ be root ideals, and let w′, w ∈ Sn be Ψ-tame +elements such that X(w′) ⊂ X(w). The natural restriction map +H0(T ∗ +ΨX(w), OXΨ(w)(λ)) −→ H0(T ∗ +Ψ′X(w′), OXΨ′ (w′)(λ)) +λ ∈ P+ +is surjective. +Proof. Note that w′ is Ψ′-tame. +By Proposition 5.1, we have the following +commutative diagram +H0(T ∗ +ΨX(w), OXΨ(w)(λ)) +� H0(T ∗ +Ψ′X(w′), OXΨ(w′)(λ)) +H0(XΨ(w), OXΨ(w)(λ + m̟n)) ⊗ km̟n +�� +� +� � H0(XΨ′(w′), OXΨ′ (w′)(λ + m̟n)) ⊗ km̟n +�� +� +for each λ ∈ P+ and m ≥ 0. Thus, Theorem 5.2 yields the assertion. +Corollary 5.5. Let Ψ′ ⊂ Ψ ⊂ ∆+ be root ideals. Let w′, w ∈ Sn be such that +X(w′) ⊂ X(w). The natural restriction map +H0(T ∗ +ΨX(w), OXΨ(w)(λ)) −→ H0(T ∗ +Ψ′X(w′), OXΨ′ (w′)(λ)) +λ ∈ P+ +is surjective. +Proof. Note that w0 is Ψ-tame. By Corollary 4.7 and Corollary 5.4, we find a +commutative diagram +H0(T ∗ +ΨX(w0), OXΨ(w0)(λ)) +� � +�� +H0(T ∗ +Ψ′X(w0), OXΨ(w0)(λ)) +�� +H0(T ∗ +ΨX(w), OXΨ(w)(λ)) +� H0(T ∗ +Ψ′X(w′), OXΨ(w′)(λ)) +. +Now the commutativity forces the bottom arrow to be a surjection. +42 + +In the rest of this section, we work over k = C. For a root ideal Ψ ⊂ ∆+ +and λ, µ ∈ P+, we set +Kλ,µ,Ψ(q) := +� +m∈Z +qm dim HomG×Grot +m (Vλ⊠C−mδ, H0(T ∗ +ΨX, OXΨ(w0)(µ))∗) ∈ Z[[q]]. +Corollary 5.6. Let Ψ′ ⊂ Ψ ⊂ ∆+ be root ideals, and let k = C. We have +Kλ,µ,Ψ′(q) ≤ Kλ,µ,Ψ(q) +λ, µ ∈ P+. +Proof. Since k = C, the rational representations of (G × Grot +m ) are completely +reducible. Hence, Kλ,µ,Ψ(q) counts the graded multiplicities of Vλ in +H0(T ∗ +ΨX(w0), OXΨ(w0)(λ))∗. +Therefore, the w = w′ = w0 case of Corollary 5.4 yields the assertion. +Remark 5.7. By Corollary 5.4, we find that the composition map +Spec H0(T ∗ +ΨX, OT ∗ +ΨX) → Spec H0(T ∗X, OT ∗X) ⊂ sl(n) +defines an irreducible closed subscheme1, that must be the closure of a nilpotent +orbit that we denote by OΨ. +Taking account into the fact that ̟n is the determinant character of G, we +find that the line bundle π∗ +ΨOX(̟n) is (non-equivariantly) trivial on T ∗ +ΨX for +each Ψ ⊂ ∆+. +From these, we have +Kk̟n,µ,Ψ′(q) ≤ Kk̟n,µ,Ψ(q) +if +k ∈ Z, µ ∈ P+ +if Ψ, Ψ′ ⊂ ∆+ satisfies OΨ′ ⊂ OΨ. +This is (a generalization of) the contents of [40, Conjecture 13] since the +dominance ordering appearing there corresponds to OΨ′ ⊂ OΨ, and their con- +dition µ = (kn) corresponds to employ OT ∗ +ΨX ⊗ C−k̟n. In a similar manner, +Corollary 5.6 provides a wide extension the speculations in [40, §2.10], including +[40, Conjecture 12]. +6 +Example: G = GL(3) +Keep the setting of the previous section. Assume that k is algebraically closed +and char k = p > 0. We have +W = S3 = ⟨s1, s2⟩ ⊂ Waf = ⟨s0, s1, s2⟩ . +a) The case Ψ = ∆+. +We have w∆+ +0 += s2. Our variety X∆+ admits a map to G/P ∼= P2 with its +fiber X∆+(w∆+ +0 +), where B ⊂ P is the parabolic subgroup corresponding to the +1This feature no longer holds if we replace G with other type even if we employ an equivari- +ant vector subbundle of T ∗(G/B) corresponding to the pullback of T ∗(G/P ) for a parabolic +subgroup P ⊂ G (see e.g. [12, 36]). +43 + +root ±α2. The variety X∆+(w∆+ +0 +) admits an action of �B and is isomorphic to +a P2-fibration over P2, where the base is given as +P2 ∼= P(V2) ⊂ P(L(Λ2)), where V1 = kvΛ2 ⊕ kvs2Λ2 ⊕ kvs0s2Λ2, +and its fiber is given as +P2 ∼= P(V0) ⊂ P(L(Λ0)), where V0 = kvΛ0 ⊕ kvs0Λ0 ⊕ kvs1s0Λ0 +(6.1) +that admits the action of �B. Here, the V0 is stable under the action of �B, as +well as the subgroup SL(2, k) ⊂ G((z)) corresponding to the simple root ±α2. +Therefore, if we consider the subgroup +SL(2, k) ∼= G1(k) ⊂ G((z)) +corresponding to the non-simple root ±(α0 + α2), then the closed algebraic +subgroup B1 ⊂ G1 with B1(k) = �B(k) ∩ G1(k) preserves V0. In particular, +Z := G1 ×B1 ([vΛ1] × P(V0)) ⊂ P(L(Λ2)) × P(L(Λ0)) +defines a closed subvariety of X∆+(w∆+ +0 +) that is B-stable. +We set X♯ +∆+ := +G ×B Z to obtain maps +X∆+ = G.Z ←− X♯ +∆+ ≡ G ×B Z −→ X = G/B. +Note that V0 ∼= V0(0)⊕V0(1)⊕V0(2), where V0(r) (r = 0, 1, 2) are the degree +r characters of B1. Namely, we have +V0(0) = kvs0Λ0, V0(1) = kvΛ0, and V0(2) = kvs1s0Λ0. +Therefore, Z has two boundary divisors D1 and D2: The divisor D1 is the +pullback of the T -fixed point in (G1/B1) \ (B1/B1) through +Z ∼= G1 ×B1 ([vΛ1] × P(V0)) → G1/B1. +The divisor D2 is the translation of the divisor on the fiber P(V0(0) ⊕ V0(2)) ⊂ +P(V0) by the G1-action. +We have an isomorphism +Z \ (D1 ∪ D2) ∼= n ∼= (nz−1)v0, +computed in P(L(Λ0)). +By Proposition 2.10, we know that P(V0) admits a +Frobenius splitting compatible with the boundary divisor P(V0(0)⊕V0(2)), that +induces a Frobenius splitting of Z compatible with the boundary divisors D1 +and D2. +b) The case Ψ = {ǫ1 − ǫ3, ǫ2 − ǫ3}. +We have wΨ +0 = e. We have +XΨ = X♯ +Ψ = G ×B P(kvΛ0 ⊕ kvs0Λ0 ⊕ kvs1s0Λ0). +c) The case Ψ = {ǫ1 − ǫ3}. +44 + +We have wΨ +0 = e. We have +XΨ = X♯ +Ψ = G ×B P(kvΛ0 ⊕ kvs0Λ0). +d) The case Ψ = ∅. +We have X∅ = X♯ +∅ = G/B. +Frobenius splitting assertions of X♯ +Ψ (compatible with the boundary of the +fibers) of the cases b),c),d) are straight-forward. +e) The case Ψ = {ǫ1 − ǫ2, ǫ1 − ǫ3}. +We have wΨ +0 = s2. We have +XΨ = G ×P Y +, +and +X♯ +Ψ = G ×B Y, +where Y + and Y are described as follows: We consider the space +Y + := �P0 �P2([vΛ2] × P(kvΛ0 ⊕ kvs0Λ0)) ⊂ P(L(Λ2)) × P(L(Λ0)). +It is easy to see that �P2 preserves Y +. We consider the condition that the +coordinate corresponding to vs2Λ2 is zero (that defines a linear hyperplane in +P2). This yields a B-stable subvariety Y ⊂ Y + that is smooth and satisfies +Y + = �P2Y . +In fact, Y is the Hirzebruch surface of degree two. +Frobenius +splitting assertion of Y + follows from a repeated application of Corollary 2.9. +The Frobenius splitting assertion of Y can be seen from by applying the SL(2)- +action instead of the action of �P0 �P2 in the definition of Y +. 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Zeit., +201(no.1):19–31, 1989. +47 + diff --git a/VNAyT4oBgHgl3EQf8vqZ/content/tmp_files/load_file.txt b/VNAyT4oBgHgl3EQf8vqZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e064c35f713b5b7310c39dfcd15fbc0f3bdc0b26 --- /dev/null +++ b/VNAyT4oBgHgl3EQf8vqZ/content/tmp_files/load_file.txt @@ -0,0 +1,1818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf,len=1817 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='00862v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='AG] 2 Jan 2023 Catalan varieties∗ Syu Kato† January 4, 2023 Abstract We construct two smooth projective algebraic varieties XΨ and X♯ Ψ that compactify an equivariant vector subbundle of the cotangent bundle of the flag variety of GL(n) (corresponding to a root ideal Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The variety XΨ carries natural class of line bundles whose spaces of global sections realize the Catalan functions defined in Chen-Haiman [thesis, UCBerke- ley] and studied in Blasiak-Morse-Pun-Summers [J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We prove the vanishing conjectures of Chen-Haiman and Blasiak- Morse-Pun [arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='04952] (in their full generalities), as well as the monotonicity conjectures of Shimozono-Weyman [Electronic J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' (2000)] using the geometry of X♯ Ψ and XΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Introduction In search of better understanding of the internal structure of Macdonald poly- nomials ([34]) after Haiman’s solution ([18]) of the Macdonald positivity con- jecture, LaPointe-Lascoux-Morse [33] proposed the notion of k-Schur functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' They are shown to represent Schubert classes of affine Grassmannians ([31]), and hence has a rˆole in quantum cohomology of flag variety X of G = GL(n, C) ([39, 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' However, we still do not fully understand its precise relationship with the Macdonald polynomials (without specializations) and computations in quantum cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Chen-Haiman [11] made remarkable conjectures on the internal structure of k-Schur functions and its generalizations, sometimes called the Catalan func- tions, by offering their geometric interpretation using certain vector bundles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Their conjectures include the conjectures posed by Broer [7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='16] (for type A) and Shimozono-Weyman [40] as their particular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Although the numer- ical portion of their conjectures are established by Blasiak-Morse-Pun-Summers [4, 3], we still do not know some cohomology vanishing conjectures, that is fur- ther refined in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Since these conjectures are central in the geometric picture in [11] and also in the logic in the monotonicity conjectures in [40, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='10], we might say they are the final missing pieces to establish the framework anticipated by many people for more than 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In this paper, we define and study two compactifications XΨ and X♯ Ψ of a G-equivariant vector subbundle T ∗ ΨX ⊂ T ∗X employed by Chen-Haiman [11] ∗MSC2010: 14N15,20G44 †Department of Mathematics, Kyoto University, Oiwake Kita-Shirakawa Sakyo Kyoto 606- 8502 JAPAN E-mail:syuchan@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='jp 1 indexed by a Dyck path Ψ of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let a(Ψ) denote the area statistic of the Dyck path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let Pn denote the set of partitions of length ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The set of irreducible polynomial representations of G is parametrized by Pn as Pn ∋ λ �→ Vλ (up to isomorphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The character of Vλ is the Schur polynomial sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' By recording the character of C× by q•, we have the notion of the graded character gch V of a rational (G × C×)-module V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Theorem A ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='= Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='14 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='5 + Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The ex- ists two varieties XΨ and X♯ Ψ equipped with (G × C×)-actions and (G × C×)- equivariant morphisms XΨ ηΨ ←− X♯ Ψ πΨ −→ X with the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The varieties XΨ and X♯ Ψ are smooth of dimension 2 dim X − a(Ψ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We have an embedding T ∗ ΨX ⊂ X♯ Ψ, and the restriction of ηΨ to the image of T ∗ ΨX is an isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For each λ ∈ Pn, we have (G × Gm)-equivariant line bundles OXΨ(λ) and OX♯ Ψ(λ) := η∗ ΨOXΨ(λ) such that H>0(XΨ, OXΨ(λ)) = 0 = H>0(X♯ Ψ, OX♯ Ψ(λ)) and � HΨ(λ) � q�→q−1 = gch H0(XΨ, OXΨ(λ))∗ = gch H0(X♯ Ψ, OX♯ Ψ(λ))∗, where HΨ(λ) is the Catalan polynomial ([4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='2)]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' There is a (G × C×)-equivariant effective Cartier divisor ∂ supported on X♯ Ψ \\ T ∗ ΨX such that we have H>0(X♯ Ψ, OX♯ Ψ(λ + m∂)) = 0 λ ∈ Pn, m ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In particular, we have H>0(T ∗ ΨX, OT ∗ ΨX(λ)) = � m≥0 H>0(XΨ, OX♯ Ψ(λ + m∂)) = 0 λ ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Theorem A 4) resolves [11, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='3 2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In conjunction with [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='18], this establishes [11, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='3] in its full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' As [11, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='3] include conjectures by Broer [7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='16] (for type A, in the form that the desired supports are always empty) and Shimozono-Weyman [40, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='4, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='10], our consideration resolves (the geometric portions of) these conjectures as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' As a corollary of Theorem A, we find: Corollary B ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='= Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' There exists an action of GL(n, C[[z]]) ⋊ Gm on XΨ such that H0(T ∗ ΨX, OT ∗ ΨX(λ))∨ −→ → H0(XΨ, OXΨ(λ))∨ is a quotient as (a graded) cyclic representations of gl(n, C) ⊗ C[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' 2 In case T ∗ ΨX ∼= T ∗X, Corollary B tells us the exact relation between two realizations of Hall-Littlewood functions via the geometry of X ([17]) and via representation theory of gl(n, C[[z]]) ([35, 9]), in conjunction with the existing relations ([10, 23, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let B ⊂ G be the subgroup consisting of upper triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The set of B-orbits of X is parametrized by the symmetric group Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For w ∈ Sn, let us denote by X(w) the closure of the corresponding B-orbit (a Schubert variety of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set X♯ Ψ(w) := π−1 Ψ (X(w)) and T ∗ ΨX(w) := T ∗ ΨX ∩ X♯ Ψ(w) ⊂ X♯ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' This is the restriction of the vector bundle T ∗ ΨX to X(w), together with its compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' By a similar logic as in Theorem A 4), we deduce: Theorem C ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='= Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For each λ ∈ Pn and w ∈ Sn, we have H>0(T ∗ ΨX(w), OT ∗ ΨX(w)(λ)) = 0, where OT ∗ ΨX(w)(λ) is the line bundle on T ∗ ΨX(w) obtained as the restriction of OXΨ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Theorem C resolves [3, Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='4 (ii)], that generalizes many previous results and conjectures (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' As a bonus of our consideration, we have: Corollary D ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='= Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let Ψ′ ⊂ Ψ be an inclusion of Dyck paths that yields T ∗ Ψ′X ⊂ T ∗ ΨX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For each λ ∈ Pn, the restriction map H0(T ∗ ΨX, OT ∗ ΨX(λ)) −→ H0(T ∗ Ψ′X, OT ∗ Ψ′X(λ)) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Corollary D resolves conjectures in [40, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='10] as explained in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Our constructions of the varieties XΨ and X♯ Ψ are explicit constructions of their coordinate rings, as well as their blowing-ups along explicit loci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The main technical portion of the vanishing results is to afford suitable Frobenius D-splittings on the positive characteristic analogues of XΨ and X♯ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In order to carry this out, we tweak standard results in compatible Frobenius splittings enforcing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The organization of this paper is as follows: We fix notations and record basic results §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We present some basic materials and our construction of Frobenius splittings in §2 by assuming the characteristic is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In §3, we work over an arbitrary algebraically closed field and construct our variety XΨ, and establish a part of Theorem A and Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In §4, we construct our variety X♯ Ψ, and establish the remaining part of Theorem A, as well as Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We discuss consequences of our results including Corollary D in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The last section is devoted to example calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' So far, we have introduced algebraic varieties that are natural geometric counterparts of the Catalan functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' An obvious question is how to place them in the context of topological field theories and geometric realizations of Macdonald polynomials arising from G = GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We hope to answer these questions in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' 3 1 Preliminaries We work over an algebraically closed field k unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' A variety means a separated (integral) normal scheme of finite type over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For a scheme X over a ring S and a ring map S → R, we denote by X(R) the set of R-valued points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For a k-vector space V , let S•V = � i≥0 SiV denote its symmetric power ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let L be an abelian free monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' A L-graded k-vector space V is a k- vector space V equipped with a direct sum decomposition V = � a∈L Va such that dimk Va < ∞ for each a ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For a L-graded vector space V = � a∈L Va, we set V ∨ := � a∈L V ∗ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' A L-graded ring R is an unital k-algebra that is a L-graded k-vector space such that k1 = R0 and Ra · Ra′ ⊂ Ra+a′ (a, a′ ∈ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content='1 Algebraic Groups We fix an integer n > 0 and consider the algebraic group G = GL(n) ⊂ Mn(k) = Mn ∼= An2 k over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Here we omit k when the meaning is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We also define an algebraic group G = GL(n, k[[z]]) over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We also set G((z)) := GL(n, k((z))) and regard it as a (topological) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let Eij ∈ Mn(k) (1 ≤ i, j ≤ n) be the matrix unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let T ⊂ G be the diagonal torus and let B ⊂ G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' B− ⊂ G) be the upper (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' lower) triangular part of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The group N := [B, B] ⊂ B is the group of upper unitriangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We have the evaluation map ev0 : G −→ G z �→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set B := ev−1 0 (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For each 1 ≤ i < n, we set Pi ⊂ G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Pi ⊂ G) as the (algebraic or pro-algebraic) subgroup generated by B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' I) and Id + kEi+1,i inside G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set P0 as the (pro-algebraic) group generated by I and Id + kz−1E1,n inside G((z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Note that we have the extra loop rotation Gm-action on each of I ⊂ Pi and G (that we denote by Grot m ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We denote by �I, �Pi, and �G the semi-direct products of I, Pi, and G with Grot m , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In addition, the group G((z)) admits a central extension by k×, that induces a (trivial) central extension �Pi (0 ≤ i < n) of �Pi by Gm (that we denote by Gc m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' In particular, we have extended tori �T := T × Grot m × {1} ⊂ T × Grot m × Gc m =: �T such that �B := �B× Gc m contains �T such that �B∩ �T = �T and �Pi ∩ �Pj = �B when i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For each 0 ≤ i < n, we define a map of algebraic groups ui : Ga(k) = k ∋ x �→ � 1 + xEi,i+1 (i ̸= 0) 1 + xzEn,1 (i = 0) ∈ �B(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' 4 We also have �G((z)) := Grot m (k) ⋉ G((z)) ⋉ Gc m(k) as a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let �B− ⊂ �G((z)) be the subgroup generated by �T, the lower trian- gular part of G, and Id+kz−1E1,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let �G− ⊂ �G((z)) be the subgroup generated by �B− and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We warn that the groups �G((z)), �B−, and �G− are not algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For 1 ≤ i ≤ n, we have a(n algebraic) character ǫi : T → Gm that extracts the i-th (diagonal) entry of T (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set P := �n i=1 Zǫi and consider its subsets P := � i=1 Z≥0ǫi, P+ := { n � i=1 miǫi ∈ P | m1 ≥ m2 ≥ · · · ≥ mn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For λ = �n i=1 λiǫi ∈ P, we set |λ| := �n i=1 λi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The permutation of indices define Sn-actions on P and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set P+ := (P+ ∩ P) and identify it with the set of partitions with its length at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The semi-group P+ is generated by ̟i := ǫ1 + · · · + ǫi 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' For λ ∈ P+, we may write λ ≫ 0 whenever all the expansion coefficients of ̟1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' , ̟n are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We may regard ̟i as a character of �T through the projection to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set ρ := �n i=1(n + 1 − i)̟i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Let ℘ and δ denote the degree one character of Gc m and Grot m extended to �T trivially, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We define another (non-standard) lift of ̟i to �T as: Λi := ̟i + ℘ (1 ≤ i < n), ̟n + ℘ (i = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set Iaf := {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=', (n − 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We frequently identify 0 with n in the se- quel, and hence {̟i}i is indexed by Iaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' Note that {̟i}i∈Iaf and {Λi}i∈Iaf corresponds to each other by restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' We set Paf := � n � i=1 Z̟i � ⊕ Z℘ ⊕ Zδ and P+ af := ( � i∈Iaf Z≥0Λi) + Z̟n + Zδ ⊂ Paf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The set Paf is the character group of �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf'} +page_content=' The set of positive roots ∆+ of G is ∆+ := {ǫi − ǫj}1≤i 0. The absolute mass scale and the Dirac CP phase have not +been determined while θ23 can still be in the first quadrant. From the theoretical side, the +mechanism behind neutrino mass together with the nature of the mass, Dirac or Majorana +(including quasi-Dirac), remains an open question. +Treating the SM as an effective field theory, Majorana mass for neutrinos arise from +the unique dimension-5 Weinberg operator [3, 4]. +This is the minimal scenario without +additional light degrees of freedom. In order to have Dirac mass, additional light degrees +of freedom are needed to be the Dirac partners of the SM neutrinos. In either cases, it is +not necessary that there is unitary violation or nonunitarity in the sense that the matrix +Uαi which relates the flavor (with index α) and mass (with index i) eigenstates of neutrinos +when sum over the kinematically accessible mass eigenstates or over the three SM flavors +are not unitary +accessible +� +i +UαiU∗ +βi ̸= δαβ, +� +α=e,µ,τ +UαiU∗ +αj ̸= δij. +(1) +In this work, we will focus on nonunitary scenario when the relations above hold true and +contrast it to the three-flavor paradigm when unitarity is preserved. +We aim to give a more complete theoretical discussion of nonunitarity in neutrino +oscillation [5–22]. We start by discussing how nonunitary can arise in Section II, highlighting +two distinct scenarios: high scale nonunitary scenario where new physics resides beyond the +energy scale of neutrino oscillation experiments and low scale nonunitary scenario where +new fermionic states (sterile neutrinos) mix with the SM neutrinos and are accessible in +neutrino oscillation experiments. In Section III, we derive analytical solution for neutrino +2 + +FIG. 1. The probability of νµ → νe for neutrino crossing the entire Earth as a function of neutrino +energy Eν. As the modulus of (UU †)eµ decreases, the probability approaches that of the standard +three-flavor unitary scenario shown as the black solid line. +oscillation without assuming unitarity and in Section IV, we discuss how the high scale and +low scale nonunitary effects can show up and be distinguished in experiments. Finally we +present some concluding remarks in Section V. While many excellent discussions are there +in previous work for e.g. [5], we will present some new results. In particular, we will prove +a theorem in Section III that if nonunitary is only diagonal +accessible +� +i +UαiU∗ +βi = cαδαβ, +(2) +with cα > 0, then neutrino oscillation probability in an arbitrary matter potential is +indistinguishable from the unitary scenario. An important implication is that high scale +nonunitary effects are proportional to the off-diagonal elements of UU† and can only be +confirmed in appearance experiments, in contrast to low scale nonunitary scenario where the +effects can remain. To illustrate this point, in Figure 1, we show the probability of νµ → νe +3 + +Earth-crossing neutrinos, NO +0.7 +-×-- (UUf)eu = 0.001 +0.6 +--(UUt) +ew=0.005 +--(UUt)e +=0.01 +★ +(UUf)ew = 0.02 +0.5 +(UUt) +u= 0.03 +(UUt) +Jet=0.04 +0.4 +0.3 +0.2 +0.1 +0.0 +100 +101 +E[GeV]as a function of neutrino energy Eν in high scale nonunitary scenario for neutrino passing +through the Earth using the public code NuProbe [23, 24] with a simplified (Preliminary +Reference Earth Model) PREM model [25]. Denoting U as a 3 × 3 mixing matrix, we have +fixed (UU †)ee = (UU †)µµ = 0.96, (UU †)µµ = 1, (UU †)eτ = (UU †)µτ = 0 while varying +(UU †)eµ from 0.001 to 0.04 (dashed lines) and the rest of the standard parameters have +been fixed to the NO global best fit values from [1, 2]. As the modulus of (UU †)eµ decreases, +the probability approaches that of the standard three-flavor unitary scenario (solid black +line). For reader interested in experimental probe of nonunitary scenarios, he or she can +jump straight to Section IV. +II. +MODELS +Here we will focus on models for neutrino oscillation assuming that the center-of-mass +energy involved is below the electroweak symmetry breaking E < vEW ≡ 174 GeV. We +will consider only Majorana mass term for neutrinos though the discussions below are +independent of whether we have a Majorana or Dirac mass term. While neutrino oscillation +cannot distinguish between strictly Majorana and Dirac mass, it is possible to distinguish +them from quasi-Dirac scenario in which both types of mass terms exist while the Majorana +mass term is much smaller than the Dirac one [26–30]. We will consider this interesting +scenario in a future publication. +A. +High scale nonunitary +Assuming that for E < vEW, we only have three SM neutrinos (νe, νµ and ντ are the +SM left-handed neutrino flavor states), the general neutrino Lagrangian allowed by the SM +electromagnetic gauge symmetry U(1)EM in the charged lepton flavor basis is given by +Lν = 1 +2 +� +iνα/∂Dαβνβ − νc +αmαβνβ + h.c. +� +− +�g +2W − +µ ℓαγµPLνα + +g +√ +2 cos θW +ZµναγµPLνα + h.c. +� +, +(3) +where the flavor indices are α, β = e, µ, τ, D is a dimensionless Hermitian matrix while m +is a symmetric matrix with mass dimension. In the second line, g is the SU(2)L gauge +coupling, θW is the Weinberg or weak angle, PL = +1 +√ +2 (1 − γ5) is the left-handed projector, +4 + +{ℓe, ℓµ, ℓτ} ≡ {e−, µ−, τ −} are the charged leptons and W ∓ and Z are the charged and neutral +weak bosons, respectively. Without additional light degrees of freedom, only Majorana mass +term is possible.1 Treating the Standard Model as an effective field theory, the neutrino mass +and the modified kinetic terms come respectively from dimension-5 [3, 4] and dimension-6 +operators [31, 42, 43] +O5 = λαβ +Λ5 +� +Lc +αϵH +� +(LβϵH) , +(4) +O6 = ηαβ +Λ2 +6 +� +LαϵH∗� /∂ (LβϵH) , +(5) +where L and H are the SU(2)L lepton and Higgs doublets, respectively, λ and η are +dimensionless symmetric and Hermitian matrices, respectively, and Λ5 and Λ6 are effective +scales below which the operators O5 and O6 are valid. Implicitly, we have assumed E ≪ +Λ5, Λ6. Not all ultraviolet models which generate O5 also generate O6. For instance, type- +I and type-III seesaw models generate both O5 and O6 while type-II seesaw model only +generates O5. +In order to obtain canonical normalized kinetic term, we can first diagonalize the kinetic +term as D = Y † ˆDY where Y is unitary and ˆD is real and diagonal. Defining the normalized +neutrino field as �ν ≡ +� +ˆDY ν, eq. (3) becomes +Lν = 1 +2 +� +i�να/∂�να − �νc +α �mαβ�νβ + h.c. +� +− +� +g +2W − +µ ℓαγµPL +� +Y †� +ˆD +−1� +αβ +�νβ + +g +√ +2 cos θW +Zµ +� +ˆD−1� +αα �ναγµPL�να + h.c. +� +, (6) +where we have defined +�m ≡ +� +ˆD +−1 +Y ∗mY †� +ˆD +−1 +. +(7) +The symmetric mass matrix above can be diagonalized by a unitary matrix V as �m = V ∗ ˆmV † +where ˆm is real and diagonal. Defining the neutrino field in the mass basis as ˆν = V †�ν, we +have [5] +Lν = 1 +2 +� +iˆνi/∂ˆνi − ˆνc +i ˆmiiˆνi + h.c. +� +− +�g +2WµℓαγµPLUαiˆνi + +g +√ +2 cos θW +Zµˆνi +� +U †U +� +ij γµPLˆνj + h.c. +� +, +(8) +1 In order to write down a Dirac mass term, new light degrees of freedom are required such that we can +write ν′ +fm′ +fβνβ where ν′ +f are some new fermion fields which do not participate in weak interactions. In +this case, m′ +fβ is a general complex matrix with mass dimension. Besides the fact that the mass term +should be diagonalized by two unitary matrices, the discussion will remain the same since the effect of +unitary rotation of ν′ which do not feel the weak force, is not observable. +5 + +where we denote i, j = 1, 2, 3 to be the indices in mass basis and we have defined +Uαi ≡ +� +Y †� +ˆD +−1 +V +� +αi +. +(9) +Notice that +UU † = Y † ˆD−1Y, +U †U = V † ˆD−1V. +(10) +Only if ˆD = I is the 3 × 3 identity matrix, unitary is restored in which U †U = UU † = I. +We denote the general case with UU † ̸= I and U †U ̸= I, high scale nonunitary scenario. +B. +Low scale nonunitarity +Assuming that for E < vEW, besides the three SM neutrinos (νe, νµ and ντ), we also +have additional neutral fermions fields (νs1, νs2,..., νsN) which do not participate in weak +interactions but mix with the SM neutrinos through the mass term. The mixing between +the SM and the additional fermions are the Dirac mass term. Here we will focus on mostly +Majorana scenario where the Majorana mass term for new fermions is somewhat larger than +the Majorana mass for the SM neutrinos as well as the Dirac mass term.2 +In order to +highlight the distinction from the high scale unitary violation scenario, we further assume +that the kinetic terms for all the fermions are canonical. In this case, the U(1)EM-invariant +neutrino Lagrangian in the charged lepton flavor basis is given by +Lν = 1 +2 +� +iνα/∂να − νc +αmαβνβ + h.c. +� +− +�g +2W − +µ ℓαγµPLνα + +g +√ +2 cos θW +ZµναγµPLνα + h.c. +� +, +(11) +where α, β, ... = e, µ, τ, s1, s2, ..., sN. The symmetric mass matrix m can be diagonalized by +a unitary matrix U as m = U∗ ˆmU† where ˆm is real and diagonal. Here and in the following, +we use a boldface U to denote a (3 + N) × (3 + N) matrix while U is reserved for a 3 × 3 +matrix. Defining the neutrino field in the mass basis as ˆν = U†�ν, we have +Lν = 1 +2 +� +iˆνi/∂ˆνi − ˆνc +i ˆmiiˆνi + h.c. +� +− +�g +2WµℓαγµPLUαiˆνi + +g +√ +2 cos θW +ZµˆνiγµPLˆνj + h.c. +� +, +(12) +2 Depending on the relative size of the corresponding mass terms, one can obtain the mostly Majorana, +strictly Majorana, strictly Dirac or quasi-Dirac scenario. Strictly Dirac scenario cannot be distinguished +from strictly Majorana scenario and will not also not result in apparent unitary violation. Mostly Majorana +scenario will result in apparent unitary violation allowed by current measurements. Finally, though there +is no apparent unitary violation, quasi-Dirac scenario will result in distinguished signatures [26–30] and +will be considered in a future work. +6 + +where i, j, ... = 1, 2, ..., 3+N. What distinguish this from the high scale nonunitary scenario +is that U†U = UU† = I where I is the (3 + N) × (3 + N) identity matrix. +Strictly speaking, there is no unitary violation in this case. +However, since only the +mixing involving the SM neutrinos can be measured, one have +� +α=e,µ,τ +UαiU∗ +αj ̸= δij. +(13) +Furthermore, if mi>3 ≫ +� +∆m2 +atm such that oscillations involving νi>3 can be averaged out, +at leading order in small unitary violating parameter, the mixing elements involved are those +of i = 1, 2, 3 which sum to [14, 17] +3 +� +i=1 +UαiU∗ +βj ̸= δαβ. +(14) +We denote this as low scale nonunitary scenario. +If some additional states are not +kinematically allowed in the process, one should describe it as in the high scale nonunitary +scenario discussed previously, with possible enlargement of flavor beyond 3 to accommodate +additional kinematically accessible states. +III. +GENERIC 3 + N NEUTRINO OSCILLATIONS +Here we will develop a generic 3+N neutrino oscillation framework which can be applied +to both unitary and nonunitary neutrino oscillation. In general, the neutrino flavor states +|να⟩ are related to the mass eigenstates |νi⟩ through a matrix U +|να⟩ = +1 +� +(UU†)αα +� +i +U∗ +αi |νi⟩ , +(15) +where α, β, ... = e, µ, τ, s1, s2, ..., sN (νe, νµ and ντ are the SM left-handed neutrino flavor +states) and i, j, ... = 1, 2, ..., 3 + N. In general, U does not have to be unitary UU† ̸= I +and U†U ̸= I. Since the mass eigenstates are orthogonal ⟨νj|νi⟩ = δji, the flavor states are +properly normalized ⟨να|να⟩ = 1 though they are not necessarily orthogonal but equal to +⟨νβ|να⟩ = +� +UU†� +βα +� +(UU†)αα (UU†)ββ +. +(16) +In other words, there is nonzero overlap between different flavor states and there is a +probability of flavor changing even at zero distance. The set of all {|νi⟩} is complete and +7 + +from the orthogonality condition, it further satisfies the completeness relation +� +i +|νi⟩ ⟨νi| = I. +(17) +From eq. (15), we can write the inverse relation3 +|νi⟩ = +� +α +� +U∗,−1� +iα +� +(UU†)αα |να⟩ . +(18) +One can verify the orthogonality condition +⟨νj|νi⟩ = +� +α,β +� +U−1� +jβ +� +U∗,−1� +iα ⟨νβ|να⟩ += +� +U−1UU†U†,−1� +ji = δji, +(19) +where in the second equality, we have used eq. (16). In the rest of the article, to avoid +expressions crowded with normalization factors, we will define +Uαi ≡ +Uαi +� +(UU†)αα +. +(20) +Starting from an initial state|να (0)⟩ = |να⟩, the time-evolved state |να (t)⟩ is described +by the Schrödinger equation +i d +dt |να (t)⟩ = H |να (t)⟩ , +(21) +where the Hamiltonian is H = H0 + HI with H0 the free Hamiltonian +H0 |νi⟩ = Ei |νi⟩ , +Ei = +� +⃗p2 +i + m2 +i , +(22) +and HI the interaction Hamiltonian with matrix elements +⟨νβ| HI |να⟩ = Vβα, +(23) +where V ∗ +βα = Vαβ since H† +I = HI. +Assuming relativistic neutrinos, we trade t = x and the amplitude of the transition +να → νβ at distance x is then given by Sβα (x) ≡ ⟨νβ|να (x)⟩ and the probability of an initial +state |να⟩ being detected as |νβ⟩ at distance x is +Pβα (x) = |Sβα (x)|2 . +(24) +3 We can prove that inverse exists. Supposing that +|νi⟩ = +� +α +Viα |να⟩ , +and using orthogonality condition ⟨νj|νi⟩ = δji, we have +δij = +� +α +Viα ⟨νj|να⟩ = +� +α +Viα +1 +� +(UU†)αα +U∗ +αj, +and hence Viα = +� +(UU†)αα +� +U∗,−1� +iα. +8 + +From (21), we can write the evolution equation of Sβα (x) as +i d +dxSβα (x) = ⟨νβ| H0 + HI |να (t)⟩ += +� +i +⟨νβ| H0 + HI |νi⟩ ⟨νi|να (t)⟩ += +� +η +�� +i +UβiEi +� +U +−1� +iη + +� +γ +Vβγ +� +UU +†�−1 +γη +� +Sηα (x) , +(25) +where in the second equality, we have inserted the completeness relation eq. (17) and in the +last equality, we have used eqs. (22), (23), (18) and (16). Considering relativistic neutrinos +E ≫ mi and expanding Ei ≃ E + m2 +i +2E , we obtain, in matrix notation +idS (x) +dx += +� +U∆U +−1 + V +� +UU +†�−1� +S (x) , +(26) +where +∆ ≡ 1 +2E diag +� +m2 +1, m2 +2, ..., m2 +3+N +� += diag (∆1, ∆2, ..., ∆3+N) . +(27) +We have dropped the constant E which is an overall phase in S(x) and hence does not affect +the probability in eq. (24). +A. +Vacuum mass basis +From eq. (26), the Hamiltonian in the flavor basis given by +H ≡ U∆U +−1 + V +� +UU +†�−1 +, +(28) +is not Hermitian H† ̸= H nor normal H†H ̸= HH†. In the following, we will prove that +they can be diagonalized with real eigenvalues. +Let us change the Hamiltonian to the vacuum mass basis in which the free Hamiltonian +is diagonal +�H ≡ U +−1HU = ∆ + U +−1V U +†,−1. +(29) +Notice that eq. (29) is Hermitian �H† = �H. Assuming V = V † is constant in the interval of +interest 0 ≤ x < x1, we can diagonalize the �H with a unitary matrix X +�H = X ˆHX†, +(30) +9 + +where X is unitary and ˆH = diag (λ1, λ2, ..., λ3+N) is diagonal and real. Since eqs. (28) and +(29) are related by similar transformation, they have the same eigenvalues �HX = X ˆH =⇒ +HUX = UX ˆH and hence we can also write +H = UX ˆH +� +UX +�−1 , +(31) +where the nonnormal H is diagonalized by a nonunitary UX. +We see explicitly that +despite the Hamiltonian in flavor basis H appears to be non-Hermitian, the eigenvalues +remain real while the source of nonunitarity comes nonunitary U resulting in nonorthogonal +flavor basis. Although UX is nonunitary, one can formally solve for +� +UX +� +αi +� +UX +�−1 +iβ in +terms of eigenvalues and Hamiltonian elements using the same method as in refs. [23, 32]. +Nevertheless, as we will see in the next subsection, the combination which appears in neutrino +oscillation probability is not +� +UX +� +αi +� +UX +�−1 +iβ and we will solve for XikX∗ +jk instead. +As shown in refs. [23, 32], by raising eq. (30) to the power of 1, 2, ..., 2 + N and taking +into account the unitary relation XX† = I, one can form a set of 3+N linearly independent +equations for XikX∗ +jk where the coefficients form a Vandermonde matrix4 which can be +inverted to give [23] (see also the pioneering work of Kimura, Takamura and Yokomakura +who applied similar method for 3-flavor scenario [33, 34]) +XikX∗ +jk = +2+N +� +p=0 +(−1)p ( �Hp)ijc2+N−p,k +Zk +, +(32) +where we have defined +Zk ≡ +� +p̸=k +(λp − λk) , +(33) +cp,k ≡ +� +{q̸=r̸=...}̸=k +λqλr... +� �� � +p +, +(34) +with [ ˜H0]ij = δij and c0,k = 1. The sum in cp,k is over all possible unordered combinations +of p distinct eigenvalues λqλr... where none of them is equal to λk and hence with 3 + N +neutrino flavors, cp,k has +� 2+N +p +� += +(2+N)! +p!(2+N−p)! terms in the sum. As shown in ref. [35], the +4 The equations obtained with power in λi greater than 2 + N are not linearly independent since they +can be rewritten in term of lower power using the characteristic equation of �H. Suppose we have d + 1 +degenerate eigenvalues λl = λk for l = k, ..., k + d, we only need to solve for the combination � +l XilX∗ +jl +corresponding to λk, i.e. 3 + N − d linear equations can be obtained from raising eq. (30) to the power +of 1, 2, ..., 2 + N − d, including XX† = I. +10 + +numerator of eq. (32) can be written in mathematically equivalent form in terms of elements +of the adjugate of λkI − �H i.e. [Adj(λkI − �H)]ij though we have found that that numerically +evaluating the analytical expression eq. (32) is about three times faster than evaluating the +expression with adjugate. +B. +Oscillation probability +There is one subtle but crucial point regarding �S in the vacuum mass basis which satisfies +id�S (x) +dx += �H �S (x) , +(35) +and the solution is given by +�S (x) = Xe−i ˆHxX†. +(36) +How do we relate �S (x) of the vacuum mass basis and S (x) in the flavor basis? Fixing the +initial conditions �S (0) = I and S (0) = UU +† which follow from the orthogonality of mass +eigenstates and the nonorthogonality of flavor eigenstates eq. (16), respectively, we have +�S (x) ≡ U +−1S (x) U +†,−1. +(37) +Hence we can write +S (x) = UXe−i ˆHx � +UX +�† . +(38) +It follows from eq. (24) that the neutrino oscillation probability for να → νβ at a distance +0 ≤ x < x1 is5 +Pβα (x) = +����� +� +i,j,k +UβiU +∗ +αjXikX∗ +jke−iλkx +����� +2 +. +(39) +which has exactly the same form as the unitary case despite that U does not have to be +unitary. +One can generalize the solution in eq. (38) to the case when V is x-dependent by splitting +x into intervals small enough that V (x) is approximately constant. Considering 0 = x0 < +x1 < x2 < ... where V (x) is equal to constant Va for each interval xa−1 < x < xa, we obtain +S (x) = T +� +a=1 +S(a) (x) , +(40) +5 The oscillation probability P βα (x) for antineutrino να → νβ is obtained by taking Uαi → U +∗ +αi and since +our Universe consists only of matter, we should also take V → −V in eq. (35). +11 + +where we have defined +S(a) (x) ≡ +� +UX(a)� +e−i ˆH(a)x(a) � +UX(a)�† , +(41) +x(a) ≡ [(x − xa−1) θ (xa − x) + (xa − xa−1) θ (x − xa)] θ (x − xa−1) , +(42) +with θ (x ≥ 0) = 1, θ (x < 0) = 0 and T denotes the space ordering of the matrix +multiplication such that the a term is always to the left of a − 1 term. +Furthermore, +ˆH(a) = diag +� +λ(a) +1 , λ(a) +2 , ..., λ(a) +3+N +� +and X(a) denote respectively the matrix of eigenvalues and +unitary matrix which diagonalizes �H as �H(a) = X(a)† ˆH(a)X(a) in the interval xa−1 < x < xa. +The neutrino oscillation probability can be calculated by substituting eq. (40) into eq. (24). +We will end this section by proving the following theorem. +Theorem. If +� +UU†� +αα ̸= 1 and +� +UU†� +αβ = 0 for all α ̸= β, then the neutrino oscillation +probability in an arbitrary matter potential is indistinguishable from the unitary scenario. +The proof is as follows +� +i +UβiU +∗ +αi = +1 +� +(UU†)αα (UU†)ββ +� +i +UβiU∗ +αi = +� +UU†� +αα δαβ +� +(UU†)αα (UU†)ββ += δαβ. +(43) +From the above, it follows that +� +UU +†�−1 += I +=⇒ +U +†,−1U +−1 = I +=⇒ +U +†U = I and +hence U is unitary. With unitary U, the Hamiltonians in the vacuum mass basis in eq. (29) +reduces and coincides with the unitary one and hence the solution in eq. (41) will coincide +with the unitary scenario as well. This result holds for an arbitrary matter potential V (x) +since one can always construct the full solution as in (40). Let us denote this scenario as +the hidden nonunitary scenario. +C. +Identities +The combination that appears in the oscillation amplitude in the flavor basis (38) is +�Uβk �U∗ +αk ≡ +� +i,j +UβiU +∗ +αjXikX∗ +jk. +(44) +Substituting eq. (32) into the equation above, we obtain +�Uβk �U∗ +αk = +2+N +� +p=0 +(−1)p � +H +p� +βα c2+N−p,k +Zk +, +(45) +12 + +where we have defined +H ≡ U �HU +† = HUU +† = U∆U +† + V, +(46) +and we have used eq. (29) and eq. (28) to arrive at the last equality. It is important to note +that H is not equal to the Hamiltonian in the flavor basis H but they only coincide with +each other if U is unitary. Furthermore, H, not being related to �H and H by similarity +transformation, does not have to have the same eigenvalues as �H and H. +Under trace transformation H → H+c I with c any real constant or phase transformation +H → ΦHΦ† with Φ = diag +� +eiφ1, eiφ2, ... +� +, the probabilities (observables) (24) remain +invariant. +By observing that the following trace and phase transformation invariant +combinations [36] +Iαβ = Im +�� +H2� +αβ H∗ +αβ +� +, +(47) +Rαβ = |Hαβ|2 , +α ̸= β, +(48) +should be independent of matter potential if the matter potential is diagonal in the flavor +basis, several matter invariant identities are derived. The first one results in the Naumov- +Harrison-Scott (NHS) identity [36, 37] or their generalized versions [35] while the second +one results in further matter-invariant identities [35, 36]. In the nonunitary scenario, this +is no longer true since the matter potential in the flavor basis (28) is no longer diagonal +but given by V +� +UU +†�−1 +. +Incidentally, this also shows that once we have nondiagonal +NonStandard neutrino Interaction (NSI), eqs. (47) and (48) are no longer invariant under +matter potential [23, 35]. As shown in ref. [23], NSI is still distinct from nonunitary scenario +since the latter further breaks the unitary relations that we will discuss next. +Let us define the Jarlskog combinations by taking the imaginary part of the combination +above [38] +Jjk +βα ≡ Im +� +�Uβj �U∗ +αj �U∗ +βk �Uαk +� +. +(49) +If �U is unitary, one must have +� +k +Jjk +βα = +� +k +Im +� +�Uβj �U∗ +αj �U∗ +βk �Uαk +� += Im +� +�Uβj �U∗ +βj +� += 0. +(50) +Let us look at the modification due to nonunitarity. Since all the terms with p = q are real +and we are only left with terms of p ̸= q +Jjk +βα = +2+N +� +p̸=q;0≤p 0 +1 +p = 0 +, +(52) +we arrive at +� +k +Jjk +βα = +2+N +� +q=1 +(−1)q+1 c2+N−q,j +Zj +Im +�� +UU +†� +βα +� +H +q�∗ +βα +� +, +(53) +where we have used H +0 = UU +†. For unitary U +=⇒ +� +UU +†� +βα = δβα and since H +q is +Hermitian, the diagonal elements are real and we recover eq. (50). This is consistent with +the theorem we have proven and indeed, if +� +UU†� +αβ = 0 for all α ̸= β, the right hand side +of eq. (53) vanishes. In the vacuum, H +q = +� +U∆U +†�q +and one recover +� +k +Jjk +βα = −Im +�� +UU +†� +βα UαjU +∗ +βj +� +, +(54) +as can also be derived directly from eq. (49). So by measuring these relations above, we can +uncover unitarity violation in the matter (53) or in the vacuum (54). +IV. +HIGH VERSUS LOW SCALE NONUNITARITY +A. +In vacuum +In the absence of matter Xij = δij, eq. (39) becomes +Pβα (x) = +����� +� +i +UβiU +∗ +αie− +im2 +i x +2E +����� +2 += +1 +(UU†)αα (UU†)ββ +����� +� +i +UβiU∗ +αie− +im2 +i x +2E +����� +2 +. +(55) +For the high scale nonunitary scenario, +� +UU†� +αα = +� +UU †� +αα ̸= 1 and we obtain +P high +βα (x) = +1 +(UU †)αα (UU †)ββ +����� +3 +� +i=1 +UβiU ∗ +αie− +im2 +i x +2E +����� +2 +, +(56) +where U spans over three flavors. According to the theorem proved in the previous section, +in the hidden unitary scenario +� +UU †� +αβ = 0 for all α ̸= β, one cannot distinguish it from +unitary scenario since U will be unitary. This implies that nonunitary effect is proportional +to +� +UU †� +αβ for α ̸= β as encapsulated in eq. (54), making it more challenging to distinguish +14 + +it from the unitary scenario if the modulus of +� +UU †� +αβ is small (this conclusion also holds +in an arbitrary matter potential as we will discuss next). +Next, let us contrast the high scale nonunitary scenario to the low scale nonunitary +scenario. +(i) For the low scale nonunitary scenario, U is unitary with +� +UU†� +αα = 1 and we have +P low +βα (x) = +����� +3+N +� +i=1 +UβiU∗ +αie− +im2 +i x +2E +����� +2 +. +(57) +The most direct way to discover low scale nonunitary scenario is to have an experiment +with m2 +i>3 ∼ E/x such that oscillations involving new fermions can be measured.6 The +main challenge is, a priori, we do not know mi>3 or even if νi>3 exist but one can design +an experiment with identical detectors at various baselines to cover as large range of +E/x as possible. In the scenario, unitary relations (50) is satisfied in contrast to high +scale nonunitary scenario which satisfies (54). +(ii) If mi>3 ≫ +� +∆m2 +atm such that the oscillations involving them are averaged out, we +obtain [14, 17] +P low,ave +βα +(x) = Cαβ + +����� +3 +� +i=1 +ˆUβi ˆU ∗ +αie− +im2 +i x +2E +����� +2 +, +(58) +where we use ˆU to denote 3 × 3 submatrix from U and Cαβ is an additional constant +term, also known as the probability leaking term +Cαβ = +3+N +� +i=4 +|Uαi|2 |Uβi|2 . +(59) +It is bounded from above and below [14] +1 +N dαβ ≤Cαβ≤ dαβ, +(60) +where dαβ ≡ +� +1 − �3 +i=1 +��� ˆUαi +��� +2� � +1 − �3 +j=1 +��� ˆUβj +��� +2� +. In principle, a measurement of +Cαβ will allow us to obtain information about the number N of additional fermions. +6 For example, a recent short baseline reactor experiment STEREO [39] rules out the existence of a sterile +neutrino with mass in the eV range and mixing element of the order of 0.4 and larger. +15 + +Notice that for N = 1, Cαβ is completely fixed by ˆU. In practice, it is a challenging +task since this term is expected to be small, being fourth order in unitarity violation +parameter +ϵαβ ≡ +� +� +� +� +�����δαβ − +3 +� +i=1 +ˆUβi ˆU ∗ +αi +�����. +(61) +It is easier to distinguish eq. +(58) and eq. +(56) from the normalization term +� +UU †� +αα +� +UU †� +ββ since this is an effect of ϵ2. We can also measure this deviation +by looking at the deviation from unitary relations (50). With oscillations involving +νi>3 averaged out, one cannot have direct access to the Jjk +βα with j, k > 3 in eq. (49) +and hence we have +� +k +Jjk +βα = −Im +�� +ˆU ˆU †� +βα +ˆUαj ˆU ∗ +βj +� +, +(62) +in contrast to eq. (54) for the high scale nonunitary scenario. +In Figure 2, we plot the oscillation probability for νµ → νe as a function of neutrino energy +Eν fixing the baseline x = 1300 km, for the standard three-flavor unitary scenario (solid black +line), high scale (dotted lines) and low scale (dashed lines) nonunitary scenarios. Here and +in the following, the standard parameters are always set to the global best fit values for +NO from [1, 2]. For the high scale nonunitary scenario, we set +� +UU †� +ee = +� +UU †� +µµ = 0.96, +� +UU †� +ττ = 1, +� +UU †� +eτ = +� +UU †� +µτ = 0 and +� +UU †� +eµ = {10−3, 10−2, 0.03}. To compare +with the low scale nonunitary scenario where oscillations involving νi>3 are averaged out, we +also set ( ˆU ˆU †)ee = ( ˆU ˆU †)µµ = 0.96, ( ˆU ˆU †)ττ = 1, ( ˆU ˆU †)eτ = ( ˆU ˆU †)µτ = 0 and ( ˆU ˆU †)eµ = +{10−3, 10−2, 0.03}. With this choice, the leaking term is bounded as 0.0016/N < Ceµ ≤ +0.0016 and we have set Cαβ = 0.0016 for illustration. As we can see explicitly, as +� +UU †� +eµ +decreases, the high scale nonunitary scenario approaches the unitary scenario while for the +low scale nonunitary scenario, this does not happens. To illustrate the effect of Ceµ, we plot +in Figure 3 by setting Ceµ = 0 (dotted lines) in comparison with the case with Ceµ = 0.0016 +(dashed lines). +In the neutrino experiments, the number of observed of neutrinos at the detector can be +16 + +FIG. 2. Comparison of the probability of νµ → νe at x = 1300 km in the vacuum as a function +of neutrino energy Eν between the high scale nonunitary scenario (dotted lines) and low scale +nonunitary scenario (dashed lines) with (UU †)ee = (UU †)µµ = ( ˆU ˆU †)ee = ( ˆU ˆU †)µµ = 0.96. The +solid black line is the standard three-flavor unitary scenario. +written as7 +Nβα = σβPβα (x) φα, +(63) +where φα is the να neutrino flux at production and σβ is the detection cross section of νβ +and the energy dependence of all the terms are left implicit. In order to determine Pβα (x) +from Nβα, it is crucial to have precise determination of σβ ×φα. To mitigate the uncertainty +in flux determination, one can take the ratio of measurement in a far detector placed at x1 +and a near detector placed at x0 +Nβα (x1) +Nαα (x0) = σβPβα (x1) φα +σαPαα (x0) φα += σβPβα (x1) +σαPαα (x0). +(64) +7 This is a theorist’s expression that we have not included experimental effects like detection efficiency and +energy reconstruction. +17 + +c = 1300 km, NO, p = 0 g/cm3 +0.12 +(UUt)eu= 0.001 +: (UUt)eμ= 0.01 +0.10 +(UUt)eμ= 0.03 +- (UUt)eμ= 0.001 +0.08 +(UUt)eμ = 0.01 +(Utt)eμ = 0.03 +0.04 +0.02 +0.00 +100 +101 +E,[GeV]FIG. 3. The probability of νµ → νe at x = 1300 km in the vacuum as a function of neutrino energy +Eν for low scale nonunitary scenario with ( ˆU ˆU †)ee = ( ˆU ˆU †)µµ = 0.96 by setting the leaking term +to be the maximum Ceµ = 0.0016 (dashed lines) and the minimum Ceµ = 0 (dotted lines). The solid +black line is the standard three-flavor unitary scenario. +For the high and low scale unitary scenarios with x0 ≪ E/m2 +i for all i, eqs. (56) and (57) +are +P high +αα (x0) ≃ P low +αα (x0) ≃ 1 − O +� +x2 +0m4 +i /E2� +, +(65) +which give +Nβα (x1) +Nαα (x0) ≃ σβ +σα +P high,low +βα +(x1) . +(66) +With dedicated measurements of σβ and σα, the two scenarios can be distinguished from +eqs. (56) and (57).8In the low scale unitary scenario with x0 ≫ E/m2 +i for i > 3 such that +8 In the high scale unitary scenario, the additional factor which appears in the cross section in comparison +with the SM expectation σα = σSM +α +� +UU †� +αα. +This factor should already be included in dedicated +measurement. +18 + +c = 1300 km, NO, p = 0 g/cm3 +0.12 +(UUt)eμ = 0.001,Ceμ = 0 +...X... +(UUt)eμ = 0.01, Ceμ = 0 +0.10 +(UUt)eμ = 0.03,Ceμ = 0 +(UUt)eμ = 0.001, Ceμ = 0.0016 +0.08 +(UUt)eμ = 0.01, Ceμ = 0.0016 +(UUt)eμ = 0.03,Ceμ = 0.0016 +10.06 +0.04 +0.02 +0.00 +100 +101 +E,[GeV]the fast oscillations can be averaged out, eq. (58) gives +P low,ave +αα +(x0) ≃ Cαα + +����� +3 +� +i=1 +ˆUαi ˆU ∗ +αi +����� +2 +− O +� +x2 +0m4 +i /E2� +. +(67) +In this case, one has +Nβα (x1) +Nαα (x0) ≃ σβ +σα +P low,ave +βα +(x1) +Cαα + +����3 +i=1 ˆUαi ˆU ∗ +αi +��� +2. +(68) +which can be differentiated from eq. (66). This type of arrangement has been planned in +the upcoming neutrino experiments DUNE [40] and T2HK [41]. +Besides through neutrino oscillation experiments, the synergy with constraints from the +electroweak precision measurements is needed to discover high scale nonunitarity [42, 43] +or low scale nonunitarity [44]. If +� +UU†� +ee = +� +UU†� +µµ = +� +UU†� +ττ ̸= 1, one can measure +this deviation by comparing the leptonic weak processes with the hadronic weak processes. +For instance, while absolute lifetimes of µ±, π±, K± and K0 will be affected, the leptonic +branching ratios will be the same. +For K± and K0, the branching ratios to hadronic +and leptonic channels will be modified. If +� +UU†� +ee ̸= +� +UU†� +µµ ̸= +� +UU†� +ττ ̸= 1, lepton +universality is broken and one can measure this by studying different leptonic weak processes. +The reader can refer to refs. [42–44] for more details. +B. +In matter +Now let us consider the scenario with matter effect. For high scale nonunitary scenario, +eq. (29) is just +�Hhigh = ∆ + U +−1V U +†,−1, +(69) +where U spans over 3 flavor. According to theorem proved earlier, in the hidden unitary +scenario +� +UU †� +αβ = 0 for all α ̸= β, U is unitary and hence �Hhigh is indistinguishable from +the unitary scenario. Moreover, this result holds for an arbitrary potential V (x) since one +can always split x into intervals small enough that V (x) is constant and then construct the +full solution as in eq. (40). So even in matter, nonunitary effect is proportional to +� +UU †� +αβ +for α ̸= β in the high scale nonunitary scenario as encapsulated in eq. (53). +In low scale nonunitary scenario, U +−1 = U†, eq. (29) becomes +�Hlow = ∆ + U†V U. +(70) +19 + +First of all, notice that eqs. (47) and (48) remain matter invariant as long as V is diagonal +and hence the resulting matter invariant identities hold. Next, besides the two possibilities +discussed in the vacuum case, now we have a new handle: with the matter effect, the +difference appear at leading order in small unitary violating parameter ϵ in which the leading +Hamiltonian �Hlow is given by [17] +�Hlow,0 = ∆ + ˆU †V ˆU. +(71) +Comparing the �Hhigh and �Hlow,0, the difference is proportional to +�Hlow,0 − �Hhigh = ˆU †V ˆU − U −1κV κU †,−1, +(72) +where we have written U = κ−1U with κ ≡ diag +�� +(UU †)ee, +� +(UU †)µµ, +� +(UU †)ττ +� +. +Let us suppose that ˆU = U, i.e., the nonunitary effect results in an identical 3 × 3 +submatrix. Then �Hlow,0 − �Hhigh ̸= 0 and matter effects will result in different eigenvalues +and eigenvectors. For 0 ≤ x < x1 where V is constant, we can solve for the oscillation +amplitude S in the flavor basis (38) as follows +Shigh +βα += κ−1 +β κ−1 +α +� +i,j,k +UβiU ∗ +αjXhigh +ik +Xhigh,∗ +jk +e−iλhigh +k +x, +(73) +Slow,0 +βα += +� +i,j,k +UβiU ∗ +αjXlow +ik Xlow,∗ +jk +e−iλlow +k +x. +(74) +Hence, besides the amplitudes, the frequencies are different due to different eigenvalues λk. +To minimize the difference in matter potential, let us make a different choice ˆU = κU †,−1 +such that �Hlow,0 = �Hhigh. In this case, �Hhigh and �Hlow,0 will have exactly the same eigenvalues +λk and are diagonalized by the same unitary matrix X. So, we have +Shigh +βα += κ−1 +β κ−1 +α +� +i,j,k +UβiU ∗ +αjXikX∗ +jke−iλkx, +(75) +Slow,0 +βα += κβκα +� +i,j,k +� +U †,−1� +βi +� +U †,−1�∗ +αj XikX∗ +jke−iλkx, +(76) +where the differences are only in the amplitudes. +In Figure 4, we show identical situation with Figure 2 except with a constant matter +density of 3 g/cm3. Overall, the matter effect enhances the differences between the high +scale and low scale nonunitary scenarios as expected. We also observe that with decreasing +modulus of +� +UU †� +eµ, the high scale nonunitary scenario approaches the unitary scenario +(the black line) while this does not happen for the low scale nonunitary scenario. In Figure +20 + +FIG. 4. Same as Figure 2 but in a constant matter density of 3 g/cm3. +5, we consider earth core-crossing neutrinos in a simplified PREM model (see appendix A +of [23]) with high scale nonunitary parameters +� +UU †� +ee = +� +UU †� +µµ = 0.96, +� +UU †� +ττ = 1, +� +UU †� +eτ = +� +UU †� +µτ = 0 and +� +UU †� +eµ = {10−3, 10−2, 0.03}. +For low scale nonunitary +parameters, on the left plot, we set ˆU = U while on the right plot, we set ˆU = κU †,−1. +These represent the two extreme cases where for ˆU = U, the difference in matter between +the two scenarios is maximal while for ˆU = κU †,−1, the matter effect is identical. +V. +CONCLUSIONS +In this article, we have derived analytical oscillation probability amplitude for arbitrary +flavors of neutrinos without assuming a unitary U. With this result, we have proven a +theorem that as long as +� +UU†� +βα = 0 for all α ̸= β, the scenario is indistinguishable +from a unitary scenario in an arbitrary matter potential. +We further derive a general +identity (53) which reduces to (54) in the vacuum and vanishes in the unitary scenario. +21 + + = 1300 km, NO, p= 3 g/cm3 +0.12 +(UUf)eu = 0.001 +. (UUt)eu = 0.01 +0.10 +(UUt)eμ= 0.03 +- (UUt)eμ = 0.001 +0.08 +(UUt)eμ = 0.01 +(Utt)eμ = 0.03 +0.06 +0.04 +0.02 +0.00 +100 +101 +E,[GeV]FIG. 5. Comparison of the probability of νµ → νe as a function of neutrino energy Eν using the +simplified PREM model for neutrino crossing through the Earth core between high scale nonunitary +scenario (dotted lines) and low scale nonunitary scenario (dashed lines). We have shown the two +extreme choices ˆU = U (left plot) and ˆU = κU †,−1 (right plot). See text for further explanation. +The solid black line is the standard three-flavor unitary scenario. +We have highlighted the differences between high scale and low scale nonunitary scenarios +in neutrino oscillations, which are to be expected since in the former case, all new physics are +integrated out while in the latter, new states are accessible though not necessarily remain +coherent to result in oscillations. On the one hand, although high scale nonunitary scenario +is model-independent (all the effects are fully captured by a nonunitary U), nonunitary +effects are proportional to +� +UU †� +βα for α ̸= β and hence they can be discovered only +through appearance experiments νβ → να. On the other hand, while low scale nonunitary +scenario is model-dependent (depending on the properties of the new states), an almost +model-independent scenario can be obtained if oscillations involving the new states can be +averaged out and nonunitary effects can be captured by a nonunitary ˆU and a leaking term +Cαβ. Furthermore, we have shown that low scale nonunitary effects remain even in the limit +of vanishing ( ˆU ˆU †)βα for all α ̸= β. +High scale and low scale nonunitary scenarios are +distinct, one should test them both in the neutrino oscillation experiments. +22 + +Earth-crossing neutrinos, NO +0.7 +— (UUt)eu= 0.001 +(UUt)eμ = 0.01 +0.6 +- (UUt)eμ = 0.03 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +金宝金金金 +100 +101 +E[GeV]Earth-crossing neutrinos, NO +0.7 + (UUt)eu=0.001 +(UUt)eμ = 0.01 +0.6 +- (UUt)eμ = 0.03 +0.5 +0.4 +0.3 +0.2 +0.1 +: +: +0.0 +金金 +100 +101 +E,[GeV]VI. +ACKNOWLEDGMENTS +C.S.F. acknowledges the support by grant 2019/11197-6 and 2022/00404-3 from São Paulo +Research Foundation (FAPESP), and grant 301271/2019-4 and 407149/2021-0 from National +Council for Scientific and Technological Development (CNPq). +[1] Ivan Esteban, M. C. 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Rev. +D 93, 033005 (2016), arXiv:1511.00683 [hep-ph]. +26 + diff --git a/VtFOT4oBgHgl3EQf6zSP/content/tmp_files/load_file.txt b/VtFOT4oBgHgl3EQf6zSP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6783f76a64f5d7c30a466537203210be95b97c54 --- /dev/null +++ b/VtFOT4oBgHgl3EQf6zSP/content/tmp_files/load_file.txt @@ -0,0 +1,680 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf,len=679 +page_content='Theoretical Aspect of Nonunitarity in Neutrino Oscillation Chee Sheng Fong1, ∗ 1Centro de Ciências Naturais e Humanas Universidade Federal do ABC, 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='210-170, Santo André, SP, Brazil Abstract Nonunitarity can arise in neutrino oscillation when the matrix with elements Uαi which relate the neutrino flavor α and mass i eigenstates is not unitary when sum over the kinematically accessible mass eigenstates or over the three Standard Model flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We review how high scale nonunitarity arises after integrating out new physics which is not accessible in neutrino oscillation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We contrast this to the low scale nonunitary scenario in which there are new states accessible in neutrino oscillation experiments but the oscillations involving these states are fast enough such that they are averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Then we derive analytical formula for the neutrino oscillation probability amplitude for an arbitrary flavor of neutrinos without assuming unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' This result allows us to prove a theorem that if � UU†� αβ = 0 for all α ̸= β, then the neutrino oscillation probability in an arbitrary matter potential is indistinguishable from the unitary scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The main implication is that nonunitary effects are proportional � UU†� αβ with α ̸= β and disappearance experiments νβ → να are necessary for their discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Independently of matter potential, while nonunitary effects for high scale nonunitary scenario disappear as � UU†� αβ → 0 for all α ̸= β, low scale nonunitary effects remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' ∗ sheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='fong@ufabc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='br 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='12960v1 [hep-ph] 30 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' INTRODUCTION In the Standard Model (SM), there are three neutrinos which participate in the weak interactions and we have detected all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Despite they should be massless in the SM, experimentally, we have determined two nonzero mass-squared differences among them, showing that at least two of them are massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Great experimental progress has been made in pinning down the neutrino parameters in the three-flavor paradigm with the current global best fit values given by [1, 2]: two mass splitting ∆m2 sol ≡ m2 2 − m2 1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='4 × 10−5 eV2, ∆m2 atm ≡ |m2 3 − m2 1| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='5 × 10−3 eV2, and three mixing angles θ12 = 33◦, θ23 = 49◦, and θ13 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='5◦ determined to precision of a few percents with a preference of Normal mass Ordering (NO) m2 3 − m2 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The absolute mass scale and the Dirac CP phase have not been determined while θ23 can still be in the first quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' From the theoretical side, the mechanism behind neutrino mass together with the nature of the mass, Dirac or Majorana (including quasi-Dirac), remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Treating the SM as an effective field theory, Majorana mass for neutrinos arise from the unique dimension-5 Weinberg operator [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' This is the minimal scenario without additional light degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In order to have Dirac mass, additional light degrees of freedom are needed to be the Dirac partners of the SM neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In either cases, it is not necessary that there is unitary violation or nonunitarity in the sense that the matrix Uαi which relates the flavor (with index α) and mass (with index i) eigenstates of neutrinos when sum over the kinematically accessible mass eigenstates or over the three SM flavors are not unitary accessible � i UαiU∗ βi ̸= δαβ, � α=e,µ,τ UαiU∗ αj ̸= δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (1) In this work, we will focus on nonunitary scenario when the relations above hold true and contrast it to the three-flavor paradigm when unitarity is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We aim to give a more complete theoretical discussion of nonunitarity in neutrino oscillation [5–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We start by discussing how nonunitary can arise in Section II, highlighting two distinct scenarios: high scale nonunitary scenario where new physics resides beyond the energy scale of neutrino oscillation experiments and low scale nonunitary scenario where new fermionic states (sterile neutrinos) mix with the SM neutrinos and are accessible in neutrino oscillation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In Section III, we derive analytical solution for neutrino 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The probability of νµ → νe for neutrino crossing the entire Earth as a function of neutrino energy Eν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' As the modulus of (UU †)eµ decreases, the probability approaches that of the standard three-flavor unitary scenario shown as the black solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' oscillation without assuming unitarity and in Section IV, we discuss how the high scale and low scale nonunitary effects can show up and be distinguished in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Finally we present some concluding remarks in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' While many excellent discussions are there in previous work for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' [5], we will present some new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In particular, we will prove a theorem in Section III that if nonunitary is only diagonal accessible � i UαiU∗ βi = cαδαβ, (2) with cα > 0, then neutrino oscillation probability in an arbitrary matter potential is indistinguishable from the unitary scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' An important implication is that high scale nonunitary effects are proportional to the off-diagonal elements of UU† and can only be confirmed in appearance experiments, in contrast to low scale nonunitary scenario where the effects can remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' To illustrate this point, in Figure 1, we show the probability of νµ → νe 3 Earth-crossing neutrinos, NO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='7 ×-- (UUf)eu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='6 --(UUt) ew=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='005 --(UUt)e =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='01 ★ (UUf)ew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='5 (UUt) u= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='03 (UUt) Jet=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='0 100 101 E[GeV]as a function of neutrino energy Eν in high scale nonunitary scenario for neutrino passing through the Earth using the public code NuProbe [23, 24] with a simplified (Preliminary Reference Earth Model) PREM model [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Denoting U as a 3 × 3 mixing matrix, we have fixed (UU †)ee = (UU †)µµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='96, (UU †)µµ = 1, (UU †)eτ = (UU †)µτ = 0 while varying (UU †)eµ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='001 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='04 (dashed lines) and the rest of the standard parameters have been fixed to the NO global best fit values from [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' As the modulus of (UU †)eµ decreases, the probability approaches that of the standard three-flavor unitary scenario (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' For reader interested in experimental probe of nonunitary scenarios, he or she can jump straight to Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' MODELS Here we will focus on models for neutrino oscillation assuming that the center-of-mass energy involved is below the electroweak symmetry breaking E < vEW ≡ 174 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We will consider only Majorana mass term for neutrinos though the discussions below are independent of whether we have a Majorana or Dirac mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' While neutrino oscillation cannot distinguish between strictly Majorana and Dirac mass, it is possible to distinguish them from quasi-Dirac scenario in which both types of mass terms exist while the Majorana mass term is much smaller than the Dirac one [26–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We will consider this interesting scenario in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' High scale nonunitary Assuming that for E < vEW, we only have three SM neutrinos (νe, νµ and ντ are the SM left-handed neutrino flavor states), the general neutrino Lagrangian allowed by the SM electromagnetic gauge symmetry U(1)EM in the charged lepton flavor basis is given by Lν = 1 2 � iνα/∂Dαβνβ − νc αmαβνβ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � − �g 2W − µ ℓαγµPLνα + g √ 2 cos θW ZµναγµPLνα + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � , (3) where the flavor indices are α, β = e, µ, τ, D is a dimensionless Hermitian matrix while m is a symmetric matrix with mass dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In the second line, g is the SU(2)L gauge coupling, θW is the Weinberg or weak angle, PL = 1 √ 2 (1 − γ5) is the left-handed projector, 4 {ℓe, ℓµ, ℓτ} ≡ {e−, µ−, τ −} are the charged leptons and W ∓ and Z are the charged and neutral weak bosons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Without additional light degrees of freedom, only Majorana mass term is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='1 Treating the Standard Model as an effective field theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' the neutrino mass and the modified kinetic terms come respectively from dimension-5 [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 4] and dimension-6 operators [31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 43] O5 = λαβ Λ5 � Lc αϵH � (LβϵH) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (4) O6 = ηαβ Λ2 6 � LαϵH∗� /∂ (LβϵH) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (5) where L and H are the SU(2)L lepton and Higgs doublets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' λ and η are dimensionless symmetric and Hermitian matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' and Λ5 and Λ6 are effective scales below which the operators O5 and O6 are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Implicitly, we have assumed E ≪ Λ5, Λ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Not all ultraviolet models which generate O5 also generate O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' For instance, type- I and type-III seesaw models generate both O5 and O6 while type-II seesaw model only generates O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In order to obtain canonical normalized kinetic term, we can first diagonalize the kinetic term as D = Y † ˆDY where Y is unitary and ˆD is real and diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Defining the normalized neutrino field as �ν ≡ � ˆDY ν, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (3) becomes Lν = 1 2 � i�να/∂�να − �νc α �mαβ�νβ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � − � g 2W − µ ℓαγµPL � Y †� ˆD −1� αβ �νβ + g √ 2 cos θW Zµ � ˆD−1� αα �ναγµPL�να + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � , (6) where we have defined �m ≡ � ˆD −1 Y ∗mY †� ˆD −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (7) The symmetric mass matrix above can be diagonalized by a unitary matrix V as �m = V ∗ ˆmV † where ˆm is real and diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Defining the neutrino field in the mass basis as ˆν = V †�ν, we have [5] Lν = 1 2 � iˆνi/∂ˆνi − ˆνc i ˆmiiˆνi + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � − �g 2WµℓαγµPLUαiˆνi + g √ 2 cos θW Zµˆνi � U †U � ij γµPLˆνj + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � , (8) 1 In order to write down a Dirac mass term, new light degrees of freedom are required such that we can write ν′ fm′ fβνβ where ν′ f are some new fermion fields which do not participate in weak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In this case, m′ fβ is a general complex matrix with mass dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Besides the fact that the mass term should be diagonalized by two unitary matrices, the discussion will remain the same since the effect of unitary rotation of ν′ which do not feel the weak force, is not observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 5 where we denote i, j = 1, 2, 3 to be the indices in mass basis and we have defined Uαi ≡ � Y †� ˆD −1 V � αi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (9) Notice that UU † = Y † ˆD−1Y, U †U = V † ˆD−1V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (10) Only if ˆD = I is the 3 × 3 identity matrix, unitary is restored in which U †U = UU † = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We denote the general case with UU † ̸= I and U †U ̸= I, high scale nonunitary scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Low scale nonunitarity Assuming that for E < vEW, besides the three SM neutrinos (νe, νµ and ντ), we also have additional neutral fermions fields (νs1, νs2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', νsN) which do not participate in weak interactions but mix with the SM neutrinos through the mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The mixing between the SM and the additional fermions are the Dirac mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Here we will focus on mostly Majorana scenario where the Majorana mass term for new fermions is somewhat larger than the Majorana mass for the SM neutrinos as well as the Dirac mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='2 In order to highlight the distinction from the high scale unitary violation scenario, we further assume that the kinetic terms for all the fermions are canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In this case, the U(1)EM-invariant neutrino Lagrangian in the charged lepton flavor basis is given by Lν = 1 2 � iνα/∂να − νc αmαβνβ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � − �g 2W − µ ℓαγµPLνα + g √ 2 cos θW ZµναγµPLνα + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � , (11) where α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' = e, µ, τ, s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', sN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The symmetric mass matrix m can be diagonalized by a unitary matrix U as m = U∗ ˆmU† where ˆm is real and diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Here and in the following, we use a boldface U to denote a (3 + N) × (3 + N) matrix while U is reserved for a 3 × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Defining the neutrino field in the mass basis as ˆν = U†�ν, we have Lν = 1 2 � iˆνi/∂ˆνi − ˆνc i ˆmiiˆνi + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � − �g 2WµℓαγµPLUαiˆνi + g √ 2 cos θW ZµˆνiγµPLˆνj + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � , (12) 2 Depending on the relative size of the corresponding mass terms, one can obtain the mostly Majorana, strictly Majorana, strictly Dirac or quasi-Dirac scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Strictly Dirac scenario cannot be distinguished from strictly Majorana scenario and will not also not result in apparent unitary violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Mostly Majorana scenario will result in apparent unitary violation allowed by current measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Finally, though there is no apparent unitary violation, quasi-Dirac scenario will result in distinguished signatures [26–30] and will be considered in a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 6 where i, j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', 3+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' What distinguish this from the high scale nonunitary scenario is that U†U = UU† = I where I is the (3 + N) × (3 + N) identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Strictly speaking, there is no unitary violation in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' However, since only the mixing involving the SM neutrinos can be measured, one have � α=e,µ,τ UαiU∗ αj ̸= δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (13) Furthermore, if mi>3 ≫ � ∆m2 atm such that oscillations involving νi>3 can be averaged out, at leading order in small unitary violating parameter, the mixing elements involved are those of i = 1, 2, 3 which sum to [14, 17] 3 � i=1 UαiU∗ βj ̸= δαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (14) We denote this as low scale nonunitary scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' If some additional states are not kinematically allowed in the process, one should describe it as in the high scale nonunitary scenario discussed previously, with possible enlargement of flavor beyond 3 to accommodate additional kinematically accessible states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' GENERIC 3 + N NEUTRINO OSCILLATIONS Here we will develop a generic 3+N neutrino oscillation framework which can be applied to both unitary and nonunitary neutrino oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In general, the neutrino flavor states |να⟩ are related to the mass eigenstates |νi⟩ through a matrix U |να⟩ = 1 � (UU†)αα � i U∗ αi |νi⟩ , (15) where α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' = e, µ, τ, s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', sN (νe, νµ and ντ are the SM left-handed neutrino flavor states) and i, j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', 3 + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In general, U does not have to be unitary UU† ̸= I and U†U ̸= I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Since the mass eigenstates are orthogonal ⟨νj|νi⟩ = δji, the flavor states are properly normalized ⟨να|να⟩ = 1 though they are not necessarily orthogonal but equal to ⟨νβ|να⟩ = � UU†� βα � (UU†)αα (UU†)ββ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (16) In other words, there is nonzero overlap between different flavor states and there is a probability of flavor changing even at zero distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The set of all {|νi⟩} is complete and 7 from the orthogonality condition, it further satisfies the completeness relation � i |νi⟩ ⟨νi| = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (17) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (15), we can write the inverse relation3 |νi⟩ = � α � U∗,−1� iα � (UU†)αα |να⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (18) One can verify the orthogonality condition ⟨νj|νi⟩ = � α,β � U−1� jβ � U∗,−1� iα ⟨νβ|να⟩ = � U−1UU†U†,−1� ji = δji, (19) where in the second equality, we have used eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In the rest of the article, to avoid expressions crowded with normalization factors, we will define Uαi ≡ Uαi � (UU†)αα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (20) Starting from an initial state|να (0)⟩ = |να⟩, the time-evolved state |να (t)⟩ is described by the Schrödinger equation i d dt |να (t)⟩ = H |να (t)⟩ , (21) where the Hamiltonian is H = H0 + HI with H0 the free Hamiltonian H0 |νi⟩ = Ei |νi⟩ , Ei = � ⃗p2 i + m2 i , (22) and HI the interaction Hamiltonian with matrix elements ⟨νβ| HI |να⟩ = Vβα, (23) where V ∗ βα = Vαβ since H† I = HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Assuming relativistic neutrinos, we trade t = x and the amplitude of the transition να → νβ at distance x is then given by Sβα (x) ≡ ⟨νβ|να (x)⟩ and the probability of an initial state |να⟩ being detected as |νβ⟩ at distance x is Pβα (x) = |Sβα (x)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (24) 3 We can prove that inverse exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Supposing that |νi⟩ = � α Viα |να⟩ , and using orthogonality condition ⟨νj|νi⟩ = δji, we have δij = � α Viα ⟨νj|να⟩ = � α Viα 1 � (UU†)αα U∗ αj, and hence Viα = � (UU†)αα � U∗,−1� iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 8 From (21), we can write the evolution equation of Sβα (x) as i d dxSβα (x) = ⟨νβ| H0 + HI |να (t)⟩ = � i ⟨νβ| H0 + HI |νi⟩ ⟨νi|να (t)⟩ = � η �� i UβiEi � U −1� iη + � γ Vβγ � UU †�−1 γη � Sηα (x) , (25) where in the second equality, we have inserted the completeness relation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (17) and in the last equality, we have used eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (22), (23), (18) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Considering relativistic neutrinos E ≫ mi and expanding Ei ≃ E + m2 i 2E , we obtain, in matrix notation idS (x) dx = � U∆U −1 + V � UU †�−1� S (x) , (26) where ∆ ≡ 1 2E diag � m2 1, m2 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', m2 3+N � = diag (∆1, ∆2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', ∆3+N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (27) We have dropped the constant E which is an overall phase in S(x) and hence does not affect the probability in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Vacuum mass basis From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (26), the Hamiltonian in the flavor basis given by H ≡ U∆U −1 + V � UU †�−1 , (28) is not Hermitian H† ̸= H nor normal H†H ̸= HH†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In the following, we will prove that they can be diagonalized with real eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Let us change the Hamiltonian to the vacuum mass basis in which the free Hamiltonian is diagonal �H ≡ U −1HU = ∆ + U −1V U †,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (29) Notice that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (29) is Hermitian �H† = �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Assuming V = V † is constant in the interval of interest 0 ≤ x < x1, we can diagonalize the �H with a unitary matrix X �H = X ˆHX†, (30) 9 where X is unitary and ˆH = diag (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', λ3+N) is diagonal and real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Since eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (28) and (29) are related by similar transformation, they have the same eigenvalues �HX = X ˆH =⇒ HUX = UX ˆH and hence we can also write H = UX ˆH � UX �−1 , (31) where the nonnormal H is diagonalized by a nonunitary UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We see explicitly that despite the Hamiltonian in flavor basis H appears to be non-Hermitian, the eigenvalues remain real while the source of nonunitarity comes nonunitary U resulting in nonorthogonal flavor basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Although UX is nonunitary, one can formally solve for � UX � αi � UX �−1 iβ in terms of eigenvalues and Hamiltonian elements using the same method as in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' [23, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Nevertheless, as we will see in the next subsection, the combination which appears in neutrino oscillation probability is not � UX � αi � UX �−1 iβ and we will solve for XikX∗ jk instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' As shown in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' [23, 32], by raising eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (30) to the power of 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', 2 + N and taking into account the unitary relation XX† = I, one can form a set of 3+N linearly independent equations for XikX∗ jk where the coefficients form a Vandermonde matrix4 which can be inverted to give [23] (see also the pioneering work of Kimura, Takamura and Yokomakura who applied similar method for 3-flavor scenario [33, 34]) XikX∗ jk = 2+N � p=0 (−1)p ( �Hp)ijc2+N−p,k Zk , (32) where we have defined Zk ≡ � p̸=k (λp − λk) , (33) cp,k ≡ � {q̸=r̸=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='}̸=k λqλr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � �� � p , (34) with [ ˜H0]ij = δij and c0,k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The sum in cp,k is over all possible unordered combinations of p distinct eigenvalues λqλr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' where none of them is equal to λk and hence with 3 + N neutrino flavors, cp,k has � 2+N p � = (2+N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='(2+N−p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' terms in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' As shown in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' [35], the 4 The equations obtained with power in λi greater than 2 + N are not linearly independent since they can be rewritten in term of lower power using the characteristic equation of �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Suppose we have d + 1 degenerate eigenvalues λl = λk for l = k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', k + d, we only need to solve for the combination � l XilX∗ jl corresponding to λk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 3 + N − d linear equations can be obtained from raising eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (30) to the power of 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', 2 + N − d, including XX† = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 10 numerator of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (32) can be written in mathematically equivalent form in terms of elements of the adjugate of λkI − �H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' [Adj(λkI − �H)]ij though we have found that that numerically evaluating the analytical expression eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (32) is about three times faster than evaluating the expression with adjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Oscillation probability There is one subtle but crucial point regarding �S in the vacuum mass basis which satisfies id�S (x) dx = �H �S (x) , (35) and the solution is given by �S (x) = Xe−i ˆHxX†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (36) How do we relate �S (x) of the vacuum mass basis and S (x) in the flavor basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Fixing the initial conditions �S (0) = I and S (0) = UU † which follow from the orthogonality of mass eigenstates and the nonorthogonality of flavor eigenstates eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (16), respectively, we have �S (x) ≡ U −1S (x) U †,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (37) Hence we can write S (x) = UXe−i ˆHx � UX �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (38) It follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (24) that the neutrino oscillation probability for να → νβ at a distance 0 ≤ x < x1 is5 Pβα (x) = ����� � i,j,k UβiU ∗ αjXikX∗ jke−iλkx ����� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (39) which has exactly the same form as the unitary case despite that U does not have to be unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' One can generalize the solution in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (38) to the case when V is x-dependent by splitting x into intervals small enough that V (x) is approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Considering 0 = x0 < x1 < x2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' where V (x) is equal to constant Va for each interval xa−1 < x < xa, we obtain S (x) = T � a=1 S(a) (x) , (40) 5 The oscillation probability P βα (x) for antineutrino να → νβ is obtained by taking Uαi → U ∗ αi and since our Universe consists only of matter, we should also take V → −V in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' 11 where we have defined S(a) (x) ≡ � UX(a)� e−i ˆH(a)x(a) � UX(a)�† , (41) x(a) ≡ [(x − xa−1) θ (xa − x) + (xa − xa−1) θ (x − xa)] θ (x − xa−1) , (42) with θ (x ≥ 0) = 1, θ (x < 0) = 0 and T denotes the space ordering of the matrix multiplication such that the a term is always to the left of a − 1 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Furthermore, ˆH(a) = diag � λ(a) 1 , λ(a) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=', λ(a) 3+N � and X(a) denote respectively the matrix of eigenvalues and unitary matrix which diagonalizes �H as �H(a) = X(a)† ˆH(a)X(a) in the interval xa−1 < x < xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The neutrino oscillation probability can be calculated by substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (40) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' We will end this section by proving the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' If � UU†� αα ̸= 1 and � UU†� αβ = 0 for all α ̸= β, then the neutrino oscillation probability in an arbitrary matter potential is indistinguishable from the unitary scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The proof is as follows � i UβiU ∗ αi = 1 � (UU†)αα (UU†)ββ � i UβiU∗ αi = � UU†� αα δαβ � (UU†)αα (UU†)ββ = δαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (43) From the above, it follows that � UU †�−1 = I =⇒ U †,−1U −1 = I =⇒ U †U = I and hence U is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' With unitary U, the Hamiltonians in the vacuum mass basis in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (29) reduces and coincides with the unitary one and hence the solution in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (41) will coincide with the unitary scenario as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' This result holds for an arbitrary matter potential V (x) since one can always construct the full solution as in (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Let us denote this scenario as the hidden nonunitary scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Identities The combination that appears in the oscillation amplitude in the flavor basis (38) is �Uβk �U∗ αk ≡ � i,j UβiU ∗ αjXikX∗ jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (44) Substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (32) into the equation above, we obtain �Uβk �U∗ αk = 2+N � p=0 (−1)p � H p� βα c2+N−p,k Zk , (45) 12 where we have defined H ≡ U �HU † = HUU † = U∆U † + V, (46) and we have used eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (29) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (28) to arrive at the last equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' It is important to note that H is not equal to the Hamiltonian in the flavor basis H but they only coincide with each other if U is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Furthermore, H, not being related to �H and H by similarity transformation, does not have to have the same eigenvalues as �H and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Under trace transformation H → H+c I with c any real constant or phase transformation H → ΦHΦ† with Φ = diag � eiφ1, eiφ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' � , the probabilities (observables) (24) remain invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' By observing that the following trace and phase transformation invariant combinations [36] Iαβ = Im �� H2� αβ H∗ αβ � , (47) Rαβ = |Hαβ|2 , α ̸= β, (48) should be independent of matter potential if the matter potential is diagonal in the flavor basis, several matter invariant identities are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' The first one results in the Naumov- Harrison-Scott (NHS) identity [36, 37] or their generalized versions [35] while the second one results in further matter-invariant identities [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' In the nonunitary scenario, this is no longer true since the matter potential in the flavor basis (28) is no longer diagonal but given by V � UU †�−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Incidentally, this also shows that once we have nondiagonal NonStandard neutrino Interaction (NSI), eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (47) and (48) are no longer invariant under matter potential [23, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' As shown in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' [23], NSI is still distinct from nonunitary scenario since the latter further breaks the unitary relations that we will discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Let us define the Jarlskog combinations by taking the imaginary part of the combination above [38] Jjk βα ≡ Im � �Uβj �U∗ αj �U∗ βk �Uαk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (49) If �U is unitary, one must have � k Jjk βα = � k Im � �Uβj �U∗ αj �U∗ βk �Uαk � = Im � �Uβj �U∗ βj � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' (50) Let us look at the modification due to nonunitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content=' Since all the terms with p = q are real and we are only left with terms of p ̸= q Jjk βα = 2+N � p̸=q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf'} +page_content='0≤p 70%, the over-confidence pseudo labels would easily +lead the network to confirmation bias (Arazo et al. 2020). +Sensitivity Analysis of hyper-parameter α +Besides, to +the trade-off hyper-parameter α, the experimental results are +shown in Fig.4. From these results, we observe that the per- +formance will not fluctuate greatly when α is varying. This +demonstrates that our CCGC is insensitive to α. Moreover, +CCGC is also insensitive to the layer number t of Laplacian +filters. Experimental evidences can be found in Fig. 1 in Ap- +pendix. +Visualization Analysis +In this part, we visualize the distribution of the learned em- +beddings of six baselines and CCGC to show the superiority +of CCGC on CORA and AMAP datasets via t-SNE algo- +rithm (Van der Maaten and Hinton 2008). As shown in Fig. +CORA +EAT +UAT +AMAP +BAT +CITESEER +Figure 5: Sensitivity analysis of the hyper-parameter τ on +six datasets. +3, we can conclude that CCGC better reveals the intrinsic +clustering structure compared with other baselines. +Conclusion +In this work, we propose a Cluster-guided Contrastive deep +Graph Clustering network termed CCGC to improve the +quality of positive and negative samples. To be specific, +we firstly construct two views with the un-shared param- +eters Siamese encoders to avoid semantic drift caused by +the inappropriate graph data augmentations. Besides, the +proposed positive and negative samples construction strate- +gies improve the discriminative capability and reliability of +samples by mining the supervision information in the high- +confidence clustering pseudo labels. Extensive experiments +on six datasets demonstrate the effectiveness of our pro- +posed method. + +70 +60 +50 +40 +ACC +F1 +100 +10.0 +NMI +1.0 +ARI +0.1 +Metrics +0.0160 +50 +Score +40 +30 +20 +10 +ACC +F1 +100 +10.0 +NMI +1.0 +ARI +0.1 +Metrics +0.0160 +50 +Score +40 +30 +20 +10 +ACC +F1 +100 +10.0 +NMI +1.0 +ARI +0.1 +Metrics +0.0180 +70 +Score +60 +50 +40. +ACC +F1 +100 +10.0 +NMI +1.0 +ARI +0.1 +Metrics +0.01 +αr80 +70 +Score +60 +50 +40 +30 +ACC +F1 +100 +10.0 +NMI +1.0 +ARI +0.1 +Metrics +0.01 +Q70 +60 +Score +50 +40 +30 +ACC +F1 +100 +10.0 +NMI +1.0 +ARI +0.1 +Metrics +0.0170 +60 +50, +40 +ACC +F1 +90% +70% +NMI +50% +ARI +30% +Metrics +10% +T60 +50 +Score +40 +30 +20 +ACC +F1 +90% +70% +NMI +50% +ARI +30% +Metrics +10% +T60 +40 +20 +0 +ACC +F1 +%06 +70% +NMI +50% +ARI +30% +Metrics +10% +T80 +70 +60 +50 +ACC +F1 +%06 +70% +NMI +%09 +ARI +30% +Metrics +10% +T80 +70 +Score +60 +50 +40 +ACC +F1 +%06 +70% +NMI +%09 +ARI +30% +Metrics +10% +T70 +60 +Score +50 +40 +30 +ACC +F1 +%06 +70% +NMI +%09 +ARI +30% +Metrics +10% +TAcknowledgments +This work was supported by the National Key R&D +Program +of +China +(project +no. +2020AAA0107100, +2021YFB3100700) and the National Natural Science +Foundation of China (project no. 61922088, 61976196, +62006237, and 61872371). +References +Arazo, E.; Ortego, D.; Albert, P.; O’Connor, N. 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In +Proceedings of the Web Conference 2021, 2069–2080. + diff --git a/XtAzT4oBgHgl3EQfKfuP/content/tmp_files/load_file.txt b/XtAzT4oBgHgl3EQfKfuP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6f347c7dd44e8ba7411ca9e79fdbea58605788a --- /dev/null +++ b/XtAzT4oBgHgl3EQfKfuP/content/tmp_files/load_file.txt @@ -0,0 +1,2290 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf,len=2289 +page_content='Cluster-guided Contrastive Graph Clustering Network Xihong Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='1* Yue Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='1*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Sihang Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='2 Siwei Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='1 Wenxuan Tu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='1 Qun Zheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='3 Xinwang Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='1† Liming Fang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='4 En Zhu1† 1College of Computer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' National University of Defense Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Changsha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' China 2College of Intelligence Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' National University of Defense Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Changsha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' China 3University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 4 Nanjing University of Aeronautics and Astronautics {yangxihong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' yueliu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' wangsiwei13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' twx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' xinwangliu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' enzhu}@nudt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='cn, sihangjoe@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='com, zhengqun@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='cn, fangliming@nuaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='cn Abstract Benefiting from the intrinsic supervision information ex- ploitation capability, contrastive learning has achieved promising performance in the field of deep graph cluster- ing recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' However, we observe that two drawbacks of the positive and negative sample construction mechanisms limit the performance of existing algorithms from further improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 1) The quality of positive samples heavily depends on the carefully designed data augmentations, while inappropri- ate data augmentations would easily lead to the semantic drift and indiscriminative positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2) The constructed negative samples are not reliable for ignoring important clus- tering information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To solve these problems, we propose a Cluster-guided Contrastive deep Graph Clustering network (CCGC) by mining the intrinsic supervision information in the high-confidence clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Specifically, instead of conducting complex node or edge perturbation, we con- struct two views of the graph by designing special Siamese encoders whose weights are not shared between the sibling sub-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Then, guided by the high-confidence cluster- ing information, we carefully select and construct the positive samples from the same high-confidence cluster in two views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Moreover, to construct semantic meaningful negative sam- ple pairs, we regard the centers of different high-confidence clusters as negative samples, thus improving the discrimi- native capability and reliability of the constructed sample pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Lastly, we design an objective function to pull close the samples from the same cluster while pushing away those from other clusters by maximizing and minimizing the cross- view cosine similarity between positive and negative sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Extensive experimental results on six datasets demon- strate the effectiveness of CCGC compared with the existing state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The code of CCGC is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='com/xihongyang1999/CCGC on Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Introduction Thanks to the strong representation learning capacity of the graph data, Graph Neural Networks (GNNs) have been suc- cessfully applied to various applications, such as node clas- sification (Kipf and Welling 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Veliˇckovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020, 2021d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b), graph classification (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Equal contribution †Corresponding author (a) GCA (c) Ours (b) SCAGC (d) Ground truth Figure 1: Visualization of the positive sample pairs selected by (a) GCA (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021), (b) SCAGC, (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d) and (c) the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The red dots denote the gener- ated sample pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Specifically, if a point (i, j) is selected as positive, the i-th sample from the first view and the j-th sam- ple from the second view are integrated as a positive sam- ple pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (d) is the ground-truth cluster indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The sample order is rearranged to make samples from the same cluster beside each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' From the figures, we can find that our proposed positive sample extraction mechanism is more dis- criminative than the existing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' As a consequence, the learned network is also more informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021), time series analysis(Liu and Liu 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Liu, Wu, and Liu 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022), knowledge graph(Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022a,b), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Among all the directions in graph learning, deep graph clustering is a fundamental yet chal- lenging unsupervised task, which has become a hot research spot recently (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='01098v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='LG] 3 Jan 2023 用2022b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Contrastive learning, which could capture the supervision information implicitly without human annotations, has be- come a prominent technique in deep graph clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Al- though promising performance has been achieved, we ob- serve two issues in the contrastive sample-pair construction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 1) The quality of positive samples heavily depends on the carefully selected graph data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' How- ever, inappropriate graph data augmentations, like random attribute permutation and random edge drop-out, would eas- ily lead to semantic drift (Lee, Lee, and Park 2021), and indiscriminative positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2) The constructed neg- ative samples are not reliable enough since the existing al- gorithms neglect to exploit the important clustering infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Concretely, the existing methods randomly select negative samples, which loosely assign negative labels to samples from the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To improve the quality of negative samples, GDCL (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021) and SCAGC (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d) randomly select samples from the dif- ferent clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Although verified to be effective, the current clustering-result-based methods heavily rely on the carefully designed graph data augmentation and the well pre-trained model, thus limiting the clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To solve these issues, we propose a novel Cluster-guided Contrastive deep Graph Clustering method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', CCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Concretely, to construct two node views with different se- mantics, we take advantage of the Siamese encoders and make the parameters un-shared between two sub-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this way, complex structure- and attribute-level data aug- mentations are avoided while the semantic drift problem has also been solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' After that, we carefully select and construct the positive samples from the same cluster in two views according to high-confidence clustering pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this manner, we improve the discriminative capacity of the positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 1, we visualize the positive sample pairs constructed by (a) GCA (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021), (b) SCAGC (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d), (c) our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' It is clearly observed that our constructed positive samples could better reveal the ground truth compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Meanwhile, we regard the centers of different high-confidence clusters as the negative sample pairs, which are more reliable and semantic mean- ingful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Moreover, we design an objective function to pull close the samples from the same cluster and push away those from different clusters by maximizing and minimizing the cross-view cosine similarity between positive and negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The key contributions of this paper are listed as follows: We propose a cluster-guided contrastive deep graph clus- tering network termed CCGC to improve the quality of positive and negative samples by mining the high- confidence clustering information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Instead of using carefully designed complex graph data augmentation, we conduct two views by designing spe- cial un-shared parameters Siamese encoders, thus avoid- ing semantic drift caused by inappropriate graph data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Extensive experimental results on six benchmark datasets demonstrate the effectiveness of the proposed method against the existing state-of-the-art deep graph clustering competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Related Work Deep Graph Clustering Clustering is a fundamental yet challenging task, which aims to learn node semantic representations and divide nodes into different clusters (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022c,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Deep learning methods also attract attention (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Among those methods, deep graph clustering has been a hot research spot in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' According to the learning mechanism, the existing methods can be roughly grouped into three classes including generative methods, adversarial methods, and contrastive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Our survey paper (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022c) summarizes the detailed information about the fast-growing deep graph clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' CCGC is categorized into the last one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', contrastive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' This section re- views the generative methods and adversarial methods, and the contrastive methods will be detailed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Inspired by the success of graph-auto-encoder (GAE) (Kipf and Welling 2016), the pioneer MGAE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2017) first encodes the nodes with the graph-encoder (Kipf and Welling 2016) and then performs clustering on the latent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' After that, DAEGC (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019) adopts the attention mechanisms (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Veliˇckovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2017) in early works to improve the clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Furthermore, ARGA (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019) and AGAE (Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019) improve the discriminative capability of samples by adversarial mechanisms (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In addition, SDCN (Bo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020) alleviates the over-smoothing prob- lem by integrating GAE and auto-encoder into the unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' More recently, R-GAE (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d) en- hances the existing GAE-based methods by alleviating the feature randomness and feature drift issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Although verified to be effective, since most of these methods adopt a distribution alignment loss function (Xie, Girshick, and Farhadi 2016) to force the learned node embeddings to have the minimum distortion against the pre-learned cluster centers, their clustering performance is highly dependent on good initial cluster centers, thus lead- ing to manual trial-and-error pre-training (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Bo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' As a conse- quence, the performance consistency, as well as the imple- menting convenience, is largely decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Different from them, several contrastive methods (Hassani and Khasahmadi 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Pan and Kang 2021) replace the clus- tering guided loss function by the contrastive loss, thus get- ting rid of trial-and-error pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Contrastive Deep Graph Clustering Contrastive learning has achieved great success in the fields of computer vision (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022a) and graph learning (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022a) in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Inspired by their success, con- trastive deep graph clustering methods (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b,f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021) are increasingly proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Figure 2: Illustration of the Cluster-guided Contrastive Graph Clustering (CCGC) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In our proposed algorithm, we firstly encode the two-view node embeddings with the proposed parameter un-shared Siamese encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Then, we perform K- means on the fused node embeddings and obtain the clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Subsequently, based on the high-confidence clustering results, we improve the quality of positive and negative samples by the discriminative positive sample construction strategy and reliable negative sample construction strategy in section .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Lastly, we design an objective function to pull close the samples from the same cluster while pushing away different high-confidence cluster centers, thus enhancing the discriminative capability of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The fashions of the data augmentations and the positive- negative sample pair construction are two crucial factors to determine the performance of the contrastive deep graph clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this section, we review the existing contrastive methods from these two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The technique of data augmentation plays an important role in contrastive deep graph clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Specifically, the existing methods construct different views of the graph by applying distinct augmentations to the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' For example, the graph diffusion matrix would be regarded as one of the augmented graphs in MVGRL (Hassani and Khasahmadi 2020), GDCL (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021), and DCRN (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Differently, SCAGC (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d) randomly adds or drops edges to perturb the structure of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' From the feature perspective, DCRN and SCAGC conduct augmentations on node attributes by attribute cor- ruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Although verified to be effective, the promising per- formance of these methods highly depends on the carefully selected data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Some works (Lee, Lee, and Park 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021a) point out that the inappropriate data augmentations would easily lead to semantic drift and the similar conclusion could be found in section .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To over- come the issue, we propose a novel augmentation fashion to construct different graph views by setting the parameters of Siamese encoders to be un-shared, thus avoiding the seman- tic drift caused by inappropriate augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Positive and negative sample pair construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Another crucial component in contrastive methods is the fashion of the positive and negative sample pair construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Specif- ically, contrastive methods pull together positive samples while pushing away negative ones, thus the quality of pos- itive and negative samples determines the performance of contrastive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Concretely, MVGRL (Hassani and Khasahmadi 2020) regards different augmented views of the same node and generates the negative samples by randomly shuffling the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Besides, DCRN (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b) pulls together the same node in different views while pushing away different nodes under the feature decorrelation con- strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Moreover, SCAGC (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d) and GDCL (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021) improve the quality of negative samples by randomly selecting samples from the different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Although verified to be effective, they still rely on a well pre-trained model to select high-quality positive and neg- ative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To solve this problem, we propose a high- confidence clustering information guided fashion of positive and negative sample construction, thus enhancing the dis- criminative capability and reliability of the sample pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Method In this section, we propose a novel Cluster-guided Con- trastive deep Graph Clustering algorithm (CCGC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The over- all framework of CCGC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In the following sections, we will introduce the proposed CCGC in specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Notations and Problem Definition In an undirected graph G = {X, A}, let V = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' , vN} be a set of N nodes with K classes and E be a set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=" X ∈ RN×D and A ∈ RN×N de- Parameter Un-shared Siamese Encoders Cluster-guided Contrastive Learning X x Encoder, Neighbour Information' Un-Share Aggregation Parameters 002 Encoder2 X Attribute Matrix x Smoothed Attributes High-confidence Node E Embeddings Cluster Centernote the attribute matrix and the original adjacency ma- trix, respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The degree matrix is formulated as D = diag(d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' , dN) ∈ RN×N and di = � (vi,vj)∈E aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The graph Laplacian matrix is defined as L = D − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' With the renormalization trick �A = A + I, the symmet- ric normalized graph Laplacian matrix is denoted as �L = I − �D − 1 2 �A�D − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Parameter Un-shared Siamese Encoders In this section, following SCGC (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d), we em- bed the nodes into the latent space and construct two dif- ferent sample views by designing a parameter un-shared Siamese encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Before encoding, we adopt a widely-used Laplacian filter (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020) to conduct neighbour information aggrega- tion as follows: �X = (I − �L)tX, (1) where �L is the symmetric normalized graph Laplacian ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' t denotes the layer number of the Laplacian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' �X is the smoothed attribute matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Then we encode �X with MLP encoders as follows: Ev1 = Encoder1(�X), Ev2 = Encoder2(�X), (2) where Ev1 and Ev2 denotes the first and second view of the node embeddings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' For the encoders, we design them to have the same architecture but un-shared learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Subsequently, we normalize Ev1 and Ev2 with ℓ2-norm: Ev1 = Ev1 ||Ev1||2 , Ev2 = Ev2 ||Ev2||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (3) By this setting, we construct two node views with differ- ent semantic, thus avoiding semantic drift caused by inap- propriate data augmentations on the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Experimental evidence could be found in section .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Cluster-guided Contrastive Learning In this section, we propose the Cluster-guided Contrastive Learning (CCL) to improve the discriminative capability and reliability of samples by mining the high-confidence cluster- ing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To be specific, we firstly fuse the two views of the node embeddings as follows: E = 1 2(Ev1 + Ev2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (4) Then we perform K-means on E and obtain the clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In order to generate more reliable clustering infor- mation (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022f), we define the confidence score CONFi of i-th sample as formulated: CONFi = e−||Ei−Cp||2, (5) where Ei denotes the i-th node embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Besides, Cp(p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', K) denotes the center of the cluster, which contains the i-th sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Subsequently, based on CONF, we denote the high-confidence sample indexes h as follows: h = {h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', hi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='}, (6) where the element hi indicates that hi-th sample belongs to top τ high-confidence sample set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Based on these high-confidence samples and their clus- tering pseudo labels, we propose two sample construction strategies including Discriminative Positive sample con- struction Strategy (DPS) and Reliable Negative sample con- struction Strategy (RNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Discriminative Positive Sample Construction Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this part, we design DPS to enhance the discriminative capa- bility of positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The proposed DPS contains three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Firstly, we select the high-confidence samples of two views with high-confidence indexes h as follows: Hv1 = Ev1 [h,:], Hv2 = Ev2 [h,:].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (7) Then, according to the corresponding pseudo labels, we group Hv1 and Hv2 into K disjoint clusters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', Bv1 p (p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', K), and Bv2 q (q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Subsequently, the positive samples will be selected and constructed from the same high-confidence clusters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this setting, the high-confidence clustering pseudo labels could be utilized as the supervisory information to improve the discriminative capability of the positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Reliable Negative Sample Construction Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' For the negative sample construction, the existing works (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b) directly regard all other non-positive samples as negative samples, easily bringing false-negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To alleviate this issue, we propose RNS, which contains two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Concretely, we first calculate the centers of high-confidence samples in two views: CENv1 p = avg(Bv1 p ), p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', K, CENv2 q = avg(Bv2 q ), q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', K, (8) where avg is the average function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Then we regard the dif- ferent high-confidence centers as the negative samples in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this manner, RNS would enhance the reliability of negative samples, thus reducing the possibility of false- negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In summary, the proposed CCL would guide our network to mine the supervisory information in the high-confidence clustering pseudo labels, thus improving the discriminative capability and reliability of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Objective Function The proposed method jointly optimizes two objectives in- cluding the positive sample loss Lpos and the negative sam- ple loss Lneg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In detail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Lpos is the Mean Squared Error (MSE) loss between the normalized cross-view positive sample embed- dings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' as formulated: Lpos = 1 K K � p=1 np � i=1 ���Bv1 p[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=':] − Bv2 p[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=':] ��� 2 2 = 1 K K � p=1 np � i=1 � 2 − 2 � Bv1 p[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=':],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Bv2 p[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=':] �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (9) MGAE DAEGC ARGA SDCN DFCN AGE MVGRL AutoSSL AGC-DRR AFGRL GDCL ProGCL CCGC Dataset Metrix CIKM 17 IJCAI 19 IJCAI 19 WWW 20 AAAI 21 SIGKDD 20 ICML 20 ICLR 22 IJCAI 22 AAAI 22 IJCAI 21 ICML 22 Ours ACC 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='38±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='11 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='36 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='25 35.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='99 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='52±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='09 UAT F1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='95±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='52 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='33±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='24±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='69 Table 1: The average clustering performance of ten runs on six benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The performance is evaluated by four metrics with mean value and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The red and blue values indicate the best and the runner-up results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' where Bv1 p[i,:] and Bv2 p[i,:] denotes the i-th normalized node embedding in the p-th cluster of the first and second view, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Besides, np is the number of high-confidence samples in the p-th cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this manner, the positive samples are pulled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Besides, we define Lneg as the cosine similarity between different centers of the high- confidence embeddings: Lneg = 1 K2 − K K � p=1 K � q=1 � CENv1 p , CENv2 q � ∥CENv1 p ∥2 · ∥CENv2 q ∥2 , p ̸= q, (10) where CENv1 p is the p-th high-confidence center in the first view and CENv2 q is the q-th high-confidence center in the second view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' By setting this, we push the negative samples away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In summary, the total loss of the proposed CCGC is cal- culated as: L = Lpos + αLneg, (11) where α is a trade-off between Lpos and Lneg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The detailed learning process of CCGC is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Experiments Benchmark Datasets The experiments are conducted on six widely-used bench- mark datasets, including CORA (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020), CITE- SEER (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020), BAT (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Mrabah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021), EAT (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d), UAT (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d), AMAP (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The summarized information is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Experiment Setup The experimental environment contains one desktop com- puter with the Intel Core i7-7820x CPU, one NVIDIA GeForce RTX 2080Ti GPU, 64GB RAM, and the PyTorch Dataset Type Sample Dimension Edge Class CORA Graph 2708 1433 5429 7 CITESEER Graph 3327 3703 4732 6 AMAP Graph 7650 745 119081 8 BAT Graph 131 81 1038 4 EAT Graph 399 203 5994 4 UAT Graph 1190 239 13599 4 Table 2: Statistics summary of six datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' deep learning platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The max training epoch number is set to 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' We minimize the total loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (11) with widely-used Adam optimizer (Kingma and Ba 2014) and then perform K-means over the learned embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To ob- tain reliable clustering, we adopt a two-stage training strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The discriminative capacity of the model can be im- proved in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In the second stage, the contrastive learning mechanism can be enhanced by the high-confidence clustering pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Ten runs are conducted for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' For the baselines, we adopt their source with orig- inal settings and reproduce the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The hyper-parameter settings are summarized in Table 1 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The clustering performance is evaluated by four metrics includ- ing ACC, NMI, ARI, and F1 (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Performance Comparison In this subsection, we compare the clustering performance of our proposed algorithm with baselines on six datasets with four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Among these methods, five classical deep graph clustering methods (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Bo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020) utilize the graph auto- encoder to learn the node representation for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Be- sides, seven contrastive deep graph clustering methods (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Hassani and Khasahmadi 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Lee, Lee, and Park 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b) improve the discriminative capability of samples by the contrastive strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' DAEGC MVGRL SDCN AutoSSL AFGRL GDCL Ours Figure 3: 2D visualization on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The first row and second row correspond to CORA and AMAP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Algorithm 1: CCGC Input: The input graph G = {X, A};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The iteration number I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Hyper-parameters τ, t, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Output: The clustering result R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 1: Obtain the smoothed attributes �X with Eq (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2: for i = 1 to I do 3: Encode �X into two views with parameter un-shared Siamese encoders with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 4: Normalize the embeddings Ev1, Ev2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 5: Perform K-means on E to obtain the clustering result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 6: Fuse Ev1 and Ev2 to obtain E with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 7: Obtain high-confidence samples Hv1 and Hv2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 8: Construct positive and negative samples by DPS and RNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 9: Calculate the positive samples loss Lpos with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 10: Calculate the negative samples loss Lneg with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 11: Update the whole network by minimizing L in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 12: end for 13: Perform K-means on E to obtain the final clustering result R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 14: return R From the results in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='1, we find that CCGC obtains better performance compared with the classical deep graph clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The reason is that contrastive learning could assist the model capture the supervision information implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Besides, the contrastive methods achieve sub- optimal performance compared to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' This is because we improve the discriminative capability and reliability of sam- ples with the important clustering information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In summary, our method outperforms most of them on six datasets with four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Taking the results on EAT dataset for exam- ple, CCGC exceeds the runner-up by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='06%, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='92%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='46%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='34% with respect to ACC, NMI, ARI, and F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Besides, due to the limitation of the space, we conduct additional com- parison experiments with nine baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Those results are shown in Table 2 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The results could also demonstrate the superiority of CCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Ablation Studies In this section, we first verify the effectiveness of two pro- posed sample construction strategies with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Be- sides, we demonstrate the effect of parameter un-shared en- coders and analyze the sensitivity of hyper-parameters in CCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Effectiveness of DPS and RNS To verify the effect of the proposed Discriminative Positive sample construction Strat- egy (DPS) and Reliable Negative sample construction Strat- egy (RNS), we conduct extensive experiments as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' For simplicity, we denote “(w/o) DPS” and “(w/o) RNS” as replacing DPS and RNS in our model with the reg- ular positive and negative sample construction fashion (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=', regarding the same samples in two view as positive samples while considering other samples as neg- ative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' From the observations, we conclude that the per- formance will decrease without any one of DPS and RNS, revealing that both strategies make essential contributions to boosting the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In addition, the quality of posi- tive and negative sample pairs is improved compared with the regular sample construction fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Overall, the exper- imental results have verified the effectiveness of DPS and RNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Effectiveness of Parameter Un-shared Encoders To avoid the complex augmentations on graphs, we design the un-shared Simases encoders to conduct two-node views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In this part, we compare our view construction method with other classical graph data augmentations including edge dropping (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d), edge adding (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2022d), graph diffusion (Hassani and Khasahmadi 2020), and feature masking (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Concretely, in Ta- ble 3, we first make the encoders in CCGC to share pa- rameter and then adopt the data augmentation as randomly dropping 20% edges (“Drop Edges”), or randomly adding 20% edges (“Add Edges”), or graph diffusion (“Diffusion”) with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='20 teleportation rate, or randomly masking 20% fea- tures (“Mask Features”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' From the results, we observe that the commonly used graph augmentations might lead to se- mantic drift (Lee, Lee, and Park 2021), thus undermining the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' In summary, expensive experiments have demonstrated the effectiveness of the proposed parameter un-shared encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Hyper-parameter Analysis Sensitivity Analysis of hyper-parameter threshold τ We investigate the influence of the hyper-parameter thresh- old τ on six datasets as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' From the results, we observe that the model obtains promising performance when τ ∈ [50%, 70%].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' The reasons are as follows: 1) When 6:6F: 2 3 6 3 5Dataset Metric (w/o) Positive (w/o) Negitive Drop Edges Add Edges Diffusion Mask Feature Ours ACC 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='03±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='28 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='86 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='95±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='32 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='89±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='16 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='57±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='95 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='40±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='76 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='88±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='20 NMI 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='33±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='97 53.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='97 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='17±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='16 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='24±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='69 Table 3: Ablation studies of CCGC on six datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' CORA EAT UAT AMAP BAT CITESEER Figure 4: Sensitivity analysis of the hyper-parameter α on six datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' τ < 50%, the discriminative capacity of the network is limited due to few number of positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2) When τ > 70%, the over-confidence pseudo labels would easily lead the network to confirmation bias (Arazo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Sensitivity Analysis of hyper-parameter α Besides, to the trade-off hyper-parameter α, the experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' From these results, we observe that the per- formance will not fluctuate greatly when α is varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' This demonstrates that our CCGC is insensitive to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Moreover, CCGC is also insensitive to the layer number t of Laplacian filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Experimental evidences can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 1 in Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Visualization Analysis In this part, we visualize the distribution of the learned em- beddings of six baselines and CCGC to show the superiority of CCGC on CORA and AMAP datasets via t-SNE algo- rithm (Van der Maaten and Hinton 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' CORA EAT UAT AMAP BAT CITESEER Figure 5: Sensitivity analysis of the hyper-parameter τ on six datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 3, we can conclude that CCGC better reveals the intrinsic clustering structure compared with other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Conclusion In this work, we propose a Cluster-guided Contrastive deep Graph Clustering network termed CCGC to improve the quality of positive and negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' To be specific, we firstly construct two views with the un-shared param- eters Siamese encoders to avoid semantic drift caused by the inappropriate graph data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Besides, the proposed positive and negative samples construction strate- gies improve the discriminative capability and reliability of samples by mining the supervision information in the high- confidence clustering pseudo labels.' 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2020AAA0107100, 2021YFB3100700) and the National Natural Science Foundation of China (project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' 61922088, 61976196, 62006237, and 61872371).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' References Arazo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Ortego, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf'} +page_content=' Albert, P.' 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0000000000000000000000000000000000000000..7d740bcd554191b3de7af045d046db94efa663b1 --- /dev/null +++ b/_dFLT4oBgHgl3EQfDy7e/content/tmp_files/2301.11981v1.pdf.txt @@ -0,0 +1,1995 @@ +Unearthing InSights into Mars: +unsupervised source separation with limited data +Ali Siahkoohi 1 Rudy Morel 2 Maarten V. de Hoop 1 Erwan Allys 2 Gr´egory Sainton 3 Taichi Kawamura 3 +Abstract +Source separation entails the ill-posed problem +of retrieving a set of source signals observed +through a mixing operator. Solving this problem +requires prior knowledge, which is commonly +incorporated by imposing regularity conditions +on the source signals or implicitly learned in su- +pervised or unsupervised methods from existing +data. While data-driven methods have shown +great promise in source separation, they are of- +ten dependent on large amounts of data, which +rarely exists in planetary space missions. Consid- +ering this challenge, we propose an unsupervised +source separation scheme for domains with lim- +ited data access that involves solving an optimiza- +tion problem in the wavelet scattering represen- +tation space—an interpretable low-dimensional +representation of stationary processes. We present +a real-data example in which we remove transient +thermally induced microtilts, known as glitches, +from data recorded by a seismometer during +NASA’s InSight mission on Mars. Owing to the +wavelet scattering covariances’ ability to capture +non-Gaussian properties of stochastic processes, +we are able to separate glitches using only a few +glitch-free data snippets. +1. Introduction +Source separation is a problem of fundamental importance +in the field of signal processing, with a wide range of +applications in various domains such as telecommunica- +tions (Chevreuil & Loubaton, 2014; Gay & Benesty, 2012; +Khosravy et al., 2020), speech processing (Pedersen et al., +2008; Chua et al., 2016; Grais et al., 2014), biomedical sig- +nal processing (Adali et al., 2015; Barriga et al., 2003; Hasan +et al., 2018) and geophysical data processing (Ibrahim & +Sacchi, 2014; Kumar et al., 2015; Scholz et al., 2020). +1Rice University 2 ´Ecole Normale Sup´erieure 3Institut de +Physique du Globe de Paris. Correspondence to: Ali Siahkoohi +. +Preprint. +Figure 1. Unsupervised removal of background noise and ther- +mally induced microtilts (glitches) from a marsquake recorded +by the InSight landers’s seismometer on February 03, 2022 (In- +Sight Marsquake Service, 2023). Approximately 14 hours of +raw data (with no marsquakes) from the U component is used +for background noise separation without any prior knowledge on +marsquakes or glitches. Horizontal axis is in UTC time zone. +Source separation arises when multiple source signals of +interest are combined through a mixing operator. The goal +is to estimate the original sources with minimal prior knowl- +edge of the mixing process or the source signals themselves. +This makes source separation a challenging problem, as the +number of sources is usually unknown, and the sources are +often non-Gaussian, nonstationary, and multiscale. +Classical signal-processing based blind source separation +methods (Cardoso, 1989; Jutten & Herault, 1991; Bingham +& Hyv¨arinen, 2000; Nandi & Zarzoso, 1996; Cardoso, 1998; +Jutten et al., 2004) while being extensively studied and well +understood, often make simplifying assumptions regard- +ing the sources that might negatively bias the outcome of +source separation (Cardoso, 1998; Parra & Sajda, 2003). To +partially address the shortcomings of classical approaches, +deep learning methods have been proposed as an alternative +approach for source separation, which exploit the informa- +arXiv:2301.11981v1 [cs.LG] 27 Jan 2023 + +Raw data +P-wave start +S-wave start +08:08 +08:10 +08:12 +08:14 +08:16 +08:18Marsquake +P-wave start +S-wave start +08:08 +08:10 +08:12 +08:14 +08:16 +08:1808:08 +08:10 +08:12 +08:14 +08:16 +08:18Unsupervised source separation with limited data +tion in existing datasets to learn prior information about the +sources. In particular, supervised learning methods (Jang & +Lee, 2003; Hershey et al., 2016; Ke et al., 2020; Kameoka +et al., 2019; Wang & Chen, 2018) commonly rely on exis- +tence of labeled training data and perform source separation +using an end-to-end training scheme. However, since they +require access to ground truth source signals for training, +supervised methods are limited to domains in which labeled +training data is available. +On the other hand, unsupervised source separation methods +(F´evotte et al., 2009; Drude et al., 2019; Wisdom et al., 2020; +Liu et al., 2022; Denton et al., 2022; Neri et al., 2021) do +not rely on the existence of labeled training data and instead +attempt to infer the sources based on the properties of the ob- +served signals. These methods make minimal assumptions +about the underlying sources, which make them a suitable +choice for realistic source separation problems. Despite +their success, unsupervised source separation methods often +require tremendous amount of data during training (Wisdom +et al., 2020), which is often infeasible in certain applica- +tions such as problem arising in planetary space missions, +e.g., because of challenges associated with data acquisi- +tion. Moreover, generalization concerns preclude the use of +data-driven methods trained on synthetic data in real-world +applications due to the discrepancies between synthetic and +real data. +To address these challenges, we propose an unsupervised +source separation method applicable to domains with lim- +ited access to data. In order to achieve this, we embed in- +ductive biases into our approach through the use of domain +knowledge from time series analysis and signal processing +via the scattering transform (Bruna & Mallat, 2013). As a +means of capturing non-Gaussian and multiscale character- +istics of the sources, we extract second-order information +of scattering coefficients, known as the wavelet scattering +covariance representation (Morel et al., 2022). We perform +source separation by solving an optimization problem over +the unknown sources that entails minimizing multiple care- +fully selected and normalized loss functions in the wavelet +scattering covariance representations space. These loss func- +tion are designed to: (1) ensure data-fidelity, i.e., enforce +the recovered sources to explain the observed (mixed) data; +(2) incorporate prior knowledge in the form of limited (e.g., +≈ 50) training examples from one of the sources; and (3) +impose a notion of statistical independence between the re- +covered sources. Our proposed method does not require any +labeled training data, and can effectively separate sources +even in scenarios where access to data is limited. +As a motivating example, we apply our approach to data +recorded by a seismometer on Mars during NASA’s In- +terior Exploration using Seismic Investigations, Geodesy +and Heat Transport (InSight) mission (Giardini et al., 2020; +Golombek et al., 2020; Knapmeyer-Endrun & Kawamura, +2020). The InSight lander’s seismometer has been detect- +ing marsquakes (Horleston et al., 2022; Ceylan et al., 2022; +Panning et al., 2023; InSight Marsquake Service, 2023) and +transient atmospheric signals, such as wind and temperature +changes, that provide information about the Martian atmo- +sphere (Stott et al., 2022) and enable studying the interior +structure and composition of the Red Planet (Beghein et al., +2022). The signal recorded by the InSight seismometer +is heavily influenced by atmospheric activity and surface +temperature (Lognonn´e et al., 2020; Lorenz et al., 2021), +resulting in a distinct daily pattern and a pronounced non- +stochastic character. Among different types of noise, tran- +sient thermally induced microtilts, commonly referred to +as glitches (Scholz et al., 2020; Barkaoui et al., 2021), are +a significant component of the noise and one of the most +frequent recorded events. These glitches, hinder the down- +stream analysis of the data if left uncorrected (Scholz et al., +2020). We show that our method is capable of removing +glitches from the recorded data by only using a few snippets +of glitch-free data. +In the following sections, we first described several related +work that similarly address the challenges of unsupervised +source separation in limited data regimes. Next, we intro- +duce wavelet scattering covariance as a domain-knowledge +rich representation for analyzing time series and provide jus- +tification for their usage in the context of source separation. +As a means to perform source separation in domains with +limited data, we introduce our source separation approach +that involves solving an optimization problem with loss func- +tions defined in the wavelet scattering covariance space. We +present two numerical experiments: (1) a synthetic setup in +which we can quantify the accuracy of our method; and (2) +examples regarding separating glitches from data recorded +by InSight lander’s seismometer. +2. Related work +Regaldo-Saint Blancard et al. (2021) introduced the notion +of components separation through a gradient descent in sig- +nal space with indirect constraints with applications to to +the separation of an astrophysical emission (polarized dust +emission in microwave) and instrumental noise. In an exten- +sive study, Delouis, J.-M. et al. (2022) attempts to separate +the full sky observation of the dust emission with instru- +mental noise using similar techniques via wavelet scattering +covariance representations. Authors take the nonstationarity +of the signal into account by constraining statistics on sev- +eral sky masks. Contrarily to a usual denoising approach, +both of these works focus primarily on recovering the statis- +tics of the signal of interest. In a related approach, Jeffrey +et al. (2022) use a scattering transform generative model to +perform source separation in a Bayesian framework. While + +Unsupervised source separation with limited data +very efficient, this approach requires training samples from +each component, which are often not available. Finally, +Xu et al. (2022) similarly aim to remove glitches and they +develop a supervised learning based on deglitched data ob- +tained by existing glitch removal tools. As a result, the +accuracy of their result is limited to the accuracy of the +underlying data processing tool, which our method avoid +by being unsupervied. As we show in our examples, we are +able to detect and remove glitches that were undetected by +the main deglitching software (Scholz et al., 2020) devel- +oped closely by the InSight team. +3. Wavelet scattering covariance +In order to enable unsupervised source separation with +limited quantities of data, we propose to design a low- +dimensional, domain-knowledge rich representation of data +using which we perform source separation. This is par- +tially motivated by recent success of self-supervised learn- +ing methods in natural language processing where high- +performing representations of data—obtained through pre- +trained Transformers (Vaswani et al., 2017; Baevski et al., +2020; Gulati et al., 2020; Zhang et al., 2020)—are used in +place of raw data to successfully perform various down- +stream tasks (Polyak et al., 2021; Gulati et al., 2020; +Baevski et al., 2020; Zhang et al., 2020; Chung et al., 2021; +Siahkoohi et al., 2022). +Since we are operating in a limited-data regime, we cannot +afford self-supervised learning with Transformers in order +to obtain high-performing features. Instead, we propose to +use wavelet scattering covariances (Morel et al., 2022) as +means to transfer data to a suitable representation space for +source separation. This transform is based on scattering +networks (Bruna & Mallat, 2013) that provide interpretable +representation of data and are able to characterize a wide +range of non-Gaussian properties of multiscale stochastic +processes (Morel et al., 2022)—a type of signals that are +considered in this paper. The wavelet scattering covariance +generally does not require any pretraining and its weights, +i.e., wavelets in the scattering network, are often chosen +beforehand (see Seydoux et al. (2020) for a data-driven +wavelet choice) according to the time-frequency properties +of data. In the next section, we introduce the construction +of this representation space by first describing scattering +networks. +3.1. Wavelet transform and scattering networks +The main ingredient of the wavelet scattering covariance +representation is a scattering network (Bruna & Mallat, +2013) that consists of a cascade of wavelet transforms fol- +lowed by a nonlinear activation function (akin to a typi- +cal convolutional neural network). In this network archi- +tecture, the wavelet transform, denoted by a linear oper- +ator W, is a convolutional operator with predefined ker- +nels, i.e., wavelet filters. +These filters include a low- +pass filter ϕJ(t) and J complex-valued band-pass filters +ψj(t) = 2−jψ(2−jt), 1 ≤ j ≤ J, which are obtained by +the dilation of a mother wavelet ψ(t) and have zero-mean +and a fast decay away from t = 0. The wavelet transform +is often followed by the modulus operator in scattering net- +works. The output of a two-layer scattering network S can +be written as, +S(x) := +� +Wx +W|Wx| +� +, +(1) +where Wx := x ⋆ ψj(t) denotes the wavelet transform that +extracts variations of the input signal x(t) around time t at +scale 2j, and |·| is the modulus activation function (Bruna & +Mallat, 2013). The second component W|Wx| computes +the variations at different time and scales of the wavelet co- +efficients Wx. The scattering transform yields features that +characterize time evolution of signal envelopes at different +scales. Even though such representation has many success- +ful applications, e.g., intermittency analysis (Bruna et al., +2015), clustering (Seydoux et al., 2020), event detection +and segmentation (Rodr´ıguez et al., 2021) (with learnable +wavelets), it is not sufficient to build accurate models of mul- +tiscale processes as it does not capture crucial dependencies +across different scales (Morel et al., 2022). +3.2. Capturing non-Gaussian characteristics of random +processes +The dependencies across different scales in scattering trans- +form coefficients are crucial in characterizing and discrimi- +nating non-Gaussian signals (Morel et al., 2022). To capture +them, we explore the matrix scattering coefficients outer +product S(x)S(x)⊤: +� +Wx (Wx)⊤ +Wx (W|Wx|)⊤ +W|Wx| (Wx)⊤ +W|Wx| (W|Wx|)⊤ +� +. +(2) +This matrix contains three types of coefficients: +• The correlation coefficients Wx (Wx)⊤ across scales +form a quasi-diagonal matrix, because separate scales +do not correlate due to phase fluctuation (Morel et al., +2022). We thus only keep its diagonal coefficients, +which correspond to the wavelet power spectrum; +• The correlation coefficients Wx (W|Wx|)⊤ cap- +ture signed interaction between wavelet coefficients. +In particular, they detect sign-asymmetry and time- +asymmetry in x (Morel et al., 2022). We also consider +a diagonal approximation to this matrix. For the same +reason as Wx (Wx)⊤, this matrix is quasi-diagonal, +and we only keep coefficients that correlate same scale +channels on the second wavelet operator; + +Unsupervised source separation with limited data +• Finally coefficients W|Wx| (W|Wx|)⊤ capture +cross-envelope correlation at different scales. They cap- +ture intermittency and time-asymmetry (Morel et al., +2022). Again, we only keep coefficients that correlate +same scale channels on the second wavelet operator. +We denote diag +� +S(x)S(x)⊤� +as such diagonal approxi- +mation of the full sparse matrix S(x)S(x)⊤. The wavelet +scattering covariance representation is obtained by comput- +ing the time average (average pool, denoted by Ave) of this +diagonal approximation: +Φ(x) := Ave +�� +S(x) +diag +� +S(x)S(x)⊤� +�� +. +(3) +Non-Gaussian properties of x can be detected through +non-zero coefficients of Φ. Indeed, let us separate real +coefficients and potentially complex coefficients Φ(x) = +� +Φreal(x), Φcomplex(x) +� +, with Φreal(x) being the real coef- +ficients Ave +� +|Wx|, |Wx|2, |W|Wx||2� +and Φcomplex(x) +being the remaining potentially complex coefficients, that +is the cross-layer correlations Ave +� +Wx +� +W|Wx| +�⊤� +or +the second layer correlations Ave +� +W|Wx| +� +W|Wx| +�⊤� +with different scale correlation on the first wavelet operator. +Proposition 3.1. If x is Gaussian then Φcomplex(x) ≈ 0. +If x is time-symmetric, then ImΦcomplex(x) ≈ 0. +More precisely, beyond detecting non-Gaussianity through +non-zero coefficients up to estimation error, Φ(x) is able +to quantify different non-Gaussian behaviors, which will +be crucial for source separation. Appendix A.3 presents a +dashboard that visualizes Φ(x) and can be used to interpret +signal non-Gaussian properties such as sparsity, intermit- +tency, and time-asymmetry. +The dimensionality of the wavelet scattering covariance rep- +resentation depends on the number of scales J considered +i.e. the number of wavelet filters of W. In order for largest +scale coefficients to be well estimated, one should choose +J ≪ log2(d) where d is input data dimension. The max- +imum number of coefficients in Φ is smaller than log3 +2(d) +for d ≥ 3 (Morel et al., 2022). Contrary to higher dimen- +sional representations or higher order statistics, scattering +covariance Φ(x) are low-dimensional low-order statistics +that can be efficiently estimated on a single realization of a +source and does not require tremendous amount of data for +estimation to converge. In other word, Φ is a low-variance +representation. This point is key for our source separation +algorithm to be applied on limited data. Wavelet scatter- +ing covariance Φ extracts average and correlation features +from a 2-layer CNN with predefined wavelet filters. It is +analogous to the features extracted in Gatys et al. (2015) +for generation, that considers however a pretrained con- +volutional neural network. In the following we will also +make use of the scattering cross-covariance representation +Φ(x, y) = Ave diag +� +S(x)S(y)⊤� +that captures scale de- +pendencies across two signals x and y. In particular, if +x and y are statistically independent then one has, up to +estimation error, Φ(x, y) ≈ 0, which will be useful when it +comes to separating independent sources. +4. Unsupervised source separation +To enable high-fidelity source separation in domains in +which access to training data—supervised or unsupervised— +is limited, we cast source separation as an optimization prob- +lem in a suitable feature space. Owing to wavelet scattering +covariance representation’s ability to capture non-Gaussian +properties of multiscale stochastic processes without any +training, we perform source separation by solving an op- +timization problem over the unknown sources using loss +functions over wavelet scattering covariance representations. +Due to the inductive bias embedded in the design of this rep- +resentation space, we gain access to interpretable features, +which could further inform us regarding the quality of the +source separation process. +4.1. Problem setup +Consider a linear mixing of unknown sources s∗ +i (t), i = +1, . . . , N via a mixing operator A, +x(t) = As∗(t) + ν(t) = a⊤ +1 s∗ +1(t) + n(t), +(4) +with +s∗(t) = [s∗ +1(t), . . . , s∗ +N(t)]⊤ , A = +� +a⊤ +1 · · · a⊤ +N +� +, +n(t) = ν(t) + +N +� +i=2 +a⊤ +i s∗ +i (t). +(5) +In the above expressions, x(t) represents the observed data, +and ν(t) is the measurement noise. Here we capture the +noise and the mixture of all the sources except for s∗ +1(t) +through the mixing operator in n(t) that does not longer de- +pends on s∗ +1(t). Note that x(t) and s∗(t) are in fact matrices +and a⊤ +i s∗ +i (t) is of the same size as x(t). +Objective. The aim is to obtain a point estimate s1(t) +given a single observation x(t) with the assumption that +a1 is known and that we have access to a few realizations +{nk(t)}K +k=1 as a training dataset. For example, in the case +of removing glitches from InSight seismometer’s recordings, +we will consider nk(t) to be snippets of glitch-free data and +a1 to encodes information regarding polarization. We will +drop the time dependence of the quantities in equations (4) +and (5) for convenience. +4.2. Principle of the method +The inverse problem of estimating s1 from the given ob- +served data x, as presented in equation (4), is ill-posed + +Unsupervised source separation with limited data +since the solution is not unique. To constrain the solution +space of the problem, we incorporate prior knowledge in +the form of realizations {nk}K +k=1. We achieve this through +a loss function that emphasizes the wavelet scattering co- +variance representation of x − a⊤ +1 s1 to be close to that of +nk, k = 1, . . . , K: +Lprior (s1) := 1 +K +K +� +k=1 +���Φ +� +x − a⊤ +1 s1 +� +− Φ +� +nk +���� +2 +2. +(6) +In the above expression, Φ is the wavelet scattering co- +variance mapping. With the prior loss defined, we impose +data-consistency via: +Ldata (s1) := 1 +K +K +� +k=1 +���Φ +� +a⊤ +1 s1 + nk +� +− Φ +� +x +���� +2 +2. +(7) +The data consistency loss function Ldata promotes estima- +tions of s1 that for any training example from {nk}K +k=1 the +wavelet scattering covariance representation of a⊤ +1 s1 + nk +is close to that of the observed data. +In order to further constrain this under-determined source +separation problem, we penalize cross-scale dependencies +across two quantities a⊤ +1 s1 and nk. We formulate this by +Lcross(s1) := 1 +K +K +� +k=1 +���Φ +� +a⊤ +1 s1, nk +���� +2 +2, +(8) +where Φ(·, ·) is the scattering cross-covariance representa- +tion (see section 3.2). +4.3. Loss normalization +The losses described previously do not contain any weight- +ing term for the different coefficients of the scattering covari- +ance representation. We introduce in this section a generic +normalization scheme, based on the estimated variance of +certain scattering covariance distributions. This normal- +ization, which has been introduced in Delouis, J.-M. et al. +(2022), allows to interpret the different loss terms in a stan- +dard form, and to include them additively in the total loss +term without overall loss weights. Let us consider first +the loss term given by equation (6), which compares the +distance between x − a⊤ +1 s1 and available training sam- +ples {nk}K +k=1 in the wavelet scattering representation space. +Specifying explicitly the sum on the M wavelet scattering +covariance coefficients Φm, m = 1, . . . , M, it yields +Lprior (s1) = +1 +MK +M +� +m=1 +K +� +k=1 +���Φm +� +x − a⊤ +1 s1 +� +− Φm +� +nk +���� +2 +. +Let us consider the second sum in this expression. In the +limit where Φm +� +x − a⊤ +1 s1 +� +is drawn from the same distri- +bution as {Φm +� +nk +� +}K +k , the difference Φm +� +x − a⊤ +1 s1 +� +− +Φm +� +nk +� +, seen as a random variable, should have zero mean, +and the same variance as the distribution {Φm +� +nk +� +}K +k up to +a factor 2. Denoting σ2� +Φm +� +nk +�� +as this variance, which +can be estimated from {Φm +� +nk +� +}K +k , this gives a natural +way of normalizing the loss: +Lprior (s1) = +1 +MK +M +� +m=1 +K +� +k=1 +���Φm +� +x − a⊤ +1 s1 +� +− Φm +� +nk +���� +2 +σ2� +Φm +� +nk +�� +or in a compressed form +Lprior (s1) = 1 +K +K +� +k=1 +���Φ +� +x − a⊤ +1 s1 +� +− Φ +� +nk +���� +2 +2 +σ2� +Φ +� +nk +�� +, +(9) +which takes into account the expected standard deviation of +each coefficient of the scattering covariance representation. +This normalization allows for two things. First, it removes +the normalization inherent to the multiscale structure of Φ. +Indeed, coefficients involving low frequency wavelets tend +to have a larger norm. Second, it allows to interpret the loss +value, which is expected to be at best of order unity and to +sum different loss terms of same magnitude. +We can introduce a similar normalization for the other loss +terms. +Loss term (7) should be normalized by the M- +dimensional vector σ2� +Φ +� +a⊤ +1 s1 + nk +�� +that we approxi- +mate by σ2� +Φ +� +x + nk +�� +, in order to have a normalization +independent on s1, yielding +Ldata (s1) := 1 +K +K +� +k=1 +���Φ +� +a⊤ +1 s1 + nk +� +− Φ +� +x +���� +2 +2 +σ2� +Φ +� +x + nk +�� +. +(10) +Finally, +loss +term +(8) +should +be +normalized +by +σ2� +Φ +� +a⊤ +1 s1, nk +�� +that we approximate by σ2� +Φ +� +x, nk +�� +Lcross(s1) = 1 +K +K +� +k=1 +���Φ +� +a⊤ +1 s1, nk +���� +2 +2 +σ2� +Φ +� +x, nk +�� , +(11) +We can now sum the normalized loss terms (9),(10),(11) +to get the final optimization problem to perform source +separation +�s1 := arg min +s1 +� +Ldata(s1) + Lprior(s1) + Lcross(s1) +� +. (12) +Due to the delicate normalization of the three terms, we ex- +pect that further weighting of the three losses using weight- +ing hyperparameters is not necessary. We propose to initial- +ize the optimization problem in equation (12) with s1 := 0. +Such choice means that n = x − a⊤ +1 s1 is initialized to x, +which contains crucial information on the sources, as will +be explained in the next section. + +Unsupervised source separation with limited data +We have observed that as soon as we know the statistics +of Φ(n) our algorithm retrieves the unknown statistics of +the other source Φ(a⊤ +1 s∗). In other words the algorithm +successfully separates the sources in the scattering covari- +ance space. Of course, in many cases as we will see in +the next section, our algorithm retrieves point estimates of +s1(t) which is stronger, but this constitutes a convergence +result that can be proven in not so simplified assumptions. +Essentially, when the source n is statistically characterized +by its scattering covariance descriptors the algorithm is able +to retrieve the scattering covariance of other sources. This +emphasizes the choice of a representation Φ that can ap- +proximate efficiently the stochastic structure of multiscale +processes (Morel et al., 2022). +5. Numerical experiments +The main goal of this paper is to derive a unsupervised ap- +proach to source separation that is applicable in domain with +limited access to training data, thanks to the wavelet scat- +tering covariance representation. To provide a quantitative +analysis to the performance of our approach, we first con- +sider a stylized synthetic example that resembles challenges +of real-world data. To illustrate how our method performs +in the wild, we apply our method to data recorded on Mars +during the InSight mission. We aim to remove transient +thermally induced microtilts, i.e., glitches (Scholz et al., +2020; Barkaoui et al., 2021), from the recorded data by the +InSight lander’s seismometer. Code to partially reproduce +the results is available at GitHub. +5.1. Stylized example +We consider the problem of separating glitch-like sig- +nals from increments of a multifractal random work pro- +cess (Bacry et al., 2001). This process is a typical non- +Gaussian noise exhibiting long-range dependencies and +showing bursts of activity, e.g., see Figure 11 for several +realizations of this process. The second source signal is +composed of several peaks with exponentially decaying am- +plitude, with possibly different decay parameters on the left +than on the right. To obtain synthetic observed data, we sum +increments of a multifractal random walk realization, which +plays the role of n in equation (4), with a realization of the +second source. The top three images in Figure 2 are the +signal of interest, secondary added signal, and the observed +data, respectively. +In order to retrieve the multifractal random walk realization, +we solve the optimization problem in equation (12) using +the L-BFGS optimization algorithm (Liu & Nocedal, 1989) +using 500 iterations. We use a training dataset of 100 realiza- +tions of increments of a multifractal random walk, {nk}100 +k=1. +The architecture we use for wavelet scattering covariance +computation is two-layer scattering network with J = 8 +Figure 2. Unsupervised source separation applied to the multifrac- +tal random walk data. The vertical axis is the same for all the +plots. +Figure 3. Signal-to-noise ratio of the predicted multifractal random +walk data versus number of unsupervised samples. Shaded area +indicates the 90% interval of this quantity for ten random source +separation instances. +different octaves with Q = 1 wavelet per octave. We use +the same scattering network architecture throughout all the +numerical experiments in the paper. Given an input signal +dimension of d = 2048, this choice of parameters yields a +174-dimensional wavelet scattering covariance space. The +bottom two images in Figure 2 summarizes the results. We +are able to recover the ground-truth multifractal random +walk realization up to small, mostly incoherent, and seem- +ingly random error. To see the effect of number of training +realizations on the signal recovery, we repeated the above +examples and used varying number of training samples. Fig- +ure 3 shows that, as expected, the signal-to-noise ratio of +the recovered sources increases the more training samples +we have. +To show our method can also separate sources that are not +localized in time, we consider contaminating the multifrac- +tal random walk data with a turbulent signal (see second +image from the top in Figure 4. Without any prior knowl- + +TrueAdded signalObservedPredictedError10 +6 +8 +7 +0 +100 +200 +300 +400 +500 +600 +Number of samplesUnsupervised source separation with limited data +Figure 4. Unsupervised source separation applied to the multifrac- +tal random walk data with a turbulent additive signal. The vertical +axis is the same for all the plots. +Figure 5. Glitch-free snippets of the seismic data from Mars (U +component), +edge regarding this turbulent signal and by only using 100 +realizations of increments of a multifractal random walk as +training samples, we are able to recover the signal of interest +with arguably low error: juxtapose the ground truth and pre- +dicted multifractal random walk realization in Figure 4. The +algorithm correctly removes the low frequencies content of +the turbulent jet, and makes a small, uncorrelated, random +error at high frequencies. In this case the two signals having +different power spectra helps disentangling them at high +frequencies. In the above synthetic examples, the signal +low frequencies are well separated and the algorithm in- +fers correctly the high frequencies. In the earlier example, +the presence of time localized sources would facilitate the +algorithm to ”interpolate” the background noise knowing +its scattering covariance representation. This case makes +it more evident that the initialization s1 = 0 informs the +algorithm of the trajectory of the unknown source. +5.2. Application to data from the InSight mission +InSight lander’s seismometer, SEIS, is exposed to heavy +wind and temperature fluctuations. As a result, it is subject +to background noise. Glitches are a widely occurring family +of noise caused by a variety of causes (Scholz et al., 2020). +These glitches often appear as one-sided pulses in seismic +data and significantly affect the analysis of the data (Scholz +et al., 2020). In this section we will explore the application +of our proposed method in separating glitches and back- +ground noise from the recorded seismic data on Mars. +5.2.1. REMOVING GLITCHES +We propose to consider glitches as the source of interest +s1 in the context of equation (4). To perform source sep- +aration using our technique, we need snippets of data that +do not contain glitches. We select these windows of data +using an existing catalog and glitches (Scholz et al., 2020) +and by further eye examination to ensure no glitch contami- +nates our dataset. In total, we collect 50 windows of length +102.4 s during sol 187 (6 June 2019) for the U component. +We show four of these windows of data in Figure 5. We +perform optimization for glitch removal using the same +underlying scattering network architecture as the previous +example using 50 training samples and 1000 L-BFGS itera- +tions. Figure 6 summarizes the results. The top-left image +shows the raw data. Top-right image is the baseline (Scholz +et al., 2020) prediction for the glitch signal. Finally, the bot- +tom row (from left to right) shows our predicted deglitched +data and the glitch signal separated by our approach. As +confirmed by experts at the InSight team, indeed our ap- +proach has removed a glitch that the baseline has ignored +(most likely due the spike right at the beginning of the glitch +signal). This is one of the benefits of our unsupervised ap- +proach as the method—based on the statistics of the training +data—identifies and removes events that do not seem to be- +long to the training data distribution. See more deglitching +examples in Figures 12– 15. +Thanks to the interpretability of wavelet scattering covari- +ance representations, we can perform a source separation +quality control in domain where there is no access to +ground truth source—as in our example. Figure 7 compares +the power spectra of the reconstructed background noise +(recorded data), a deglitched realization of the background +noise and the mixed signal (observed data). It can be seen +that the power spectrum of the background noise is correctly +retrieved. In fact, the scattering covariance statistics, which +extend the power spectrum, are correctly retrieved, which is +due to the loss term in equation (6). + +TrueAdded signalObservedPredictedError4000 +-4100 +4200 +-4300 +4400 +-45004000 +4100 +4200 +4300 +4400 +-45004000 +-4100 +4200 +4300 +4400 +45004000 +-4100 +4200 +4300 +4400 +-4500Unsupervised source separation with limited data +Figure 6. Unsupervised source separation for glitch removal. Juxtapose the predicted glitches on the right. Our approach is able to remove +a glitch whereas the baseline approach fails to detect it. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Frequency (Hz) +10−4 +10−2 +100 +102 +104 +106 +˜n +n +x +Figure 7. Power spectrum of the observed signal x, the background +noise n and the reconstructed background noise ˜n. We see that the +reconstructed component statistically agrees with a Mars seismic +background noise n. The algorithm efficiently removed the low- +pass component of the signal corresponding to a glitch. +5.2.2. MARSQUAKE BACKGROUND NOISE REMOVAL +Marsquakes are of significant importance as they provide +useful information regarding the Mars subsurface, enabling +the study of Mars’ interior (Knapmeyer-Endrun et al., 2021; +St¨ahler et al., 2021; Khan et al., 2021). Recordings by the +InSight lander’s seismometer are susceptible to background +noise and transient atmospheric signals, and here we ap- +ply our proposed unsupervised source separation approach +to separate background noise from a marsquake (InSight +Marsquake Service, 2023). To achieve this, we select about +14 hours of raw data (except for a detrending step)—from +the U component with a 20Hz sampling rate—to fully char- +acterize various aspects of the background noise through +the wavelet scattering covariance representation. Next, we +window the data and use the windows as training samples +from background noise (nk in the context of equation (4)) +with the goal of retrieving the marsquake recorded at Febru- +ary 3, 2022 (InSight Marsquake Service, 2023). We use the +same network architecture as previous examples to setup +the wavelet scattering covariance representation. We use a +window size of 204.8 s and solve the optimization problem +in equation (12) with 200 L-BFGS iterations. The results +are depicted in Figure 1. There are clearly two glitches that +we have successfully separated, along with the background +noise. This results is obtained merely by using 14 hours of +raw data, allowing us to identify the marsquake as a separate +source due to differences in wavelet scattering covariance +representation. +6. Conclusions +For source separation to be effective, prior knowledge con- +cerning unknown sources is necessary. Data-driven methods +of source separation extract this information from existing +datasets during pretraining. In most cases, these methods re- +quire a large amount of data, which means that they are not +suitable for planetary science missions. To address the chal- +lenge posed by limited data, we proposed an approach based +on wavelet scattering covariances. We reaped the benefits +of the inductive bias built into the scattering covariances, +which enabled us to obtain low-dimensional data represen- +tations that characterize a wide range of non-Gaussian prop- +erties of multiscale stochastic processes without pretraining. +Using a wavelet scattering covariance space optimization +problem, we were able to separate thermally induced mi- +crotilts (glitches) from data recorded by the InSight lander’s +seismometer with only a few glitch-free data samples. In +addition, we applied the same strategy to clean a marsquake +from background and glitches using only a few hours of +data that had no recorded marsquake. Our approach did not +require any knowledge regarding glitches or marsquakes, +and it was more robust in separating glitches from recorded +data than existing signal-processing techniques. An impor- +tant characteristic of our approach is that it can be used +as an exploratory approach to unsupervised learning when +exploring challenging and real-world datasets. +7. Acknowledgments +Maarten V. de Hoop acknowledges support from the Simons +Foundation under the MATH + X program, the National +Science Foundation under grant DMS-2108175, and the cor- +porate members of the Geo-Mathematical Imaging Group +at Rice University. + +-3500 +4000 +4500 +-5000 glitch +500 +Baseline +0 +-500-3500 +Predicted +-4000 +4500 +-5000l glitch +500 +0 +-500Unsupervised source separation with limited data +A. Appendices +A.1. Wavelet filters +A wavelet ψ(t) has a fast decay away from t = 0, poly- +nomial or exponential for example, and a zero-average +� +ψ(t) dt = 0. We normalize +� +|ψ(t)| dt = 1. The wavelet +transform computes the variations of a signal x at each +dyadic scale 2j with +Wx(t, j) = x ⋆ ψj(t) where ψj(t) = 2−jψ(2−jt). +We use a complex wavelet ψ having a Fourier transform +�ψ(ω) = +� +ψ(t) e−iωt dt which is real, and whose energy +is mostly concentrated at frequencies ω ∈ [π, 2π]. It re- +sults that �ψj(ω) = �ψ(2jω) is non-negligible mostly in +ω ∈ [2−jπ, 2−j+1π]. +t = 0 +Re(ψ) +Im(ψ) +ω = 0 +π +2π +cψ +Figure 8. Left: complex Battle-Lemari´e wavelet ψ(t) as a function +of t. Right: Fourier transform �ψ(ω) as a function of ω. +We impose that the wavelet ψ satisfies the following energy +conservation law called Littlewood-Paley equality +∀ω > 0 , ++∞ +� +j=−∞ +| �ψ(2jω)|2 = 1. +(13) +A Battle-Lemari´e wavelet, see Figure 8, is an example of +such wavelet. The wavelet transform is computed up to a +largest scale 2J which is smaller than the signal size d. The +signal lower frequencies in [−2−Jπ, 2−Jπ] are captured by +a low-pass filter ϕJ(t) whose Fourier transform is +�ϕJ(ω) = +� ++∞ +� +j=J+1 +| �ψ(2jω)|2�1/2 +. +(14) +One can verify that it has a unit integral +� +ϕJ(t) dt = 1. +To simplify notations, we write this low-pass filter as a last +scale wavelet ψJ+1 = ϕJ, and Wx(t, J+1) = x⋆ψJ+1(t). +By applying the Parseval formula, we derive from (13) that +for all x with ∥x∥2 = +� +|x(t)|2 dt < ∞ +∥Wx∥2 = +J+1 +� +j=−∞ +∥x ⋆ ψj∥2 = ∥x∥2. +The wavelet transform W preserves the norm and is there- +fore invertible, with a stable inverse. +A.2. Scattering network architecture +A scattering network is a convolutional neural network with +wavelet filters. In this paper we choose a simple 2-layer +architecture with modulus non-linearity: +Sx = +� +Wx, W|Wx| +� +. +The wavelet operator W is the same at the two layers, it +uses J = 8 predefined Battle-Lemari´e complex wavelets +that are dilated from the same mother wavelet by powers of +2 (yielding one wavelet per octave). +The first layer extracts J +1 scale channels x⋆ψj(t) (corre- +sponding to J band-pass and 1 low-pass wavelet filters). The +second layer is W|Wx|(t; j1, j2) = |x ⋆ ψj1| ⋆ ψj2(t). It +is non-negligible only if j1 < j2. Indeed, the Fourier trans- +form of |X⋆ψj1| is mostly concentrated in [−2−j1π, 2−j1π]. +If j2 ≤ j1 then it does not intersect the frequency interval +[2−j2π, 2−j2+1π] where the energy of �ψj2 is mostly con- +centrated, in which case SX(t; j1, j2) ≈ 0. +Instead of the modulus |·| we could use another non-linearity +that preserves the complex phase, however it does not im- +prove significantly the results in this paper. +A.3. Scattering Covariance dashboard +The +wavelet +scattering +covariance +Φ(x) +(3) +contains +four +types +of +coefficients +Φ(x) += +� +Φ1(x), Φ2(x), Φ3(x), Φ4(x) +� +. +The first family pro- +vides J order 1 moment estimators, corresponding to +wavelet sparsity coefficients +Φ1(x)[j] = Ave |x ⋆ ψj(t)|. +(15) +The J + 1 second order wavelet spectrum associated to x +are computed by +Φ2(x)[j] = Ave +� +|x ⋆ ψj(t)|2� +. +(16) +There are J(J + 1)/2 wavelet phase-modulus correlation +coefficients for a > 0, +Φ3(x)[j; a] = Ave +� +x ⋆ ψj(t) |x ⋆ ψj−a(t)| +� +. +(17) +Finally, in total the scattering covariance includes J(J + +1)(J + 2)/6 scattering modulus coefficients for a ≥ 0 and +b < 0, +Φ4(x)[j; a, b] = Ave +� +|x⋆ψj|⋆ψj−b(t) |x⋆ψj−a|⋆ψ∗ +j−b(t) +� +. +(18) +These coefficients extend the standard wavelet power spec- +trum Φ2(x). After appropriate normalization and reduction +that we describe below, scattering covariances can be vi- +sualized, and they provide a dashboard that displays non- +Gaussian properties of x, which is shown for example in +Figure 10. + +Unsupervised source separation with limited data +The power spectrum Φ2(x) is plotted in a standard way, +it is the energy of the scale channels of x ⋆ ψj(t). This +energy affects the other coefficients Φ1(x), Φ3(x), Φ4(x). +To deduct this influence, we normalize these coeffi- +cients by the power spectrum, Φ1(x)[j]/ +� +Φ2(x)[j], +Φ3(x)[j; a]/ +� +Φ2(x)[j]Φ2(x)[j − a] +and +Φ4(x)[j; a, b]/ +� +Φ2(x)[j]Φ2(x)[j − a]. +Finally, +we +average Φ3(x) and Φ4(x) on j, in order to plot scaling +invariant quantities, which reduces the number of coefficient +to visualize. The dashboard is shown on Figure 10. +−1 +−2 +−3 +−4 +−5 +−6 +−7 +−8 +−j +28 +210 +212 +214 +216 +Wavelet Spectrum Φ2 +reconstruction +true statistics +1 +3 +5 +7 +a +0.00 +0.25 +0.50 +0.75 +1.00 +×10−1 +|Φ3| +reconstruction +true statistics +−7 +−5 +−3 +−1 +b +0.0 +0.5 +1.0 +1.5 +2.0 +|Φ4| +reconstruction +−7 +−5 +−3 +−1 +b +0.0 +0.5 +1.0 +1.5 +2.0 +true statistics +−1 +−2 +−3 +−4 +−5 +−6 +−7 +−8 +−j +2−2 +2−1 +20Sparsity factors Φ1 +reconstruction +true statistics +1 +3 +5 +7 +a +−π +−π +2 +0 +π +2 +π +Arg Φ3 +−7 +−5 +−3 +−1 +b +−π +4 +−π +8 +0 +π +8 +π +4 +Arg Φ4 +−7 +−5 +−3 +−1 +b +−π +4 +−π +8 +0 +π +8 +π +4 +Figure 9. Scattering covariance visualization of the reconstructed +deglitched Mars background noise compared with a true Mars +background noise. This plots shows that beyond the wavelet power +spectrum, other non-Gaussian properties of the background noise +such as sparsity, long-range correlations match, up to a estimation +error. +−1 +−2 +−3 +−4 +−5 +−6 +−7 +−8 +−j +2−4 +2−2 +20 +22 +Wavelet Spectrum Φ2 +White noise +Mars background +1 +3 +5 +7 +a +−2 +−1 +0 +1 +2 +×10−2 +|Φ3| +White noise +Mars background +−7 +−5 +−3 +−1 +b +0.0 +0.5 +1.0 +1.5 +2.0 +|Φ4| +White noise +−7 +−5 +−3 +−1 +b +0.0 +0.5 +1.0 +1.5 +2.0 +Mars background +−1 +−2 +−3 +−4 +−5 +−6 +−7 +−8 +−j +2−2 +2−1 +20Sparsity factors Φ1 +White noise +Mars background +1 +3 +5 +7 +a +−π +−π +2 +0 +π +2 +π +Arg Φ3 +−7 +−5 +−3 +−1 +b +−π +4 +−π +8 +0 +π +8 +π +4 +Arg Φ4 +−7 +−5 +−3 +−1 +b +−π +4 +−π +8 +0 +π +8 +π +4 +Figure 10. Scattering covariance visualization of the Mars back- +ground noise (no glitch) compared with a white noise. Estimation +is performed on the same amount of data. +A.4. 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Pushing the limits of semi- +supervised learning for automatic speech recognition. +arXiv preprint arXiv:2010.10504, 2020. + diff --git a/_dFLT4oBgHgl3EQfDy7e/content/tmp_files/load_file.txt b/_dFLT4oBgHgl3EQfDy7e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70308c3efcfc317f095d5e17581c80021b36821d --- /dev/null +++ b/_dFLT4oBgHgl3EQfDy7e/content/tmp_files/load_file.txt @@ -0,0 +1,1574 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf,len=1573 +page_content='Unearthing InSights into Mars: unsupervised source separation with limited data Ali Siahkoohi 1 Rudy Morel 2 Maarten V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' de Hoop 1 Erwan Allys 2 Gr´egory Sainton 3 Taichi Kawamura 3 Abstract Source separation entails the ill-posed problem of retrieving a set of source signals observed through a mixing operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Solving this problem requires prior knowledge, which is commonly incorporated by imposing regularity conditions on the source signals or implicitly learned in su- pervised or unsupervised methods from existing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' While data-driven methods have shown great promise in source separation, they are of- ten dependent on large amounts of data, which rarely exists in planetary space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Consid- ering this challenge, we propose an unsupervised source separation scheme for domains with lim- ited data access that involves solving an optimiza- tion problem in the wavelet scattering represen- tation space—an interpretable low-dimensional representation of stationary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' We present a real-data example in which we remove transient thermally induced microtilts, known as glitches, from data recorded by a seismometer during NASA’s InSight mission on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Owing to the wavelet scattering covariances’ ability to capture non-Gaussian properties of stochastic processes, we are able to separate glitches using only a few glitch-free data snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Introduction Source separation is a problem of fundamental importance in the field of signal processing, with a wide range of applications in various domains such as telecommunica- tions (Chevreuil & Loubaton, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Gay & Benesty, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Khosravy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2020), speech processing (Pedersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Chua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Grais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2014), biomedical sig- nal processing (Adali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Barriga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2018) and geophysical data processing (Ibrahim & Sacchi, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Scholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' 1Rice University 2 ´Ecole Normale Sup´erieure 3Institut de Physique du Globe de Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfDy7e/content/2301.11981v1.pdf'} +page_content=' Correspondence to: Ali Siahkoohi 0 and (n1, n2) is a pair of orthogonal unit vectors in the plane orthogonal to n, oriented +so that n = n1 × n2.6 In the local frame (n1, n2, n), b is represented as +b = b1n1 + b2n2. +(5) +By use of the following identity, +2q2 = tr(∇n)2 + 1 +2T 2 − 1 +2S2, +(6) +we can easily give (2) the equivalent form +WOF(n, ∇n) = 1 +2(K11 − K24)S2 + 1 +2(K22 − K24)T 2 + 1 +2K33B2 + 2K24q2, +(7) +where B2 := b · b = b2 +1 + b2 +2. Since (S, T, b1, b2, q) are all independent distortion characteristics, it +readily follows from (7) that WOF is positive semi-definite whenever +K11 ≧ K24 ≧ 0, +(8a) +K22 ≧ K24 ≧ 0, +(8b) +K33 ≧ 0, +(8c) +which are the celebrated Ericksen’s inequalities [11]. If these inequalities are satisfied in strict form, +the global ground state of WOF is attained on the uniform director field, characterized by +S = T = B = q = 0. +(9) +As already mentioned in the Introduction, inequality (8b) must be violated for the ground state of +WOF to be different from (9), involving a non-vanishing T. +6It is argued in [40] that q should be given the name tetrahedral splay, to which we would actually prefer octupolar splay for +the role played by a cubic (octupolar) potential on the unit sphere [41] in representing all scalar measures of distortion, but T. +4 + +The class of uniform distortions was defined in [10] as the one comprising all director fields for +which the distortion characteristics are constant in space. Equivalently said, a uniform distortion +is a director field that can fill three-dimensional space. It was proven that there are two distinct +families of uniform distortions, characterized by the following conditions [10], +S = 0, +T = ±2q, +b1 = ±b2 = b, +(10) +where q and b are arbitrary parameters. +The general director field corresponding to (10) is the heliconical ground state of twist-bend +nematic phases,7 in which n makes a fixed cone angle with a given axis in space (called the helix +axis), around which n precesses periodically [10].8 The special instance in which b = 0 corresponds +to the single twist that characterizes cholesteric liquid crystals. +The distortion for which all characteristics vanish, but T, is a double twist.9 It is not uniform +and cannot fill space; it can possibly be realized locally, but not everywhere. In words, we say that +it is a frustrated ground state. As shown in [12], a double twist is indeed attained exactly only on +the symmetry axis of cylinders enforcing degenerate planar anchoring on their lateral boundary. +2.2. Quartic Twist Energy +The essential feature of the quartic twist theory proposed in [20] is to envision a double twist with +two equivalent chiral variants as ground state of CLCs in three-dimensional space, +S = 0, +T = ±T0, +B = 0, +q = 0. +(11) +The degeneracy of the ground double twist in (11) arises from the achiral nature of the molecu- +lar aggregates that constitute these materials, which is reflected in the lack of chirality of their +condensed phases. +The elastic stored energy must equally penalize both ground chiral variants. Our minimalist +proposal to achieve this goal was to add a quartic twist term to the Oseen-Frank stored-energy +density, and so take W = WQT, with +WQT(n, ∇n) := 1 +2(K11 − K24)S2 + 1 +2(K22 − K24)T 2 + 1 +2K23B2 + 1 +2K24(2q)2 + 1 +4K22a2T 4, (12) +where a is a characteristic length. Unlike WOF, WQT is bounded below whenever +K11 ≧ K24 ≧ 0, +(13a) +K24 ≧ K22 ≧ 0, +(13b) +K33 ≧ 0. +(13c) +If these inequalities hold, as we shall assume here, then WQT is minimum at the degenerate double- +twist (11) characterized by +T0 := 1 +a +� +K24 − K22 +K22 +. +(14) +7With opposite chiralities, one for each sign in (10). +8In opposite senses, according to the sign of chirality. +9Here we adopt the terminology proposed by Selinger [40] (see also [42]) and distinguish between single and double twists, the +former being uniform and the latter not. +5 + +The parameter a encodes the bare length scale over which distortions would be locally stored in +the ground state.10 As to the physical size of such a length scale, it may be comprised in a wide +range. While at the lower end we may place the persistence length of the molecular order, which +characterizes the flexibility of CLC aggregates,11 the upper end is hard to make definite. We expect +that a would be exposed to the same indeterminacy that affects many (if not all) supramolecular +structures in lyotropic systems. The most telling example is perhaps given by cholestric liquid +crystals, which give rise to a chiral structure (characterized by a single twist T = ±2q) starting +from chiral molecules. If the macroscopic pitch were determined by the molecular chirality,12 it +would result several orders of magnitude smaller than the observed ones.13 Here, we shall treat a as +a phenomenological parameter, to be determined experimentally. An estimate derived in [20] from +a comparison with published data placed a in the order of microns. +3. Twisted Hedgehog +So far we have presented, mostly on equal terms, two elastic theories for chromonics, one quadratic +and the other quartic in the director gradient. Here we see how these theories can be differentiated +on the basis of the different structures they predict for the core of hedgehogs, the most common +of nematic defects in three space dimensions. Many mathematical details needed to follow our +development are collected in Appendix A. +We first discuss the distortion of a trial director field within a ball of radius R enforcing +homeotropic anchoring on its boundary. This is a field with a point defect at the center of the +ball, potentially rich in twist, as would seem fit for a material with small K22 constant. We shall +then see the analytical implications and the potential experimental significance of this field. +The point defects that we shall study are a special family of hedgehogs, which place themselves in +between the most common defects in liquid crystal science, the radial and the hyperbolic hedgehogs. +The former is represented by the director field +nR(x) := x − x0 +|x − x0|, +(15) +which has a point defect at x0, while the latter is formally obtained by the following transformation +of nR, +nH := R(π)nR, +(16) +where +R(π) := −I + 2e ⊗ e +(17) +is the special orthogonal tensor describing a rotation by angle π about a unit vector e ∈ S2. Figure 1 +illustrates the field lines of both nR and nH. +10In the elastic model proposed in [10] for twist-bend nematics, a quartic free energy was posited that admits as ground state +either of two families of uniform heliconical fields with opposite chirality. There too, a length scale appears in the equilibrium +pitch. The distortion state characterized by this length is the same everywhere. +11The persistence length of a flexible aggregate is the shortest length over which unit vectors tangent to the aggregate’s contour +lose correlation. For CLCs, it is estimated on the order of tens to hundreds of nm [43] +12Via the naive geometric argument that represents chiral molecules as cylindrical screws and derives the pitch of their assemblies +by close packing them so as to fit grooves with grooves. +13For lyotropic cholesterics, the mismatch between microscopic and macroscopic pitches, which has recently received new +experimental evidence in systems of basic living constituents [44,45], is still debated. Interesting theories based on either +molecular shape fluctuations [46,47] or surface charge patterns [48] have met with some experimental disagreement [49]. +6 + +(a) Radial hedgehog: N(nR) = +1 +(b) Hyperbolic hedgehog: N(nH) = +1 +Figure 1. +Field lines of nR and nH in (15) and (16), representing a radial and a hyperbolic hedgehog, respectively. Field lines +are drawn on the equatorial plane (in black) and on a meridian plane (in red). The whole picture is obtained by rotating these +lines around the polar axis. By their definitions, both nR and nH share the same topological charge N as introduced in (18). +The topological charge of a unit vector field n with a point defect at x0 is defined as +N(n) := 1 +4π +� +S +n · (∇sn)∗ν dA, +(18) +where S is a any surface enclosing x0, ∇s denotes the surface gradient on S , ν is the unit normal +to S , the operation (· · · )∗ takes the cofactor of a tensor,14 and dA is the area element. N(n) is an +integer of Z independent of S , provided the latter embraces x0, and so N(n) can be attributed +to x0 itself. The absolute value |N(n)| indicates the number of times n restricted to S covers the +unit sphere S2; the sign of N(n) tells whether S2 is covered coherently or not with the orientation +of the unit normal ν. Historically, we learn from [51] that the representation in (18) for N(n) was +first derived in [52]. +N(n) is additive: if the surface S encloses more than one defect, the topological charge com- +puted on it through (18) is the algebraic sum of the topological charges computed on surfaces +enclosing the single defects comprised in S . As pointed out in Sect. VII.E.3 of [53], a defect with +topological charge N(n) can be transformed continuously into a defect with opposite topological +charge, thus making |N(n)|, and not N(n) itself, a topological invariant apt to classify point defects +for director fields on S2. +The mapping +nR �→ nR := −nH +(19) +was described in [54] as a parity transformation, as it changes the sign of the topological charge,15 +N(nR) = −N(nR) = −1. +(20) +This was meant to identify nR as an anti radial hedgehog, which would neutralize the topological +14This is a tensor whose representative matrix is the cofactor matrix of the matrix representing the original tensor, see [50, +p. 22] for a formal definition. +15This equation follows from Appendix A.2 and the general property of (18) stating that N(−n) = −N(n), which stems from +being (∇sn)∗ even in n. +7 + +charge of the radial hedgehog and annihilate it when combined together in a director field on S2 +with zero total topological charge.16 +Here, instead, as shown in Appendix A.2, +N(nH) = N(nR) = +1, +(21) +so that nR and nH not only belong to the same topological class, but also have the same topological +charge. +We consider as domain B a ball BR(x0) with radius R and center at x0. We study a trial twisted +hedgehog field nT, which “interpolates” in space between nR and nH. Formally, nT is obtained by +acting on the radial hedgehog nR(x) in (15) with a rotation R(α) of variable angle α = α(r) about +a fixed axis e ∈ S2, where r is the distance of x from the defect at x0, +nT(x) := R(α(r))nR(x), +(22a) +R(α) := I + sin αW(e) + (1 − cos α)W(e)2. +(22b) +In (22b), W(e) is the skew-symmetric tensor associated with e, whose action on any vector v is +given by W(e)v = e × v. The field nT reduces to the radial hedgehog nR for α ≡ 0 and to the +hyperbolic hedgehog nH for α ≡ π. As shown in Appendix A.2, the topological charge of nT equals +that of both nR and nH, irrespective of the function α, +N(nT) = +1. +(23) +We shall call α the twist angle. +3.1. Inversion Ring +A peculiar property of the field nT is illustrated by letting e be the polar axis of a system of +spherical coordinates (r, ϑ, ϕ) with origin at x0. On the equatorial plane ϑ = π +2 , in the coordinate +frame (er, eϑ, eϕ), nT reduces to (see (A9)) +nT = cos αer + sin αeϕ, +(24) +and so it lies entirely on the equatorial plane. If, for some r∗, α(r∗) = π +2 , then nT is tangent to +the circle of radius r∗ around x0. Where α(r) < π +2 the field nT spirals outward (relative to x0), +where α(r) > π +2 it spiral inward. Figure 2 illustrates this feature within the ball BR(x0) when the +condition +α(R) = 0 +(25) +is enforced, so that nT = nR on the boundary ∂BR(x0). The ring at r = r∗ separates two opposite +spiraling regimes; there, the field lines of nT appear to coalesce in a ring, which looks like a +disclination, but is instead regular, as it bears no discontinuity of the director. We shall call the +ring at r = r∗, if present, an inversion ring, as it marks the inversion of the spiraling sense. +It is perhaps the seminal work of Lavrentovich and Terentjev [32] where a first experimental +evidence of an inversion ring within a twisted hedgehog was ever found and documented in ordinary +16Actually, in [54], nH was defined to be precisely nR, so that, being opposite to the field in (16), would form with it a +defect-anti-defect pair, as would also be clear from Fig. 1 once the field lines orientation in panel (b) are reversed. +8 + +Figure 2. +Field lines of nT in (22) within the ball BR(x0) enforcing condition (25), so that nT = nR on ∂BR(x0). An +inversion ring is present, which is depicted in blue. Black lines are field lines lying on the equatorial plane; red lines are field +lines coming out of the equatorial plane. As in Fig. 1, the whole 3D picture is obtained ny rotating this drawing about the polar +axis. +nematics.17 +Here, we shall use nT as a trial field to describe the twisted distortion that replaces nR in BR(x0) +when nR becomes unstable. We shall determine the function α subject to (25) that minimizes the +elastic free energy F in (1) with B = BR(x0). We shall do so for either W = WOF in (7) and +W = WQT in (12) to see whether the quadratic and quartic elastic theories for chromonics recalled +in Sect. 2 can be distinguished on the basis of the predictions they make about the occurrence of +a twisted hedgehog and its inversion ring. +We start by considering under what conditions nR is locally stable for either theory. +3.2. Local Stability of Radial Hedgehog +First, we observe that nR is a universal solution, as it solves the equilibrium equation for all possible +elastic free-energy functionals F in (1) associated with a frame-indifferent density W = W(n, ∇n), +see [55]. Thus, nR is an equilibrium configuration for F, irrespective of W. Moreover, it was proved +in [56] and [57] that, when W = WOF, nR is a local minimizer of F in the admissible class of director +fields n with finite energy in BR(x0) and such that +n|∂BR(x0) = nR, +(26) +provided that the following inequality is satisfied,18 +0 < k1 < 1 + k3 +8 . +(27) +17In a temperature regime where the twist constant K22 is sufficiently small. +18A result which was independently rediscovered in [58]. +9 + +Here and below, the elastic constants will be scaled to K22, +k1 := K11 +K22 +, +k3 := K33 +K22 +, +and +k24 := K24 +K22 +with +K22 > 0. +(28) +This local stability result is based on the study of the second variation of F at n = nR; the latter +is the same for both WOF and WQT, as these only differ by a quartic term that does not affect the +second variation of F at nR, see Appendix A.3. +It is remarked in [59] that when (27) is violated the free-energy functional F with W = WOF +subject to (26) admits a continuum of minimizers, all sharing the same energy. Since the proof of +this result is based on frame-indifference only, it also holds within our quartic twist theory where +W = WQT. +Figure 3 illustrates inequality (27) for k1 > 1, which is the situation that applies to chromonics, +as also shown by the dot representing data for SSY. Hereafter, we assume that +Figure 3. +Regions of interest for the local stability of the radial hedgehog nR in CLCs. In the pink region, nR is a local +minimizer of F subject to (26) when B = BR(x0), for either W = WOF and W = WQT. In the blue region, nR is no longer a +minimizer; there is a continuum of minimizers, all with the same energy. Bulk elastic constants of chromonics fall in the region +of instability; the red dot represents data for SSY, k1 ≈ 6.1 and k3 ≈ 8.7, taken from [60]. +k1 > 1 + k3 +8 . +(29) +The special family of twisted director configurations described by nT are parameterized by the +scalar function α and the symmetry axis e ∈ S2. Once, for a given e, α is chosen so as to minimize +F, letting e vary in S2 potentially embodies the continuum of minimizers expected to arise when +the radial hedgehog nR is no longer locally stable. +3.3. Minimum Problem +Here, we study the problem of minimizing the functional F in (1) for B = BR(x0) and W = WQT +subject to (26). We introduce the change of variables +r �→ ρ := r +R, +(30) +10 + +SSY +not a minimizer +ki = 1＋k3/8 +local minimizerwhich maps [0, R] onto [0, 1]. In the new variable, (25) becomes19 +α(1) = 0. +(31) +Standard computations (deferred to Appendix A.1) show that for B = BR(x0) and W = WQT the +functional F in (1) can be given the following scaled form +Fλ[α] := 15F[nT] +8πK22R − FR += +� 1 +0 +� +g(α(ρ))(ρα′(ρ))2 + 2(k1 − 1)f0(α(ρ)) + 32 +7 +λ2 +ρ2 +� +4 +� +n=1 +fn(α(ρ))(−ρα′(ρ))n +�� +dρ, +(32) +where a prime ′ denotes differentiation, FR := 15(k1 − k24) is the scaled energy of the radial +hedgehog, so that +Fλ[0] = 0, +(33) +and the functions g, and fn are defined as +g(α) = 2k1 sin2 α + 2 +7(1 − cos α)2 + k3 +14 +� +24 cos2 α + 8 cos α + 3 +� +, +(34a) +f0(α) = 2 cos2 α + cos α − 3, +(34b) +f1(α) = 3(1 − 8 cos α) sin3 α, +(34c) +f2(α) = (1 − cos α)2 sin2 α, +(34d) +f3(α) = 2 +11(1 − cos α)3 sin α, +(34e) +f4(α) = +2 +143(1 − cos α)4. +(34f) +Fλ[α] is invariant under the change of α into −α, for any α. Thus, every non-trivial equilibrium +solution αλ would be accompained by its parity conjugate −αλ. The corresponding fields nT differ +as they have opposite chirality, but they have one and the same energy. +For λ > 0, integrability of the quartic term in Fλ in (32) requires that the limiting value α(0) +of α at ρ = 0 be either 0 or π. Under the assumption that, to within parity conjugacy, Fλ has a +unique minimizer subject to (31), the choice α(0) = 0 would lead us to α ≡ 0, that is, to nR, which +is a contradiction since the radial hedgehog is unstable when (27) applies. Thus, we shall enforce +the condition +α(0) = π +for +λ > 0. +(35) +For λ = 0, α(0) is instead free to vary, as in (32) integrability is guaranteed by the integrability of +α′.20 +3.3.1. Equilibrium Solutions +Here, we specialize the analysis to positive solutions of the equilibrium equation for Fλ: we assume +that αλ ≧ 0 since the minimizer of Fλ is not expected to change sign. Clearly, this positive branch +19We shall continue to adopt the same old symbol for the function α, even if it is expressed in the new variable. +20Which requires that ρα′(ρ) be bounded as ρ → 0+. +11 + +of solutions remains associated with the conjugate negative branch, which has equal energy. The +equilibrium equation is too complicated to lend itself to analytic solutions; it was symbolically +manipulated and will be conventionally called (E).21 +We could establish the asymptotic behaviour of the solutions αλ of (E) near ρ = 0 and ρ = 1, +for every λ > 0. As shown in Appendix A.4, for k1 > 1 +αλ(ρ) = π(1 − Bρ) + O +� +ρ2� +for +ρ → 0+, +(36) +where +B = +� +7 +32 +1 +λ +√ +58058√21k1 + 19k3 − 5 +1421π +. +(37) +Similarly, +αλ(ρ) ≈ C +�1 +ρ − 1 +� +as +ρ → 1, +(38) +where C is a positive constant to be determined. +3.3.2. Energy Minimizers +Here we explore numerically the minimizers αλ of Fλ, focusing on the positive equilibrium branch +(thus selecting one chirality for nT). For λ = 0, this problem is solved in [33] by reinterpreting +F0 as an infinite-horizon action functional associated with an equivalent autonomous dynamical +system in two-dimensional phase space. For λ > 0, a similar reinterpretation for Fλ is still viable, +but the associated dynamical system is not autonomous; it is studied numerically in Appendix B +and contrasted with the autonomous system associated with F0. +The major difference between these dynamical systems resides in their equilibrium points; in +the language of the twist angle α, this translates into two different asymptotic values at the center +of the ball BR(x0), +αλ(0) = +� +�α0 := arccos(−1/4) +for +λ = 0, +π +for +λ > 0. +(39) +One may say that the classical quadratic theory (λ = 0) predicts that nR and nH are not +completely bridged inside the confining ball BR(x0), whereas the quartic theory (λ > 0) predicts +that they are. Hence we could possibly use hedgehogs in chromonics confined within a ball to +discriminate these theories from one another. However, although this is a qualitative difference, +its observation might be experimentally precluded. A further, quantitative feature must be called +upon; this is the size (relative to ball’s radius R) of the inversion ring r∗ associated with the stable +twisted hedgehogs predicted by both theories, as by (39) an inversion ring is present in both cases. +For definiteness, we consider a specific case, which was suggested by the experimental study in +[31]. This is the case of chromonic liquid crystal SSY in an aqueous solution (at a wt/wt concen- +tration of 30% and a temperature of 25 ◦C) confined within a spherical cavity produced inside a +polymeric matrix enforcing homeotropic anchoring for the director on its boundary (see Fig. 5). +Material constants are derived from [60] and deliver k1 ≈ 6.1 and k3 ≈ 8.7,22 which, as shown in +21It is equivalent to the equation of motion (B6) for the effective dynamical system described in Appendix B. +22The absolute measured values are K11 ≈ 4.3 pN, K22 ≈ 0.7 pN, and K33 ≈ 6.1 pN. +12 + +Fig. 3, locate the radial hedgehog in its unstable domain. The radius of the spherical cavity in Fig. 5 +is R ≈ 40.4µm. For the same SSY solution in the same physical conditions, in [20] we estimated +a ≈ 6.4µm, thus here we take λ = 0.16. +The profile of the minimizing twist angle αλ corresponding to these parameters is shown in +Fig. 4 (red curve) against the minimizing profile α0 for λ = 0 (blue curve). It is apparent that the +Figure 4. +Plots against ρ of the minimizer αλ of Fλ (red curve) and the minimizer α0 of F0, for k1 = 6.1, k3 = 8.7, and +λ = 0.16. As in (39), �α0 := arccos(−1/4). The broken lines reproduce the asymptotic behaviours predicted by (36) and (38), +respectively, with B .= 2.68, in agreement with (37), and C .= 0.54. The dotted line drawn at α = π +2 intercepts the graphs of αλ +and α0 at values ρ∗ of ρ that designate the scaled radius r∗ of the inversion ring in the two cases. It is apparent how ρ∗ +λ ≈ 0.2 +is appreciably larger than ρ∗ +0 ≈ 0.03, see also Fig. 5 below. +inversion rings associated with these solutions are appreciably different. +Figure 5 reproduces a spherical cavity (in a polymeric matrix) observed in [31]; there we also +draw the inversion rings predicted by both classical and quadratic theories. Judging from this single +comparison and taking for granted that the defect shown here is indeed a twisted hedgehog, we +may say that the quartic theory seems to capture better the size of the inner structure enclosed by +the inversion ring. This core structure will be further detailed in Sect. 4. +Letting ρ∗ := r∗/R designate the scaled radius of the inversion ring, we explored the dependence +of ρ∗ on λ. The plot in Fig. 6 summarizes the outcomes of this analysis; it shows how ρ∗ saturates +to ρ∗ +∞ ≈ 0.82 as λ grows indefinitely. +Not only does the inversion ring size increase monotonically with λ, but also the defect core +inside the inversion ring is qualitatively different for λ = 0 and λ > 0. These differences will be +highlighted in the following section. +13 + +Figure 5. +Reproduction of Fig. 5b of [31] showing a spherical cavity (in a polymeric matrix) of radius R ≈ 40.4µm enclosing a +SSY solution in water with concentration 30% (wy/wt) and temperature 25 ◦C. The homeotropic anchoring on the boundary of +the sphere induces a (presumably twisted) hedgehog at the center exhibiting the typical Maltese cross when observed between +crossed polarizers. The larger (green) and smaller (blue) circles superimposed to the figure are the inversion rings predicted +by the quartic and quadratic theories, respectively. In absolute terms, with a ≈ 6.4µm (from [20]), that is, λ ≈ 0.16, we have +r∗ +0 = ρ∗ +0R ≈ 1µm and r∗ +λ = ρ∗ +λR ≈ 8.1µm. +4. Spiraling Cores +Here we go into deeper details of the twisted hedgehog nT that minimizes the elastic free energy +Fλ; we are especially interested in the behaviour if its field lines within the defect core, which is +conveniently identified with a sphere of radius r∗, the radius of the inversion ring. We shall again +study primarily the distortion afforded by the quartic theory with λ > 0; this case will also be +contrasted against the case λ = 0 of the classical quadratic theory. We shall see that the differences +between the two cases are both qualitative and quantitative. +We split our analysis in two steps; in the first, we study the field lines of nT on the equatorial +plane of BR(x0) (orthogonal to the symmetry axis); in the second, we see how these lines behave +away from that plane. +4.1. Equatorial Field Lines +In a spherical coordinate system (r, ϑ, ϕ) with polar angle ϑ ∈ [0, π], the equatorial plane is described +by ϑ = π +2 and r ≧ 0, ϕ ∈ [0, 2π). Scaling lengths to the radius R of the spherical cavity and letting +ρ be still defined as in (30) above, we see from (24) that the field lines of nT on the equatorial +plane are the solutions (ϕ(τ), ρ(τ)) to the differential system +dϕ +dτ = 1, +(40a) +dρ +dτ = +ρ(τ) +tan(αλ(ρ(τ))), +(40b) +subject to +ϕ(0) = 0 +and +ρ(0) = ρ0 +with +0 < ρ0 < 1, +(40c) +14 + +Figure 6. +Plot of ρ∗ = r∗/R as a function of λ computed on the minimizer αλ of Fλ; the graph saturates at ρ∗ +∞ ≈ 0.82, +while ρ∗ +0 ≈ 0.03 is the limiting value as λ → 0. The red dot marks the inversion ring predicted for λ = 0.16, corresponding to +the spherical cavity shown in Fig. 5. +where τ is a parameter. The curves solving (40) may be winding several times around the origin as +τ → +∞; the appropriate solution of (40a) is then +ϕ = τ +mod 2π. +(41) +It follows from (40) that every field line that starts inside or outside the inversion ring, remains +inside or outside that ring, respectively. The inversion ring at ρ = ρ∗ is a field line itself, since ρ ≡ ρ∗ +is a solution of (40b). Moreover, a field line that starts from ρ0 < ρ∗ keeps spiraling (clockwise) +around the point defect at the origin, while a field line that starts from ρ0 > ρ∗ is soon bent +(anti-clockwise) towards the equator of BR(x0), where it points radially away from the defect (see +Fig. 7). +Another qualitative feature of the field lines of nT is revealed by (40). Given the monotonicity +of ρ(τ) both inside and outside the inversion ring, this function is invertible; a straightforward +integration yields the following formula for its inverse, +τ(ρ) = +�� ρ +ρ0 +tan αλ(ξ) +ξ +dξ +for +ρ > ρ∗, +� ρ0 +ρ +tan αλ(ξ) +ξ +dξ +for +ρ < ρ∗. +(42) +Two noteworthy consequences follow from (42). First, from the divergence of both integrals as +ρ → ρ∗ (from above and from below, respectively), we see that the field lines of nT wind infinitely +many times around the inversion ring, no matter which elastic theory is employed to describe a +twisted hedgehog. Second, by taking the limit as ρ → 0+ in the second integral, we see that this +diverges or not, depending on the limiting value αλ(0). Since the latter depends on being λ = 0 +or λ > 0, the two theories being compared here afford different qualitative predictions. According +15 + +* +8 +0.8 +0.6- +* +0 +0.4- +0.2- +po +0 +1 +0 +2 +4 +6 +8 +10(a) Quadratic theory (with λ = 0): The +inversion ring has (scaled) radius ρ∗ ≈ +0.03. Zooming inside the inversion ring +reveals the logarithmic nature of the +asymptotic spirals. +(b) Quartic theory (with λ = 0.16): +The inversion ring has (scaled) radius +ρ∗ ≈ 0.2. Zooming inside the inversion +ring reveals the Archimedean nature of +the asymptotic spirals. +Figure 7. +Field lines of nT in the equatorial plane of BR(x0) according to the two elastic theories considered here. Material +constants correspond to SSY in the same conditions that apply to both Figs. 4 and 5. +to the quadratic theory, for which α0 = arccos(−1/4), the second integral in (42) diverges and the +field lines of nT wind infinite many times around the point defect at the origin; asymptotically, +they are logarithmic spirals. On the contrary, according to the quartic theory, for which αλ = π +for all λ > 0, by (36), the second integral in (42) converges and the field lines of nT wind a finite +number of times around the defect; asymptotically, they are Archimedean spirals. +In Fig. 7, the field lines of nT in the equatorial plane are contrasted for the two theories, when +the twist angle is given by the functions α0 and αλ whose graphs are shown in Fig. 4. In both cases +the inversion ring is zoomed in to highlight the different nature of the asymptotic spirals around +the point defects. +4.2. Field Lines in Space +As is easily seen from (A9), the field lines of nT away from the equatorial plane of BR(x0) are +described in spherical coordinates (r, ϑ, ϕ) by the solutions to the following differential system +dρ +dτ = ρ(τ)1 + (cos αλ(ρ(τ)) − 1) sin2 ϑ +sin αλ(ρ(τ)) +, +(43a) +dϑ +dτ = (cos αλ(ρ(τ)) − 1) cos ϑ sin ϑ +sin αλ(ρ(τ)) +, +(43b) +dϕ +dτ = 1, +(43c) +where τ is a parameter chosen again so that (41) holds. +16 + +The flow described by (43) is mirror-symmetric with respect to the equatorial plane (ϑ = π +2 ) and, +as shown in Fig. 8, possesses two families of negatively invariant sets, balls and circular cylinders +Figure 8. +Field lines of nT away from the equatorial plane of BR(x0), for the same choice of parameters in both Figs. 4 and +5. Only the two limiting negatively invariant sets, the ball Br∗ and the cylinder Cr∗ built on the inversion ring, are shown. Field +lines are back inside Cr∗ and red outside. The zoomed region on the right is the ball of radius r∗; two field lines are drawn that +start near the boundary of Br∗, one inside Cr∗ (black) and the other outside (red). +with radii larger than the radius r∗ of the inversion ring. This means that field lines of nT may +only leave the regions enclosed by these sets and never enter them. +To prove this qualitative property, we denote by Br and Cr these families of balls and cylinders, +respectively, and by ν their outer unit normal. It readily follows from (A9) that +nT · ν|∂Br = sin2 ϑ cos α + cos2 ϑ, +(44a) +nT · ν|∂Cr = sin ϑ cos α, +(44b) +which are both non-negative for all ϑ ∈ [0, π] whenever α ≦ +π +2 , that is, for r > r∗. A further +geometric illustration of this property is given in Fig. 9. +5. Conclusion +In [20], we proposed a quartic twist theory for the curvature elasticity of chromonic liquid crystals, +for which we have been seeking corroborating evidence. This theory introduces a phenomenological +length a, which in [20] was estimated to be of the order of microns by fitting published data for +chromonics filling cylinders with degenerate planar anchoring on their lateral boundary. These data +could also be interpreted by use of the classical quadratic Oseen-Frank theory [6,7], which however +would be unable to predict stable shapes for the tactoidal droplets observed in the biphasic region +of these materials [14]. +17 + +Figure 9. +For the same field nT in Fig. 8, the director profiles are shown on two parallel sections of BR(x0) with planes +parallel to the equator: one cuts the ball Br∗ at mid-height, z = r∗/2, while the latter cuts the ball BR(x0) at mid-height, +z = R/2. +We turned to hedgehog defects and their core structure to find an instance where the two +theories would afford different predictions, which could serve to differentiate them. We considered +a spherical cavity of radius R enforcing homeotropic anchoring on its boundary, like those produced +in [31], and studied the twisted hedgehogs predicted by both theories in the region in parameter +space where the radial hedgehog would be unstable. +The defect core of a twisted hedgehog director field nT is characterized by an inversion ring +that encloses the defect core. Two properties of the defect core are predicted in stark contrast by +the two theories: one is qualitative, the other quantitative. +We start with the latter. The radius r∗ of the inversion ring depends only on the elastic +anisotropy for the quadratic theory and also on the ratio λ = a/R for the quartic theory. For +SSY in the same physical conditions as in [31], taking a from [20], we estimated r∗ to be nearly +an order of magnitude larger for the quartic theory compared to the quadratic one, 8.1 µm against +1 µm. +On the qualitative side, we showed that the field lines of nT spiral differently around the point +defect according to which theory is adopted: in the quadratic theory, they are logarithmic spirals; +in the quartic theory, they are instead Archimedean spirals. +We may perhaps say that the defect core of twisted defects, with its distinctive quantitative and +qualitative features, could be the hallmark of a quartic elastic theory for chromonics. However, such +a clear distinction between quadratic and quartic theories rests on being a in the order of microns; +were it much smaller, the differences highlighted here could not be appreciated. A thorough study +with direct observations of the core structure of twisted hedgehogs would be desirable. +Another critical issue that deserves further research concerns the splay constant K11. If the re- +cent theoretical estimate for the elastic constants in [61] is to be confirmed by different, independent +approaches, not only K22, but also K11 would be smaller than K24 for chromonics. This, as shown +in [14], would ignite the instability of chromonic droplets in an isotropic fluid environment enforcing +18 + +homeotropic anchoring at the interface. The defects studied in this paper inhabit a spherical cavity +of fixed shape, and so they are saved from that instability. However, should homeotropic anchor- +ing be realistic for chromonic droplets, if K11 < K24, our quartic twist theory could not prevent +such a shape instability, as it would be driven by a concentration of splay. Thus, were homeotropic +chromonic droplets actually observed, our elastic theory would need to be amended. +Appendix A. Trial Twisted Hedgehog +This Appendix contains ancillary results instrumental to our analysis in the main text. +A.1. Useful Computations +Identifying the the unit vector e designating in (22b) the symmetry axis of nT as the polar axis ez +of standard spherical coordinates (r, ϑ, ϕ), where ϑ ∈ [0, π] is the polar angle and ϕ ∈ [0, 2π) is the +azimuthal angle, we represent the gradient of the trial twisted field through the formula +∇nT =1 +r +� +Pr + sin αWz + (1 − cos α)W2 +z +� ++ +� +α′ cos α − 1 +r sin α +� +Wzer ⊗ er ++ +� +α′ sin α − 1 +r(1 − cos α) +� +W2 +zer ⊗ er, +(A1) +where Pr := I−er ⊗er is the projection onto the plane orthogonal to er, Wz is the skew-symmetric +tensor with axial vector ez, and a prime ′ denotes differentiation with respect to r. +The following expressions for the traditional measures of distortion of nT in (22a) are conse- +quences of (A1); they are written in the local frame (er, eϑ, eϕ) of spherical coordinates: +div nT = 1 +r +� +−(rα′) sin α sin2 ϑ + 1 − (1 − cos α) cos2 ϑ + cos α +� +, +(A2a) +curl nT = 1 +r +� +2 sin α cos ϑer − sin ϑ +� +(rα′) cos α + sin α +� +eϑ ++ cos ϑ sin ϑ +� +−(rα′) sin α + (1 − cos α) +� +eϕ +� +, +(A2b) +nT · curl nT = 1 +r +� +cos ϑ +� +−(rα′)(1 − cos α) sin2 ϑ + 2 sin α +�� +, +(A2c) +nT × curl nT = 1 +r{sin2 ϑ[(rα′) sin α(1 − (1 − cos α) sin2 ϑ) +− (1 − cos α)2 cos2 ϑ + sin2 α]er ++ sin ϑ[−(rα′) cos α(1 − (1 − cos α) sin2 ϑ) ++ sin α(−1 + (1 − cos α)(1 + cos2 ϑ))]eϑ} ++ sin ϑ cos ϑ[(rα′) sin α(1 − (1 − cos α) sin2 ϑ) +− (1 − cos α)(1 − (1 − cos α) sin2 ϑ) + 2 sin2 α]eϕ, +(A2d) +tr(∇nT)2 − (div nT)2 = −2(cos2 ϑ − (rα′) sin α sin2 ϑ + cos α sin2 ϑ). +(A2e) +Making use of (A2) and (31) in the free energy density WQT in (12), and integrating over B = +BR(x0), we arrive at the following scaled form for F in (1), +15F[nT] +8πK22R =: Fλ[α] + FR, +(A3) +19 + +where Fλ[α] and FR are given by (32) and +FR = 15(k1 − k24), +(A4) +respectively. +A.2. Topological Charge +Here we compute the topological charge N(nT) of the twisted hedgehog nT in (22). To this end, +we first note that +∇snT = (∇n)Pr = 1 +r[Pr + sin αWzPr + (1 − cos α)W2 +zPr], +(A5) +where use has been made of (A1). In the frame (er, eϑ, eϕ), ∇snT in (A5) is also represented as +∇snT = 1 +r{(sin2 ϑ + cos2 ϑ cos α)eϑ ⊗ eϑ − cos ϑ sin αeϑ ⊗ eϕ ++ cos ϑ sin αeϕ ⊗ eϑ + cos αeϕ ⊗ eϕ +− sin ϑ cos ϑ(1 − cos α)er ⊗ eϑ − sin ϑ sin αer ⊗ eϕ}, +(A6) +where we have employed the identity +eϑ = +1 +sin ϑ(cos ϑer − ez), +(A7) +having identified e and ez, as above. +It is now a simple matter to compute in the basis (er, eϑ, eϕ) the tensor (∇snT)∗, as it is +represented by the cofactor matrix of the matrix representing ∇snT in (A6). A tedious, but simple +calculation delivers +(∇snT)∗ = 1 +r2 {sin ϑ cos ϑ(cos α−1)eϑ⊗er +sin ϑ sin αeϕ⊗er +(sin2 ϑ cos α+cos2 ϑ)er ⊗er}. (A8) +Since it follows from (22) and (A7) that +nT = cos ϑ sin ϑ(cos α − 1)eϑ + sin ϑ sin αeϕ + (sin2 ϑ cos α + cos2 ϑ)er, +(A9) +it is easily concluded that +nT · (∇snT)∗er = 1 +r2 . +(A10) +Taking S in (18) to be a sphere of radius r and center at x0, we readily obtain that N(nT) = +1, +for any function α, which is precisely (23) in the main text. In particular, by taking α ≡ 0 or α ≡ π, +we recover (21). +20 + +A.3. Second Variation +We let F4 denote the quartic term contribution to F in (1) arising from WQT in (12), +F4[n] := 1 +4K22a2 +� +B +(n · curl n)4 dV. +(A11) +By applying to F4 the method illustrated in [12], we readily see that the second variation δ2F4 of +F4 at n can be given the general form +δ2F4(n)[v] = K22a2 +� +B +� +(n · curl n)2� +3(v · curl n + n · curl v)2 ++ 2(n · curl n)(v2n · curl n + v · curl v) +�� +dV, +(A12) +which is a quadratic functional in the perturbation field v subject to the orthogonality condition +n · v ≡ 0. +(A13) +It is a very simple matter to check that δ2F4(nR) ≡ 0, as curl nR ≡ 0. +A.4. Asymptotic Behaviours +Here we give a few details about the derivation of the asymptotic behaviours in (36) and (38) of +the equilibrium solutions αλ for Fλ. We renounce writing the equilibrium equation of Fλ since it is +too complicated; as in the main text, it will denoted by (E) and manipulated by symbolic calculus. +An equivalent form of (E) will be encountered in Appendix B below. +When ρ is near 0, we write αλ as +αλ(ρ) ≈ π +� +1 − Bρβ� +, +(A14) +which satisfies (35), and seek B > 0 and β > 1/4, assumed to exist, the latter requirement being +a direct consequence of the integrability of Fλ. In our asymptotic method, which is an adaptation +of the classical method of Frobenious (see, for example, p. 396 of [62]), we determine both β and B +by requiring that the dominant term of (E) vanishes. The first two powers of (E) near ρ = 0 are as +follows +B2π2λ2P4(β)ρ−2+3β + 143 +64 P2(β)ρβ, +(A15a) +where the polynomials P2 and P4 are defined as +P2(β) := +�19k3 +42 ++ 8 +21 +� +β2 + +�19k3 +42 ++ 8 +21 +� +β + k1 − 1, +(A15b) +P4(β) := β4 + 4β3 + 13 +3 β2 − 143 +48 β − 1287 +128 . +(A15c) +The first power in (A15a) is dominant over the second for ρ → 0 if 1 +4 < β < 1, and so β should +be chosen as a real root of P4 in that interval, which however fails to exist. On the other hand, if +β > 1, the second power in (A15a) becomes dominant and β should be chosen as a real root of P2 +21 + +in that range, which too fails to exist whenever k1 > 1. Thus, (A14) could be the asymptotic form +of αλ only if β = 1, which makes (E) take the asymptotic form +�1421 +192 π2λ2B2 − 2717 +672 k3 − 143 +32 k1 + 715 +672 +� +ρ + O +� +ρ3� += 0. +(A16) +Requiring the dominant power of (A16) to vanish determines B as in (37). +In a similar, but perhaps more customary way, by linearizing (E) about α = 0, we obtain that +α′(ρ)ρ + 2α(ρ) ≈ 0. +(A17) +By solving it subject to (31), we readily arrive at (38) in the main text. +Appendix B. Equivalent Dynamical System +In this Appendix we construct a dynamical analogy for the positive branch of equilibrium solutions +αλ for Fλ in (32) and give a phase space representation for them. +We reinterpret Fλ as the action of a dynamical system by introducing the effective time +t := − ln ρ. +(B1) +Thus, in particular, the center of the ball BR(x0) at ρ = 0 is approached in the new variable +when t → +∞, while the initial time t = 0 corresponds to the boundary ∂BR(x0) at ρ = 1. +Correspondingly, the twist angle α becomes a function on [0, ∞), which is defined by +a(t) := α +� +e−t� +(B2) +and by (31) satisfies +a(0) = 0. +(B3) +Fλ[α] thus acquires the form of an infinite-horizon action, +Aλ[a] := +� ∞ +0 +Lλ(a, ˙a, t) dt, +(B4) +where the Lagrangian Lλ is defined as +Lλ(a, ˙a, t) := e−t � +g(a)˙a2 + 2(k1 − 1)f0(a) +� ++ 32 +7 λ2et +� 4 +� +n=1 +fn(a)˙an +� +(B5) +and a superimposed dot denotes differentiation with respect to t. The orbits of the system are +22 + +solutions of the equation of motion for Lλ, +d +dt +∂Lλ +∂ ˙a − ∂Lλ +∂a = += e−t � +−γ′(a)˙a2 − 2g(a)¨a + 2g(a)˙a + 2(k1 − 1)f′ +0(a) +� +− 32 +7 λ2et +� 4 +� +n=2 +(n − 1)f′ +n(a)˙an + ¨a +� 4 +� +n=2 +n(n − 1)fn(a)˙an−2 +� ++ +4 +� +n=1 +nfn(a)˙an−1 +� += 0, +(B6) +where a prime ′ denotes differentiation. +We are interested in the orbits that start from the initial condition (B3)(and arbitrary ˙a(0)) +and whose action Aλ is bounded and a minimum. To this end, we first identify the critical points +of the dynamical system; these are obtained when both ˙a ≡ 0 and ¨a ≡ 0 in (B6), i.e., whenever +2(k1 − 1)f′ +0(a) − (λet)2f1(a) = 0. +(B7) +For λ > 0, they are +a = kπ +with +k ∈ Z. +(B8) +For λ = 0, which is the case studied in [33], they are instead +a = kπ +and +a = ± arccos(−1/4) + 2kπ +with +k ∈ Z. +(B9) +The trajectory a(t) ≡ 0 represents the radial hedgehog with action Aλ[0] = 0. On the other +hand, if for λ > 0 there is a trajectory aλ = aλ(t) such that limt→∞ aλ(t) = π and the action Aλ[aλ] +is finite; it remains to be seen whether Aλ[aλ] < 0, to decide whether the orbit aλ minimizes the +action. Multiplying both sides of (B6) by ˙a, we get +2g(a)˙ae−t−32 +7 λ2et +4 +� +n=1 +nfn(a)˙an = += et d +dt +� +g(a)˙a2 − 2(k1 − 1)f0(a) +� ++ 32 +7 λ2et d +dt +� 4 +� +n=2 +(n − 1)fn(a)˙an +� +, +(B10) +whose integration with respect to t ∈ [0, ∞) gives the following expression for the action of aλ: +Aλ[aλ] = −g(0)˙aλ(0)2 + 32 +7 λ2 +� +lim +t→∞ +� +et +4 +� +n=2 +(n − 1)fn(aλ)˙an +λ +� ++ 2 +� ∞ +0 +et +� 4 +� +n=1 +fn ˙an +λ +� +dt +� +, (B11) +under the assumption that the limit exists. +We are interested in bounded orbits aλ with bounded action Aλ[aλ]. We call these orbits admis- +sible. For a solution aλ of (B6) to be an admissible orbit, the initial value ˙aλ(0) must be chosen so +as to ensure convergence of the orbit aλ(t) to π as t → ∞. +The asymptotic behaviour in (38) here translates into +˙aλ(t) = aλ(t) + C +for +t ≈ 0, +(B12) +23 + +where C > 0 is a constant to be determined. One can show that, for λ > 0 and material constants +k3 and k1 chosen in the pink region of Fig. 3, there is a positive C for which an admissible orbit aλ +exists, but it has positive action Aλ. Thus, a twisted hedgehog nT exists, but it has more energy +than the radial hedgehog nR, which is locally stable. Hereafter we assume that (29) holds, so that +material constants are chosen in the blue region of Fig. 3. +In the phase plane (x, y), where +x(t) = a(t), +y(t) = ˙a(t), +(B13) +(B6) can be rewritten as +˙x = y, +(B14a) +˙y = += +� +−g′(x)y2 + 2g(x)y + 2(k1 − 1)f′ +0(x) − 32 +7 (λet)2 ��4 +n=2(n − 1)f′ +n(x)yn + �4 +n=1 nfn(x)yn−1�� +� +2g(x) + 32 +7 (λet)2 �4 +n=2 n(n − 1)fn(x)yn−2 +� +, +(B14b) +a system which we next study in some detail. +B.1. Asymptotically Autonomous Limit +The two-dimensional dynamical system described by (B14) is not autonomous23 and this makes it +more difficult to predict the qualitative properties of its orbits, as the standard phase plane portraits +(such as those discussed, for example, Chapt. 2 of [63]) do not apply here. However, system (B14) +has the interesting property of reducing to an autonomous system in the limit as t → ∞. For this +reason, it is called asymptotically autonomous.24 +For orbits that do not intersect either of the axes of the (x, y) phase plane, the autonomous +asymptotic limit of (B14) is +˙x = y, +(B15a) +˙y = +�4 +n=2(n − 1)f′ +n(x)yn + �4 +n=1 nfn(x)yn−1 +�4 +n=2 n(n − 1)fn(x)yn−2 +. +(B15b) +For any λ ≧ 0, its equilibrium points are +p1 = (arccos(1/8), 0) +and +p2 = (π, 0) +(B16) +and their periodic replica. The eigenvalues of the linear approximation of (B15) near these points +are +Λ± +1 = −1 +2 ± 5 +√ +119 +14 +i, +Λ± +2 = 59 +44 ± +√ +10015 +44 +, +(B17) +respectively. Thus p1 is a stable spiral node, while p2 is a saddle. A phase portrait for the asymptotic +limit (B15) is shown in Fig. B1 along with the equilibrium points in (B16).25 +23It is explicitly dependent on time. +24Asymptotically autonomous dynamical systems have an interesting literature, recalled for example in Chapt. 17 of [64]. +25See, for example, Sect. 2.2.2 of [63]. +24 + +Figure B1. +Phase portrait for the asymptotic autonomous limit (B15) around the equilibrium points in (B16), a stable spiral +node and a unstable saddle. Three exemplary orbits are drawn: the green orbit spirals about the node, the blue orbit approaches +the saddle along its stable invariant manifold (tangent to the broken straight line), and the black orbit, which is not connected +with any equilibrium point, is unbounded. +The correspondence between the solutions to an asymptotically autonomous system and those +to its autonomous asymptotic limit is a delicate one and has not been completely characterized, even +in the two-dimensional case, for which a larger number of results are available (see, for example, +[65–67]). In particular, a result of Markus [65] (see his Theorem 7) applies to our system: it says +that the ω-limit set26 of a solution to (B14) either contains the equilibria of (B15) or is the union +of periodic orbits of (B15). +Since (B15) has no periodic orbits, we conclude that the ω-limit set of any bounded solution +of (B14) must contain the equilibria of (B15). Among these latter, only p2 can be reached by an +admissible orbit (according to our definition); this justifies our numerical search for a trajectory +in phase space starting from a point (0, y0) of the y-axis and approaching in infinite time the +equilibrium point p2. The successful outcome of this search is shown in Fig. B2, where an admissible +orbit of (B14) for λ > 0 is contrasted against that obtained in [33] for λ = 0, which is when (B14) +becomes autonomous. The solutions illustrated in Fig. B2 are the same as those in Fig. 4 in the +main text; actually, the latter were generated from the former by inverting the change of variables +in (B1) and (B2). +References +[1] Lydon J. Chromonic liquid crystal phases. 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E 91, 050501(R) (2015)]. Phys Rev E. +2015;92:019905. +28 + diff --git a/bdE_T4oBgHgl3EQfzRz6/content/tmp_files/load_file.txt b/bdE_T4oBgHgl3EQfzRz6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35acea0e0b9c89658bd437f32d6cc1595c9728f9 --- /dev/null +++ b/bdE_T4oBgHgl3EQfzRz6/content/tmp_files/load_file.txt @@ -0,0 +1,956 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf,len=955 +page_content='Spiraling Defect Cores in Chromonic Hedgehogs Silvia Paparinia and Epifanio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Virgab abDipartimento di Matematica, Universit`a di Pavia, Via Ferrata 5, 27100 Pavia, Italy ARTICLE HISTORY Compiled January 23, 2023 Abstract An elastic quartic twist theory has recently been proposed for chromonic liquid crystals, intended to overcome the paradoxical conclusions encountered by the classical Oseen-Frank theory when applied to droplets submerged in an isotropic fluid environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' However, available experimental data for chromonics confined to cylindrical cavities with degenerate planar anchoring on their lateral boundary can be explained equally well by both competing theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This paper identifies a means to differentiate these theories both qualitatively and quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' They are shown to predict quite different core defects for the twisted hedgehogs that chromonics generate when confined to a fixed spherical cavity with homeotropic anchoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In the quartic twist theory, the defect core is estimated to be nearly one order of magnitude larger (tens of microns) than in the other and, correspondingly, the director field lines describe Archimedean spirals instead of logarithmic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' KEYWORDS Chromonic liquid crystals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Hedgehog defects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Core structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Elastic theories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Curvature elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Introduction The classical theory of nematic curvature elasticity is based on the assumption that in the ground state the director n has everywhere the same orientation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' the energy stored in a distortion then measures the work done to produce it starting from the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The classical Oseen-Frank theory posits a stored energy quadratic in ∇n and features four elastic constants, one for each elementary distortional mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Chromonic liquid crystals (CLCs) are lyotropic materials, which include Sunset Yellow (SSY), a popular dye in food industry, and disodium cromoglycate (DSCG), an anti-asthmatic drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In these materials, molecules stuck themselves in columns, which in aqueous solutions develop a nematic orientational order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In CLCs, n designates the average direction in space of the constituting supra- molecular aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A number of reviews have progressively become available in the last few years [1–5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' we refer the interested reader to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Experiments have been performed with these materials in capillary tubes, with either circular [6, 7] or rectangular [8] cross-sections, as well as on cylindrical shells [9], all enforcing degenerate planar anchoring, which allows constituting columns to glide freely on the anchoring surface, provided they remain tangent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' These experiments revealed a tendency of CLCs to acquire spontaneously a double twist configuration in cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Due to the lack of chirality in the molecular aggregates constituting CLCs, spontaneous double twists come equally likely in two variants with opposite chiralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' CONTACT Epifanio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Virga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Email: eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='virga@unipv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='it arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='08323v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='soft] 19 Jan 2023 Despite the lack of uniformity in the ground state of these phases,1 their curvature elasticity has been modeled by the Oseen-Frank theory, albeit with an anomalously small twist constant K22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' To accommodate the experimental findings and justify the (double) twisted ground state, this constant has to be smaller than the saddle-splay constant K24, in violation of one of the inequalities Ericksen [11] had put forward to guarantee that the Oseen-Frank stored energy be bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Actually, as shown in [12], the violation of one Ericksen’s inequality does not prevent the twisted ground state from being locally stable in a cylinder enforcing degenerate planar anchoring on its lateral boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The same conclusion was reached in [13] on different grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' But, as shown in [14], free-boundary problems may reveal noxious consequences of violating Ericksen’s inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' If K22 < K24, a CLC droplet, tactoidal2 in shape and surrounded by an isotropic fluid environment enforcing degenerate planar anchoring for the director at the interface, is predicted to be unstable against shape perturbations: it would split indefinitely in smaller tactoids while the total free energy plummets to negative infinity (see [14], for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This prediction is in sharp contrast with the wealth of experimental observations of CLC tac- toidal droplets, stable in the biphasic region of phase space, where nematic and isotropic phases coexist in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Experiments have been carried out with a number of substances (including DSCG and SSY) stabilized by the addition of neutral (achiral) condensing agents (such as PEG and Spm) [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' These studies have consistently reported stable twisted bipolar tactoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' To resolve this contradiction, in [20] we proposed a minimalist quartic theory for CLCs, which adds to the Oseen-Frank energy density a single quartic term in the (double) twist measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' hence the name quartic twist theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We showed in [20] that indeed within this theory the total free energy of chromonic droplets subject to degenerate planar interfacial anchoring remains bounded below, even if K22 < K24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' we also used published data to prove consistency with experiments and estimated a phenomenological length introduced by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Higher-order theories are not new in liquid crystal science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Go under this name either theories that allow for higher spatial gradients of n in the energy and theories that allow for higher powers in the first gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Under the first category, which perhaps has seen its first manifestation in [21] (see also [22]), falls, for example, Dozov’s theory [23] for both twist-bend and splay-bend phases predicted long ago by Meyer [24] and more recently observed in real materials [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Under the second category falls, for example, a simple one-dimensional model for splay-bend nematics [26], then extended to incorporate a whole class of seven modulated ground states, of which twist-bend and splay-bend are just two instances [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A hybrid theory was also proposed in [28], where both higher gradients of n and higher powers of the first gradient are allowed in the stored-energy density, with spatial derivatives and their powers balanced according to a criterion motivated by a molecular model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3 Our quartic theory is much simpler than these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In [20], we showed that both the classical Oseen-Frank theory and our quartic twist theory explain experimental data for the emergence of double twist in capillaries to a comparable degree of confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Here, in our quest for qualitative and quantitative features that may allow us to discriminate between these theories, we consider the case of the most common of point defects, the hedgehog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We imagine a CLC confined within a fixed spherical cavity enforcing homeotropic anchoring on its boundary, as in a recent experiment [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Since the seminal paper of Lavrentovich and Terentjev [32] we know that for K22 sufficiently small a radial hedgehog becomes twisted and exhibits field lines spiraling about the point defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A defect core can then be easily identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 1The classification of the most general uniform distortions, which can fill the whole three-dimensional space, is given in [10] and recalled in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2Tactoids are elongated, cylindrically symmetric shapes with pointed ends as poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3It should also be noted that other theories are known as “quartic” (see, for, example, the classical paper [29] and the more recent contribution [30]), but they owe this name to an elastic term globally quartic in de Gennes’ order tensor and its derivatives, added to the commonly considered version of the Landau de Gennes theory to resolve the spay-bend elastic constant degeneracy in the reduction to the Oseen-Frank theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' These theories serve a different purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2 geometrically, it is delimited by an inversion ring where spirals invert their winding sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This is the feature under close scrutiny here: We want to mark the differences between quadratic and quartic theories in describing the defect core of a twisted hedgehog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We are interested in qualitative and quantitative differences as well, aiming to outline a setting that could possibly discriminate one theory from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2, we describe the energetics of chromonics, starting from the classical quadratic Oseen-Frank theory and then summarizing our quartic twist theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3, we describe a class of director fields intended to represent twisted hedgehogs in a ball in terms of a single twist angle depending on the radial coordinate only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We review the conditions that ensure that the radial hedgehog is unstable (according to both elastic theories) and find the energy-minimizing twist angle for the quartic twist theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' for the quadratic theory, this problem was solved in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The results for the two theories are then compared and their stark differences emerge in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Finally, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5, we summarize the conclusions of our work and comment on possible avenues for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The paper is closed by two technical appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In one, we collect a number of mathematical details concerning our representation of twisted hedgehogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In the other, we describe an equivalent dynamical system, whose orbits correspond to twisted hedgehogs in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A similar corre- spondence was used in [33], with the further advantage (lost here) that the equivalent dynamical system was autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Quadratic and Quartic Theories As customary in liquid crystal science, an elastic theory for chromonics is based on a free-energy functional F that expresses the energy stored in a region in space B containing the material as F[n] := � B W(n, ∇n) dV, (1) where W is a function of the nematic director n and its gradient ∇n, which here plays the role of a local (tensorial) measure of distortion, and dV is the volume element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We start by summarizing the classical quadratic theory for the elasticity of nematic liquid crystals, albeit formulated in a novel, equivalent way that serves better our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It will then become easier to present the quartic twist theory proposed in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Classical Quadratic Energy The classical elastic theory of liquid crystals goes back to the pioneering works of Oseen [34] and Frank [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4 In this theory, the elastic free-energy density W in (1) is chosen to be the most general frame-indifferent,5 even function quadratic in ∇n, W = WOF(n, ∇n) := 1 2K11 (div n)2 + 1 2K22 (n · curl n)2 + 1 2K33|n × curl n|2 + K24 � tr(∇n)2 − (div n)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (2) 4Also a paper by Zocher [36], mainly concerned with the effect of a magnetic field on director distortions, is often mentioned among the founding contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Some authors go to the extent of also naming the theory after him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Others, in contrast, name the theory only after Frank, as they only deem his contribution to be fully aware of the nature of n as a mesoscopic descriptor of molecular order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5This requirement amounts to assume that W(Qn, Q(∇n)QT) = W(n, ∇n), for all rotations Q in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3 Here K11, K22, K33, and K24 are elastic constants characteristic of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' They are tradition- ally referred to as the splay, twist, bend, and saddle-splay constants, respectively, by the features of four different orientation fields, each with a distortion energy proportional to a single term in (2) (see, for example, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3 of [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Recently, Selinger [38] has reinterpreted the classical formula (2) by decomposing the saddle- splay mode into a set of other independent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The starting point of this decomposition is a novel representation of ∇n (see also [39]), ∇n = −b ⊗ n + 1 2TW(n) + 1 2SP(n) + D, (3) where b := −(∇n)n = n × curl n is the bend vector, T := n · curl n is the twist, S := div n is the splay, W(n) is the skew-symmetric tensor that has n as axial vector, P(n) := I − n ⊗ n is the projection onto the plane orthogonal to n, and D is a symmetric tensor such that Dn = 0 and tr D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' By its own definition, D ̸= 0 admits the following biaxial representation, D = q(n1 ⊗ n1 − n2 ⊗ n2), (4) where q > 0 and (n1, n2) is a pair of orthogonal unit vectors in the plane orthogonal to n, oriented so that n = n1 × n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='6 In the local frame (n1, n2, n), b is represented as b = b1n1 + b2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (5) By use of the following identity, 2q2 = tr(∇n)2 + 1 2T 2 − 1 2S2, (6) we can easily give (2) the equivalent form WOF(n, ∇n) = 1 2(K11 − K24)S2 + 1 2(K22 − K24)T 2 + 1 2K33B2 + 2K24q2, (7) where B2 := b · b = b2 1 + b2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Since (S, T, b1, b2, q) are all independent distortion characteristics, it readily follows from (7) that WOF is positive semi-definite whenever K11 ≧ K24 ≧ 0, (8a) K22 ≧ K24 ≧ 0, (8b) K33 ≧ 0, (8c) which are the celebrated Ericksen’s inequalities [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' If these inequalities are satisfied in strict form, the global ground state of WOF is attained on the uniform director field, characterized by S = T = B = q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (9) As already mentioned in the Introduction, inequality (8b) must be violated for the ground state of WOF to be different from (9), involving a non-vanishing T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 6It is argued in [40] that q should be given the name tetrahedral splay, to which we would actually prefer octupolar splay for the role played by a cubic (octupolar) potential on the unit sphere [41] in representing all scalar measures of distortion, but T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4 The class of uniform distortions was defined in [10] as the one comprising all director fields for which the distortion characteristics are constant in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Equivalently said, a uniform distortion is a director field that can fill three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It was proven that there are two distinct families of uniform distortions, characterized by the following conditions [10], S = 0, T = ±2q, b1 = ±b2 = b, (10) where q and b are arbitrary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The general director field corresponding to (10) is the heliconical ground state of twist-bend nematic phases,7 in which n makes a fixed cone angle with a given axis in space (called the helix axis), around which n precesses periodically [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='8 The special instance in which b = 0 corresponds to the single twist that characterizes cholesteric liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The distortion for which all characteristics vanish, but T, is a double twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='9 It is not uniform and cannot fill space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' it can possibly be realized locally, but not everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In words, we say that it is a frustrated ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' As shown in [12], a double twist is indeed attained exactly only on the symmetry axis of cylinders enforcing degenerate planar anchoring on their lateral boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Quartic Twist Energy The essential feature of the quartic twist theory proposed in [20] is to envision a double twist with two equivalent chiral variants as ground state of CLCs in three-dimensional space, S = 0, T = ±T0, B = 0, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (11) The degeneracy of the ground double twist in (11) arises from the achiral nature of the molecu- lar aggregates that constitute these materials, which is reflected in the lack of chirality of their condensed phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The elastic stored energy must equally penalize both ground chiral variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Our minimalist proposal to achieve this goal was to add a quartic twist term to the Oseen-Frank stored-energy density, and so take W = WQT, with WQT(n, ∇n) := 1 2(K11 − K24)S2 + 1 2(K22 − K24)T 2 + 1 2K23B2 + 1 2K24(2q)2 + 1 4K22a2T 4, (12) where a is a characteristic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Unlike WOF, WQT is bounded below whenever K11 ≧ K24 ≧ 0, (13a) K24 ≧ K22 ≧ 0, (13b) K33 ≧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (13c) If these inequalities hold, as we shall assume here, then WQT is minimum at the degenerate double- twist (11) characterized by T0 := 1 a � K24 − K22 K22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (14) 7With opposite chiralities, one for each sign in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 8In opposite senses, according to the sign of chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 9Here we adopt the terminology proposed by Selinger [40] (see also [42]) and distinguish between single and double twists, the former being uniform and the latter not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5 The parameter a encodes the bare length scale over which distortions would be locally stored in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='10 As to the physical size of such a length scale, it may be comprised in a wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' While at the lower end we may place the persistence length of the molecular order, which characterizes the flexibility of CLC aggregates,11 the upper end is hard to make definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We expect that a would be exposed to the same indeterminacy that affects many (if not all) supramolecular structures in lyotropic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The most telling example is perhaps given by cholestric liquid crystals, which give rise to a chiral structure (characterized by a single twist T = ±2q) starting from chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' If the macroscopic pitch were determined by the molecular chirality,12 it would result several orders of magnitude smaller than the observed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='13 Here, we shall treat a as a phenomenological parameter, to be determined experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' An estimate derived in [20] from a comparison with published data placed a in the order of microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Twisted Hedgehog So far we have presented, mostly on equal terms, two elastic theories for chromonics, one quadratic and the other quartic in the director gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Here we see how these theories can be differentiated on the basis of the different structures they predict for the core of hedgehogs, the most common of nematic defects in three space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Many mathematical details needed to follow our development are collected in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We first discuss the distortion of a trial director field within a ball of radius R enforcing homeotropic anchoring on its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This is a field with a point defect at the center of the ball, potentially rich in twist, as would seem fit for a material with small K22 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We shall then see the analytical implications and the potential experimental significance of this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The point defects that we shall study are a special family of hedgehogs, which place themselves in between the most common defects in liquid crystal science, the radial and the hyperbolic hedgehogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The former is represented by the director field nR(x) := x − x0 |x − x0|, (15) which has a point defect at x0, while the latter is formally obtained by the following transformation of nR, nH := R(π)nR, (16) where R(π) := −I + 2e ⊗ e (17) is the special orthogonal tensor describing a rotation by angle π about a unit vector e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Figure 1 illustrates the field lines of both nR and nH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 10In the elastic model proposed in [10] for twist-bend nematics, a quartic free energy was posited that admits as ground state either of two families of uniform heliconical fields with opposite chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' There too, a length scale appears in the equilibrium pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The distortion state characterized by this length is the same everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 11The persistence length of a flexible aggregate is the shortest length over which unit vectors tangent to the aggregate’s contour lose correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For CLCs, it is estimated on the order of tens to hundreds of nm [43] 12Via the naive geometric argument that represents chiral molecules as cylindrical screws and derives the pitch of their assemblies by close packing them so as to fit grooves with grooves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 13For lyotropic cholesterics, the mismatch between microscopic and macroscopic pitches, which has recently received new experimental evidence in systems of basic living constituents [44,45], is still debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Interesting theories based on either molecular shape fluctuations [46,47] or surface charge patterns [48] have met with some experimental disagreement [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 6 (a) Radial hedgehog: N(nR) = +1 (b) Hyperbolic hedgehog: N(nH) = +1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Field lines of nR and nH in (15) and (16), representing a radial and a hyperbolic hedgehog, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Field lines are drawn on the equatorial plane (in black) and on a meridian plane (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The whole picture is obtained by rotating these lines around the polar axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' By their definitions, both nR and nH share the same topological charge N as introduced in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The topological charge of a unit vector field n with a point defect at x0 is defined as N(n) := 1 4π � S n · (∇sn)∗ν dA, (18) where S is a any surface enclosing x0, ∇s denotes the surface gradient on S , ν is the unit normal to S , the operation (· · · )∗ takes the cofactor of a tensor,14 and dA is the area element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' N(n) is an integer of Z independent of S , provided the latter embraces x0, and so N(n) can be attributed to x0 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The absolute value |N(n)| indicates the number of times n restricted to S covers the unit sphere S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' the sign of N(n) tells whether S2 is covered coherently or not with the orientation of the unit normal ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Historically, we learn from [51] that the representation in (18) for N(n) was first derived in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' N(n) is additive: if the surface S encloses more than one defect, the topological charge com- puted on it through (18) is the algebraic sum of the topological charges computed on surfaces enclosing the single defects comprised in S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' As pointed out in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3 of [53], a defect with topological charge N(n) can be transformed continuously into a defect with opposite topological charge, thus making |N(n)|, and not N(n) itself, a topological invariant apt to classify point defects for director fields on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The mapping nR �→ nR := −nH (19) was described in [54] as a parity transformation, as it changes the sign of the topological charge,15 N(nR) = −N(nR) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (20) This was meant to identify nR as an anti radial hedgehog, which would neutralize the topological 14This is a tensor whose representative matrix is the cofactor matrix of the matrix representing the original tensor, see [50, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 22] for a formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 15This equation follows from Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2 and the general property of (18) stating that N(−n) = −N(n), which stems from being (∇sn)∗ even in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 7 charge of the radial hedgehog and annihilate it when combined together in a director field on S2 with zero total topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='16 Here, instead, as shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2, N(nH) = N(nR) = +1, (21) so that nR and nH not only belong to the same topological class, but also have the same topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We consider as domain B a ball BR(x0) with radius R and center at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We study a trial twisted hedgehog field nT, which “interpolates” in space between nR and nH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Formally, nT is obtained by acting on the radial hedgehog nR(x) in (15) with a rotation R(α) of variable angle α = α(r) about a fixed axis e ∈ S2, where r is the distance of x from the defect at x0, nT(x) := R(α(r))nR(x), (22a) R(α) := I + sin αW(e) + (1 − cos α)W(e)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (22b) In (22b), W(e) is the skew-symmetric tensor associated with e, whose action on any vector v is given by W(e)v = e × v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The field nT reduces to the radial hedgehog nR for α ≡ 0 and to the hyperbolic hedgehog nH for α ≡ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' As shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2, the topological charge of nT equals that of both nR and nH, irrespective of the function α, N(nT) = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (23) We shall call α the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Inversion Ring A peculiar property of the field nT is illustrated by letting e be the polar axis of a system of spherical coordinates (r, ϑ, ϕ) with origin at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' On the equatorial plane ϑ = π 2 , in the coordinate frame (er, eϑ, eϕ), nT reduces to (see (A9)) nT = cos αer + sin αeϕ, (24) and so it lies entirely on the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' If, for some r∗, α(r∗) = π 2 , then nT is tangent to the circle of radius r∗ around x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Where α(r) < π 2 the field nT spirals outward (relative to x0), where α(r) > π 2 it spiral inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Figure 2 illustrates this feature within the ball BR(x0) when the condition α(R) = 0 (25) is enforced, so that nT = nR on the boundary ∂BR(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The ring at r = r∗ separates two opposite spiraling regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' there, the field lines of nT appear to coalesce in a ring, which looks like a disclination, but is instead regular, as it bears no discontinuity of the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We shall call the ring at r = r∗, if present, an inversion ring, as it marks the inversion of the spiraling sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It is perhaps the seminal work of Lavrentovich and Terentjev [32] where a first experimental evidence of an inversion ring within a twisted hedgehog was ever found and documented in ordinary 16Actually, in [54], nH was defined to be precisely nR, so that, being opposite to the field in (16), would form with it a defect-anti-defect pair, as would also be clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 1 once the field lines orientation in panel (b) are reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Field lines of nT in (22) within the ball BR(x0) enforcing condition (25), so that nT = nR on ∂BR(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' An inversion ring is present, which is depicted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Black lines are field lines lying on the equatorial plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' red lines are field lines coming out of the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 1, the whole 3D picture is obtained ny rotating this drawing about the polar axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='17 Here, we shall use nT as a trial field to describe the twisted distortion that replaces nR in BR(x0) when nR becomes unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We shall determine the function α subject to (25) that minimizes the elastic free energy F in (1) with B = BR(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We shall do so for either W = WOF in (7) and W = WQT in (12) to see whether the quadratic and quartic elastic theories for chromonics recalled in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2 can be distinguished on the basis of the predictions they make about the occurrence of a twisted hedgehog and its inversion ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We start by considering under what conditions nR is locally stable for either theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Local Stability of Radial Hedgehog First, we observe that nR is a universal solution, as it solves the equilibrium equation for all possible elastic free-energy functionals F in (1) associated with a frame-indifferent density W = W(n, ∇n), see [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Thus, nR is an equilibrium configuration for F, irrespective of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Moreover, it was proved in [56] and [57] that, when W = WOF, nR is a local minimizer of F in the admissible class of director fields n with finite energy in BR(x0) and such that n|∂BR(x0) = nR, (26) provided that the following inequality is satisfied,18 0 < k1 < 1 + k3 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (27) 17In a temperature regime where the twist constant K22 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 18A result which was independently rediscovered in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 9 Here and below, the elastic constants will be scaled to K22, k1 := K11 K22 , k3 := K33 K22 , and k24 := K24 K22 with K22 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (28) This local stability result is based on the study of the second variation of F at n = nR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' the latter is the same for both WOF and WQT, as these only differ by a quartic term that does not affect the second variation of F at nR, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It is remarked in [59] that when (27) is violated the free-energy functional F with W = WOF subject to (26) admits a continuum of minimizers, all sharing the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Since the proof of this result is based on frame-indifference only, it also holds within our quartic twist theory where W = WQT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Figure 3 illustrates inequality (27) for k1 > 1, which is the situation that applies to chromonics, as also shown by the dot representing data for SSY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Hereafter, we assume that Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Regions of interest for the local stability of the radial hedgehog nR in CLCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In the pink region, nR is a local minimizer of F subject to (26) when B = BR(x0), for either W = WOF and W = WQT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In the blue region, nR is no longer a minimizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' there is a continuum of minimizers, all with the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Bulk elastic constants of chromonics fall in the region of instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' the red dot represents data for SSY, k1 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1 and k3 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='7, taken from [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' k1 > 1 + k3 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (29) The special family of twisted director configurations described by nT are parameterized by the scalar function α and the symmetry axis e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Once, for a given e, α is chosen so as to minimize F, letting e vary in S2 potentially embodies the continuum of minimizers expected to arise when the radial hedgehog nR is no longer locally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Minimum Problem Here, we study the problem of minimizing the functional F in (1) for B = BR(x0) and W = WQT subject to (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We introduce the change of variables r �→ ρ := r R, (30) 10 SSY not a minimizer ki = 1＋k3/8 local minimizerwhich maps [0, R] onto [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In the new variable, (25) becomes19 α(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (31) Standard computations (deferred to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1) show that for B = BR(x0) and W = WQT the functional F in (1) can be given the following scaled form Fλ[α] := 15F[nT] 8πK22R − FR = � 1 0 � g(α(ρ))(ρα′(ρ))2 + 2(k1 − 1)f0(α(ρ)) + 32 7 λ2 ρ2 � 4 � n=1 fn(α(ρ))(−ρα′(ρ))n �� dρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (32) where a prime ′ denotes differentiation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' FR := 15(k1 − k24) is the scaled energy of the radial hedgehog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' so that Fλ[0] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (33) and the functions g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' and fn are defined as g(α) = 2k1 sin2 α + 2 7(1 − cos α)2 + k3 14 � 24 cos2 α + 8 cos α + 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (34a) f0(α) = 2 cos2 α + cos α − 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (34b) f1(α) = 3(1 − 8 cos α) sin3 α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (34c) f2(α) = (1 − cos α)2 sin2 α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (34d) f3(α) = 2 11(1 − cos α)3 sin α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (34e) f4(α) = 2 143(1 − cos α)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (34f) Fλ[α] is invariant under the change of α into −α, for any α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Thus, every non-trivial equilibrium solution αλ would be accompained by its parity conjugate −αλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The corresponding fields nT differ as they have opposite chirality, but they have one and the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For λ > 0, integrability of the quartic term in Fλ in (32) requires that the limiting value α(0) of α at ρ = 0 be either 0 or π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Under the assumption that, to within parity conjugacy, Fλ has a unique minimizer subject to (31), the choice α(0) = 0 would lead us to α ≡ 0, that is, to nR, which is a contradiction since the radial hedgehog is unstable when (27) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Thus, we shall enforce the condition α(0) = π for λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (35) For λ = 0, α(0) is instead free to vary, as in (32) integrability is guaranteed by the integrability of α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Equilibrium Solutions Here, we specialize the analysis to positive solutions of the equilibrium equation for Fλ: we assume that αλ ≧ 0 since the minimizer of Fλ is not expected to change sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Clearly, this positive branch 19We shall continue to adopt the same old symbol for the function α, even if it is expressed in the new variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 20Which requires that ρα′(ρ) be bounded as ρ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 11 of solutions remains associated with the conjugate negative branch, which has equal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The equilibrium equation is too complicated to lend itself to analytic solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' it was symbolically manipulated and will be conventionally called (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='21 We could establish the asymptotic behaviour of the solutions αλ of (E) near ρ = 0 and ρ = 1, for every λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' As shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4, for k1 > 1 αλ(ρ) = π(1 − Bρ) + O � ρ2� for ρ → 0+, (36) where B = � 7 32 1 λ √ 58058√21k1 + 19k3 − 5 1421π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (37) Similarly, αλ(ρ) ≈ C �1 ρ − 1 � as ρ → 1, (38) where C is a positive constant to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Energy Minimizers Here we explore numerically the minimizers αλ of Fλ, focusing on the positive equilibrium branch (thus selecting one chirality for nT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For λ = 0, this problem is solved in [33] by reinterpreting F0 as an infinite-horizon action functional associated with an equivalent autonomous dynamical system in two-dimensional phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For λ > 0, a similar reinterpretation for Fλ is still viable, but the associated dynamical system is not autonomous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' it is studied numerically in Appendix B and contrasted with the autonomous system associated with F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The major difference between these dynamical systems resides in their equilibrium points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' in the language of the twist angle α, this translates into two different asymptotic values at the center of the ball BR(x0), αλ(0) = � �α0 := arccos(−1/4) for λ = 0, π for λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (39) One may say that the classical quadratic theory (λ = 0) predicts that nR and nH are not completely bridged inside the confining ball BR(x0), whereas the quartic theory (λ > 0) predicts that they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Hence we could possibly use hedgehogs in chromonics confined within a ball to discriminate these theories from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' However, although this is a qualitative difference, its observation might be experimentally precluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A further, quantitative feature must be called upon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' this is the size (relative to ball’s radius R) of the inversion ring r∗ associated with the stable twisted hedgehogs predicted by both theories, as by (39) an inversion ring is present in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For definiteness, we consider a specific case, which was suggested by the experimental study in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This is the case of chromonic liquid crystal SSY in an aqueous solution (at a wt/wt concen- tration of 30% and a temperature of 25 ◦C) confined within a spherical cavity produced inside a polymeric matrix enforcing homeotropic anchoring for the director on its boundary (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Material constants are derived from [60] and deliver k1 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1 and k3 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='7,22 which, as shown in 21It is equivalent to the equation of motion (B6) for the effective dynamical system described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 22The absolute measured values are K11 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3 pN, K22 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='7 pN, and K33 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1 pN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3, locate the radial hedgehog in its unstable domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The radius of the spherical cavity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5 is R ≈ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For the same SSY solution in the same physical conditions, in [20] we estimated a ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4µm, thus here we take λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The profile of the minimizing twist angle αλ corresponding to these parameters is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4 (red curve) against the minimizing profile α0 for λ = 0 (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It is apparent that the Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Plots against ρ of the minimizer αλ of Fλ (red curve) and the minimizer α0 of F0, for k1 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1, k3 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='7, and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' As in (39), �α0 := arccos(−1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The broken lines reproduce the asymptotic behaviours predicted by (36) and (38), respectively, with B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='68, in agreement with (37), and C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The dotted line drawn at α = π 2 intercepts the graphs of αλ and α0 at values ρ∗ of ρ that designate the scaled radius r∗ of the inversion ring in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It is apparent how ρ∗ λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2 is appreciably larger than ρ∗ 0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='03, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' inversion rings associated with these solutions are appreciably different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Figure 5 reproduces a spherical cavity (in a polymeric matrix) observed in [31];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' there we also draw the inversion rings predicted by both classical and quadratic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Judging from this single comparison and taking for granted that the defect shown here is indeed a twisted hedgehog, we may say that the quartic theory seems to capture better the size of the inner structure enclosed by the inversion ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This core structure will be further detailed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Letting ρ∗ := r∗/R designate the scaled radius of the inversion ring, we explored the dependence of ρ∗ on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 6 summarizes the outcomes of this analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' it shows how ρ∗ saturates to ρ∗ ∞ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='82 as λ grows indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Not only does the inversion ring size increase monotonically with λ, but also the defect core inside the inversion ring is qualitatively different for λ = 0 and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' These differences will be highlighted in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 13 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Reproduction of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5b of [31] showing a spherical cavity (in a polymeric matrix) of radius R ≈ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4µm enclosing a SSY solution in water with concentration 30% (wy/wt) and temperature 25 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The homeotropic anchoring on the boundary of the sphere induces a (presumably twisted) hedgehog at the center exhibiting the typical Maltese cross when observed between crossed polarizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The larger (green) and smaller (blue) circles superimposed to the figure are the inversion rings predicted by the quartic and quadratic theories, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In absolute terms, with a ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4µm (from [20]), that is, λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='16, we have r∗ 0 = ρ∗ 0R ≈ 1µm and r∗ λ = ρ∗ λR ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Spiraling Cores Here we go into deeper details of the twisted hedgehog nT that minimizes the elastic free energy Fλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' we are especially interested in the behaviour if its field lines within the defect core, which is conveniently identified with a sphere of radius r∗, the radius of the inversion ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We shall again study primarily the distortion afforded by the quartic theory with λ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' this case will also be contrasted against the case λ = 0 of the classical quadratic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We shall see that the differences between the two cases are both qualitative and quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We split our analysis in two steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' in the first, we study the field lines of nT on the equatorial plane of BR(x0) (orthogonal to the symmetry axis);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' in the second, we see how these lines behave away from that plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Equatorial Field Lines In a spherical coordinate system (r, ϑ, ϕ) with polar angle ϑ ∈ [0, π], the equatorial plane is described by ϑ = π 2 and r ≧ 0, ϕ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Scaling lengths to the radius R of the spherical cavity and letting ρ be still defined as in (30) above, we see from (24) that the field lines of nT on the equatorial plane are the solutions (ϕ(τ), ρ(τ)) to the differential system dϕ dτ = 1, (40a) dρ dτ = ρ(τ) tan(αλ(ρ(τ))), (40b) subject to ϕ(0) = 0 and ρ(0) = ρ0 with 0 < ρ0 < 1, (40c) 14 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Plot of ρ∗ = r∗/R as a function of λ computed on the minimizer αλ of Fλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' the graph saturates at ρ∗ ∞ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='82, while ρ∗ 0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='03 is the limiting value as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The red dot marks the inversion ring predicted for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='16, corresponding to the spherical cavity shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' where τ is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The curves solving (40) may be winding several times around the origin as τ → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' the appropriate solution of (40a) is then ϕ = τ mod 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (41) It follows from (40) that every field line that starts inside or outside the inversion ring, remains inside or outside that ring, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The inversion ring at ρ = ρ∗ is a field line itself, since ρ ≡ ρ∗ is a solution of (40b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Moreover, a field line that starts from ρ0 < ρ∗ keeps spiraling (clockwise) around the point defect at the origin, while a field line that starts from ρ0 > ρ∗ is soon bent (anti-clockwise) towards the equator of BR(x0), where it points radially away from the defect (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Another qualitative feature of the field lines of nT is revealed by (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Given the monotonicity of ρ(τ) both inside and outside the inversion ring, this function is invertible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' a straightforward integration yields the following formula for its inverse, τ(ρ) = �� ρ ρ0 tan αλ(ξ) ξ dξ for ρ > ρ∗, � ρ0 ρ tan αλ(ξ) ξ dξ for ρ < ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (42) Two noteworthy consequences follow from (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' First, from the divergence of both integrals as ρ → ρ∗ (from above and from below, respectively), we see that the field lines of nT wind infinitely many times around the inversion ring, no matter which elastic theory is employed to describe a twisted hedgehog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Second, by taking the limit as ρ → 0+ in the second integral, we see that this diverges or not, depending on the limiting value αλ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Since the latter depends on being λ = 0 or λ > 0, the two theories being compared here afford different qualitative predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' According 15 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='6- 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2- po 0 1 0 2 4 6 8 10(a) Quadratic theory (with λ = 0): The inversion ring has (scaled) radius ρ∗ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Zooming inside the inversion ring reveals the logarithmic nature of the asymptotic spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (b) Quartic theory (with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='16): The inversion ring has (scaled) radius ρ∗ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Zooming inside the inversion ring reveals the Archimedean nature of the asymptotic spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Field lines of nT in the equatorial plane of BR(x0) according to the two elastic theories considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Material constants correspond to SSY in the same conditions that apply to both Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' to the quadratic theory, for which α0 = arccos(−1/4), the second integral in (42) diverges and the field lines of nT wind infinite many times around the point defect at the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' asymptotically, they are logarithmic spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' On the contrary, according to the quartic theory, for which αλ = π for all λ > 0, by (36), the second integral in (42) converges and the field lines of nT wind a finite number of times around the defect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' asymptotically, they are Archimedean spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 7, the field lines of nT in the equatorial plane are contrasted for the two theories, when the twist angle is given by the functions α0 and αλ whose graphs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In both cases the inversion ring is zoomed in to highlight the different nature of the asymptotic spirals around the point defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Field Lines in Space As is easily seen from (A9), the field lines of nT away from the equatorial plane of BR(x0) are described in spherical coordinates (r, ϑ, ϕ) by the solutions to the following differential system dρ dτ = ρ(τ)1 + (cos αλ(ρ(τ)) − 1) sin2 ϑ sin αλ(ρ(τ)) , (43a) dϑ dτ = (cos αλ(ρ(τ)) − 1) cos ϑ sin ϑ sin αλ(ρ(τ)) , (43b) dϕ dτ = 1, (43c) where τ is a parameter chosen again so that (41) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 16 The flow described by (43) is mirror-symmetric with respect to the equatorial plane (ϑ = π 2 ) and, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 8, possesses two families of negatively invariant sets, balls and circular cylinders Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Field lines of nT away from the equatorial plane of BR(x0), for the same choice of parameters in both Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Only the two limiting negatively invariant sets, the ball Br∗ and the cylinder Cr∗ built on the inversion ring, are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Field lines are back inside Cr∗ and red outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The zoomed region on the right is the ball of radius r∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' two field lines are drawn that start near the boundary of Br∗, one inside Cr∗ (black) and the other outside (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' with radii larger than the radius r∗ of the inversion ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This means that field lines of nT may only leave the regions enclosed by these sets and never enter them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' To prove this qualitative property, we denote by Br and Cr these families of balls and cylinders, respectively, and by ν their outer unit normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It readily follows from (A9) that nT · ν|∂Br = sin2 ϑ cos α + cos2 ϑ, (44a) nT · ν|∂Cr = sin ϑ cos α, (44b) which are both non-negative for all ϑ ∈ [0, π] whenever α ≦ π 2 , that is, for r > r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A further geometric illustration of this property is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Conclusion In [20], we proposed a quartic twist theory for the curvature elasticity of chromonic liquid crystals, for which we have been seeking corroborating evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This theory introduces a phenomenological length a, which in [20] was estimated to be of the order of microns by fitting published data for chromonics filling cylinders with degenerate planar anchoring on their lateral boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' These data could also be interpreted by use of the classical quadratic Oseen-Frank theory [6,7], which however would be unable to predict stable shapes for the tactoidal droplets observed in the biphasic region of these materials [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 17 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For the same field nT in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 8, the director profiles are shown on two parallel sections of BR(x0) with planes parallel to the equator: one cuts the ball Br∗ at mid-height, z = r∗/2, while the latter cuts the ball BR(x0) at mid-height, z = R/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We turned to hedgehog defects and their core structure to find an instance where the two theories would afford different predictions, which could serve to differentiate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We considered a spherical cavity of radius R enforcing homeotropic anchoring on its boundary, like those produced in [31], and studied the twisted hedgehogs predicted by both theories in the region in parameter space where the radial hedgehog would be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The defect core of a twisted hedgehog director field nT is characterized by an inversion ring that encloses the defect core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Two properties of the defect core are predicted in stark contrast by the two theories: one is qualitative, the other quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We start with the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The radius r∗ of the inversion ring depends only on the elastic anisotropy for the quadratic theory and also on the ratio λ = a/R for the quartic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For SSY in the same physical conditions as in [31], taking a from [20], we estimated r∗ to be nearly an order of magnitude larger for the quartic theory compared to the quadratic one, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1 µm against 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' On the qualitative side, we showed that the field lines of nT spiral differently around the point defect according to which theory is adopted: in the quadratic theory, they are logarithmic spirals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' in the quartic theory, they are instead Archimedean spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We may perhaps say that the defect core of twisted defects, with its distinctive quantitative and qualitative features, could be the hallmark of a quartic elastic theory for chromonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' However, such a clear distinction between quadratic and quartic theories rests on being a in the order of microns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' were it much smaller, the differences highlighted here could not be appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A thorough study with direct observations of the core structure of twisted hedgehogs would be desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Another critical issue that deserves further research concerns the splay constant K11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' If the re- cent theoretical estimate for the elastic constants in [61] is to be confirmed by different, independent approaches, not only K22, but also K11 would be smaller than K24 for chromonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' This, as shown in [14], would ignite the instability of chromonic droplets in an isotropic fluid environment enforcing 18 homeotropic anchoring at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The defects studied in this paper inhabit a spherical cavity of fixed shape, and so they are saved from that instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' However, should homeotropic anchor- ing be realistic for chromonic droplets, if K11 < K24, our quartic twist theory could not prevent such a shape instability, as it would be driven by a concentration of splay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Thus, were homeotropic chromonic droplets actually observed, our elastic theory would need to be amended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Trial Twisted Hedgehog This Appendix contains ancillary results instrumental to our analysis in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Useful Computations Identifying the the unit vector e designating in (22b) the symmetry axis of nT as the polar axis ez of standard spherical coordinates (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' ϑ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' ϕ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' where ϑ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' π] is the polar angle and ϕ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2π) is the azimuthal angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' we represent the gradient of the trial twisted field through the formula ∇nT =1 r � Pr + sin αWz + (1 − cos α)W2 z � + � α′ cos α − 1 r sin α � Wzer ⊗ er + � α′ sin α − 1 r(1 − cos α) � W2 zer ⊗ er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A1) where Pr := I−er ⊗er is the projection onto the plane orthogonal to er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Wz is the skew-symmetric tensor with axial vector ez,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' and a prime ′ denotes differentiation with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The following expressions for the traditional measures of distortion of nT in (22a) are conse- quences of (A1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' they are written in the local frame (er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' eϑ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' eϕ) of spherical coordinates: div nT = 1 r � −(rα′) sin α sin2 ϑ + 1 − (1 − cos α) cos2 ϑ + cos α � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A2a) curl nT = 1 r � 2 sin α cos ϑer − sin ϑ � (rα′) cos α + sin α � eϑ + cos ϑ sin ϑ � −(rα′) sin α + (1 − cos α) � eϕ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A2b) nT · curl nT = 1 r � cos ϑ � −(rα′)(1 − cos α) sin2 ϑ + 2 sin α �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A2c) nT × curl nT = 1 r{sin2 ϑ[(rα′) sin α(1 − (1 − cos α) sin2 ϑ) − (1 − cos α)2 cos2 ϑ + sin2 α]er + sin ϑ[−(rα′) cos α(1 − (1 − cos α) sin2 ϑ) + sin α(−1 + (1 − cos α)(1 + cos2 ϑ))]eϑ} + sin ϑ cos ϑ[(rα′) sin α(1 − (1 − cos α) sin2 ϑ) − (1 − cos α)(1 − (1 − cos α) sin2 ϑ) + 2 sin2 α]eϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A2d) tr(∇nT)2 − (div nT)2 = −2(cos2 ϑ − (rα′) sin α sin2 ϑ + cos α sin2 ϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A2e) Making use of (A2) and (31) in the free energy density WQT in (12), and integrating over B = BR(x0), we arrive at the following scaled form for F in (1), 15F[nT] 8πK22R =: Fλ[α] + FR, (A3) 19 where Fλ[α] and FR are given by (32) and FR = 15(k1 − k24), (A4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Topological Charge Here we compute the topological charge N(nT) of the twisted hedgehog nT in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' To this end, we first note that ∇snT = (∇n)Pr = 1 r[Pr + sin αWzPr + (1 − cos α)W2 zPr], (A5) where use has been made of (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In the frame (er, eϑ, eϕ), ∇snT in (A5) is also represented as ∇snT = 1 r{(sin2 ϑ + cos2 ϑ cos α)eϑ ⊗ eϑ − cos ϑ sin αeϑ ⊗ eϕ + cos ϑ sin αeϕ ⊗ eϑ + cos αeϕ ⊗ eϕ − sin ϑ cos ϑ(1 − cos α)er ⊗ eϑ − sin ϑ sin αer ⊗ eϕ}, (A6) where we have employed the identity eϑ = 1 sin ϑ(cos ϑer − ez), (A7) having identified e and ez, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' It is now a simple matter to compute in the basis (er, eϑ, eϕ) the tensor (∇snT)∗, as it is represented by the cofactor matrix of the matrix representing ∇snT in (A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A tedious, but simple calculation delivers (∇snT)∗ = 1 r2 {sin ϑ cos ϑ(cos α−1)eϑ⊗er +sin ϑ sin αeϕ⊗er +(sin2 ϑ cos α+cos2 ϑ)er ⊗er}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A8) Since it follows from (22) and (A7) that nT = cos ϑ sin ϑ(cos α − 1)eϑ + sin ϑ sin αeϕ + (sin2 ϑ cos α + cos2 ϑ)er, (A9) it is easily concluded that nT · (∇snT)∗er = 1 r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A10) Taking S in (18) to be a sphere of radius r and center at x0, we readily obtain that N(nT) = +1, for any function α, which is precisely (23) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In particular, by taking α ≡ 0 or α ≡ π, we recover (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Second Variation We let F4 denote the quartic term contribution to F in (1) arising from WQT in (12), F4[n] := 1 4K22a2 � B (n · curl n)4 dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A11) By applying to F4 the method illustrated in [12], we readily see that the second variation δ2F4 of F4 at n can be given the general form δ2F4(n)[v] = K22a2 � B � (n · curl n)2� 3(v · curl n + n · curl v)2 + 2(n · curl n)(v2n · curl n + v · curl v) �� dV, (A12) which is a quadratic functional in the perturbation field v subject to the orthogonality condition n · v ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A13) It is a very simple matter to check that δ2F4(nR) ≡ 0, as curl nR ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Asymptotic Behaviours Here we give a few details about the derivation of the asymptotic behaviours in (36) and (38) of the equilibrium solutions αλ for Fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We renounce writing the equilibrium equation of Fλ since it is too complicated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' as in the main text, it will denoted by (E) and manipulated by symbolic calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' An equivalent form of (E) will be encountered in Appendix B below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' When ρ is near 0, we write αλ as αλ(ρ) ≈ π � 1 − Bρβ� , (A14) which satisfies (35), and seek B > 0 and β > 1/4, assumed to exist, the latter requirement being a direct consequence of the integrability of Fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In our asymptotic method, which is an adaptation of the classical method of Frobenious (see, for example, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 396 of [62]), we determine both β and B by requiring that the dominant term of (E) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The first two powers of (E) near ρ = 0 are as follows B2π2λ2P4(β)ρ−2+3β + 143 64 P2(β)ρβ, (A15a) where the polynomials P2 and P4 are defined as P2(β) := �19k3 42 + 8 21 � β2 + �19k3 42 + 8 21 � β + k1 − 1, (A15b) P4(β) := β4 + 4β3 + 13 3 β2 − 143 48 β − 1287 128 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A15c) The first power in (A15a) is dominant over the second for ρ → 0 if 1 4 < β < 1, and so β should be chosen as a real root of P4 in that interval, which however fails to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' On the other hand, if β > 1, the second power in (A15a) becomes dominant and β should be chosen as a real root of P2 21 in that range, which too fails to exist whenever k1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Thus, (A14) could be the asymptotic form of αλ only if β = 1, which makes (E) take the asymptotic form �1421 192 π2λ2B2 − 2717 672 k3 − 143 32 k1 + 715 672 � ρ + O � ρ3� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A16) Requiring the dominant power of (A16) to vanish determines B as in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In a similar, but perhaps more customary way, by linearizing (E) about α = 0, we obtain that α′(ρ)ρ + 2α(ρ) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (A17) By solving it subject to (31), we readily arrive at (38) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Equivalent Dynamical System In this Appendix we construct a dynamical analogy for the positive branch of equilibrium solutions αλ for Fλ in (32) and give a phase space representation for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We reinterpret Fλ as the action of a dynamical system by introducing the effective time t := − ln ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (B1) Thus, in particular, the center of the ball BR(x0) at ρ = 0 is approached in the new variable when t → +∞, while the initial time t = 0 corresponds to the boundary ∂BR(x0) at ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Correspondingly, the twist angle α becomes a function on [0, ∞), which is defined by a(t) := α � e−t� (B2) and by (31) satisfies a(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (B3) Fλ[α] thus acquires the form of an infinite-horizon action, Aλ[a] := � ∞ 0 Lλ(a, ˙a, t) dt, (B4) where the Lagrangian Lλ is defined as Lλ(a, ˙a, t) := e−t � g(a)˙a2 + 2(k1 − 1)f0(a) � + 32 7 λ2et � 4 � n=1 fn(a)˙an � (B5) and a superimposed dot denotes differentiation with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The orbits of the system are 22 solutions of the equation of motion for Lλ, d dt ∂Lλ ∂ ˙a − ∂Lλ ∂a = = e−t � −γ′(a)˙a2 − 2g(a)¨a + 2g(a)˙a + 2(k1 − 1)f′ 0(a) � − 32 7 λ2et � 4 � n=2 (n − 1)f′ n(a)˙an + ¨a � 4 � n=2 n(n − 1)fn(a)˙an−2 � + 4 � n=1 nfn(a)˙an−1 � = 0, (B6) where a prime ′ denotes differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We are interested in the orbits that start from the initial condition (B3)(and arbitrary ˙a(0)) and whose action Aλ is bounded and a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' To this end, we first identify the critical points of the dynamical system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' these are obtained when both ˙a ≡ 0 and ¨a ≡ 0 in (B6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=', whenever 2(k1 − 1)f′ 0(a) − (λet)2f1(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (B7) For λ > 0, they are a = kπ with k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (B8) For λ = 0, which is the case studied in [33], they are instead a = kπ and a = ± arccos(−1/4) + 2kπ with k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (B9) The trajectory a(t) ≡ 0 represents the radial hedgehog with action Aλ[0] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' On the other hand, if for λ > 0 there is a trajectory aλ = aλ(t) such that limt→∞ aλ(t) = π and the action Aλ[aλ] is finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' it remains to be seen whether Aλ[aλ] < 0, to decide whether the orbit aλ minimizes the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Multiplying both sides of (B6) by ˙a, we get 2g(a)˙ae−t−32 7 λ2et 4 � n=1 nfn(a)˙an = = et d dt � g(a)˙a2 − 2(k1 − 1)f0(a) � + 32 7 λ2et d dt � 4 � n=2 (n − 1)fn(a)˙an � , (B10) whose integration with respect to t ∈ [0, ∞) gives the following expression for the action of aλ: Aλ[aλ] = −g(0)˙aλ(0)2 + 32 7 λ2 � lim t→∞ � et 4 � n=2 (n − 1)fn(aλ)˙an λ � + 2 � ∞ 0 et � 4 � n=1 fn ˙an λ � dt � , (B11) under the assumption that the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We are interested in bounded orbits aλ with bounded action Aλ[aλ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' We call these orbits admis- sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For a solution aλ of (B6) to be an admissible orbit, the initial value ˙aλ(0) must be chosen so as to ensure convergence of the orbit aλ(t) to π as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The asymptotic behaviour in (38) here translates into ˙aλ(t) = aλ(t) + C for t ≈ 0, (B12) 23 where C > 0 is a constant to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' One can show that, for λ > 0 and material constants k3 and k1 chosen in the pink region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3, there is a positive C for which an admissible orbit aλ exists, but it has positive action Aλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Thus, a twisted hedgehog nT exists, but it has more energy than the radial hedgehog nR, which is locally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Hereafter we assume that (29) holds, so that material constants are chosen in the blue region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In the phase plane (x, y), where x(t) = a(t), y(t) = ˙a(t), (B13) (B6) can be rewritten as ˙x = y, (B14a) ˙y = = � −g′(x)y2 + 2g(x)y + 2(k1 − 1)f′ 0(x) − 32 7 (λet)2 ��4 n=2(n − 1)f′ n(x)yn + �4 n=1 nfn(x)yn−1�� � 2g(x) + 32 7 (λet)2 �4 n=2 n(n − 1)fn(x)yn−2 � , (B14b) a system which we next study in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Asymptotically Autonomous Limit The two-dimensional dynamical system described by (B14) is not autonomous23 and this makes it more difficult to predict the qualitative properties of its orbits, as the standard phase plane portraits (such as those discussed, for example, Chapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2 of [63]) do not apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' However, system (B14) has the interesting property of reducing to an autonomous system in the limit as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' For this reason, it is called asymptotically autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='24 For orbits that do not intersect either of the axes of the (x, y) phase plane, the autonomous asymptotic limit of (B14) is ˙x = y, (B15a) ˙y = �4 n=2(n − 1)f′ n(x)yn + �4 n=1 nfn(x)yn−1 �4 n=2 n(n − 1)fn(x)yn−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' (B15b) For any λ ≧ 0, its equilibrium points are p1 = (arccos(1/8), 0) and p2 = (π, 0) (B16) and their periodic replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The eigenvalues of the linear approximation of (B15) near these points are Λ± 1 = −1 2 ± 5 √ 119 14 i, Λ± 2 = 59 44 ± √ 10015 44 , (B17) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Thus p1 is a stable spiral node, while p2 is a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' A phase portrait for the asymptotic limit (B15) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' B1 along with the equilibrium points in (B16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='25 23It is explicitly dependent on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 24Asymptotically autonomous dynamical systems have an interesting literature, recalled for example in Chapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 17 of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 25See, for example, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='2 of [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 24 Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Phase portrait for the asymptotic autonomous limit (B15) around the equilibrium points in (B16), a stable spiral node and a unstable saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Three exemplary orbits are drawn: the green orbit spirals about the node, the blue orbit approaches the saddle along its stable invariant manifold (tangent to the broken straight line), and the black orbit, which is not connected with any equilibrium point, is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The correspondence between the solutions to an asymptotically autonomous system and those to its autonomous asymptotic limit is a delicate one and has not been completely characterized, even in the two-dimensional case, for which a larger number of results are available (see, for example, [65–67]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' In particular, a result of Markus [65] (see his Theorem 7) applies to our system: it says that the ω-limit set26 of a solution to (B14) either contains the equilibria of (B15) or is the union of periodic orbits of (B15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Since (B15) has no periodic orbits, we conclude that the ω-limit set of any bounded solution of (B14) must contain the equilibria of (B15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Among these latter, only p2 can be reached by an admissible orbit (according to our definition);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' this justifies our numerical search for a trajectory in phase space starting from a point (0, y0) of the y-axis and approaching in infinite time the equilibrium point p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The successful outcome of this search is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' B2, where an admissible orbit of (B14) for λ > 0 is contrasted against that obtained in [33] for λ = 0, which is when (B14) becomes autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The solutions illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' B2 are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 4 in the main text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' actually, the latter were generated from the former by inverting the change of variables in (B1) and (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' References [1] Lydon J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Chromonic liquid crystal phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Curr Opin Colloid Interface Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='3(5):458–466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 26The ω-limit set of a forward solution to a dynamical system is the collection of all limiting points attained by the solution on any diverging time sequence (see, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' 242 of [64] for a formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=') 25 10 8- y 444 6 4444 44444 44444 X X 2- K X 0 0 3 元 5元 3元 7元 9元 元 元 元 元 8 4 8 2 8 4 8 8Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Admissible orbits of (B14) in phase space for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content='16 (red line) and λ = 0 (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' The former approaches in infinite time the equilibrium point p2 along the stable invariant manifold (tangent to the broken straight line), which differs from the stable invariant manifold of p2 for the asymptotic autonomous limit system (B15) reproduced here from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' B1 (red thin line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE_T4oBgHgl3EQfzRz6/content/2301.08323v1.pdf'} +page_content=' Both orbits start from points (0, y0) on the y-axis;' metadata={'source': 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Richard, A. Siebert, E. Lagadec, N. Lagarde, O. Venot, J. Malzac, J.-B. Marquette, M. N’Diaye, D. Briot (eds) +COMMISSION FEMMES ET ASTRONOMIE DE LA SF2A : +WOMEN PARTICIPATION IN FRENCH ASTRONOMY +Rhita-Maria Ouazzani1, Caroline Bot2, Sylvie Brau-Nogu´e3, Danielle Briot4, Patrick de Laverny5, +Nad`ege Lagarde6, Nicole Nesvadba7, Julien Malzac8, Isabelle Vauglin9 and Olivia Venot10 +Abstract. +The Commission Femmes et Astronomie conducted a statistical study that aims at mapping the +presence of women in French professional Astronomy today, and set a starting point for studying its evolution +with time. +For the year 2021, we proceeded with a sub-set of 8 astronomy and astrophysics institutes, +hosting a total of 1060 employees, among which PhD students, post-doctoral researchers, and academic, +technical, and administrative staff, representing around 25% of the community. We have investigated how +the percentage of women vary with career stage, level of responsibility, job security, and level of income. The +results of this preliminary study seem to illustrate the leaky pipeline, with one major bottleneck being the +access to permanent positions. It appears that the proportion of women steadily decreases with the security +of jobs, with the career stage, with the qualification level and with the income level. +Keywords: +Astronomy & Astrophysics, Gender inequalities, Career +1 +Introduction +The Commission Femmes et Astronomie of the SF2A (Soci´et´e Fran¸caise d’Astronomie et d’Astrophysique, +French astronomical Society) was created in 2020 to form an instance where questions related to gender equality +can be addressed within the French astronomical community. The Commission has ten members, of which six +members are currently –or were at some point– also part of the SF2A Council: Caroline Bot, Sylvie Brau- +Nogu´e, Danielle Briot*, Patrick de Laverny*, Nad`ege Lagarde*, Rhita-Maria Ouazzani*, Nicole Nesvadba, +Julien Malzac*, Isabelle Vauglin, and Olivia Venot∗. The main goals of the Commission are to promote gender +equality in Astronomy & Astrophysics in France, fight against sexual and gender-based violence, support gender- +focused outreach actions, . . . +Before the commission was created, different efforts to do a census of the status of women in astronomy in +France were conducted. In particular, a survey was conducted through the SF2A to probe the future of doctors +who obtained their PhD in Astronomy and Astrophysics between 2007 and 2017. The results presented in Bern´e +& Hilaire (2020) showed, among other points, that women were less likely to be offered permanent positions +than men. That same year, Bot & Buat (2020) did a census of the percentage of women on permanent positions +in France, finding that 23% percents of permanent positions at that time were held by women. Looking at two +different age classes, they found that the number of women seemed to be decreasing for university positions +1 LESIA, Observatoire de Paris, Universit´e PSL, Sorbonne Universit´e, Universit´e Paris Cit´e, CNRS, 5 place Jules Janssen, 92195 +Meudon, France +2 Universit´e de Strasbourg, CNRS, Observatoire Astronomique de Strasbourg, UMR7550, F–67000 Strasbourg, France +3 IRAP, Universit´e de Toulouse, CNRS, UPS, CNES, Toulouse, France +4 Observatoire de Paris, 61 avenue de l’Observatoire, 75014 Paris, France +5 Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange,F-06304 Nice, France +6 Laboratoire d’Astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, all´ee Geoffroy Saint-Hilaire, 33615 Pessac, France +7 Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange,F-06304 Nice, France +8 IRAP, Universit´e de Toulouse, CNRS, UPS, CNES, Toulouse, France +9 Univ Lyon, Universit´e Lyon1, ENS de Lyon, CNRS, CRAL UMR5574, F-69230 Saint-Genis-Laval, France +10 Universit´e de Paris Cit´e and Univ Paris Est Creteil, CNRS, LISA, F-75013 Paris, France +∗members of the SF2A Council +© Soci´et´e Fran¸caise d’Astronomie et d’Astrophysique (SF2A) 2022 +arXiv:2301.03658v1 [astro-ph.IM] 9 Jan 2023 + +238 +SF2A 2022 +Fig. 1. Left: Proportion of women for the overall sample, among permanent staff, and non-permanent staff. Right: Pro- +portion of women among the researchers (from PhD to Emeritus, 687 individuals), and among administrative, technical +and engineering staff (a.k.a ITA, 373 individuals). +while increasing for astronomer positions (CNAP) and that no evidence of a glass ceiling effect was observed. +While both studies were important and necessary, they gave an instantaneous glimpse of the status of women +in astronomy in 2019-2020, they were limited to the information requested or available and were biased by the +surveyed population (young researchers for Bern´e & Hilaire 2020 or permanent positions for Bot & Buat 2020). +In this context, the Commission Femmes et Astronomie decided to conduct a statistical study that aims at +mapping the presence of women in French professional Astronomy today, and set a starting point for studying +its evolution with time. +As a first step, we would like to address general questions such as: +• What is the percentage of women in French Astronomical institutes? +• How does their number vary with their level of seniority –i.e. career stage–? +• What is the percentage of women at different levels of responsibility? +• How does the number of women depend on income level? +The perimeter of this study is restricted to the research units (institutes) depending of the Astronomy & +Astrophysics (AA) section of the National Institute for Universe Sciences (INSU), with the exception of the +LISA institute, which was included in this study although it was not labeled AA, as part of this institute’s +activities are related to astronomy and astrophysics. The approach adopted consists in collecting data directly +from the institutes heads, through their administrative services. The data is anonymised upfront, to comply +with privacy policies. Once anonymised it is distributed to the members of the committee, who are the only +persons authorized to manipulate them using secured tools. +For the year 2021, taken as the starting point for the evolutionary sequence, we proceeded with a sub-set of +institutes of the INSU-AA, as a proof of concept, with the aim of extending the study to all the INSU-AA +institutes in the near future. +2 +Participation of women in Astronomy in 2021 +2.1 +The 2021 study set-up +We were able to collect data from eight research institutes within France: GEPI, IRAP, Lagrange, LESIA, LISA, +LUTh, ObAS and the SYRTE, which include 1060 individuals, representing around 25% of the community. +For all these institutes, the data contained the following entries: +- Gender, +- Date of Birth, +- Employer (CNRS/CNAP/University/else), +- Status (students/post-doc/researcher/engineers, administrative or technical staff a.k.a. ITA), + +Proportionofwomenin2021 (8institutes) +80% +69,4% +71,5% +60% +64,8% +40% +30,6% +35,2% +20% +28,5% +0% +Overall population +Permanent staff +Non-permanent staff +Women +MenDistribution researcher/ITA +80,0% +71,3% +60,0% +66,0% +40,0% +34,0% +28,7% +20,0% +0,0% +Researchers +ITAs +Women +MenCounting women +239 +Fig. 2. Proportion of women at each stage of the research career. In solid lines are superimposed linear fits of the +proportion of women (light orange) and men (light green), the quality of the fit is indicated by a value of R2 = 0.486. +- and for the public servants: the category (known as grade in french: IR/IE/CR/DR/AA/A/MdC/PU/. . . ). +2.2 +General results and job security +The first number presented in Fig. 1 (left, overall population), gives the proportion of women, regardless of +their status, age or position. This number represents the likelihood of crossing paths with a woman in a corridor +when walking through a French astronomy institute: one in three. +The workforce in French academia is composed of permanent staff, among which we count public servants +–this includes persons in research, engineering, technical or administrative positions–, as well as very few (but +nevertheless growing number of) persons employed on corporate-like permanent positions. As for non-permanent +positions, are counted PhD students, post-doctoral researchers, teaching assistants, apprenticeships, and holders +of a short-term contract (on engineering, technical or administrative jobs). +Looking at the distribution of +women among permanent and non-permanent positions (Fig.1, left), we see that women (35.2%) are most likely +employed on temporary contracts than men (28.5%). If we restrict this comparison to research positions, 25.0% +of researchers on permanent positions are women, whereas it is 34.9% for non-permanent positions (28.7% for +the overall population). +One potential source of variability in these numbers is expected to come from the type of position (research +or not). That is what is explored on the right panel of Fig.1. We are aware that persons hired on ITA positions +also contribute to the research that is produced in these institutes, but different social and economical values +are attributed to research and ITA positions, and that is what, we believe, is determining here. Among women +working in the 8 institutes included in the study, 39.2% are hired as ITA, while for men the percentage is 33.4%. +For the following discussions, we consider separately the population of researchers (in the broad sense: from +PhD to Emeritus) on the one hand, and the population of Engineering, Technical and Administrative staff +altogether (ITA) on the other hand. +2.3 +Research career +Concerning researchers, we have sorted the population (687 individuals) according to the category of their +position. From the youngest to the most seniors, we have listed: PhD students, post-doctoral researchers, and +positions equivalent to associate professors (CR, ASTA, PHYSA, MC), to second-class professors (DR2, PU2, +AST2), to first-class professors (DR1, PU1, AST1), to professor of exceptional class (DRCE, PUCE, ASTCE), + +Distribution of researchers according to category +76,6% +76,9% +78.8% +80,0% +70,6% +72.7% +64,9% +64.3% +60,0% +40,0% +35;1% +35,7% +29,4% +27,3% +23.4% +23.1% +21,2% +20,0% +0,0% +PhD students +Post-docs +CR -ASTA - PHYSA DR2 -PU2-AST2 DR1 - PU1 -AST1 +DRCE- PUCE - +Emeritus +- MC +ASTCE +Women + Linear trend for women R² = 0,486 +Men Linear trend for Men R? = 0,486240 +SF2A 2022 +Fig. 3. Proportion of women in each category of ITA jobs. From left to right they are ordered by qualification and +income level. The solid lines give the optimal linear fits of these distributions, with a value of R2 = 0.661. +and Emeritus. The result is illustrated in Fig. 2. In order to emphasize the general trend, a linear fit has been +performed (R2 = 0.486, p = 0.08), which shows that the proportion of women decreases with the career stage. +This overall trend seems to be mostly caused by the first drop in the distribution: when women represent around +35% of the population in non-permanent positions (PhD students and post-doctoral researchers), the proportion +decreases to 23.4% for the first level of permanent employment. After this first bottleneck, the number varies +slightly between around 23% and around 29%, with two noticeable increases: one at the second-class professor +level, and another one at the Emeritus level. +It is important to bear in mind that this study gives a snapshot of the distribution for 2021. Temporal or +causal relations between one stage to another are delicate to establish, and could properly be addressed only if +this snapshot is renewed every year. However, concerning the clear increase of proportion of women between the +associate professor level and the second-class professor level, one can wonder if it is not a stellar-main-sequence +effect. Astronomers know that the large majority of observable stars are currently in their main sequence. +That is because the main sequence, during which they burn hydrogen in their core, is the longest of all stellar +evolution stages. Hence, we are led to wonder if this increase of women is not due to the fact that once they are +promoted to second-class professorship, they spend a particularly long time in that stage before being promoted +further up, if at all. +2.4 +ITA careers +Concerning the population of ITA, composed here of 373 individuals, their distribution by career level is illus- +trated in Fig. 3. It is worth mentioning that the histogram presents broad categories of ITA (known as corps in +French) ordered by levels of income and qualification, but contrary to Fig. 2, the progression from one category +to the other is very little or none. Furthermore, the sources of variability are much more numerous than in the +research career case. Firstly because of the variety of jobs it encompasses (administrative, technical, R&D . . . ). +The jobs that fall into the ATR-T and AI categories have an dominant administrative component, whereas +IE and IR are mostly scientific and technical jobs. Another source of variability are the qualifications needed +to apply for these different kinds of positions: some require a PhD (IR), and others the High School Leaving +Certificate (Baccalaur´eat). In general, it is safe to assert that women are present in higher proportion in jobs +that have a clear administrative component, and have lower levels of qualifications and lower income. As for +men, they dominate in more technical jobs, where the level of qualification can be higher, and access higher +income. In summary, we can conclude from Fig. 3 that as the qualification and the level of income increases, +the number of women decreases (illustrated by the linear fit, with a p value of p = 0.1). A finer analysis would + +DistributionofITAsbycategory +80% +74,7% +60% +64,8% +56,7% +51,4% +48,6% +40% +43,3% +35,2% +20% +25,3% +0% +ATR-T +AI +IE +IR +Women +MenCounting women +241 +Fig. 4. Proportion of women and men at each income level give in Table 1, increasing from left to right. The quality of +the fit is given by a value of R2 = 0.662. +level +category +L1 +AJT, TCN, TCS +L2 +TCE, AI +L3 +IECN, IR2 +L4 +IEHC, IR1, CRCN, MC, PRAGCN, ASTA, PHYSA +L5 +IRHC, CR1, CRHC, MCHC, DR2, PU2, AST2 +L6 +DR1, DRCE, PU1, PUCE, AST1, ASTCE, Emerites +Table 1. Scale of income for all the workers in the sample, researchers and ITA altogether. In terms of gross salary, it +starts from about 1590 euros and reaches around 6200 euros. +require to blow up each of the histogram stick into finer category (grade in French), but with the current data +at hand, we risk ending up with the small numbers statistics issue. +2.5 +The sinews of war +We also addressed the crux of the matter: salary levels. For this purpose, we bring together again the whole +sample (1060 individuals), and sort them out simply by level of income. To do so, we chose to use a scale of +income set up by S. Brau-Nogu´e (see https://www.irap.omp.eu/egalite/bilan-social-et-parite-2022/) +in her work to document gender inequalities in her institute (IRAP), work which has largely inspired this study. +The scale is presented in Table 1. Far from being a motivation for working in the academic sector, we believe +that the level of income is a good indicator of the social value associated to a given position. Concerning the first +two levels, they encompass mainly technical and administrative positions that are known to be positions which +are very gender specific. But once we reach level L3, we observe that there is a clear and regular decrease of the +proportion of women as the income increases. In solid lines are given the optimal linear fits of the distributions +for men and women, with a p value of p = 0.05. +3 +Preliminary conclusions and perspectives +We present the results of our first statistical study on the participation of women in French Astronomical +Institutes. Our sample was composed of 1060 individuals, belonging to 8 institutes, making up for around 25% + +Distributionbylevelofincome +74,6% +76,7% +80,0% +71,4% +72,9% +58,5% +60.0% +52.1% +47,9% +41,5% +40,0% +28;6% +27,1% +25,4% +23,3% +20,0% +0,0% +L1 +L2 +L3 +L4 +L5 +L6 +Incomelevel +omer +LMen242 +SF2A 2022 +of the targeted population. Although the results are still preliminary, it appears that the proportion of women +steadily decreases with the security of jobs / the career stage / the qualification level / the income level. This +seems to illustrate the well known leaky pipeline issue, but needs further confirmation. +In particular, the sample for this study suffered from a number of shortcomings. Some institutes included +in the study cover topics which go beyond Astrophysics, such as the LISA and the SYRTE. Looking at the p +values associated to the trends determined, we wish to improve the statistical robustness of the inferences. We +aim at solving this issue by extending the sample to all the French Astronomical Institutes in the coming years. +Moreover, the snap-shot nature of this study prevents from drawing strong conclusions about the evolutionary +trends. Even if it can be very tempting to get a sense of evolution by looking at different generations of workers, +one should keep in mind that state policies, or even the culture can change from one generation to the other. +Appendix: Index of jobs in A&A +Here we give some elements for understanding the zoo of jobs that one can find in the astronomical institutes +in France. In general all the positions are divided into category (a.k.a. corps in french), and class (a.k.a. grade +in french). +Academic careers: The three main job providers are the CNRS (French national institute for research), Univer- +sities, and the CNAP (Conseil National des Astronomes et Physiciens). According to the hiring institution, we +can define different career paths, all of which contain 2 categories and each category can be split into 2 or 3 +classes: +• CNRS: Charg´es de Recherche (research associates; 2 classes: +CRCN and CRHC) → Directeurs de +Recherche (research directors; 3 classes: DR2, DR1, DRCE) +• CNAP: Astronomes Associ´es (associate astronomers; 2 classes: ASTA, PHYSA) → Astronomes (as- +tronomers; 3 classes: AST2, AST1, ASTCE) +• University: Maitres de Conf´erence (associate professors, 2 classes: MC, MCHC) → Professeurs (professors; +3 classes: PU2, PU1, PUCE) +Engineering, technical, administrative careers (ITA): For ITA, the positions are divided into 5 categories, which +are themselves divided into several classes (ordered by level of qualification): +• Adjoints techniques de la recherche (AJT) +• Techniciens de la recherche (TCN, TCS, TCE) +• Assistants ing´enieurs (AI) +• Ing´enieurs d’´etudes (IECN, IEHC) +• Ingenieurs de recherche (IR2, IR1, IRHC) +The members of the Commission Femmes et Astronomie would like to thank all institutes directors and colleagues who kindly +agreed to providing the data that made this study possible, and the administrative staff who worked on their compilation and +anonymisation. +References +Bern´e, O. & Hilaire, A. 2020, Nature Astronomy, 4, 296 +Bot, C. & Buat, V. 2020, http://sf2a.eu/Bot_Buat.pdf + diff --git a/h9E0T4oBgHgl3EQfpgF0/content/tmp_files/2301.02540v1.pdf.txt b/h9E0T4oBgHgl3EQfpgF0/content/tmp_files/2301.02540v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..95e0f22ab39ad8220cbc1b966e17c1fff1c45303 --- /dev/null +++ b/h9E0T4oBgHgl3EQfpgF0/content/tmp_files/2301.02540v1.pdf.txt @@ -0,0 +1,1416 @@ +Draft version January 9, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Cross-Correlation Forecast of CSST Spectroscopic Galaxy and MeerKAT Neutral Hydrogen +intensity Mapping Surveys +Yu’er Jiang,1, 2 Yan Gong*,1, 3, 4 Meng Zhang,1, 2 Qi Xiong,1, 2 Xingchen Zhou,1, 2 Furen Deng,1, 2 +Xuelei Chen,1, 2, 5, 6, 4 Yin-Zhe Ma,7, 4 and Bin Yue1 +1National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China +2School of Astronomy and Space Sciences, University of Chinese Academy of Science (UCAS), +Yuquan Road NO. 19A, Beijing 100049, China +3Science Center for China Space Station Telescope, National Astronomical Observatories, Chinese Academy of Sciences, +20A Datun Road, Beijing 100101, China +4NAOC-UKZN Computational Astrophysics Centre (NUCAC), University of KwaZulu-Natal, Durban, 4000, South Africa +5Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China +6Center for High Energy Physics, Peking University, Beijing 100871, China +7School of Chemistry and Physics, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, +Durban 4000, South Africa +ABSTRACT +Cross-correlating the data of neutral hydrogen (HI) 21cm intensity mapping with galaxy surveys is an +effective method to extract astrophysical and cosmological information. In this work, we investigate +the cross-correlation of MeerKAT single-dish mode HI intensity mapping and China Space Station +Telescope (CSST) spectroscopic galaxy surveys. We simulate a survey area of ∼ 300 deg2 of MeerKAT +and CSST surveys at z = 0.5 using Multi-Dark N-body simulation. The PCA algorithm is applied to +remove the foregrounds of HI intensity mapping, and signal compensation is considered to solve the +signal loss problem in HI-galaxy cross power spectrum caused by the foreground removal process. We +find that from CSST galaxy auto and MeerKAT-CSST cross power spectra, the constraint accuracy of +the parameter product ΩHIbHIrHI,g can reach to ∼ 1%, which is about one order of magnitude higher +than the current results. After performing the full MeerKAT HI intensity mapping survey with 5000 +deg2 survey area, the accuracy can be enhanced to < 0.3%. This implies that the MeerKAT-CSST +cross-correlation can be a powerful tool to probe the cosmic HI property and the evolution of galaxy +and the Universe. +Keywords: intensity mapping, large-scale structure, cosmological constraint +1. INTRODUCTION +Probing the large-scale structure (LSS) of the Uni- +verse has always been one of the main missions of cos- +mological observations. +Constraining the property of +dark matter and dark energy, recovering the primordial +fluctuations and testing gravity theories are all in need +of cosmological surveys with large survey area and wide +redshift coverage. To achieve this target, line intensity +mapping (LIM) has been proposed and proven to be an +efficient technique. Line intensity mapping makes use +Corresponding author: Yan Gong +Emial: gongyan@bao.ac.cn +of the emission lines from the energy level transition of +atoms or molecules, such as HI 21cm, CII, CO, Lyα, Hα, +[OIII], etc. (see e.g. Visbal & Loeb 2010; Carilli 2011; +Lidz et al. 2011; Gong et al. 2011, 2012, 2013, 2014; Silva +et al. 2013, 2015; Pullen et al. 2014; Uzgil et al. 2014; +Fonseca et al. 2017; Gong et al. 2017, 2020). These lines +can reflect different properties and progresses of galaxy +evolution, and could be good tracers of the LSS. +Instead of the traditional observations targeting the +resolvable sources, intensity mapping probe accumula- +tive intensity of all sources in a spatial volume (voxel) +defined by survey spatial and frequency resolutions. So +even though some sources are too faint to be detected in +traditional sky surveys, in principle, their signals can be +probed in intensity mapping. And the frequency shifts +arXiv:2301.02540v1 [astro-ph.CO] 6 Jan 2023 + +2 +of the emission lines are the natural probe of redshift. +So intensity mapping is expected to be a powerful tool +to obtain cosmic 3D matter structure information traced +by emission lines from galaxies with high efficiency and +relatively low cost. Among various emission lines, HI +21cm line from atomic hydrogen is the most widely stud- +ied in intensity mapping research. +Besides as a main +probe of epoch of reionization, neutral hydrogen 21cm +line has tight connection with star formation and galaxy +evolution, and it can trace galaxy and hence dark matter +distribution in low and high redshifts. +While many experiments about HI intensity mapping +are proposed or already running, the foreground con- +tamination problem is still one of the biggest challenges, +that the foregrounds can be as large as five orders of +magnitude higher than the signal. +The high bright- +ness temperature of the Galactic emission and other +sources makes HI signal hardly be detected from auto- +correlations. +In principle, cross-correlating the 21cm +observation with an optical galaxy survey in the same +survey area is a good method to reduce the foreground +contamination and instrumental noise, and extract the +signal (e.g. Chang et al. 2010). The signal-to-noise ra- +tio (SNR) can be significantly improved since the fore- +grounds and instrumental noise of different wave bands +in different surveys are barely correlated. +However, in practice, the cross-correlation result is not +fully satisfied due to the complex components of the +foreground. So the foreground removal algorithms are +still in need in cross-correlations. +Various algorithms +have been purposed, including the blind foreground +removal techniques like principal component analysis +(PCA) (Davis et al. 1985a) and independent component +analysis (ICA) (Wolz et al. 2014) which make use of dif- +ferent frequency smoothness of foreground and signal, +the polynomial/parametric-fitting method which fits the +physical properties of the foreground (Bigot-Sazy et al. +2015), and machine learning (ML) methods (Li & Wang +2022), etc. Although signal loss and foreground residual +are usually inevitable, foreground removal techniques do +make progress and are necessary in cross-correlation de- +tection. +Currently, positive results on HI abundance and HI- +galaxy correlation have been obtained by several exper- +iments. The Green Bank Telescope (GBT) has imple- +mented their HI intensity mapping correlation detection +with Deep2 optical redshift survey (Chang et al. 2010), +WiggleZ Dark Energy Survey (Masui et al. 2013), and +eBOSS survey (Wolz et al. 2022). +And Parkes radio +telescope also presented their work of correlating HI +intensity mapping with 2dF galaxy survey (Anderson +et al. 2018). Recently, MeerKAT accomplished HI in- +tensity mapping correlation detection with WiggleZ sur- +vey (Cunnington et al. 2022a). They all constrain the +HI-galaxy correlation parameter product ΩHIbHIrHI,g at +different redshifts, where ΩHI, bHI, and rHI,g are the HI +energy density parameter, HI bias, and correlation co- +efficient of HI and galaxy, respectively. +In this work, +we will determine the constraint power on neutral hy- +drogen parameters by the observations of MeerKAT and +the next-generation galaxy survey of China Space Sta- +tion Telescope (CSST). +MeerKAT is a pathfinder project of SKA and in fu- +ture will become a part of SKA-mid (Santos et al. 2017; +Bacon et al. 2020). It is a state-of-the-art intensity map- +ping instrument which is capable of complementing and +extending cosmological measurements at a wide range of +wavelengths. While MeerKAT is a large interferometric +array which can access small scales of cosmic structure, +single-dish mode is preferred in intensity mapping ex- +periments. We plan to perform MeerKAT HI intensity +mapping cross-correlation with the China Space Station +Optical Survey (CSS-OS) (Zhan 2011, 2021; Cao et al. +2018; Gong et al. 2019). CSS-OS is the major obser- +vation project of CSST, and it will cover 17500 deg2 +sky area in 10 years’ working time. And the spectro- +scopic survey of CSS-OS will provide large amount of +galaxy catalog with verified redshift using slitless grat- +ings. CSST is planned to start its observation around +2024, while MeerKAT will still be on full-time job be- +fore SKA which will begin full operations in 2028, and +these two surveys would have large overlapping survey +area. Thus we believe MeerKAT HI intensity mapping +and CSST galaxy survey would make promising cross- +correlation detection in the coming future. +This paper is organized as follow: in Section 2, we in- +troduce our method of creating mock data of MeerKAT +HI intensity mapping and CSST spectroscopic galaxy +surveys; in Section 3, we apply PCA algorithm to re- +move the foreground in HI intensity map; in Section 4, +we calculate the galaxy auto and HI-galaxy cross power +spectra, and discuss the signal compensation method +for cross power spectrum; in Section 5 we forecast the +constrains on relevant cosmological parameters; we con- +clude our work and have discussion in Section 6. +2. MOCK DATA +We generate MeerKAT intensity maps and CSST +spectroscopic galaxy survey data using MultiDark cos- +mological simulations (Klypin et al. 2016). MultiDark is +a suite of N-body cosmological simulations which have +been carried out by L-GADGET-2 code. Most simula- +tions of this suite have 38403 particles, with box sizes +ranging from 250 Mpc/h to 2500 Mpc/h. Based on the + +3 +Table 1. Simulation parameters +for SMDPL. +Parameter +Value +Lbox +400 Mpc/h +Np +38403 +Mp +9.63 × 107M⊙/h +ϵ +1.5 kpc/h +h +0.6777 +ΩM +0.307 +Ωb +0.048 +ΩΛ +0.693 +ns +0.96 +σ8 +0.8228 +survey area and redshift of MeerKAT observation plan, +the Small MultiDark Planck simulation (SMDPL) has +been chosen in this work. The box size of SMDPL is +400 Mpc/h, and halos in SMDPL boxes are identified +through halo finding code Friends-of-Friends (FOF) with +relative linking length of 0.2 (Davis et al. 1985b). The +relevant simulation and cosmological parameters that +SMDPL adopted are listed in Table 1, and its halo cat- +alog can be required from the CosmoSim database1. +In our work, we focus on the cosmology at z = 0.5, +which is one of the main observational target redshifts +for both CSST and MeerKAT L-band. So our mock data +are generated from the snpashot70 of SMDPL, whose +redshift z ≈ 0.5. We also find that the 400 Mpc/h box +size of the snapshot70 corresponds to a survey is of ∼ +297 deg2 at z = 0.5. +2.1. HI intensity mapping with MeerKAT +Since HI can only survive from UV radiation in dense +clumps in galaxies after the epoch of reionization, we +assume that HI can only exist in halos hosting galaxies +at z = 0.5. We place the HI mass in the center of a +halo, as it has been proven to be reasonable in previous +studies (see e.g. Villaescusa-Navarro et al. 2018). Un- +der this assumption, we construct a catalogue applying +the halo HI mass function given by Villaescusa-Navarro +et al. (2018), and it takes the form as +MHI(M, z) = M0 +� M +Mmin +�α +exp +� +− +� M +Mmin +�0.35� +. +(1) +1 The data is available at https://www.cosmosim.org/ +109 +1010 +1011 +1012 +1013 +1014 +1015 +M [M +/h] +105 +106 +107 +108 +109 +1010 +1011 +1012 +MHI [M +/h] +z=0 (Villaescusa-Navarro et al.2018) +z=1 (Villaescusa-Navarro et al.2018) +z=0.5 (This work) +Figure 1. +The HI mass MHI and halo mass M relation +we use at z = 0.5 (green curve), which is derived from the +relations at z = 0 (blue dashed curve) and z = 1 (orange +dashed curve) given in Villaescusa-Navarro et al. (2018). +Here M is the halo mass, and we have three free pa- +rameters, i.e. +α, M0 and Mmin, which determine the +shape of the fitting curve at different redshifts. +In +order to get the values of these three parameters at +z=0.5, we make interpolation of the fitting values at +z = 0 and 1 given in Villaescusa-Navarro et al. (2018). +Then we find that α = 0.42, M0 = 2.50 × 1010h−1M⊙, +Mmin = 1.13 × 1012h−1M⊙ at z = 0.5. +In Figure 1, +we plot the MHI − M relation at z = 0.5 (green solid +curve), and the relations at z = 0 (blue dashed curve) +and 1 (orange dashed curve) from Villaescusa-Navarro +et al. (2018) are also shown for comparison. +Then we can calculate the HI energy density parame- +ter ΩHI, which is given by +ΩHI(z) = 1 +ρ0c +� +n(M, z)MHI(M, z)dM, +(2) +where ρ0 +c is the critical denisty of the present Universe, +and n(M, z) is the halo mass function (Sheth & Tormen +1999). We find that ΩHI = 6.73×10−4 in our simulation, +that agrees with the estimation of ΩHI −z relation given +in literatures (see e.g. Villaescusa-Navarro et al. 2018). +Next, we can create the map of HI brightness temper- +ature. The brightness temperature field δT traces the +underlying matter fluctuations δm as +δT (r, z) = T b(z)bHI(z)δm(r, z) +(3) +where T b(z) is the mean HI brightness temperature at +z, and bHI is the HI bias, which can be estimated by +bHI(z) = ΩHI(z) +ρ0c +� +n(M, z)b(M, z)MHI(M, z)dM. (4) + +4 +Figure 2. Left panel: the dark matter distribution at z = 0.5 in the simulation. Right panel: the corresponding map of HI +brightness temperature. We can see that the HI map have similar structure as dark matter, and can be a tracer for the LSS. +Here b(M, z) is the halo bias. So for a voxel with position +on the sky r and redshift z, its HI brightness tempera- +ture can be derived as +Tb(r, z) = 189 +h +E(z)ΩHI(r, z)(1 + z)2 [mK] += T0 × ΩHI(r, z) [mK], +(5) +where E(z) = H(z)/H0 represents the evolution of the +Hubble parameter. At z = 0.5, we find that the corre- +sponding mean HI brightness temperature is T b = 0.145 +mK in our simulation. The brightness temperature of HI +distribution (right panel) and the corresponding dark +matter distribution (left panel) in the simulation are +shown in Figure 2. +After obtaining the HI brightness temperature in the +simulation box, our next step is to create the HI inten- +sity maps with MeerKAT instrumental parameters and +observational effects. Since the observable of HI inten- +sity mapping is the HI brightness temperature of each +voxel in the survey volume, we divide the survey volume +(here is our simulation box) into voxels that MeerKAT +can observe. The details are as follows: +• To divide frequency bins along the line of sight +(LOS), we put the center of the box at z=0.5. As +the box length of 400 Mpc/h is known, the redshift +range of the survey volume could be calculated. +We find that the redshift range of snapshot70 is +0.415 ∼ 0.590, corresponding to the observed HI +frequency of 1004.14MHz ∼ 893.30MHz. This fre- +quency range can be observed by the MeerKAT +L-band with frequency resolution of 0.2MHz, and +it allows us to divide the survey volume into 554 +bins, that each bin width is about 0.72Mpc/h. For +simplicity, we assume that there is no redshift evo- +lution in this range. +• As for the pixels perpendicular to LOS, since we +plan to use the single-dish mode observation, res- +olution θ is defined by the FWHM of the beam of +an individual dish. Then the beam size or spatial +resolution is given by +θb = 1.02 λobs +Ddish +, +(6) +where λobs is the observed wavelength, and Ddish +is the dish aperture diameter. We find that the +spatial resolution of MeerKAT at z=0.5 is 1.36 +deg. Since the size of simulation box 400 × 400 +(Mpc/h)2 corresponding to a 297 deg2 survey area, +the number of pixels in a HI map is found to be +12×12 for MeerKAT single-dish mode observation. +The HI signal intensity map obtained by MeerKAT +at z = 0.5 is shown in the upper left panel of Figure 3. +In real observation, the HI intensity mapping will be +contaminated by different components, such as system +thermal noise, foreground emission from the Milky Way +and radio frequency interference (RFI), etc., which can +lower the signal to noise ratio (SNR). Here we model the +system thermal noise of single-dish as Gaussian noise. +Its rms noise temperature can be calculated as (Bull +et al. 2015) +σT = +Tsys +√δνttot +λ2 +θ2 +bAe +� +AS/θ2 +b, +(7) +where δν is the frequency interval, ttot is the total ob- +servation time of the survey, Ae is the effective collect- +ing area of an dish, AS is the survey area and Tsys is + +40 +10 +35 +8 +30 +6 +25 +4 +Dec. +20 +2 +15 +0 +10 +-2 +5 +-4 +0 +154 +156 +158 +160 +162 +164 +166 +168 +170 +R.A.0.25 +10 +8 +0.20 +6 +0.15 +4 +Dec. +mK +2 +0.10 +0 +-2 +0.05 +-4 +0.00 +154 +156 +158 +160 +162 +164 +166 +168 +170 +R.A.5 +155.0 +157.5 +160.0 +162.5 +165.0 +167.5 +170.0 +R.A. +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Dec. +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +HI Brightness Temperature [mK] +155.0 +157.5 +160.0 +162.5 +165.0 +167.5 +170.0 +R.A. +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Dec. +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Systematic Noise[mK] +155.0 +157.5 +160.0 +162.5 +165.0 +167.5 +170.0 +R.A. +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Dec. +4.8 +5.0 +5.2 +5.4 +5.6 +5.8 +6.0 +mK +Figure 3. The mock MeerKAT intensity maps for the central slice in the simulation at ν = 946.7 MHz or z = 0.5. The upper +left panel is the signal map of HI brightness temperature. The upper right panel shows the map of Gaussian system noise. +The bottom left panel is the total foreground map generated by GSM2016 at ν = 946.7MHz, including Galactic synchrotron +emission, free-free emission, cold and warm dust thermal emission, the CMB anisotropy, and Galactic HI emission. The bottom +right panel shows the total sky map observed by MeerKAT containing all components we consider. +the system temperature which is usually described as a +combination of four components, that gives +Tsys = Tsky(ν) + Tspill + Tatm + Trec. +(8) +The mean sky temperature can be approximated by +Tsky = 2.725+1.6(ν/GHz)−2.75, and Tspill, Tatm and Trec +represent spillover temperature, atmosphere tempera- +ture and receiver temperature, respectively. The values +of these parameters we adopte are listed in Table. 2, +and then we obtain σT = 0.102 mK at ν = 946.7 MHz +(z=0.5). +The corresponding map of Gaussian system +noise is shown in the upper right panel of Figure 3. +The foreground emission from the Milky Way is actu- +ally the main challenge to HI intensity mapping. The +brightness temperature of foregrounds can be more than +Table 2. Parameters of the MeerKAT single- +dish observations. +Parameter +Value +Number of dishes +64 +Observation mode +Single-dish +Dish diameter +13.5 m +System temperature +20 K +L-band Frequency range +856-1712 MHz +Frequency resolution +0.2 MHz +Survey time +200 hrs per dish + +6 +4 orders of magnitude brighter than HI signal, so its ef- +fect has to be seriously taken account in our forecast +of MeerKAT observation. +Here we generate the fore- +ground emission using the GSM2016 model (Zheng et al. +2017). GSM2016 is an improved model of the original +GSM. It uses an extended PCA algorithm to identify +different components in the diffuse Galactic emission. +Six components of Galactic emission that matches the +known physical emission mechanisms are obtained, i.e. +synchrotron emission, free-free emission, cold and warm +dust thermal emission, the CMB anisotropy and Galac- +tic HI emission. This algorithm allows it to make use +of 29 sky maps from 10 MHz to 5 THz, and make in- +terpolation to get full-sky map of any frequency in this +frequency range. +To apply the foreground model on our HI map, the +coordinates of the survey area have to be set. +Ac- +cording to the previous work (Wang et al. 2021), we +set our survey area at 153.38◦ < R.A. < 170.62◦ and +−5.62◦ D). +In the region where the magnetic field is non-zero, the transverse component of +the gravitational wave h12 is described by a wave packet of the form5 +hs(z + t)hr(z − t) = A sin (ωs(z + t)) sin (ωr(z − t)) +(3.1) +5As we saw in §2, the function hr(z − t) can be chosen arbitrarily. Here we make the simplifying +assumption that hr(z − t) ∝ sin(ωr(z − t)). + +Gertsenshtein... +9 +where ωr (ωs ≡ L−1) is the frequency of the rapidly (slowly) oscillating component of +the metric, A is the maximum amplitude of the wave, and for simplicity we have set +the global phase φ = 0 in (2.19). +The most general solution for the gravitational wave in region II then is a super- +position of wave packets of the form (3.1) and plane waves. Following [6], we turn +now to a complex formalism to describe these waves, with the tacit assumption that +physical solutions are always given by the real part of complex amplitudes. +Using this formalism, h12 in region II is given by (the real part of) +hII +12(z, t) = aeiωr(z−t) + be−iωr(z+t) + Ae−iωs(z+t)eiωr(z−t) +for +0 ≤ z ≤ D. +(3.2) +The first two terms represent right-traveling and left-traveling plane waves respec- +tively, while the third term is a superposition of wave packets.6 +In both regions z < 0 and z > D where the background magnetic field is zero, the +wave equation for the metric is □h12 = 0, and the corresponding vacuum solutions +are +hI +12(z, t) = ce−iωr(z+t) +for +z < 0, +(3.3) +and +hIII +12(z, t) = deiωr(z−t) +for +z > D, +(3.4) +describing a generic left-traveling (z < 0) or right-traveling (z > D) plane wave. +In order to determine the constant coefficients (a, b, c, d) in terms of A we im- +pose the usual boundary conditions demanding continuity of the functions and their +6Specifically, +the +third +term +is +a +combination +of +the +four +elementary +wave +pack- +ets cos(ωs(z + t)) cos(ωr(z − t)), +cos(ωs(z + t)) sin(ωr(z − t)), +sin(ωs(z + t)) cos(ωr(z − t)), +and +sin(ωs(z + t)) sin(ωr(z − t)). + +Gertsenshtein... +10 +derivatives at z = 0 and z = D: +hI +12(0, t) = hII +12(0, t), +∂zhI +12(0, t) = ∂zhII +12(0, t), +hII +12(D, t) = hIII +12(D, t), +∂zhII +12(D, t) = ∂zhIII +12(D, t). +(3.5) +Or, in terms of the coefficients: +c = a + b + Ae−iωst, +c = b − a − +� +1 − ωs +ωr +� +Ae−iωst, +d = a + be−2iωrD + Ae−iωs(D+t), +d = a − be−2iωrD + Ae−iωs(D+t) +� +1 − ωs +ωr +� +. +(3.6) +This is a system of four linear equations for five unknowns. In terms of A, coefficients +(a, b, c, d) are: +a = −A +� +1 − ωs +2ωr +� +e−iωst, +b = −A ωs +2ωr +eiD(2ωr−ωs)e−iωst, +c = A ωs +2ωr +� +1 − eiD(2ωr−ωs)� +e−iωst, +d = −A +� +1 − ωs +2ωr +� +(1 − e−iDωs)e−iωst. +(3.7) +The presence of the time-dependent factor exp(−iωst) means that the coefficients in +(3.7) are not truly constant. However, given our assumption that e−iωst is slowly +varying, to first approximation we can consider them time-independent: +a = −A +� +1 − ωs +2ωr +� +, +b = −A ωs +2ωr +eiD(2ωr−ωs), +c = A ωs +2ωr +(1 − eiD(2ωr−ωs)), +d = −A +� +1 − ωs +2ωr +� +(1 − e−iDωs). +(3.8) + +Gertsenshtein... +11 +One can easily check that the metric h12 with the coefficients in (3.8) satisfies the +wave equation in all three regions (in our approximation) as well as the boundary +conditions in (3.5). Hence, the full solution for h12(z, t) is +hI +12(z, t) = A ωs +2ωr +(1 − eiD(2ωr−ωs))e−iωr(z+t), +hII +12(z, t) = −A +� +1 − ωs +2ωr +� +eiωr(z−t) − A ωs +2ωr +eiD(2ωr−ωs)e−iωr(z+t) ++Ae−iωs(z+t)eiωr(z−t), +hIII +12 (z, t) = −A +� +1 − ωs +2ωr +� +(1 − e−iDωs)eiωr(z−t). +(3.9) +We can similarly derive a solution for the electromagnetic wave. From equations +(2.17) and (2.18), br = h′ +r/(2√B0) and bs = 2h′ +s/√B0. Given that hr(z−t) = eiωr(z−t) +and hs(z + t) = Ae−iωs(z+t), we have br(z − t) = iωr/(2√B0)eiωr(z−t) and bs(z + t) = +−2iAωs/√B0e−iωs(z+t), so the electromagnetic wave packet in region II is of the form +br(z − t)bs(z + t) = 1 +2Aωre−iωs(z+t)eiωr(z−t). +(3.10) +By assumption, the incident electromagnetic wave is right-moving in both regions I +and III. In addition, we need to include a produced left-moving electromagnetic wave +in region I, as well as a combination of right-moving and left-moving plane waves in +region II, as we did for the gravitational wave in (3.2). Consequently, after imposing +the relevant boundary conditions as in (3.5) and disregarding the slowly oscillating +term exp(−iωst) ≈ 1, the full solution for b(z, t) is: +bI(z, t) = Bωreiωr(z−t) + 1 +4(1 − ei(2ωr−ωs)D)Aωse−iωr(z+t), +bII(z, t) = +� +B − 1 +2A +� +1 − ωs +2ωr +�� +ωreiωr(z−t) − 1 +4ei(2ωr−ωs)DAωse−iωr(z+t) ++1 +2Aωre−iωs(z+t)eiωr(z−t), +bIII(z, t) = +� +B − 1 +2A +� +1 − ωs +2ωr +� +(1 − e−iDωs) +� +ωreiωr(z−t), +(3.11) +where B is the incident electromagnetic wave amplitude. Thus far, the coefficient B +is undetermined because we have had to solve for five coefficients but have only four + +Gertsenshtein... +12 +boundary conditions. However, one can easily check that the wave equation (2.14) +for h12 in region II is consistent with the second of (3.11) only if B = A/2. One can +also check that the resulting expressions for h12 and b conserve energy. +The solutions (3.9) and (3.11) describe the conversion of an electromagnetic wave +to a gravitational wave, catalyzed by the constant background magnetic field B0. +In region II where B0 ̸= 0, the electromagnetic and gravitational amplitudes slowly +oscillate between each other via the conversion term exp(−iωs(z + t)). Outside, in +regions I and III, the oscillation stops and the wave packets propagate independently. +The solution of [6] is an approximation of ours for ωs ≪ ωr. In fact, to leading +order in ωs, the gravitational wave solution in (3.9) becomes +hI +12(z, t) = A ωs +2ωr +(1 − e2iDωr)e−iωr(z+t), +hII +12(z, t) = A ωs +2ωr +eiωr(z−t) − A ωs +2ωr +e2iDωre−iωr(z+t) − iωszAeiωr(z−t), +hIII +12 (z, t) = −iDωsAeiωr(z−t), +(3.12) +where we have used e−iωsz ≈ 1 − iωsz for small ωs. +The coefficients of (3.12) are the same as in [6], given that the authors use a +convention in which the wave packet solution is given by λz/(2iωr) exp(iωr(z − t)), +with λ = 2ωrωsA. In this approximation, the amplitude of the gravitational wave in +region I vanishes if the condition exp(2iDωr) = 1 is satisfied, and the amplitude of +the gravitational wave in region III depends linearly upon the length and magnitude +of the static field. +Similarly, the approximate solution for the electromagnetic wave is, to leading +order in ωs, +bI(z, t) = 1 +2Aωreiωr(z−t) + 1 +4(1 − e2iDωr)Aωse−iωr(z+t), +bII(z, t) = 1 +2Aωreiωr(z−t) − 1 +4e2iDωrAωse−iωr(z+t) − 1 +2iAωrωszeiωr(z−t), +bIII(z, t) = 1 +2 (1 − iDωs) Aωreiωr(z−t). +(3.13) +If the condition e2iDωr = 1 is satisfied, the back-propagating wave amplitude vanishes + +Gertsenshtein... +13 +and only the incident wave is left in region I. The solution in this approximation +describes an incident electromagnetic wave of amplitude A and energy ωr that prop- +agates into region II and slowly converts to a gravitational wave with a rate given by +ωs ≡ L−1. When the wave enters region III, the incident electromagnetic wave ampli- +tude is effectively reduced by a factor of iDωs while the gravitational wave amplitude +is increased by that same factor: +EM : +A → A +� +1 − iD +L +� +, +GW : +0 → −iAD +L . +(3.14) +4 +Axion-Photon Conversion +Quite generally the Lagrangian density for the interaction of gravitational waves with +matter is [13] +L = −1 +2hµνT µν, +(4.1) +where in our problem T µν is given by (2.3) and (2.4). +Currently, there is a large interest in axion-photon conversion [14, 15, 16], as +axions are presently the leading candidates for dark matter. In a process much like +the Gertsenshtein mechanism, axions emerging from the sun can be transformed into +X-ray photons by a strong laboratory magnetic field and subsequently detected by +an X-ray telescope. Indeed, the axion-photon interaction Lagrangian can be written +as +L = −gaγ +4 aFµν ˜F µν, +(4.2) +where ˜F µν ≡ 1 +2εµναβFαβ is the dual electromagnetic field tensor, a is the axion ampli- +tude and gaγ is the axion-photon coupling constant. +In terms of the electric and magnetic fields, the above expression amounts to +L = gaγ a E · B , which is essentially the same Lagrangian as the one we have been + +Gertsenshtein... +14 +using except that the external magnetic field couples to the E-field of the photon +rather than the B-field. +Given the quadratic form of Lagrangians in general and the similarity of the +axion-photon Lagrangian to the gravitational interaction Lagrangian in particular, we +may conjecture that Gertshenstein-like mechanisms which cause mixing among many +different fields are “universal” and have widespread cosmological utility. In future +papers we hope to extend the effect to the Yang Mills field as well as consider the +possibility that graviton-photon mixing in the long-wavelength limit might generate +something resembling a cosmological constant. + +Gertsenshtein... +15 +References +[1] M. E. Gertsenshtein, “Wave resonance of light and gravitational waves,” Zh. Eksp. +Theor. Fiz. 41, 113-114 (1961); Soviet Physics JETP 14, 84-85 (1962). +[2] T. Rothman and S. Boughn, “Can gravitons be detected?” Foundations of Physics 36, +1801-1825 (2006). +[3] Ya. B. Zel’dovich,“Electromagnetic and gravitational waves in a stationary magnetic +field,” Zh. Eksp. Theor. Fiz. 68 1311-1315 (1973);Sov. Phys.–JETP 38, 652-653 (1974). +[4] F. J. Dyson, “Photon-Graviton Oscillations” (unpublished). +[5] F. J. Dyson, “Do gravitons exist?” Talk given at Boston University, November 8, 2005. +[6] D. Boccaletti et al., “Conversion of Photons into Gravitons and Vice Versa in a Static +Electromagnetic Field,” Nuovo Cim. 20 B, 129-146 (1970). +[7] W. K. De Logi and A. R. Mickelson, “Electrogravitational conversion cross sections in +static electromagnetic fields,” Phys. Rev. D 16, 2915-2927 (1977). +[8] H. N. Long et al. “Electromgnetic-gravitational cross-sections in external elctromag- +netic fields,” Mod. Phys. Lett. A9, 3619-3628 (1994). +[9] D. +Ejlli, +“Mixing +of +gravitons +with +photons +in +primordial +magnetic +fields,” +arXiv:1307.7883. +[10] P. +Jones +and +D. +Singleton, +“Interaction +Between +Gravitational +Ra- +diation +and +Electromagnetic +Radiation,” +Int. +J. +Mod. +Phys. +D +https://doi.org/10.1142/S0218271819300106. +[11] C. Misner, K. Thorne and J. Wheeler, Gravitation, (W.H. Freeman, New York, 1973). +[12] S. Weinberg, Gravitation and Cosmology (John Wiley, New York, 1972). +[13] S. Boughn and T. Rothman, “Aspects of Graviton Detection: Graviton Emission and +Absorption by Atomic Hydrogen,” Class.Quant.Grav. 23 5839-5852 (2006). +[14] V. Anastassopoulos et al. “New CAST Limit on the Axion-Photon Interaction,” Nature +Phys. 13 584-590 (2017). +[15] C. Adair et al. “Search for Dark Matter Axions with CAST-CAPP,” Nature Commun. +13 6180 (2022). +[16] J. Matthews et al. “How do Magnetic Field Models Affect Astrophysical Limits on +Light Axion-like Particles? An X-ray Case Study with NGC 1275,” Ap. J., in press, +arXiv:2202.08875. + diff --git a/htA0T4oBgHgl3EQfIP9i/content/tmp_files/load_file.txt b/htA0T4oBgHgl3EQfIP9i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2878d6e914b33e6ce3bc27af4b5e8f90c9150b85 --- /dev/null +++ b/htA0T4oBgHgl3EQfIP9i/content/tmp_files/load_file.txt @@ -0,0 +1,353 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf,len=352 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='02072v1 [gr-qc] 5 Jan 2023 A Simple Derivation of the Gertsenshtein Effect Andrea Palessandro ∗1 and Tony Rothman †2 1Deloitte Consulting, Artificial Intelligence and Data 2New York University, Department of Applied Physics (retired) January 6, 2023 Abstract As shown by Gertsenshtein in 1961, an external magnetic field can catalyze the mixing of graviton and photon states in a manner analogous to neutrino- flavor oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We first present a straightforward derivation of the mech- anism by a method based on unpublished notes of Freeman Dyson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We next extend his method to include boundary conditions and retrieve the results of Boccaletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' from 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We point out that, although the coupling be- tween the graviton and photons states is extremely weak, the large magnetic fields around neutron stars ∼ 1014 G make the Gertsenshtein effect a plausible source of gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Indeed, an “in principle” observable consequence would be the change of optical brightness of a neutron star between directions parallel and perpendicular to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We also point out that axion-photon mixing, a subject of active current research, is essentially the same process as the Gert- senshtein effect, and so the general mechanism may be of broad astrophysical and cosmological interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 1 Introduction In a classic 1961 paper [1], Mikhail Gertsenshtein demonstrated that electromagnetic waves passing through an external magnetic field in curved space could be transmuted into gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The result was entirely classical but in quantum mechanical language one can say that the external field “catalyzes” a resonant mixing between ∗apalessandro@deloitte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='dk †tonyrothman@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='com 1 Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 2 photon and graviton states in a manner analogous to the mixing of neutrino flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Although the coupling between the graviton and photon states is extremely weak, the large magnetic fields associated with, for example, neutron stars, make the Gertsen- shtein effect a plausible source for producing gravitons [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Zel’dovich [3], however, and Dyson [4, 5] pointed out that the Gertsenshtein mechanism requires coherence between the gravitational and electromagnetic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Large magnetic fields create electron-positron pairs, which in turn alter the index of refraction of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The speed of electromagnetic wave propagation is lowered relative to that of the gravitational wave and the Gertsenshtein mechanism is quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The Gertsenstein effect received some attention in the 1970s when Boccaletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' [6] provided a comprehensive discussion and de Logi and Mikelson [7] calculated the cross section for graviton-photon conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Cross sections have also been computed by Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' [8], and the effect has also been discussed by Ejlli [9], as well as Jones and Singleton [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Most physicists, however, apparently remain unaware of the Gertsenshtein effect, although current calculations of axion-photon conversion employ a nearly identical formalism (see §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' As it turns out, Freeman Dyson rediscovered the Gertsenshtein effect in 2005 [4], evidently unaware of the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' When informed that he was not first, he chose not to publish his notes, but gave them to one of us (TR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' As Dyson’s derivation is simpler than the others we have seen and has remained unpublished, we thought it worthwhile to make it available in some form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In the following section of this paper we present Dyson’s approach, filling in some steps and clarifying a few ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In §3 we make contact with Boccaletti et al.’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' It will be seen that Dyson’s method is much simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Finally in §4 we point out the similarity to axion-photon conversion and indicate some possible applications of the Gertsenshtein effect in the cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 3 2 Graviton-Photon Oscillations Consider the propagation of electromagnetic and gravitational waves in a flat space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' For the gravitational wave (GW), we assume as usual that it is represented by a weak perturbation hµν(x, t) ≪ 1 traveling on a Minkowski background such that the full spacetime metric is gµν = ηµν + hµν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' gµν = ηµν − hµν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' hµν ≡ ηαµηβµhαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='1) where we take the flat-space metric to be ηµν = (−1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We carry out our calculations in the standard transvere-traceless (TT), or Lorentz, gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' If the wave is taken to be propagating in the z-direction, then hµν = hµν(z, t) and the transverse components of the field are h11 = −h22 and h12 = h21, which represent the two independent polarizations of the GW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='1 In the Lorentz gauge, the linearized gravitational wave equation (see [11], chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='18 and [12], chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='10) is □hµν = −16πTµν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='2) with □ ≡ ∂µ∂µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The Gertsenstein effect assumes that the energy-momentum tensor Tµν consists entirely of the electromagnetic energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Thus 4πTµν = FµαFν α − 1 4gµνFαβF αβ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='3) where Fµν ≡ ∂Aν ∂xµ − ∂Aµ ∂xν is the usual electromagnetic field tensor and Aµ = (−φ, A) is the vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In terms of the electric and magnetic fields, the Maxwell stress tensor reads 4πTij = −(EiEj + BiBj) + 1 2δij(E · E + B · B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='4) 1We use units in which G = c = 1 and follow MTW [11] conventions throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In particular, Greek indices = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='3 are spacetime indices, while Latin indices are spatial indices = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Repeated indices in any position are summed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 4 With our conventions, Fij = Aj,i − Ai,j = εijkBk and Fi0 = A0,i − Ai,0 ≡ −∂iφ − ˙Ai = Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='2 (Here, εijk is the usual permutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=') We further assume that the electromagnetic wave obeys the vacuum Einstein- Maxwell equations: 1 √g(√gF αβ),β = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='5) which in our case amounts to (gαµgβνFµν),β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='6) With gµν given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='1), working to first order in hµν yields ηαµηβνFµν,β −ηαµhβνFµν,β −ηαµFµνhβν,β −ηβνhαµFµν,β −ηβνFµνhαµ,β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='7) However, in the Lorentz gauge hβν,β ≡ 0, so the third term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Discarding the second and fourth terms as small, but keeping the final term as potentially large leaves ηαµηβνFµν,β −ηβνhαµ,β Fµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='8) For α = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='8) gives merely ∇ · E = 0, as in flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' For α = i, the equation becomes − ˙Ei + εikℓBℓ,k +˙hijEj − hij,k εjkℓBℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='9) (Since hij = hij, it is immaterial whether one writes this equation with indices up or down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=') The ˙Ei term can be eliminated as in freshman physics to get a wave equation for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Taking the curl of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='9) yields −εrpi ˙Ei,p +εrpiεikℓBℓ,kp +εrpi(˙hijEj),p −εrpiεjkℓ(hij,k Bℓ),p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='10) By virtue of the homogeneous Maxwell equation ∇ × E = − ˙B, which is the same as in flat space, the first term is ∂2Br/∂t2 ≡ Br,tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' By virtue of the homogeneous Maxwell equation ∇ · B = 0, the second term is ∇ × ∇ × B = −∇2B ≡ −Br,ℓℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='3 2Because h11 = −h22, we must also have T11 = −T22, which is not generally true for the stress tensor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='4) unless E3 = B3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' This will be the case for transverse electromagnetic waves traveling in the z-direction (“radiation gauge”), which we also assume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 3Here we have used the vector identity ∇ × ∇ × A = ∇(∇ · A) − ∇2A Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 5 Next, we assume that the electric field E is entirely that of the electromagnetic wave, but that the magnetic field consists of the field b of the electromagnetic wave plus a background field B0, such that B = B0 + b with B0 ≫ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' B0 ≫ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Thus we can ignore the third term above as second-order small and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='10) becomes Br,tt −Br,ℓℓ −εrpiεjkℓ(hij,k Bℓ),p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='11) With the same assumptions as above and discarding terms ∼ b2, the gravitational wave equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='2) becomes □hij = 4[B0iB0j + B0ibj + biB0j − 1 2δij(B2 0i + 2B0kbk)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='12) The trick is, first, to choose B0 = constant,4 which linearizes both (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='12), and secondly, to choose B0 at right angles to b, which ensures a nonzero coupling for the h12 mode in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Assuming that the electromagnetic wave is traveling in the z-direction, we take b = by(z, t) and B0 ≡ B0x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='11): ¨by − b′′ y = h′′ 12B0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='13) where the prime (′) indicates differentiation with respect to z, and from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='12): ¨h12 − h′′ 12 = −4B0by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='14) For the given magnetic field configuration, there is no interaction with the other polarization state h11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' It remains only to solve the two coupled equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We search for “wave packet” solutions f the form f(z, t) = fr(z − t)fs(z + t), where r stands for “rapid” and s stands for “slow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The Ur-example would be f = cos(krz − ωrt) cos(ksz + ωst) = 1 2 [cos((kr + ks)z + (ωs − ωr)t) + cos((kr − ks)z − (ωr + ωs)t)] , 4Note that for i = j = 3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='12) gives □h33 ∼ B2 0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' This entails the appearance of a longitudinal component of the gravitational wave h33, induced by T33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Given that in our case □h33 ∼ B0 2 = constant, this mode does not mix with the other two transverse components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' therefore, for the purpose of our derivation, we can effectively ignore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 6 where ωr ≫ ωs and in our units kr = ωr ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' ks = ωs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Such wave packets can thus be regarded either as product functions consisting of a high and low frequency component or as a coherent superposition of waves traveling in the same direction, which can be expected to form beats as in elementary physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Dropping subscripts, we write h = hr(z − t)hs(z + t) and b = br(z − t)bs(z + t), where in our units h′ r = −˙hr and h′ s = ˙hs, with similar expressions for br and bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Inserting the trial functions into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='14) gives b′ rb′ s = −B0 4 (h′′ rhs + 2h′ rh′ s + hrh′′ s) ≈ −B0 4 h′′ rhs, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='15) and h′ rh′ s = B0brbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='16) In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='15) we have approximated h′′ as h′′ rhs, since by assumption hr is rapidly varying and hs is slowly varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The equations can now be solved by inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Pick br = a √B0 h′ r ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' h′ s = a � B0bs ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' b′ s = −B3/2 0 4a hs, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='17) with a an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' These choices satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The functions br and hr which describe the shape of the wave-packets can also be chosen arbitrarily and have no effect on the behavior of the functions bs and hs, which describe the slow oscillation of the packets between photon and graviton states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Letting a = 1/2 gives h′ s = 1 2 � B0bs ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' b′ s = −1 2B3/2 0 hs, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='18) which have solutions hs = A sin �z + t + φ L � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' bs = A � B0 cos �z + t + φ L � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='19) provided that L ≡ 2/B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Notice also that h2 s + b2 s B0 = A2 = constant, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='20) Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 7 which shows that the total energy remains constant as the packet oscillates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The frequency of oscillation is given by the magnitude of the magnetic field, as ωs ≡ L−1 = B0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We have seen that a uniform classical magnetic field B0 catalyzes a linear mixing of the gravitational and photon fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' A quantum of the mixed field oscillates between photon and graviton states with a mixing length L independent of wavelength, namely L = 2 B0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='21) where B0 is the component of the background field perpendicular to the direction of propagation of the quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' This means that if a single photon travels a distance D through the uniform field B0, it will emerge as a graviton with probability P = sin2(D/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='22) The quadratic dependence of P on D makes this process interesting as a possible astrophysical source of gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In cgs units L = 2c2 √ GB0 ≈ 2Mpc �1 Gauss B0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='23) Around a magnetar, the field can be ∼ 1014 G, leading to a mixing length of only ∼ 106 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='22) above, an observer stationed within the boundaries of such a field at a distance D ≫ L from the star could in principle observe a significant periodic change of magnetar luminosity on a timescale of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' More realistically, the extent of a neutron star’s magnetic field is ∼ 10 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In that case the brightness would be diminished by a factor ∼ 10−10 in the direction perpendicular to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In either case, beyond the field boundaries oscillations cease (§3) and an observer sees only a constant flux of photons in a given direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Conceivably two such observers could infer the presence of the Gertsenstein mechanism by comparing photon fluxes in directions parallel and perpendicular to the neutron star B-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' However, since Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 8 the effect is classical, such an observation would not constitute a direct detection of gravitons [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We also note that 1014 G is approximately the Schwinger limit, at which one expects the electromagnetic field equations to become nonlinear and electron-positron pairs to be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Pair creation would in turn alter the index of refraction of the vacuum and change the speed of light, destroying the presumed coherence of the gravitational and electromagnetic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Hence, for such strong fields one should do a more detailed calculation before drawing definite conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 3 Inclusion of Boundary Conditions In the previous section we obtained a general solution for graviton-photon oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Boccaletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' [6] employed a more conventional equation-solving method to obtain a solution in which conversion takes place within a finite region, say for 0 ≤ z ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Such a scenario is probably more relevant to graviton production in the vicinity of neutron stars, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We now employ the Dyson approach to retrieve their results for photon-to-graviton conversion in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' We assume that the constant background magnetic field B0 ≡ B0x is confined to the finite region 0 ≤ z ≤ D (region II), with a plane right-moving electromagnetic wave traveling along the z-axis incident upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' By the Gertsenshtein effect, the electromagnetic wave produces a left-traveling gravitational wave in region I (z < 0), and a right-traveling gravitational wave in region III (z > D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In the region where the magnetic field is non-zero, the transverse component of the gravitational wave h12 is described by a wave packet of the form5 hs(z + t)hr(z − t) = A sin (ωs(z + t)) sin (ωr(z − t)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='1) 5As we saw in §2, the function hr(z − t) can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Here we make the simplifying assumption that hr(z − t) ∝ sin(ωr(z − t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 9 where ωr (ωs ≡ L−1) is the frequency of the rapidly (slowly) oscillating component of the metric, A is the maximum amplitude of the wave, and for simplicity we have set the global phase φ = 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The most general solution for the gravitational wave in region II then is a super- position of wave packets of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='1) and plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Following [6], we turn now to a complex formalism to describe these waves, with the tacit assumption that physical solutions are always given by the real part of complex amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Using this formalism, h12 in region II is given by (the real part of) hII 12(z, t) = aeiωr(z−t) + be−iωr(z+t) + Ae−iωs(z+t)eiωr(z−t) for 0 ≤ z ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='2) The first two terms represent right-traveling and left-traveling plane waves respec- tively, while the third term is a superposition of wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='6 In both regions z < 0 and z > D where the background magnetic field is zero, the wave equation for the metric is □h12 = 0, and the corresponding vacuum solutions are hI 12(z, t) = ce−iωr(z+t) for z < 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='3) and hIII 12(z, t) = deiωr(z−t) for z > D, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='4) describing a generic left-traveling (z < 0) or right-traveling (z > D) plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In order to determine the constant coefficients (a, b, c, d) in terms of A we im- pose the usual boundary conditions demanding continuity of the functions and their 6Specifically, the third term is a combination of the four elementary wave pack- ets cos(ωs(z + t)) cos(ωr(z − t)), cos(ωs(z + t)) sin(ωr(z − t)), sin(ωs(z + t)) cos(ωr(z − t)), and sin(ωs(z + t)) sin(ωr(z − t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 10 derivatives at z = 0 and z = D: hI 12(0, t) = hII 12(0, t), ∂zhI 12(0, t) = ∂zhII 12(0, t), hII 12(D, t) = hIII 12(D, t), ∂zhII 12(D, t) = ∂zhIII 12(D, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='5) Or, in terms of the coefficients: c = a + b + Ae−iωst, c = b − a − � 1 − ωs ωr � Ae−iωst, d = a + be−2iωrD + Ae−iωs(D+t), d = a − be−2iωrD + Ae−iωs(D+t) � 1 − ωs ωr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='6) This is a system of four linear equations for five unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In terms of A, coefficients (a, b, c, d) are: a = −A � 1 − ωs 2ωr � e−iωst, b = −A ωs 2ωr eiD(2ωr−ωs)e−iωst, c = A ωs 2ωr � 1 − eiD(2ωr−ωs)� e−iωst, d = −A � 1 − ωs 2ωr � (1 − e−iDωs)e−iωst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='7) The presence of the time-dependent factor exp(−iωst) means that the coefficients in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='7) are not truly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' However, given our assumption that e−iωst is slowly varying, to first approximation we can consider them time-independent: a = −A � 1 − ωs 2ωr � , b = −A ωs 2ωr eiD(2ωr−ωs), c = A ωs 2ωr (1 − eiD(2ωr−ωs)), d = −A � 1 − ωs 2ωr � (1 − e−iDωs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='8) Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 11 One can easily check that the metric h12 with the coefficients in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='8) satisfies the wave equation in all three regions (in our approximation) as well as the boundary conditions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Hence, the full solution for h12(z, t) is hI 12(z, t) = A ωs 2ωr (1 − eiD(2ωr−ωs))e−iωr(z+t), hII 12(z, t) = −A � 1 − ωs 2ωr � eiωr(z−t) − A ωs 2ωr eiD(2ωr−ωs)e−iωr(z+t) +Ae−iωs(z+t)eiωr(z−t), hIII 12 (z, t) = −A � 1 − ωs 2ωr � (1 − e−iDωs)eiωr(z−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='9) We can similarly derive a solution for the electromagnetic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' From equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='18), br = h′ r/(2√B0) and bs = 2h′ s/√B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Given that hr(z−t) = eiωr(z−t) and hs(z + t) = Ae−iωs(z+t), we have br(z − t) = iωr/(2√B0)eiωr(z−t) and bs(z + t) = −2iAωs/√B0e−iωs(z+t), so the electromagnetic wave packet in region II is of the form br(z − t)bs(z + t) = 1 2Aωre−iωs(z+t)eiωr(z−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='10) By assumption, the incident electromagnetic wave is right-moving in both regions I and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In addition, we need to include a produced left-moving electromagnetic wave in region I, as well as a combination of right-moving and left-moving plane waves in region II, as we did for the gravitational wave in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Consequently, after imposing the relevant boundary conditions as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='5) and disregarding the slowly oscillating term exp(−iωst) ≈ 1, the full solution for b(z, t) is: bI(z, t) = Bωreiωr(z−t) + 1 4(1 − ei(2ωr−ωs)D)Aωse−iωr(z+t), bII(z, t) = � B − 1 2A � 1 − ωs 2ωr �� ωreiωr(z−t) − 1 4ei(2ωr−ωs)DAωse−iωr(z+t) +1 2Aωre−iωs(z+t)eiωr(z−t), bIII(z, t) = � B − 1 2A � 1 − ωs 2ωr � (1 − e−iDωs) � ωreiωr(z−t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='11) where B is the incident electromagnetic wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Thus far, the coefficient B is undetermined because we have had to solve for five coefficients but have only four Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 12 boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' However, one can easily check that the wave equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='14) for h12 in region II is consistent with the second of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='11) only if B = A/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' One can also check that the resulting expressions for h12 and b conserve energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The solutions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='11) describe the conversion of an electromagnetic wave to a gravitational wave, catalyzed by the constant background magnetic field B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In region II where B0 ̸= 0, the electromagnetic and gravitational amplitudes slowly oscillate between each other via the conversion term exp(−iωs(z + t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Outside, in regions I and III, the oscillation stops and the wave packets propagate independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The solution of [6] is an approximation of ours for ωs ≪ ωr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In fact, to leading order in ωs, the gravitational wave solution in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='9) becomes hI 12(z, t) = A ωs 2ωr (1 − e2iDωr)e−iωr(z+t), hII 12(z, t) = A ωs 2ωr eiωr(z−t) − A ωs 2ωr e2iDωre−iωr(z+t) − iωszAeiωr(z−t), hIII 12 (z, t) = −iDωsAeiωr(z−t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='12) where we have used e−iωsz ≈ 1 − iωsz for small ωs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The coefficients of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='12) are the same as in [6], given that the authors use a convention in which the wave packet solution is given by λz/(2iωr) exp(iωr(z − t)), with λ = 2ωrωsA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In this approximation, the amplitude of the gravitational wave in region I vanishes if the condition exp(2iDωr) = 1 is satisfied, and the amplitude of the gravitational wave in region III depends linearly upon the length and magnitude of the static field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Similarly, the approximate solution for the electromagnetic wave is, to leading order in ωs, bI(z, t) = 1 2Aωreiωr(z−t) + 1 4(1 − e2iDωr)Aωse−iωr(z+t), bII(z, t) = 1 2Aωreiωr(z−t) − 1 4e2iDωrAωse−iωr(z+t) − 1 2iAωrωszeiωr(z−t), bIII(z, t) = 1 2 (1 − iDωs) Aωreiωr(z−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='13) If the condition e2iDωr = 1 is satisfied, the back-propagating wave amplitude vanishes Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 13 and only the incident wave is left in region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' The solution in this approximation describes an incident electromagnetic wave of amplitude A and energy ωr that prop- agates into region II and slowly converts to a gravitational wave with a rate given by ωs ≡ L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' When the wave enters region III, the incident electromagnetic wave ampli- tude is effectively reduced by a factor of iDωs while the gravitational wave amplitude is increased by that same factor: EM : A → A � 1 − iD L � , GW : 0 → −iAD L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='14) 4 Axion-Photon Conversion Quite generally the Lagrangian density for the interaction of gravitational waves with matter is [13] L = −1 2hµνT µν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='1) where in our problem T µν is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Currently, there is a large interest in axion-photon conversion [14, 15, 16], as axions are presently the leading candidates for dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In a process much like the Gertsenshtein mechanism, axions emerging from the sun can be transformed into X-ray photons by a strong laboratory magnetic field and subsequently detected by an X-ray telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Indeed, the axion-photon interaction Lagrangian can be written as L = −gaγ 4 aFµν ˜F µν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='2) where ˜F µν ≡ 1 2εµναβFαβ is the dual electromagnetic field tensor, a is the axion ampli- tude and gaγ is the axion-photon coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In terms of the electric and magnetic fields, the above expression amounts to L = gaγ a E · B , which is essentially the same Lagrangian as the one we have been Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 14 using except that the external magnetic field couples to the E-field of the photon rather than the B-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Given the quadratic form of Lagrangians in general and the similarity of the axion-photon Lagrangian to the gravitational interaction Lagrangian in particular, we may conjecture that Gertshenstein-like mechanisms which cause mixing among many different fields are “universal” and have widespread cosmological utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' In future papers we hope to extend the effect to the Yang Mills field as well as consider the possibility that graviton-photon mixing in the long-wavelength limit might generate something resembling a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Gertsenshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 15 References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Gertsenshtein, “Wave resonance of light and gravitational waves,” Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 41, 113-114 (1961);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' Soviet Physics JETP 14, 84-85 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htA0T4oBgHgl3EQfIP9i/content/2301.02072v1.pdf'} +page_content=' 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b/jdE0T4oBgHgl3EQf7QI8/content/tmp_files/2301.02773v1.pdf.txt @@ -0,0 +1,439 @@ + + + + + + + + + + + + +Building a Parallel Corpus and Training Translation +Models Between Luganda and English +Richard Kimera, Daniela N. Rim, Heeyoul Choi + + +Abstract: Neural machine translation (NMT) has achieved great successes with +large datasets, so NMT is more premised on high-resource languages. This +continuously underpins the low resource languages such as Luganda due to the +lack of high-quality parallel corpora, so even ‘Google translate’ does not serve +Luganda at the time of this writing. In this paper, we build a parallel corpus with +41,070 pairwise sentences for Luganda and English which is based on three +different open-sourced corpora. Then, we train NMT models with hyper-parameter +search on the dataset. Experiments gave us a BLEU score of 21.28 from Luganda +to English and 17.47 from English to Luganda. Some translation examples show +high quality of the translation. We believe that our model is the first Luganda- +English NMT model. The bilingual dataset we built will be available to the public. + +Keywords : Luganda, neural machine translation, Transformer, hyper-parameter + + + +1. Introduction +Uganda has not had an official language survey since 1971, and most scholars and reports have +provided varying statistics about languages in Uganda. It is reported that there are 43 living indigenous +languages [1] down from 63 varieties identified in 1972 [2]. +Even with this diversity, the constitution provides for English as the official language and Swahili +widely spoken in East Africa as the second language [3]. English is common among the elites, and is the +language of instruction in schools and offices, serving as a unifying language in the country. However, +Swahili remains largely expressed in the border districts and is not a de-facto language in the parliament +or courts of law in Uganda [1]. Instead, Luganda is steadily positioning itself as an undeclared national +language in Uganda with the last published report of 1998 showing a total of 4,130,000 speakers [4] out +of a population of 22 million people. Recent reports indicate that over 20 million people can speak the +language due to its influence in the business domain [5]. +With the recent increase in the need for technology to deliver and access services online, +communication is a catalyst. This calls for the use of language to pass on information between the +involved parties. A case in point is using web portals to apply for and receive services like passports, +visas, business, and tax registration. An assessment of 78 Uganda’s E-government websites called for +an improvement in making the websites multilingual [6]. These are usually delivered in an official +language, English, that is not well understood by the Illiterate. Machine Translation (MT) has the power +to utilize Natural Language Processing (NLP) models [7, 8, 9, 10] to translate these sites into the local +languages. +However, since the recent NLP models need a large dataset as well as enough computation power to +train the model on the datasets, it has been challenging to utilize NLP models in Uganda for technological +advancements and service delivery due to the absence of a corpus, especially for the Luganda language. +Furthermore, as of 1st May 2022, platforms like Google translate, and IBM Watson have not yet +incorporated Luganda into their translation systems. Moreso, the current NLP research that is done on +the Luganda language has mainly focused on building small datasets [11, 12, 13]. +In this work, we build a bilingual parallel corpus for Luganda and English by combining the various +existing open datasets. We describe the data acquisition and preprocessing steps with the source of the +datasets. We also apply the Transformer [10] for Neural Machine Translation (NMT) using the built +dataset, which confirms the quality of the dataset. We train the model architectures with many different +hyper-parameter configurations and the best one is obtained by the Bayesian approach that automates +the hyper-parameter search. +In experiments, we achieved a BLEU score of 21.28 from Luganda to English and 17.47 from English +to Luganda on the test data with 1,233 pairwise sentences. Also, we present some translation examples +to show the quality of the translation. To the best of our knowledge, this is the first NMT models between +Luganda and English, and we will open our bilingual dataset to the public. +2. Background +In this section, we present the key models used in MT and why we chose to use the Transformer. +2.1. +Neural Machine Translation (NMT) +Inspired by the human brain, there are several types of Neural Networks (NNs) including feedforward, +recurrent and convolutional networks [14]. They can further be classified as Deep Neural Networks + + + +(DNNs) with an increase in the number of layers [7]. DNNs have made NMT the mainstream approach +and technology to MT systems [15, 16, 17]. +An NMT model is trained on a parallel text corpus. Using the Encoder-Decoder (E-D) architecture, a +sentence from the source language is translated into a fixed-length vector by the encoder network which +is then decoded to the target language [8]. +Recurrent Neural Networks (RNNs) are one of the architectural choices for NMT, since they can +process sequential data [7, 8, 9]. One of the setbacks of RNNs is the vanishing gradient, whereby longer +sentences tend to deteriorate the performance of the E-D architecture even with Long Short-Term +Memory (LSTM) [9, 18, 19]. Although not as popular as RNNs, Convolutional Neural Networks (CNN) +were also used in NMT [20, 21, 22]. +2.2. +Transformer +Transformer was proposed to replace the previous network architectures like RNNs and CNNs in +language processing [10]. Self-attention and feed-forward connections used together made the +Transformer architecture a key model for advancing the field of neural machine translation. It increased +the efficiency and reduced the speed of convergence. Transformer also supports parallel processing, +hence leveraging the power of modern GPUs [10]. +Generally, the Transformer architecture consists of encoder and decoder blocks. The encoder receives +an input sequence !x("), x($), . . . , x(%!)% and processes it, and the decoder generates an output sequence +!y("), . . . , y(%")%, one element at a time [9]. The encoder and decoder consist of N stacked layers that +include a multi-head attention and feed-forward layer. The decoder adds extra attention layer for the +output of the encoder. It is finally connected to the linear and SoftMax layers to output the probabilities +of the target language based on the input. +The Transformer has been widely utilized in high-resource languages like English, French, and +German. The original paper utilized datasets of 4.5M (English to German) and 36M (English to French) +sentence pairs [10]. Such a large dataset is not yet existent for the Luganda language; however, it is +possible to utilize a smaller dataset as low as 30,000 as evidenced in English – isiXhosa translation [23]. +In this paper, we use a dataset (Luganda-English pairwise corpus) focused on the use of Transformer for +translation. +3. Luganda Datasets and NMT Training +Even though MT has been highly embraced, its utilization on the Luganda language has been a +challenge due to the absence of the required quality of corpora. A many-to-many multilingual translation +for African languages, excluding Luganda, was built with a mixture of biblical [24] and non-biblical +corpora [25]. The training was based on a pre-trained variant of the transformer-based architecture mT5 +model [26]. To solve the dataset challenge, as one of their goals, the Masakhane society [27] brings +together NLP researchers in Africa to spur research in African languages. They focus on using a more +participatory approach. +There has been an effort to build a model using a Luganda monolingual dataset for text to speech [28]. +This research was limited to the usage of a small corpus. Even though the dataset was publicly published, +it was not helpful in our research due to its monolingual nature. A community-based method was used +to create a corpus of five based Ugandan languages (Ateso, Luganda, Lugbara, and Runyankole) [29]. +Religious datasets have equally been utilized by web scrapping [30] text from websites (watchtower), +magazines (awake), and the Bible [24, 31, 32]. + + + +An MT model to translate from Lumasaaba to English was trained on a Bible-based text corpus [33]. +Lumaasaba is a closely related language to Luganda, however, the use of the Bible corpus is underscored +as the language context does not usually represent the local contextual use [29]. +As far as we know, there have been small bilingual datasets published [11, 12, 13], which calls for the +expansion of their work. Additionally, NMT models had not yet been tried on any of the datasets. +3.1. +Data Acquisition +We sourced and built a corpus from research centers; three different corpora were merged, and the +quality of the translation was checked. +Corpus 1: This dataset was published in 2022 by Zenodo in collaboration with a team of researchers +from Makerere AI and data science research lab, Luganda teachers, students, and freelancers. It consisted +of a total of 1,042 English and Luganda parallel sentences [12]. The original text was downloaded as a +csv file format, and text sentences have been extracted as seen in Table 1. + +English +Luganda +The teacher taught us how to multiply numbers yesterday. Omusomesa yatusomesezza okukubisa emiwendo eggulo. +The doctor asked to see me physically because he couldn't +diagnose me without a proper checkup. +Ddokita yasaba okundaba mu buliwo kubanga yali tasobola +kumanya kinnuma nga takoze kukebera kutuufu. +Table 1. Sample sentences from Corpus 1 + +Corpus 2: This dataset consists of 15,022 English-Luganda parallel sentences published in 2021. We +show some sample text from the dataset as seen in Table 2. The corpus was built using the same +procedure as Corpus 1 [11], which was originally a csv file. + +English +Luganda +Refugees had misunderstandings between themselves. +Abanoonyiboobubudamu +b'abadde +n'obutakkaanya +wakati +waabwe. +We were urged to welcome refugees into our +communities. +Twakubirizibwa okwaniriza abanoonyiboobubudamu mu bitundu +byaffe. +Table 2. Sample sentences from Corpus 2 + +Corpus 3: This dataset had a total of 25,006 language phrases and was released in 2021 by sunbird AI +in collaboration with the Makerere AI lab. The dataset consisted of a parallel text in English with +corresponding translations in 5 local languages (Luganda, Lugbara, Runyakitara, Acholi, and Ateso). +The English text was sourced from social media, transcripts from radio, online newspapers, articles, +blogs, text contributions from Makerere University NLP community, and farmer responses from surveys +[34]. The JSON format corpus was downloaded from the company’s GitHub account [13]. Table 3 shows +a set of the sampled text in the JSON format from the dataset. +Given the three datasets in csv or JSON file formats, we merged them into an xlsx format. The new +created dataset was cleaned and formed a corpus of 41,070 pairwise sentences. We then proceeded with +data preprocessing which consisted of splitting and tokenization. The corpus was split into training, +validation, and testing portions of 94% (38,641), 3%, and 3%, respectively. This was done for each +language and a total of 6 sets were generated. After tokenizing each set, we used Byte-Pair Encoding +(BPE) to create a vocabulary of 10,000 words that were later indexed [35]. + +“{"English":"Eggplants always grow best under warm conditions.", +“Luganda”:”Bbiringanya lubeerera asinga kukulira mu mbeera ya bugumu”, + + + +"Runyankole":"Entonga buriijo zikurira omu mbeera y'obwire erikutagata", +“Ateso”:”Epoloi ebirinyanyi ojok apakio nu emwanar akwap.”, +"Lugbara":"Birinyanya eyi zo kililiru ndeni angu driza ma alia.", +"Acholi":"Bilinyanya pol kare dongo maber ka lyeto tye"}” +Table 3. One sample from Corpus 3 in the JSON format with six languages: English, Luganda, Runyankole, Ateso, +Lugbara, and Acholi + +We further went ahead to confirm the quality of the corpus manually by checking that the translation +was of a high standard. In this NMT task, we implemented a Transformer using the PyTorch library, the +scripts were written in both Python and Perl. The experiments were run on an NVIDIA GeForce RTX +2080 Ti GPU on top of a Linux-powered OS. +3.2. +Training +We trained an English-Luganda (En2Lu) model and a Luganda-English (Lu2En) model. During the +training process, Adam [36] was used as the optimizer, and the early stopping trick was implemented to +prevent the model from overfitting. We applied the trained models to check for performance using the +test datasets. The experiments were conducted on a vocabulary of 10,000 words, with 6 layers of the +encoder and 6 layers of the decoder [37]. +To find the best hyper-parameters, we interfaced with weights and biases (wandb) [38], an MLOps +online platform, we were able to keep track of our experiments as well as automate the hyper-parameter +search. We sampled hyper-parameters using the Bayes approach aimed at maximizing the BLEU score. +The Bayes approach uses the Gaussian process to model the process between parameters to optimize the +probability of improvement [39]. Table 4 shows the number lists that were used in the hyper-parameter +search. + +dim_model +tm_dim_ff +Batch_size +8, 16, 32, 64, 128, 256, 512, 1024, 2048 +8, 16, 32, 64, 128, 256, 512, 1024, 2048 +8, 16, 32, 64, 128 +Table 4. Range of the hyper-parameter search for ‘dim_model’ (dimension model in Transformer), ‘tm_dim_ff’ +(dimension in the feed forward layer), and the batch size +4. Results +4.1. +BLEU Score +We used the BLEU score to evaluate the quality of training and to select the best models. The default +hyperparameters used during the training of both models were (dim_model=512, tm_dim_ff=2048 and +Batch size=64). The best parameter configurations by hyper-parameter search are (256, 2048, 16) and +(512, 128, 32) for the En2Lu and Lu2En models, respectively, which achieve the best BLUE scores. + + +En2Lu +Lu2En +Valid (Before hyper-parameter search) +16.13 +20.59 +Valid (After hyper-parameter search) +17.60 +23.09 +Test +17.47 +21.28 +Table 5. BLEU scores of translations (the test scores were measured from the model with the best hyper-parameter) + +Table 5 shows that there is an improvement in the BLEU score of the validation set by +1.47 for En2Lu +and +2.5 for Lu2En based on the hyper-parameter search. When we applied the model to the test dataset, +the BLEU scores are 17.47 and 21.28 for En2Lu and Lu2En, respectively. + + + +To understand the effect of hyper-parameters, we present the relevancy and correlation of the hyper- +parameters to the prediction of the BLUE score in Figure 1. We used a sample configuration of 30 runs, +each of which was a combination of ‘dim_model’, ‘tm_dim_ff’, and ‘Batch_size’. For example, a +combination (16, 32, 1024) makes a BLEU score of 13.96. +In Figure 1, we can see that there is a positive correlation between the BLEU score and the dim_model +as well with tm_dim_ff, and a negative correlation with the batch size by the end of the hyper-parameter +search. In addition, using Random Forest, the importance of the BLEU score was also calculated. + + + +Figure 1. Correlation and importance of hyperparameters after training with wandb + +The results indicate that for the En2Lu model, a moderately high dim_model value results in a high +BLEU score, a low batch size results in a high BLEU score, and a slightly lower tm_dim_ff results in a +slightly high BLEU score. Also, the dim_model is very important in determining the BLEU score, +followed by the batch size and lastly tm_dim_ff. +4.2. +Quality of Translation +After the selection of the best trained and validated model, it was applied to the test dataset to check +the quality of translation. We selected some sample text from each of the languages. The first sentence +as reflected in Tables 6 and 7 had a high translation quality meaning that it was not paraphrased by the +model, whereas the second sentence had a low translation quality meaning that it was paraphrased by +the model. +In both Tables 6 and 7, the same sentences are translated between both languages by the En2Lu and +Lu2En models, respectively. The first sentence source (Src) in both tables is translated to the target +language (Output) which is exactly the same as the target sentence (Trg). The translation of the second +sentence is acceptable since the meaning of the output is quite similar to the meaning of the Trg sentence, +though it is not exactly the same. The imperfect part in translation is underlined for each table. In Table +4, we have included the English translation of the Luganda output text in the parenthesis to show that +the meaning is close to the Src sentence. + +Src (Eng) +Trg (Luganda) +Output (Luganda) +Mob Justice is highly Punished +Okutwalira amateeka mu ngalo +kibonerezebwa nnyo . +Okutwalira amateeka mu ngalo +kibonerezebwa nnyo . +People engaging in +deforestation have been arrested +Abantu abeenyigira mu kusaanyawo ebibira +bakwatiddwa +Abantu abeenyigira mu kutema emiti +bakwatiddwa +(People engaging in cutting down trees have +been arrested) +Table 6. Examples of translation from English to Luganda + +: ::: +Parameterimportance with respect to +ill bleu +Q Search +Parameters +1-3- of 3 +Config parameter +Importance@ +Correlation +dim_model +batch_size +tm_dim_ff + + +Src (Luganda) +Trg (Eng) +Output (Eng) +Okutwalira +amateeka +mu +ngalo +kibonerezebwa nnyo +Mob justice is punishable +Mob justice is punishable +Abantu abeenyigira mu kusaanyawo +ebibira bakwatiddwa +People Engaging in deforestation +have been arrested +People who engage in forestry activities have +been arrested +Table 7. Examples of translation from Luganda to English +5. Conclusion +Deep learning architectures require the use of very large datasets to train neural machine translation +models. In this paper, we built a pairwise corpus of Luganda and English with a total of 41,070 sentences. +We then trained NMT models by using the Transformer architecture with hyper-parameter search. The +results indicate that the best trained model gives us a BLEU score of 17.47 for translation from English +to Luganda and 21.28 from Luganda to English. +In the future, we intend to compare our translation with Google translate. By the time of submission +of this paper, Google justified the need for the Luganda language and included it among the languages +on the Google translate platform on May 11th, 2022. +References +[1] S. Namyalo, B. Isingoma, and C. Meierkord, “Towards assessing the space of English in Uganda’s linguistic +ecology,” Ugandan English, pp. 19–50, Oct. 2016, doi: 10.1075/veaw.g59.02nam. +[2] P. Ladefoged, R. Glick, and Clive Criper, “Language in Uganda”. London, New York, Oxford University Press, +1972. +[3] Parliament +of +the +Republic +of +Uganda. “The +Constitution +(amendment) +act,” +2005. +https://www.parliament.go.ug/cmis/views/87694c9d-0255-4d29-bf41-d89407bfc287%253B1.0 +[4] J. 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Choi, "Kor-Eng NMT using Symbolization of Proper Nouns", Journal of +KIISE, vol. 48, no. 10, pp. 1084-1089, 2021. +[38] “Weights & Biases – Developer tools for ML,” wandb.ai. https://wandb.ai +[39] “Sweep Configuration - Documentation,” Wandb.ai, 2022. https://docs.wandb.ai/guides/sweeps/configuration + + diff --git a/jdE0T4oBgHgl3EQf7QI8/content/tmp_files/load_file.txt b/jdE0T4oBgHgl3EQf7QI8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae9f44907cff22bd6edd7e342d440cceac18cf7c --- /dev/null +++ b/jdE0T4oBgHgl3EQf7QI8/content/tmp_files/load_file.txt @@ -0,0 +1,528 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf,len=527 +page_content='Building a Parallel Corpus and Training Translation Models Between Luganda and English Richard Kimera, Daniela N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Rim, Heeyoul Choi Abstract: Neural machine translation (NMT) has achieved great successes with large datasets, so NMT is more premised on high-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' This continuously underpins the low resource languages such as Luganda due to the lack of high-quality parallel corpora, so even ‘Google translate’ does not serve Luganda at the time of this writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In this paper, we build a parallel corpus with 41,070 pairwise sentences for Luganda and English which is based on three different open-sourced corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Then, we train NMT models with hyper-parameter search on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Experiments gave us a BLEU score of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='28 from Luganda to English and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='47 from English to Luganda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Some translation examples show high quality of the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We believe that our model is the first Luganda- English NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The bilingual dataset we built will be available to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Keywords : Luganda, neural machine translation, Transformer, hyper-parameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Introduction Uganda has not had an official language survey since 1971, and most scholars and reports have provided varying statistics about languages in Uganda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' It is reported that there are 43 living indigenous languages [1] down from 63 varieties identified in 1972 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Even with this diversity, the constitution provides for English as the official language and Swahili widely spoken in East Africa as the second language [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' English is common among the elites, and is the language of instruction in schools and offices, serving as a unifying language in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' However, Swahili remains largely expressed in the border districts and is not a de-facto language in the parliament or courts of law in Uganda [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Instead, Luganda is steadily positioning itself as an undeclared national language in Uganda with the last published report of 1998 showing a total of 4,130,000 speakers [4] out of a population of 22 million people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Recent reports indicate that over 20 million people can speak the language due to its influence in the business domain [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' With the recent increase in the need for technology to deliver and access services online, communication is a catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' This calls for the use of language to pass on information between the involved parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' A case in point is using web portals to apply for and receive services like passports, visas, business, and tax registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' An assessment of 78 Uganda’s E-government websites called for an improvement in making the websites multilingual [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' These are usually delivered in an official language, English, that is not well understood by the Illiterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Machine Translation (MT) has the power to utilize Natural Language Processing (NLP) models [7, 8, 9, 10] to translate these sites into the local languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' However, since the recent NLP models need a large dataset as well as enough computation power to train the model on the datasets, it has been challenging to utilize NLP models in Uganda for technological advancements and service delivery due to the absence of a corpus, especially for the Luganda language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Furthermore, as of 1st May 2022, platforms like Google translate, and IBM Watson have not yet incorporated Luganda into their translation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Moreso, the current NLP research that is done on the Luganda language has mainly focused on building small datasets [11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In this work, we build a bilingual parallel corpus for Luganda and English by combining the various existing open datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We describe the data acquisition and preprocessing steps with the source of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We also apply the Transformer [10] for Neural Machine Translation (NMT) using the built dataset, which confirms the quality of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We train the model architectures with many different hyper-parameter configurations and the best one is obtained by the Bayesian approach that automates the hyper-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In experiments, we achieved a BLEU score of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='28 from Luganda to English and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='47 from English to Luganda on the test data with 1,233 pairwise sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Also, we present some translation examples to show the quality of the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' To the best of our knowledge, this is the first NMT models between Luganda and English, and we will open our bilingual dataset to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Background In this section, we present the key models used in MT and why we chose to use the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Neural Machine Translation (NMT) Inspired by the human brain, there are several types of Neural Networks (NNs) including feedforward, recurrent and convolutional networks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' They can further be classified as Deep Neural Networks (DNNs) with an increase in the number of layers [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' DNNs have made NMT the mainstream approach and technology to MT systems [15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' An NMT model is trained on a parallel text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Using the Encoder-Decoder (E-D) architecture, a sentence from the source language is translated into a fixed-length vector by the encoder network which is then decoded to the target language [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Recurrent Neural Networks (RNNs) are one of the architectural choices for NMT, since they can process sequential data [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' One of the setbacks of RNNs is the vanishing gradient, whereby longer sentences tend to deteriorate the performance of the E-D architecture even with Long Short-Term Memory (LSTM) [9, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Although not as popular as RNNs, Convolutional Neural Networks (CNN) were also used in NMT [20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Transformer Transformer was proposed to replace the previous network architectures like RNNs and CNNs in language processing [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Self-attention and feed-forward connections used together made the Transformer architecture a key model for advancing the field of neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' It increased the efficiency and reduced the speed of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Transformer also supports parallel processing, hence leveraging the power of modern GPUs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Generally, the Transformer architecture consists of encoder and decoder blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The encoder receives an input sequence !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='x("), x($), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' , x(%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' )% and processes it, and the decoder generates an output sequence !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='y("), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' , y(%")%, one element at a time [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The encoder and decoder consist of N stacked layers that include a multi-head attention and feed-forward layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The decoder adds extra attention layer for the output of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' It is finally connected to the linear and SoftMax layers to output the probabilities of the target language based on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The Transformer has been widely utilized in high-resource languages like English, French, and German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The original paper utilized datasets of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='5M (English to German) and 36M (English to French) sentence pairs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Such a large dataset is not yet existent for the Luganda language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' however, it is possible to utilize a smaller dataset as low as 30,000 as evidenced in English – isiXhosa translation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In this paper, we use a dataset (Luganda-English pairwise corpus) focused on the use of Transformer for translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Luganda Datasets and NMT Training Even though MT has been highly embraced, its utilization on the Luganda language has been a challenge due to the absence of the required quality of corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' A many-to-many multilingual translation for African languages, excluding Luganda, was built with a mixture of biblical [24] and non-biblical corpora [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The training was based on a pre-trained variant of the transformer-based architecture mT5 model [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' To solve the dataset challenge, as one of their goals, the Masakhane society [27] brings together NLP researchers in Africa to spur research in African languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' They focus on using a more participatory approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' There has been an effort to build a model using a Luganda monolingual dataset for text to speech [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' This research was limited to the usage of a small corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Even though the dataset was publicly published, it was not helpful in our research due to its monolingual nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' A community-based method was used to create a corpus of five based Ugandan languages (Ateso, Luganda, Lugbara, and Runyankole) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Religious datasets have equally been utilized by web scrapping [30] text from websites (watchtower), magazines (awake), and the Bible [24, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' An MT model to translate from Lumasaaba to English was trained on a Bible-based text corpus [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Lumaasaba is a closely related language to Luganda, however, the use of the Bible corpus is underscored as the language context does not usually represent the local contextual use [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' As far as we know, there have been small bilingual datasets published [11, 12, 13], which calls for the expansion of their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Additionally, NMT models had not yet been tried on any of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Data Acquisition We sourced and built a corpus from research centers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' three different corpora were merged, and the quality of the translation was checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Corpus 1: This dataset was published in 2022 by Zenodo in collaboration with a team of researchers from Makerere AI and data science research lab, Luganda teachers, students, and freelancers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' It consisted of a total of 1,042 English and Luganda parallel sentences [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The original text was downloaded as a csv file format, and text sentences have been extracted as seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' English Luganda The teacher taught us how to multiply numbers yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Omusomesa yatusomesezza okukubisa emiwendo eggulo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=" The doctor asked to see me physically because he couldn't diagnose me without a proper checkup." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Ddokita yasaba okundaba mu buliwo kubanga yali tasobola kumanya kinnuma nga takoze kukebera kutuufu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Sample sentences from Corpus 1 Corpus 2: This dataset consists of 15,022 English-Luganda parallel sentences published in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We show some sample text from the dataset as seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The corpus was built using the same procedure as Corpus 1 [11], which was originally a csv file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' English Luganda Refugees had misunderstandings between themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=" Abanoonyiboobubudamu b'abadde n'obutakkaanya wakati waabwe." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We were urged to welcome refugees into our communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Twakubirizibwa okwaniriza abanoonyiboobubudamu mu bitundu byaffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Sample sentences from Corpus 2 Corpus 3: This dataset had a total of 25,006 language phrases and was released in 2021 by sunbird AI in collaboration with the Makerere AI lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The dataset consisted of a parallel text in English with corresponding translations in 5 local languages (Luganda, Lugbara, Runyakitara, Acholi, and Ateso).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The English text was sourced from social media, transcripts from radio, online newspapers, articles, blogs, text contributions from Makerere University NLP community, and farmer responses from surveys [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The JSON format corpus was downloaded from the company’s GitHub account [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Table 3 shows a set of the sampled text in the JSON format from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Given the three datasets in csv or JSON file formats, we merged them into an xlsx format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The new created dataset was cleaned and formed a corpus of 41,070 pairwise sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We then proceeded with data preprocessing which consisted of splitting and tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The corpus was split into training, validation, and testing portions of 94% (38,641), 3%, and 3%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' This was done for each language and a total of 6 sets were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' After tokenizing each set, we used Byte-Pair Encoding (BPE) to create a vocabulary of 10,000 words that were later indexed [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' “{"English":"Eggplants always grow best under warm conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' ", “Luganda”:”Bbiringanya lubeerera asinga kukulira mu mbeera ya bugumu”, "Runyankole":"Entonga buriijo zikurira omu mbeera y\'obwire erikutagata", “Ateso”:”Epoloi ebirinyanyi ojok apakio nu emwanar akwap.”, "Lugbara":"Birinyanya eyi zo kililiru ndeni angu driza ma alia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' ", "Acholi":"Bilinyanya pol kare dongo maber ka lyeto tye"}” Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' One sample from Corpus 3 in the JSON format with six languages: English, Luganda, Runyankole, Ateso, Lugbara, and Acholi We further went ahead to confirm the quality of the corpus manually by checking that the translation was of a high standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In this NMT task, we implemented a Transformer using the PyTorch library, the scripts were written in both Python and Perl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The experiments were run on an NVIDIA GeForce RTX 2080 Ti GPU on top of a Linux-powered OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Training We trained an English-Luganda (En2Lu) model and a Luganda-English (Lu2En) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' During the training process, Adam [36] was used as the optimizer, and the early stopping trick was implemented to prevent the model from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We applied the trained models to check for performance using the test datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The experiments were conducted on a vocabulary of 10,000 words, with 6 layers of the encoder and 6 layers of the decoder [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' To find the best hyper-parameters, we interfaced with weights and biases (wandb) [38], an MLOps online platform, we were able to keep track of our experiments as well as automate the hyper-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We sampled hyper-parameters using the Bayes approach aimed at maximizing the BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The Bayes approach uses the Gaussian process to model the process between parameters to optimize the probability of improvement [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Table 4 shows the number lists that were used in the hyper-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' dim_model tm_dim_ff Batch_size 8, 16, 32, 64, 128, 256, 512, 1024, 2048 8, 16, 32, 64, 128, 256, 512, 1024, 2048 8, 16, 32, 64, 128 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Range of the hyper-parameter search for ‘dim_model’ (dimension model in Transformer), ‘tm_dim_ff’ (dimension in the feed forward layer), and the batch size 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' BLEU Score We used the BLEU score to evaluate the quality of training and to select the best models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The default hyperparameters used during the training of both models were (dim_model=512, tm_dim_ff=2048 and Batch size=64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The best parameter configurations by hyper-parameter search are (256, 2048, 16) and (512, 128, 32) for the En2Lu and Lu2En models, respectively, which achieve the best BLUE scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' En2Lu Lu2En Valid (Before hyper-parameter search) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='13 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='59 Valid (After hyper-parameter search) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='60 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='09 Test 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='47 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='28 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' BLEU scores of translations (the test scores were measured from the model with the best hyper-parameter) Table 5 shows that there is an improvement in the BLEU score of the validation set by +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='47 for En2Lu and +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='5 for Lu2En based on the hyper-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' When we applied the model to the test dataset, the BLEU scores are 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='47 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='28 for En2Lu and Lu2En, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' To understand the effect of hyper-parameters, we present the relevancy and correlation of the hyper- parameters to the prediction of the BLUE score in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We used a sample configuration of 30 runs, each of which was a combination of ‘dim_model’, ‘tm_dim_ff’, and ‘Batch_size’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' For example, a combination (16, 32, 1024) makes a BLEU score of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In Figure 1, we can see that there is a positive correlation between the BLEU score and the dim_model as well with tm_dim_ff, and a negative correlation with the batch size by the end of the hyper-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In addition, using Random Forest, the importance of the BLEU score was also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Correlation and importance of hyperparameters after training with wandb The results indicate that for the En2Lu model, a moderately high dim_model value results in a high BLEU score, a low batch size results in a high BLEU score, and a slightly lower tm_dim_ff results in a slightly high BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Also, the dim_model is very important in determining the BLEU score, followed by the batch size and lastly tm_dim_ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Quality of Translation After the selection of the best trained and validated model, it was applied to the test dataset to check the quality of translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We selected some sample text from each of the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The first sentence as reflected in Tables 6 and 7 had a high translation quality meaning that it was not paraphrased by the model, whereas the second sentence had a low translation quality meaning that it was paraphrased by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In both Tables 6 and 7, the same sentences are translated between both languages by the En2Lu and Lu2En models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The first sentence source (Src) in both tables is translated to the target language (Output) which is exactly the same as the target sentence (Trg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The translation of the second sentence is acceptable since the meaning of the output is quite similar to the meaning of the Trg sentence, though it is not exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The imperfect part in translation is underlined for each table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In Table 4, we have included the English translation of the Luganda output text in the parenthesis to show that the meaning is close to the Src sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Src (Eng) Trg (Luganda) Output (Luganda) Mob Justice is highly Punished Okutwalira amateeka mu ngalo kibonerezebwa nnyo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Okutwalira amateeka mu ngalo kibonerezebwa nnyo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' People engaging in deforestation have been arrested Abantu abeenyigira mu kusaanyawo ebibira bakwatiddwa Abantu abeenyigira mu kutema emiti bakwatiddwa (People engaging in cutting down trees have been arrested) Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Examples of translation from English to Luganda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=': ::: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='Parameterimportance with respect to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='ill bleu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='Q Search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='1 3 of 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='Config parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='Importance@ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='Correlation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='dim_model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='batch_size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='tm_dim_ff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='Src (Luganda) Trg (Eng) Output (Eng) Okutwalira amateeka mu ngalo kibonerezebwa nnyo Mob justice is punishable Mob justice is punishable Abantu abeenyigira mu kusaanyawo ebibira bakwatiddwa People Engaging in deforestation have been arrested People who engage in forestry activities have been arrested Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Examples of translation from Luganda to English 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' Conclusion Deep learning architectures require the use of very large datasets to train neural machine translation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In this paper, we built a pairwise corpus of Luganda and English with a total of 41,070 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' We then trained NMT models by using the Transformer architecture with hyper-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' The results indicate that the best trained model gives us a BLEU score of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='47 for translation from English to Luganda and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content='28 from Luganda to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' In the future, we intend to compare our translation with Google translate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' By the time of submission of this paper, Google justified the need for the Luganda language and included it among the languages on the Google translate platform on May 11th, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf'} +page_content=' References [1] S.' 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sha256:be696543cc6299f3a8b47c0bf677213ff3f9aac056e8c2120fc293dea838dac5 +size 73924 diff --git a/k9E4T4oBgHgl3EQftQ0b/content/tmp_files/2301.05222v1.pdf.txt b/k9E4T4oBgHgl3EQftQ0b/content/tmp_files/2301.05222v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..37d03c42c86fa472504c07fb701c9dffada026fb --- /dev/null +++ b/k9E4T4oBgHgl3EQftQ0b/content/tmp_files/2301.05222v1.pdf.txt @@ -0,0 +1,2338 @@ +New Non-Abelian Reissner-Nordstr¨om Black Hole Solutions in the Generalized SU(2) +Proca Theory And Some Astrophysical Implications +Gabriel G´omez∗ +Departamento de F´ısica, Universidad de Santiago de Chile, +Avenida V´ıctor Jara 3493, Estaci´on Central, 9170124, Santiago, Chile +Jos´e F. Rodr´ıguez† +Escuela de F´ısica, Universidad Industrial de Santander, +Ciudad Universitaria, Bucaramanga 680002, Colombia +and +ICRANet, Piazza della Repubblica 10, 65122, Pescara PE, Italy +The Generalized SU(2) Proca theory is a vector-tensor theory of gravity whose action is in- +variant under global transformations of the SU(2) group and includes second-order derivative self- +interactions of the vector field beyond the massive Yang-Mills theory. We find, in particular, that the +presence of two Lagrangian pieces consisting of four gauge fields minimally coupled to gravity gives +rise to an exact Reissner-Nordstr¨om black hole solution endowed with two different non-Abelian +effective charges that depend on the specific combination, χ = 2χ1 + χ2, of the respective coupling +constants. After studying the spacetime structure of the black hole, which allows to characterize the +parameter space that preserves the weak cosmic censorship conjecture, some astrophysical implica- +tions of the black hole solutions are investigated. First, joint analysis of observations of the EHT’s +first images of Sagittarius A⋆ of our Galaxy and the Keck telescope, set the first serious constraint on +the free parameters of the theory beyond the theoretical bounds found. Second, we investigate the +accretion properties of spherical steady flows around this class of non-Abelian Reissner-Nordstr¨om +black hole. Specifically, we examine the general conditions under which transonic flow is allowed. +Analytical solution for critical accretion is found in terms of the coupling constant. In addition, +we explore numerically the effect of changing χ on the radial velocity and mass density, and show +how the extremal Reissner-Nordstr¨om and the standard Schwarzschild solutions as limit cases are +achieved. Lastly, working in the fully relativistic regime, an analytical expression for the critical +mass accretion rate of a polytropic fluid onto a black hole is derived. As a main result, we find that +the critical accretion rate efficiency can be noticeably improved compared to the Schwarzschild case +for a specific region of the parameter space where the non-Abelian charge becomes imaginary. +I. +INTRODUCTION +Black holes (BHs) are one of the most fascinating ob- +jects in the Universe that arise as a result of gravitational +collapse of massive objects, as predicted by general rel- +ativity (GR). Apart from the fundamental conceptions +and bizarre properties they harbor, BHs are ideal labo- +ratories, due to their intense gravitational fields, to study +high-energy astrophysical processes that take place in +neighborhoods around them and, more importantly, they +offer a direct route to study the spacetime structure it- +self which is a challenge of alternatives theories of gravity +beyond general relativity. Motivated by these concerns, +BHs have been the central target of current astrophys- +ical experiments such as the Event Horizon Telescope +(EHT) and the Very Large Telescope global networks +[1, 2], GRAVITY collaboration [3] and the LIGO and +VIRGO collaborations [4, 5] among other important sci- +entific projects. Although the results derived from these +observations are well consistent with GR, they are not +conclusive in the sense that they can be reproduced by +∗ gabriel.gomez.d@usach.cl +† jose.rodriguez2@correo.uis.edu.co +non-trivial spacetime metrics (see e.g. [6–9]). Such ob- +servations however do provide strong evidences about the +existence of BHs. +Beyond the simple Schwarzschild BH realization, BHs +can be endowed with an electric and magnetic charges +when, for instance, the Maxwell theory is coupled to +gravity. +Nevertheless, it is believed that astrophysical +BHs are electrically neutral due to charge neutralization +by astrophysical plasma, among other suitable physical +mechanisms. Another less conservative alternative is to +consider BHs carrying, for instance, U(1) charge instead +of electromagnetic charge due to early universe mecha- +nisms within the dark (hidden) sector with no coupling +to Standard-Model particles (see e.g. [10, 11]). See also +[12] for a discussion from astrophysical point of view. No +matter the underling physical process behind the charge +mechanism, this is still an open issue that have brought +recently much attention after the measurements made by +the EHT of the supermassive BH M87⋆ shadow size and +the detection of gravitational waves of compact object +binaries, which demand careful examination on the BH +charge far beyond academic considerations [13–18]. See +also [19, 20] for observational limits on the charge of the +Galactic center BH. +On the other hand, it has been confirmed by several +observations that BHs must rotate to account for some +arXiv:2301.05222v1 [gr-qc] 12 Jan 2023 + +2 +astrophysical phenomena observed as X-rays streaming +off material near BHs as a result of the formation of ac- +cretion disk (see e.g. +Ref. [21]). +Due to the technical +complications behind the Kerr solution that accounts for +the BHs angular momentum, an useful first approach to +study realistic and complex phenomena is to consider the +Reissner Nordstr¨om (RN) solution whose charge plays a +similar role as the spin does in the event horizon struc- +ture. +Indeed, the effects of the magnetic and electric +charges of the RN BH can mimic the spin parameter of +a rotating Kerr black hole in observations of the mag- +netar J1745-2900 orbiting around the supermassive BH +Sagittarius A⋆ (Sgr A⋆) [22]. In addition, a recent anal- +ysis of the motion of S-stars has constrained the spin of +the object at the center of the Milky Way to be rather +small, a/M ≲ 0.1 [23]. This implies, that even though +it is expected that BHs exhibit considerable spin, there +are objects whose angular momentum is small and can +be described by a stationary and spherically symmetric +spacetime [24]. +After the finding of (purely magnetic) static spheri- +cally symmetric non-Abelian BH solutions in the Einstein +SU(2) Yang-Mills (EYM) model [25–27], it was shown +soon after that they are perturbatively unstable [28]. +Higher order curvature terms of the gauge field [29, 30], +as well as non-trivial combinations with other theories +have been included to the gravitational sector aiming at +resolving this impasse (see e.g. Ref. [31] for former pro- +posals). Some BHs solutions with non-Abelian hairs have +been also found in theories beyond the canonical Yang- +Mills theory but still in the context of GR [29, 30, 32]. +Higher curvature terms of the metric tensor such as f(R) +gravity coupled to the Yang-Mills field admit also BH so- +lutions with single or double horizons [33]. +Interestingly, in the EYM case, there exists a RN so- +lution [34, 35], but unlike the Einstein-Maxwell case, +this is unstable. This shows that although both models +share the same spacetime configuration, they are pertur- +batively different. +The same happens in the Einstein- +Yang-Mills-Higgs, where a RN solution exists, but the +Higgs mechanism can stabilize the solution [36]. In this +work, we found an exact solution in the context of a mod- +ified gravity model. This solution is also of RN spacetime +type, but as mentioned before, it is not the same as the +EYM RN solution. +On the hand, it is possible to build a healthy theory +including higher derivative self-interactions of the SU(2) +gauge field but still propagating the correct number of +degrees of freedom. The result is the Generalized SU(2) +Proca (GSU2P) theory [37, 38], which is the non-Abelian +version of the Generalized Proca theory [39–41], and it +belongs to a class of vector-tensor theories that lies in the +spirit of Horndeski’s theory [42]. Considering particularly +some Lagrangian pieces that involves four gauge fields +minimally coupled to gravity, which arise from a system- +atic construction in the full theory, gives place to BH so- +lutions with two different non-Abelian effective charges +that depend on the coupling constants. It is worthwhile +mentioning that our findings are rooted to modified theo- +ries of gravity despite they were particularly derived from +these Lagrangian pieces. +From the astrophysical side, accretion processes of a +ideal and polytropic fluids onto black holes has been a +theme of intense study (see e.g [43–46]), as a probe of +concept, either in the heart of GR or in most general +frameworks. In particular, accretion flows in an arbitrary +spacetime have been extensively investigated as astro- +physical probes to reveal any deviation from GR [47–60]. +The paper is structured as follows. +After a concise +introduction, in Section II we describe the model and +derive in detail an exact non-Abelian RN BH solution +in terms of the coupling constants of the theory. Some +properties of the BH solutions such as the event hori- +zon, photon sphere and shadow are studied. In partic- +ular, using observational data of the EHT’s first images +of Sagittarius A⋆, places first constraints on the effective +coupling constant. We also investigate some astrophysi- +cal implications of the BH solutions. Firstly, in Section +III, a general description of the hydrodynamics equations +of accretion flow is presented; and secondly, in Section IV +the critical accretion rate for both isothermal and poly- +tropic fluids are calculated. This is done by implementing +both analytical and numerical computations. We finish +discussing the main findings of this work as well as pos- +sible extensions of it in Section V. Further observational +constraints on the theory are also discussed. Throughout +this paper we use geometrized units with c = G = 1. +II. +REISSNER NORDSTROM BLACK HOLE +WITH NON-ABELIAN CHARGE +The action of the model, which corresponds to some +Lagrangian pieces of the GSU2P theory [37], includes +quartic order self-interactions of the vector field1, +S = +1 +16π +� √−g d4x[R − FaµνF aµν ++ χ1BaµBaµBbνBbν + χ2BaµBa +νBb +µBbν], +(1) +where R is the Ricci scalar, Baµ represents the vector +fields, Faµν = ∂µBaν − ∂νBaµ + ˜gϵabcBbµBcν is the field +strength, ˜g is the gauge coupling constant and ϵabc is +the structure constant tensor of the SU(2) group. +In +geometrized units ˜g has units of inverse length, and the +free parameters χ1 and χ2 have units of inverse square +length. +1 As the inclusion of a mass term µ2AaαAaα spoils the existence +of the solution, it has been taken away from the model. This +result is similar to the classical massive vector field, where the +mass needs to vanish to guarantee regularity of the solution and +to allow therefore a vector hair to exist [61]. + +3 +The line element in a stationary and spherical sym- +metric spacetime has the following form, +ds2 = gtt(r)dt2 + grr(r)dr2 + r2dΩ2 += −e−2δNdt2 + N −1dr2 + r2dΩ2, +(2) +where N = 1 − 2m/r, δ and m are functions of the co- +ordinate r, and dΩ is the solid angle element. Regarding +the vector fields we chose the Wu-Yang monopole, +A0 = Ar = 0 +(3) +Aθ = (w/v + 1) tφ +(4) +Aφ = (v − w) sin θ tθ, +(5) +where, +tθ = cos θ cos φ t1 + cos θ sin φ t2 − sin θ t3, +(6) +tφ = − sin φ t1 + cos φ t2, +(7) +ti = −iσi/2 correspond to the vector basis of the SU(2) +algebra with σi being the Pauli matrices, w is constant +and v is an integer denoting the azimuthal winding num- +ber. We use the coupling constant ˜g to define the normal- +ized variables, ˆr = r˜g, ˆm = m˜g, ˆχ1 = χ1/˜g2 ˆχ2 = χ2/˜g2. +The form of the equations in the normalized variables +can be obtained effectively by setting ˜g = 1. Hereafter, +all the equations are normalized, but we drop the hat to +ease the notation. +The field equation obtained after varying the action +with respect to Baµ is, +(v + w) +�� +v2χ1 + v2χ2 + χ1 +� +(v + w)2 + w(v − w) +� += 0. +(8) +The solutions of this last equation are, +wschw = −v, +(9) +wI,II = v + 2vχ1 + 2v3 (χ1 + χ2) ± +� +v2 [8v2χ2 + 8 (v2 + 1) χ1 + 1] +2 (1 − v2χ1 − v2χ2 − χ1) +. +(10) +The first solution (9) is the trivial solution with vanish- +ing vector field which corresponds to the Schwarzschild +spacetime. Instead, the latter solutions (10), with two +branches I and II, allow the existence of a non-trivial +vector field we shall focus on, and constitutes, therefore, +an important outcome of this work, as will be described +in detail below. +The field equations obtained by varying the action (1) +with respect to metric are given by, +m′ − +� +v3 − vw2�2 +2r2v4 ++ +(v + w)4 � +v4χ2 + +� +v2 + 1 +�2 χ1 + χ2 +� +4r2v4 += 0 +(11) +δ′ = 0 +(12) +m′′ + v2(v − w)(v + 3w)(v + w)2 +r3v4 ++ +�� +v4 + 5 +� +χ2 + +� +v4 + 6v2 + 5 +� +χ1 +� +(v + w)4 +2r3v4 += 0. +(13) +We look for asymptotically flat solutions, i.e. when r → +∞ the functions δ → 0, m → M ≡ finite. Under these +conditions Eq. (12) is easily solved as δ = 0. On the other +hand, the solution of mass function m for the cases given +by (10) has the following Reissner-Nordstr¨om solution, +m = M − Q2 +NA +2r , +(14) +where M is the total gravitational mass and QNA is a +constant representing the effective charge and depends +on the free parameters of the theory. +Eqs. (11) and (13) must be consistent, which implies +an additional constraint between χ1, χ2 and v given by, + +4 +3 +� +v4 − 1 +� +(χ1 + χ2) +� +4v3χ2 + 3 +� +v2 [8v2χ2 + 8 (v2 + 1) χ1 + 1] + 4 +� +v3 + v +� +χ1 + 5v +�2 +[v2χ2 + (v2 + 1) χ1 − 1] 4 += 0. +(15) +One solution fixes the winding number as, +v = ±1, +(16) +with the parameters the parameters χ1 and χ2 indepen- +dent. The other solution gives a relation between the free +parameters of the action model, +χ2 = −χ1, +(17) +with the winding number now unconstrained2. +Despite the solution set by (16) gives two possible solu- +tions, it represents only one since changing the sign of the +winding number interchanges the solutions wI and wII as +can be verified in Eq. (10). In the solution given by (16) +the value of w depends on the combination χ = 2χ1 +χ2, +wI,II = 1 + 2χ ± √1 + 8χ +2 − 2χ +. +(18) +In the other solution corresponding to (17), the value of +w/v has the same functional form of (18) after making +χ �→ χ1. Therefore, the physical behavior of both cases +can be analyzed by means of (18). Hence, all subsequent +analysis will be carried out in terms of the new effective +coupling constant χ. +Consequently, the value of the effective charge Q2 +NA is +given by, +Q2 +NA,I,II = 1 − 4χ(5 + 2χ) ∓ (1 + 8χ)3/2 +2(1 − χ)3 +, +(19) +where the sign − corresponds to the branch I, and the +sign + to the branch II. The dependence of both quanti- +ties on the coupling constant is displayed in Figure 1. +There exist solutions for the interval −1/8 < χ < 1 ∪ +χ > 1. When χ = 0 the branch I solution corresponds +to the Schwarzschild solution, and the branch II solution +corresponds to the EYM charged solution with Q2 = 1. +When χ → 1 the branch I has a divergence, in contrast, +in the same limit, the branch II is finite and tends to +wII → −1/3, Q2 +NA,II → 16/27. +On the hand, if we assume that the total mass of the +black hole in normalized units is M = 1, we can find +regions where Q2 +NA > 1, corresponding to naked sin- +gularities. +These regions are given by −1/8 < χ < +(11 − 5 +√ +5)/2 ∪ 1 < χ < (11 + 5 +√ +5)/2, for the branch +2 It seems that there is another solution given by the vanishing +of the expression between braces in the numerator of Eq. (15). +Nevertheless, this does not constitute a solution because it makes +also the denominator to vanish, inducing a divergence. +I, and −1/8 < χ < 0, for the branch II (see the gray +regions in Figure 1). +Finally, it is worthwhile to mention that the en- +ergy density associated with the vector fields is ρ = +Q2 +NA/(8πr4). +Thus, when the non-Abelian charge is +imaginary the energy density is negative, and this hap- +pens only in the branch I in the interval 0 < χ < 1. +A. +Event horizon +In the stationary and spherically symmetric case, the +vanishing of the metric function gtt defines unequivocally +the horizon (see e.g. [62]). When multiple solutions exist +the greatest positive solution is identified with the event +horizon of the black hole r+ = rH. In particular, the met- +ric function associated to the Reissner-Nordstr¨om (RN) +solution has two distinct roots +r± = M ± +� +M 2 − Q2 +NA. +(20) +The internal solution r− is an apparent horizon and +the external solution corresponds to the event horizon. +Hence, any observer outside the black hole (in asymp- +totically flat spacetimes), or on the event horizon itself, +cannot see any singularity because they are protected by +an event horizon. Otherwise it is said to posses a naked +singularity at r = 0. We do not mention here all the mi- +nor details and conditions about the precise formulation +of what is called the weak cosmic censorship conjecture3. +The structure of the RN solution has been studied ex- +tensively whereby we do not pretend to make here a +detailed examination on this subject. Nevertheless, an +intriguing query arises when one asks about the impli- +cations of the coupling constant χ on the charge, and +therefore, on the event horizon. In particular, we are in- +terested in finding which values of the coupling constant +account for the convergence to both the Schwarzschild +and the extremal RN black holes, as limit cases of our +solution. +When performing numerical analysis and presenting +the corresponding general discussion, we shall work with +dimensionless variables by normalizing all physical quan- +tities by the black hole mass M, unless otherwise said. +Accordingly, we introduce, as usual, the charge to mass +ratio qNA = QNA/M and the dimensionless radial coor- +dinate x = r/M. +3 We refer reader to [63] for a general and robust formulation of +the cosmic censor conjecture. + +5 +10 +2 +100 +102 ++1/8 +3 +2 +1 +0 +1 +2 +3 +Q 2 +NA, I +wI +naked singularity +10 +2 +100 +102 ++1/8 +1.0 +0.5 +0.0 +0.5 +1.0 +Q 2 +NA, II +wII +naked singularity +FIG. 1. Values of the vector field and effective charge as a function of χ for the branch I (left panel) and for the branch II +(right panel). In both cases when χ → ∞ the charge tends to zero, thus the solution becomes the Schwarzschild spacetime. If +the mass of the black hole in normalized units is M = 1, values of Q2 +NA > 1 represent a naked singularity. These last cases are +shown as gray regions. In the branch II the charge is always real. In the branch I for 0 < χ < 1 the charge is imaginary, which +implies that the energy density is negative. Notice also that the shift χ + 1/8 in the abscissa has been done for convenience. +q NA,I +q NA,II +0.01 +0.10 +1 +10 +100 +1000 +104 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +χ+1/8 +xH +q NA,I +q NA,II +0.05 0.10 +0.50 +1 +5 +10 +0 +2 +4 +6 +8 +10 +χ+1/8 +xISCO +FIG. 2. Left panel: position of the event horizon xH as a function of the coupling constant for both charges as specified by +the legend. Right panel: innermost stable circular orbit xISCO as a function of the coupling parameter χ as well for both +non-Abelian charges. In both plots red points indicate the match with the extremal RN solution. It is evident here the two-fold +degeneracy of qNA,I and the convergence to the Schwarzschild solutions when χ → ∞ for both charges. +The event horizon is depicted in left panel of Figure 2 +as a function of the coupling constant for both charges4. +For qNA,II (purple solid line) the event horizon is a well- +behaved function of the coupling constant. +Indeed, it +covers continually the full range χ ∈ (0, ∞) where the fi- +nite extreme value corresponds to the extremal RN case, +in which case both solutions of Eq. (20) meet at xH = 1 +(red point on the purple curve); while large values lead +to the uncharged solution where xH = 2 and the appar- +ent horizon (dashed curve) coincides with the singularity. +This is hence a quite normal behavior that reproduces +plainly the standard RN solution. +4 When plotting, the shift χ + 1/8 in the abscissa is done for con- +venience. +On the contrary, the event horizon for the case qNA,I +(blue solid curves) exhibits a peculiar structure: there +is a two-fold degeneracy with respect to the constant +coupling. +It means that the limit cases, that is, the +Schwarzschild solution and the extremal RN black hole, +can be described in two distinct regions of the param- +eter space. The same degeneracy is also presented for +the apparent horizon (blue dashed curves). +This is +clearly appreciated in the regions defined by the ranges +χ ∈ (−0.0901, 0) and χ ∈ (11.0902, ∞) of left panel of +Figure 2. There is also a special region of the parameter +space χ ∈ (0, 1), where the square of the effective charge +becomes negative and gravity repulsive. Out of the men- +tioned ranges results in q2 +NA > 1, case that describes a +naked singularity. + +6 +B. +Inner most stable circular orbit +Next, we turn our attention to the motion of massive +test particles in the equatorial plane, as usual, to find the +position of the so-called innermost stable circular orbits +(ISCO). Following the standard procedure (see e.g. [64]), +one finds that the geodesic equation can be written as +�dr +ds +�2 += +� +E2 − N +� +1 + L2 +r2 +�� +, +(21) +where E and L are identified, respectively, as the energy +and angular momentum which correspond to conserved +quantities as in the classical Keplerian motion. +From +the above equation, one can also identify the effective +potential +Veff = N +� +1 + L2 +r2 +� +. +(22) +We remind that N = 1 − 2m/r, the mass function m(r) +has the form given by Eq. (14) and and the (squared) +angular momentum is defined as +L2 = +N ′(r)r3 +2N(r) − rN ′(r). +(23) +In marginally stable circular orbits, the local extremum +of the effective potential, i.e. V ′′ +eff = 0, determines the +position of the ISCO. It yields +rISCO = 2M + +4M 4 − 3M 2Q2 +NA + +� +8M 6 − 9M 4Q2 +NA + 2M 2Q4 +NA + +� +5M 8Q4 +NA − 9M 6Q6 +NA + 4M 4Q8 +NA +�2/3 +M +� +8M 6 − 9M 4Q2 +NA + 2M 2Q4 +NA + +� +5M 8Q4 +NA − 9M 6Q6 +NA + 4M 4Q8 +NA +�1/3 +. +(24) +This expression of course equals the corresponding stan- +dard RN case [64]. At this point we are able to compute +the location of the ISCO in terms of the coupling con- +stant χ with the aid of Eq. (19). The result is shown +in right panel of Figure 2. Clearly the effect of the cou- +pling constant is replicated on the ISCO structure in the +same fashion as it does on the event horizon. Hence the +same parameter space region establishes the correspond- +ing Schwarzschild solution xISCO = 6, and the extremal +RN case xISCO = 4 (red point on all curves) for both +positive square of the non-Abelian charges as can be read +from the plot. Thus, all the discussed properties for the +event horizon are preserved for the ISCO location. It is +interesting to notice, however, that for the charge qNA,I, +there appears another region where gravity is repulsive +in the interval χ ∈ (1, 11.092), in comparison to the ones +seen for the event horizon. For this charge it is admissi- +ble then to have stable circular orbits for almost all the +parameter space with the exception of its extreme values +even though there does not exist event horizon. This par- +ticular behavior is reminiscent to the fact of having stable +accretion flows onto a RN black hole even thought there +is not a physical event horizon. Hence, in a naked sin- +gularity situation it is (mathematically) possible to have +stable circular orbits. We will not discuss however this +point in detail since it is not of physical interest for the +present work. +C. +Photon sphere and shadow +A black hole has a central dark area called the shadow. +This shadow is not delimited by the event horizon, but +by the photon sphere, which is made of circular photon +orbits. The radius of the photon sphere rph is given by +solving, +rphg′ +tt(rph) − 2gtt(rph) = 0. +(25) +However, the observed shadow radius, rsh is given by the +lensed image of this surface [65], +rsh = +rph +� +gtt(rph) +. +(26) +For the Reissner-Nordstr¨om solution the shadow radius +is, +rsh +M = xsh = +√ +2 +�� +9 − 8q2 +NA + 3 +� +� +4q2 +NA+√ +9−8q2 +NA−3 +q2 +NA +. +(27) +This last equation can be inverted such that, for a given +observed shadow radius, we can constrain the charge qNA. +From this value we can constrain in turn the free param- +eters of the theory ˜g and χ. The observations of Sagittar- +ius A∗ made by the EHT collaboration give the following +constrains on the size of the shadow, which depend on +the mass-to-distance ratio [66], +4.5 ≲ xsh ≲ 5.5, +(28) +with Keck and +4.3 ≲ xsh ≲ 5.3, +(29) +with VLTI. With this information, we can place con- +straints on the parameters ˜g and χ, shown in Figure 3. + +7 +Since (19) is normalized but depends physically on the ˜g +and χ, only in this part of the analysis we put back the +units. The first conclusion is that an extremal RN black +hole is not consistent with the current observations. +If the non-Abelian charge is real, i.e., Q2 +NA > 0, the +maximum value of the shadow is 3 +√ +3M. This value cor- +responds to the Schwarzschild case χ → ∞ for both the +branches I and II. The shadow of a Schwarzschild black +hole is inside (28) and (29). Thus, for the branch II we +can give lower limits on χ (see the blue and orange curves +in the lower plot of Figure 3). +In contrast, the branch I exhibits imaginary values of +the charge, allowing a greater shadow than the one of a +Schwarzschild black hole. +Also, in this branch the al- +lowed region of values for ˜g and χ is greater than in +the branch II. For instance, when ˜gM = 0.9 the mini- +mum value of χM 2 is −0.0326136 and −0.0414562, for +mass-to-distance ratio with Keck and VLTI, respectively. +This implies a less stringent constraint in the parame- +ters than in the branch discussed above. For example, +if ˜gM = 1.01, the constraints are −0.0545704 ≲ χM 2 ≲ +0.0223263 ∪ 14.3271 ≲ χM 2, and −0.0708403 ≲ χM 2 ≲ +0.00774184 ∪ 12.6237 ≲ χM 2, for Keck and VLTI, re- +spectively. More cases can be inferred from the Figure 3. +The main purpose now is to figure out how the struc- +ture of the non-Abelian RN black hole can impact the +transonic properties of accretion flows. After formulat- +ing the basic hydrodynamics equations, this subject will +be investigated by employing both numerical and analyt- +ical treatments in the subsequent sections. +III. +SPHERICAL STEADY ACCRETION +FLOWS IN A SPHERICALLY SYMMETRIC +SPACETIME +Bondi accretion processes in a spherically symmetric +spacetime is briefly described in this section following the +general prescription presented in Ref. [47]. Accordingly, +we consider a spacetime with line element given by (2), +with m and Q2 +NA given, respectively, by (14) and (19), de- +scribing a black hole of mass M and non-Abelian charge +Q2 +NA in Schwarzschild coordinates. +Although the non- +Abelian RN solution Eq. (14) is, in principle, globally in- +distinguishable for the standard RN BH solution, it was +found, formally speaking, in the framework of modified +gravity. So, we keep as much as possible the generality in +the description5. On the other hand, we consider a steady +fluid with total density ρ, mass density ρ0 and internal +energy density ϵ, such that ρ = ρ0 + ϵ. For isentropic +fluids the pressure can be defined as P = kργ where k +is a constant and γ is the adiabatic index. For perfect +5 This general treatment serves also as a starting point to other +(non-analytical) BH solutions found in the GSU2P theory [67]. +fluids, the stress energy momentum tensor is given by +T µν = (ρ + P)uµuν + Pgµν, +(30) +where uµ = (ut, ur, 0, 0) is the four velocity of the fluid +characterized by infall radial flow. +The normalization +condition allows to obtain the relation between the com- +ponents ut = +� +grr(u2+gtt) +gtt +, where we have defined for +abbreviation u ≡ ur. From the baryon conservation and +energy momentum conservation +∇µ(ρ0uµ) = 0, +(31) +∇µT µν = 0, +(32) +one obtains two master equations, respectively +ρ′ +0 +ρ0 ++ u′ +u +Σ = 0, +(33) +uu′ + +gtt′ +2grrgtt +(1 + grru2) + g′ +rr +2grr +u2 + c2 +s +grr +(1 + grru2)ρ′ +0 +ρ0 += 0, +(34) +where prime denotes radial derivative and the quantity +Σ ≡ +(√−g)′ +√−g +has been introduced for brevity as [47]. +These equations reduce to the Schwarzschild case when +−gtt = +1 +grr = 1 − 2M +r . In finding Eq. (34), we have used +the definition of the sound speed of a medium at constant +entropy c2 +s ≡ dP +dρ , and the useful relation P ′ = (ρ+P ) +ρ0 +c2 +s ρ′ +0 +derived from the first law of thermodynamics in the form +dρ +dρ0 = ρ+P +ρ0 . Integration of Eqs. (33) and (34) gives, re- +spectively, the mass accretion rate +˙M = 4πr2uρ0, +(35) +and, after some algebraic manipulations, the relativistic +version of the Bernoulli equation (see Ref. [47]. for more +details) +gtt(1 + grru2) +�ρ + P +ρ0 +� += C, +(36) +where C is an integration constant. Once C is defined by +the boundary conditions at infinity for instance, and the +metric functions are specified, the inward radial velocity +can be computed for a given equation of state P = P(ρ0). +Our primary concern is to solve this equation to deter- +mine, in turn, the accretion rate given by Eq. (35). Before +computing this, we find critical values at which accretion +flow is regular and causality is guarantee. +IV. +ACCRETION IN A NON-ABELIAN +REISSNER-NORDSTR¨OM BLACK HOLE +A. +Critical accretion +We study in this part general conditions under which +transonic flow can take place in the vicinity of black holes. + +8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +gM +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +log10( M 2) +Branch I +xsh =4 (extremal) +xsh =4.3 +xsh =4.5 +xsh =5.19 +Keck +VLTI +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +gM +0.10 +0.05 +0.00 +0.05 +0.10 +M 2 +Branch I +xsh =4 (extremal) +xsh =4.3 +xsh =5.3 +xsh =4.5 +xsh =5.5 +xsh =3 +3 (Schw) +Keck +VLTI +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +gM +1 +0 +1 +2 +3 +log10( M 2 +1/8) +Branch II +xsh =4 (extremal) +xsh =4.3 +xsh =4.5 +xsh =5.19 +Keck +VLTI +FIG. 3. Constraints on the parameters ˜g, and χ obtained from the observation of the shadow of the object located at the +Galactic center of the Milky Way, Sagittarius A⋆. The upper plots correspond to the branch I which has been split into two +ranges of χ for better representation, and the lower plot corresponds to the branch II. The different colored curves show the +possible values of gc and χ for the lower and upper limits on xsh given by (28) and (29) as indicated by the legend. For +comparison, we have added the curves for xsh = 4, 5.19, 3 +√ +3. The regions with slanted orange lines represent the constraints +obtained from the mass-to-distance ratio with Keck, and the blue regions represent the constraints with VLTI. It can be seen +that a naked singularity is not consistent with the observational data. +Next, we shall describe both the spacetime geometry and +the fluid nature. Let us first write Eqs. (33) and (34) in +the more convenient form +u′ +u = gtt(2c2 +s(1 + u2grr)Σ − u2g′ +rr) − (1 + u2grr)g′ +tt +2grrgtt(u2 − c2s(grru2)) +, +(37) +ρ′ +0 +ρ0 += −u2gtt(−2grrΣ + g′ +rr) + (1 + u2grr)g′ +tt +2grrgtt(u2 − c2s(grru2)) +. +(38) +Imposing regular condition in both equations, implies +that both numerators must vanish simultaneously at +some critical point rc, resulting in +u2 +c = − +g′ +tt +gttg′rr + grr(−2gttΣ + g′ +tt), +(39) +c2 +s,c = +grrg′ +tt +2grrgttΣ − gttg′rr +. +(40) +Causality constraint c2 +s < 1 in the flow sets a special +point (rc) by which the flow must pass. This physical +requirement leads to the relation +u2 +c = − c2 +s,c grr +−1 + c2s,c +, +(41) +between the radial velocity and the sound speed. At large +radius, the flow is in the subsonic regime u2 < c2 +s. So far +these results are general in the sense that can be applied +to any spherically symmetric black hole solution. At this +point we must specify necessarily the metric functions to +obtain the exact forms for the critical velocity and sound +speed Eqs. (39) and (40), respectively. Thus, considering +the non-Abelian Reissner Norstr¨om BH solution Eq. (14), +leads to the critical values +u2 +c = −Q2 +NA − Mr +2r2 +, c2 +s,c = +−Q2 +NA + Mr +Q2 +NA + r(−3M + 2r). (42) + +9 +From the transonic condition that u2 +c = c2 +s,c at the +critical radius, and considering Eq. (41), one obtains un- +equivocally the critical radius +rc = +M + 3c2 +s,cM ± +� +(M + 3c2s,cM)2 − 8c2s,c(1 + c2s,c)Q2 +NA +4c2s,c +, +(43) +where the non-trivial contribution of the non-Abelian +charge to the Schwarzschild solution, rc,Sch = M(1 + +3c2 +s,c)/2c2 +s,c and u2 +c,Sch = M/2rc, is clearly manifested. +From Eq. (43) we can see that there exist, mathemati- +cally speaking, two distinct critical points but the pos- +itive branch corresponds only to the physical solution +which resides outside the event horizon, and therefore, +it is the one we shall pay our attention. Of course, the +negative branch is in a region observationally inaccessible +since it is delimited by the event horizon. Existence of +the critical radius demands +Q2 +NA +M 2 < 1 + 6c2 +s,c + 9c4 +s,c +8c2s,c + 8c4s,c +, +(44) +which in turn puts constraints on the coupling parameter +for a given non-Abelian charge. Nevertheless, the result- +ing expressions are very lengthy (and not illuminating) +to be reported here. We shall illustrate below this aspect +numerically. +In order to understand the striking structure of the +critical radius we must necessarily specify the non- +Abelian charge and the nature of the fluid which charac- +terizes the sound speed. For simplicity in the former anal- +ysis, we assume an isothermal fluid6 so that the sound +speed equals its equation of state. This simple choice will +give us, however, a profound insight about the behavior +of the critical radius in terms on the model parameters. +So the present case suffices to prove the rich structure +of the critical radius due to the non-Abelian charge. It +should be noticed, on the other hand, that for the (phys- +ical) critical radius there are two solutions due to the +existence of the non-Abelian charges QNA,I,II. +Let us first explore the effect of changing the sound +speed on the critical radius for some specific values of +the coupling constant that cover mostly the (physical) +BH solutions of interest. The behavior of the critical ra- +dius depends, however, on the non-Abelian charge cho- +sen. This is shown in Fig. 4. Here left and right pan- +els correspond, respectively, to the non-Abelian charges +qNA,I and qNA,II. In particular, for qNA,I and χ = 0, the +critical radius matches exactly the Schwarzschild solu- +tion, whereas for qNA,II the convergence is possible pro- +vided that χ → ∞. +This is the reason why we have +6 Another possibility is to take a polytropic fluid but it introduces +an extra parameter that overclouds the present analysis about +the structure of the critical radius. Nevertheless, it will be suc- +cessfully addressed in the next part for the sake of completeness +when looking for the critical mass accretion rate. +displayed the exact Schwarzschild solution (green curve) +in right panel as an asymptotic limit of large coupling +constant values. Accordingly, for the larger case shown, +χ = 20, the solution is barely distinguishable from the +uncharged case but physically different since the former +is endowed with a charge qNA,I = 0.113438. Hence, de- +spite taking the same values of χ for both non-Abelian +charges, the physical implications do not hold for the crit- +ical radius as it also happens for the event horizon and +ISCO. +To better illustrate the role of χ in the critical radius, +we take the same values of the coupling constant, as de- +scribed below, for both charges. +For instance, taking +qNA,I (see left panel) and a given coupling parameter +lead to the following physical situations: the extremal +case χ = −0.0901, the Schwarzschild solution χ = 0, +the charged solution χ = 20 (qNA,I = 0.4112) and a +naked singularity χ = 11.0222, that affect in a way differ- +ent the behavior of the critical radius. Now, taking the +same values for χ as before but for qNA,II, the previous +physical meaning is lost. +Naked singularities can take +place, for instance, for χ = −0.0901 which in the qNA,I +case occurs for χ = 11.0222 . Notice that for the for- +mer case accretion is possible only for (subsonic) sound +speeds c2 +s,c < 0.3478 while for the latter it occurs out of +the range 0.6930 < c2 +s,c < 1.5677. This explains why the +discontinuity of those curves. Hence accretion may be +possible even though the event horizon vanishes. An in- +teresting discussion about how to distinguish a BH from +a naked singularity spacetime by using the image of thin +accretion disks is addressed in [59]. +As a general trend, for subsonic sound speeds the crit- +ical radius can be located far away from the event hori- +zon, and for supersonic sound speeds it can be accommo- +dated, on the contrary, between the event horizon and +the apparent horizon. For c2 +s = 1 all critical points coin- +cide with their respective position of the event horizon as +can be easily verified. Another interesting feature is that +for the Schwarzschild case and subsonic speeds c2 +s,c < 1 +the critical radius is outside the event horizon while for +c2 +s,c > 1 the critical radius is located behind it. See purple +and green curves in the left and right panels, respectively. +This general discussion is in agreement with former stud- +ies about the accretion of perfect fluids in the RN metric +[68]. +In contrast, in Fig. 5, the coupling parameter is fixed +while the sound speed varies. Notice that for the case +qNA,I the critical radius exhibits a discontinuity as it +was also perceived for the event horizon structure. +It +should be noticed that as c2 +s,c decreases (see for instance +c2 +s,c=1/4) the critical radius can be in a region where +naked singularity takes place. As to the case qNA,II, the +critical radius is also a monotonically increasing function +of χ, but it is well-behaved until the condition Eq. (44) +is broken for small values of χ and given squared sound +speeds. This discussion provides, in complement to the +one made around Fig. 4, a full picture on the general +conditions under which the critical radius exists in terms + +10 +of the coupling constant through Eq. (44). +Now, we are in a more grounded position to investigate +the effect of changing the coupling parameter on the ac- +cretion properties, concretely on the radial velocity and +on the mass density around this class of black hole. This +will be carried out first numerically for an isothermal +fluid, and later with the aid of some analytical treat- +ments for polytropic fluids, to allow a more robust and +complete exploration of the involved parameters. These +two cases will be then treated separately in the next sec- +tions. +B. +Isothermal test fluid +Before describing accretion flows for a more general +fluid, let us first consider a simplistic but useful isother- +mal test fluid. It will provide us some physical insights +on how the coupling constant influences the behavior of +the infall radial velocity and mass density. This inquiry +is complementary to the discussion on the critical point +realized previously. +We focus for comparison reasons in a range of the +coupling constant that resembles the RN BH and the +Schwarzschild solution and leave the (allowed) range that +provides q2 +NA,I < 0 out of this analysis. It implies that +the corresponding parameter space of χ for a given non- +Abelian charge, provide the same physics whereby we +focus on qNA,I. +To illustrate this point we show only +a limit case for qNA,II to see the convergence. The full +range of χ will be, however, considered in the calculation +of the critical accretion rate for a polytropic fluid. The +above are also advantageous for numerical facilities since +we have many variables involved. +Accordingly, the equation of state is of the form P = +κρ, with κ being a constant, from which the simple rela- +tion for the sound speed c2 +s,c = κ is derived. +Let us start our analysis by considering a stiff fluid +κ = 1. As we already discussed, for this case (c2 +s,c = 1) +critical points coincide with the event horizons no mat- +ter the value of the coupling constant. The latter spans +the allowed region of the parameter space χ ∈ (−0.09, 0) +for the charge qNA,I, as can be seen in the bar legend +of Figure. 6. We are not considering the other possible +range of values χ ∈ (11.0902, ∞) because the same phys- +ical properties are replicated in the already shown range. +For the other cases describing an ultra-relativistic fluid +κ = 1/2, a radiation fluid κ = 1/3 and a sub-relativistic +fluid κ = 1/4; all critical points move out the BH as κ +decreases according to Eq. (43). As a general trend, all +transonic solutions are delimited from below to the ex- +tremal RN solution χ = −0.0901 and from above to the +Schwarzschild case. This latter case is attained whenever +the coupling constant increases until it reaches the max- +imum value showed here (χ = 0). We have also included +the solution χ = 11.09 (dashed magenta curve) for the +charge qNA,II which matches the RN solution of the qNA,I +case. What is of physical interest here is that all infall +radial velocities pass through the corresponding critical +points marked as points on the curves and computed from +Eq. (43), guaranteeing thus the transonic flow of all solu- +tions, otherwise a stellar wind is generated. This selects +an unique solution with constant inward mass flux that +connects the subsonic with the supersonic regimes, as ex- +pected in Bondi-type accretion [43, 44]. Far beyond the +BH influence, that is, in the non-relativistic regime, the +radial velocities of the particles are too low such that +the inflow rate is decreased compared with particles with +high-speed velocities. +For the mass density distribution due to the BH grav- +itational potential, the RN solution now delimits all so- +lutions from above, as seen in Figure. 7. As in the radial +velocity case, χ covers the same range of values for each +value of the constant κ considered. We can observe that +as κ reduces, the mass density is more spread out along +the radial coordinate. +So far we have obtained the expected behavior for the +radial velocity and mass density within the steady-state, +spherical accretion scenario for the discussed range of χ- +values. As in previous sections, this has been very useful +to understand the role of the coupling constant on the +transonic flows and how our solutions approach to the ex- +tremal RN and Schwarzschild solutions. Making a more +robust description of the fluid, in next section we will +perform analytical calculations of the critical accretion +rate in the fully relativistic regime along with numerical +computations for the entire range of values of χ and both +non-Abelian charges. +C. +Polytropic fluid +Once the properties of the steady flows are known at +the sonic point, this is, radial velocity, sound speed and +critical radius, we can proceed to express such quantities +in terms of the boundary conditions, with the help of the +Bernoulli equation, as commonly done. The purpose of +doing so is to calculate the accretion rate explicitly. It +is necessary also to adopt an equation of state for the +gas. So we study a non-relativistic baryonic gas with a +polytropic equation +P = Knγ, +(45) +where γ is the adiabatic index and K is a constant. With +this and from the first of law of thermodynamic one can +get [69] +ρ = mn + +K +γ − 1nγ. +(46) +1. +Relativistic regime +Surprisingly, the fully relativistic accretion rate for the +Reissner Nordstr¨om solution has not been thoroughly +treated in the literature, where most of existing works + +11 +χ=-0.0901 +χ=0 +χ=20 +χ=11.0222 +qNA,I +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +4 +5 +6 +cs +2 +xc +Sch. +χ=-0.0901 +χ=0 +χ=20 +χ=11.0222 +qNA,II +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +4 +5 +6 +cs +2 +xc +FIG. 4. Left panel: critical radius xc ≡ rc/M (43) for the non-abelian charge qNA,I for a given coupling parameters that +correspond to the extremal case χ = −0.0901, Schwarzschild solution χ = 0, charged solution χ = 20 (qNA,I = 0.4112) and a +naked singularity χ = 11.0222 (qNA,I = 1.01). Right panel: critical radius for the non-abelian charge qNA,II for the same values +of parameters as left panel. The exact Schwarzschild solution (green curve) has been included here for comparison. Notice that, +on the contrary, the solutions χ = −0.0901, χ = 0, χ = 20 and χ = 11.0222 correspond, respectively, to a naked singularity +(qNA,II = 1.11352), a extremal case and charge BH cases qNA,II = 0.113438 and qNA,II = 0.17345. +cs +2=1/4 +cs +2=1/3 +cs +2=1/2 +cs +2=1 +qNA,I +0.1 +1 +10 +100 +0 +2 +4 +6 +8 +χ+1/8 +xc +cs +2=1/4 +cs +2=1/3 +cs +2=1/2 +cs +2=1 +qNA,II +0.1 +1 +10 +100 +0 +1 +2 +3 +4 +5 +6 +χ+1/8 +xc +FIG. 5. Left panel: critical radius for the non-abelian charge qNA,I for certain critical sound speeds, as described in the legend, +as a function of the coupling parameter. Right panel: critical radius for the non-abelian charge qNA,II for the same parameters +as left panel. +have focused on the non-relativistic limit. It leaves un- +doubtedly an incomplete comprehension of the full pic- +ture. +We first derive some analytical expressions and +show some numerical examples to illustrate better the +role of the coupling constant on the mass accretion rate. +We follow closely Ref. [45], where Bondi accretion of +steady spherical gas flow onto a Schwarzschild black hole +has been studied. We extend this work to the charged +case. +With the aid of the polytropic equation, it is possible +to relate the sound speed with the mass density +c2 +s = +γkργ−1 +0 +1 + γkργ−1 +0 +/(γ − 1) +. +(47) +This expression can be evaluated at the critical point and +in the asymptotic region to provide the useful relation +ρ0,s = ρ0,∞ +� +c2 +s,c +c2s,∞ +� +1 +γ−1 � +γ − 1 − c2 +s,∞ +γ − 1 − c2s,c +� +1 +γ−1 +. +(48) +This closed expression requires the knowledge of the crit- +ical sound speed that can be extracted from the relativis- +tic Bernoulli equation +(1 + 3c2 +s,c) +� +1 − c2 +s,c +γ − 1 +�2 += +� +1 − c2 +s,∞ +γ − 1 +�2 +. +(49) +So, once the sound speed at infinity is specified, the +sound speed and the mass density at the critical point +are uniquely determined. The Bernoulli equation is ac- +tually a cubic equation for c2 +s,c with one real solution for + +12 +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +κ = 1 +0 +-0.09 +χ=11.09 +0 +1 +2 +3 +4 +5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +x +u r +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +χ=11.09 +0 +-0.09 +κ = 1/2 +0 +1 +2 +3 +4 +5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +x +u r +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +κ = 1/3 +-0.09 +0 +χ=11.09 +0 +1 +2 +3 +4 +5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +x +u r +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +κ = 1/4 +-0.09 +0 +χ=11.09 +0 +1 +2 +3 +4 +5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +x +u r +FIG. 6. Infall radial velocity for some specific values of κ describing a stiff fluid (κ = 1), ultra-relativistic fluid (κ = 1/2), +radiation fluid (κ = 1/3) and sub-relativistic fluid (κ = 1/4) while the coupling parameter χ spans the allowed range as depicted +by the bar legend. The dotted magenta curve that delimits all the possible transonic solutions from below corresponds to the +extremal case χ = 11.09 for qNA,I (or χ = −0.09 for qNA,II), while χ → 0 resembles the Schwarzschild solution from above. +the range 1 < γ < 5/3. There are some procedures to +solve analytically this equations as, for instance, a stan- +dard root-finding schema which we implement to. +Having expressed all quantities at the critical radius in +terms of the boundary conditions, the critical accretion +rate +˙M = 4πρ0,susr2 +s, +(50) +can be computed easily +˙M = 4π +� M +c2s,∞ +�2 +c2 +s,∞ ρ0,∞ λNA +RN, +(51) +with the accretion rate eigenvalue +λNA +RN ≡ +� +c2 +s,c +c2s,∞ +� 5−3γ +γ−1 � +γ − 1 − c2 +s,∞ +γ − 1 − c2s,c +� +1 +γ−1 (1 + 3c2 +s,c)3/2 +4 +β, +(52) +and the β factor, containing information of the non- +Abelian charge, is +β = 1 +4 +� +1 + +� +1 − 8c2s,c(1 + c2s,c)q2 +NA +(1 + 3c2s,c)2 +�2 +. +(53) +This quantity clearly accounts for the deviation from the +Schwarzschild case. In what follows, we quantify such a +deviation by computing the ratio of both accretion rates +˙M NA +RN +˙MSch += λNA +RN +λSch += β. +(54) +As it is known, the electric charge of the RN black hole +reduces the accretion rate compared to the Schwarzschild +black hole. Our case may be, however, different in the +sense that if the imaginary charge of the black hole is +allowed, i.e. when q2 +NA = q2 +NA,I < 0 and χ ∈ (0, 1). Un- +der this choice, +˙M NA +RN > +˙MSch as can be verified in all +left panels of Figure 8 where the accretion rate has been +plotted as a function of the coupling constant χ for differ- +ent adiabatic indices as indicated. Notice that all curves +meet in the corresponding position of the event horizon +χ → 0, ∞. Out of the mentioned range, the expected be- +havior of the Reissner Nordstr¨om black hole is displayed +just as the case qNA = qNA,II (right panels). Even though +the non-Abelian charge case is effectively distinguishable +from its electric counterpart in the small range χ ∈ (0, 1) +for qNA,I, these numerical computations allow, besides, +to understand better the multiplicity of the non-Abelian + +13 +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +κ = 1 +χ=11.09 +0 +-0.09 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +7 +x +ρ +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +κ = 1/2 +0 +-0.09 +χ=11.09 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +x +ρ +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +κ = 1/3 +χ=11.09 +-0.09 +0 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +x +ρ +χ +-0.08 +-0.07 +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +κ = 1/4 +0 +-0.09 +χ=11.09 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +x +ρ +FIG. 7. Mass density distribution for some specific values of κ describing a stiff fluid (κ = 1), ultra-relativistic fluid (κ = 1/2), +radiation fluid (κ = 1/3) and sub-relativistic fluid (κ = 1/4) while the coupling parameter χ spans the allowed range as depicted +by the bar legend. The dotted magenta curve that delimits all the possible transonic solutions from below corresponds to the +extremal case χ = 11.09 for qNA,I (or χ = −0.09 for qNA,II), while χ → 0 (or χ → ∞) resembles the Schwarzschild solution +from above. +Reissner Nordstr¨om black solution and its implications +in the accretion rate, in particular for the range of val- +ues of χ which leads to a significant enhancement of the +accretion flow. +As a last remark about the boundary conditions, the +larger the boundary sound speed, the shorter the accre- +tion rate variations are among the polytropic fluid con- +sidered. +The stiff case γ = 1 is more sensible to the +increase of the boundary sound speed: notice how the +dashed black curves in right panels meet the other curves +for low χ or, which is equivalent, in the non-Abelian ex- +tremal case qNA → 1. +2. +Asymptotic limit +We do not describe here in detail all derivations con- +cerning the asymptotic limit of the mass accretion rate +since it can be found, for instance, in reference [69]. The +purpose of this part is to check the consistency with the +well-known Newtonian limit and the corresponding de- +pendence on the non-Abelian charge. From the Bernoulli +equation, one can derive the useful relation +c2 +s,c ≈ +2c2 +s,∞ +(5 − 3γ), +(55) +under the non-relativistic condition cs,c ≪ 1 which holds +for reasonable large radius, r ≫ rc, far away from the +BH gravitational influence. The same condition leads to +the simple relation +c2 +s,c ≈ Kγ ργ−1 +0,c , +(56) +between the sound speed and the mass density at the +critical point. This implies that the mass density can be +expressed in terms of the sound speed at the infinity in +view of Eq. (55) to yield +ρ0,c ≈ ρ0,∞ +� +c2 +s,c +c2s,∞ +� +1 +γ−1 +≈ +� +2 +5 − 3γ +� +1 +γ−1 +. +(57) +As in the uncharged case, physical solutions require +γ < 5/3 in this. At this point all above is standard and + +14 +γ=4/3 +γ=5/3 +γ=1 +qNA,I +0.1 +1 +10 +100 +0.5 +1 +5 +10 +χ+1/8 +M + +RN +NA +M + +Sch +γ=4/3 +γ=5/3 +γ=1 +qNA,II +0.1 +1 +10 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +χ+1/8 +M + +RN +NA +M + +Sch +γ=4/3 +γ=5/3 +γ=1 +qNA,I +0.1 +1 +10 +100 +0.5 +1 +5 +10 +χ+1/8 +M + +RN +NA +M + +Sch +γ=4/3 +γ=5/3 +γ=1 +qNA,II +0.1 +1 +10 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +χ+1/8 +M + +RN +NA +M + +Sch +γ=4/3 +γ=5/3 +γ=1 +qNA,I +0.1 +1 +10 +100 +0.5 +1 +5 +10 +χ+1/8 +M + +RN +NA +M + +Sch +γ=4/3 +γ=5/3 +γ=1 +qNA,II +0.1 +1 +10 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +χ+1/8 +M + +RN +NA +M + +Sch +FIG. 8. Ratio of mass accretion rate in the non-Abelian RN BH to the accretion rate in the Schwarzschild BH for different +adiabatic indices as indicated in the legend. Left and right panels correspond to the cases qNA = qNA,I and qNA = qNA,II, +respectively. In tops panels the boundary condition approaching to the non-relativistic regime c2 +s,∞ = 0.001 has been taken, +whereas in middle and bottom panels a relativistic boundary sound speed c2 +s,∞ = 0.1 and c2 +s,∞ = 0.5 have been chosen, +respectively, in contrasting. In all cases the condition uc > cs,∞ is guaranteed, ensuring thus the transonic flow. +these quantities do not receive contributions from the ef- +fective charge QNA at lowest order in cs,c. This is not +the case however for the critical radius Eq. (43) where, +at leading order, QNA already appear explicitly as second +order power +rc ≈ +M +2c2s,c ++3M 2 − 2Q2 +NA +2M ++2(M 2Q2 +NA − Q4 +NA) +M 3 +c2 +s,c+O(c4 +s,c). +(58) +Keeping only higher order contributions in cs,c and using +Eq. (55), the critical radius can be approximated to +rc ≈ (5 − 3γ) +4c2s,∞ +Mη(QNA), +(59) +where the dimensionless correction factor due to the ef- +fective charge has been defined as +η(QNA) = 1 + +c2 +s,∞ +(5 − 3γ) +� +6 − 4Q2 +NA +M 2 +� +. +(60) + +15 +With all this, the critical accretion mass rate can be writ- +ten in the familiar form +˙M ≈ 4πρ0,∞M 2c−3 +s,∞ +�1 +2 +� +γ+1 +2(γ−1) �5 − 3γ +4 +� 3γ−5 +2(γ−1) +η(QNA)2, +(61) +which agrees with [70]. Finally, the uncharged Newtonian +limit is straightforward achieved in the limit QNA → 0 +and, appropriately, c2 +c,∞ ≪ 1. +V. +DISCUSSION AND CONCLUSIONS +Within the framework of modified theories of gravity, +in particular in the context of vector-tensor theories that +follow the same spirit of Horndeski’s theory, a kind of +RN black hole solution with two different non-Abelian +effective charges have been found in terms of the cou- +pling constants of the involved Lagrangian pieces of the +generalized SU(2) Proca theory. Such new solutions cor- +respond to genuine non-Abelian RN BH solutions in the +sense they were derived from a theory where the vec- +tor fields belong to the Lie algebra of the SU(2) group. +These objects then carry a dark charge because the mag- +netic charge is not of electromagnetic origin. There do +exist other solutions coming from other more involved +Lagrangians of the theory [37], but they do not posse ef- +fective charges in the asymptotic limit, which means that +they converge formally speaking to the Schwarzschild so- +lution. This will be reported in a separated paper. +Even though these solutions retain the main global +properties of the standard RN BH derived from the +Einstein-Maxwell theory, the solutions found exhibit an +appealing structure due to the non-trivial dependence on +the coupling constants. Interestingly, the solution with +qNA,I is characterized by having a negative square of the +effective charge for a certain range of values of the cou- +pling constant χ, similar to the tidal charge of brane BHs +[71–73] and BH solutions in Horndeski theory [74]. Pre- +venting the naked singularity from forming in both non- +Abelian RN BHs leads to discard a small region of the +parameter space. This happens particularly for the solu- +tions with qNA,I since the solution with qNA,II is always +real and well-behaved in the sense that no divergences +are present in the event horizon and ISCO structures. +Some phenomenological implications of the BH solu- +tions were also investigated in the astrophysical setting in +order to constrain the parameter space in a joint way with +the aforementioned theoretical considerations. They are +summarized as follows. +• Observations of the EHT’s first images of Sagittar- +ius A⋆ of our Galaxy, along with Keck telescope +results, set the first serious constraint on the free +parameters of the theory (˜g, χ), leaving almost all +the available parameter space of the non-Abelian +RN BHs basically unconstrained. For a given ˜g, a +lower limit on χ is determined as can be inferred +from the parameter space Fig. 3. As in the electric +RN BH case, these observational constraints also +rule out regions of the parameter space of the BH +solutions associated with naked singularities and +with the extremal BH. On the contrary, the corre- +sponding region of the parameter space for which +q2 +NA,I < 0, i.e., imaginary non-Abelian charge, is +allowed. +• As a first step towards a more realistic and elabo- +rated description of accretion processes, a fully rel- +ativistic treatment of spherical accretion of isother- +mal and polytropic fluids onto this class of BHs has +been performed to quantify the effect of the non- +Abelian charge and, therefore, of the coupling con- +stant, on the critical accretion rate. Interestingly, +we have found some dissimilarities in the accretion +process with respect to the standard electric RN +case that can serve as a potential observational sig- +nature to test the theory. Concretely, the critical +accretion rate efficiency can be noticeably improved +compared to the Schwarzschild case (and also to the +electric RN case), that is, +˙M NA +RN > +˙MSch, provided +that χ ∈ (0, 1) and qNA = q2 +NA,I < 0 which is, as +discussed above, admitted form the observational +side. In this regard, we have examined carefully, +with the aid of numerical computations for differ- +ent adiabatic indices of a polytropic fluid, the role +of the coupling parameter on the transonic proper- +ties of steady flows. +• As a way of probing the consistency of the non- +Abelian BH solutions, the Schwarzschild solution +and the extremal RN BH solution are recovered in +our solutions, as limit cases of the theory, for cer- +tain values of χ. This is a probe of concept of how +the BH solutions found behave in extreme regimes +of the parameter space. +An immediate theoretical extension of this work is +to implement the Newman-Janis algorithm to find ro- +tating non-Abelian charge BH solutions. Effectively, a +Kerr-(non-Abelian) Newman BH solution is naturally ex- +pected. Although the applicability of this algorithm must +be taken with great care [75], the absence of direct cou- +plings of the gauge fields to curvature terms guarantee +the viability of this future work. Then, we plan to study +the main properties of the image of the resulting BHs, +such as the shadows and photon rings, surrounded by an +optically and geometrically thin accretion disk and the +subsequent comparison with current observations. In this +regard, it is imperative to use observational constraints +from the shadow of the supermassive BH galaxy M87⋆, +as was recently done for the tidal charge [73]. Gravita- +tional and electromagnetic waveforms for charged black +hole binaries can be used to estimate the charges of BHs +in current and future gravitational wave experiments as +has been discussed recently [14–18]. This is another in- +teresting way to assess the effect of the coupling constants + +16 +in the strong-field regime in the vicinity of BHs. Hence, +gravitational wave observations have also the potential +of putting constraints on the coupling constants of the +theory. +ACKNOWLEDGMENTS +G. G. acknowledges financial support from Agencia +Nacional de Investigaci´on y Desarrollo (ANID) through +the FONDECYT postdoctoral Grant No. +3210417. +J.F.R. thanks financial support from the Patrimonio +Aut´onomo - Fondo Nacional de Financiamiento para la +Ciencia, la Tecnolog´ıa y la Innovaci´on Francisco Jos´e de +Caldas (MINCIENCIAS - COLOMBIA) under the grant +No. 110685269447 RC-80740–465–2020, project 69553. +[1] K. Akiyama et al. (Event Horizon Telescope), Astrophys. +J. 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D 88, 104020 +(2013), arXiv:1308.6631 [gr-qc]. + diff --git a/k9E4T4oBgHgl3EQftQ0b/content/tmp_files/load_file.txt b/k9E4T4oBgHgl3EQftQ0b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3d7b803f1e93885f22af1d3fc0bc74c2c55da80 --- /dev/null +++ b/k9E4T4oBgHgl3EQftQ0b/content/tmp_files/load_file.txt @@ -0,0 +1,1327 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf,len=1326 +page_content='New Non-Abelian Reissner-Nordstr¨om Black Hole Solutions in the Generalized SU(2) Proca Theory And Some Astrophysical Implications Gabriel G´omez∗ Departamento de F´ısica, Universidad de Santiago de Chile, Avenida V´ıctor Jara 3493, Estaci´on Central, 9170124, Santiago, Chile Jos´e F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Rodr´ıguez† Escuela de F´ısica, Universidad Industrial de Santander, Ciudad Universitaria, Bucaramanga 680002, Colombia and ICRANet, Piazza della Repubblica 10, 65122, Pescara PE, Italy The Generalized SU(2) Proca theory is a vector-tensor theory of gravity whose action is in- variant under global transformations of the SU(2) group and includes second-order derivative self- interactions of the vector field beyond the massive Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We find, in particular, that the presence of two Lagrangian pieces consisting of four gauge fields minimally coupled to gravity gives rise to an exact Reissner-Nordstr¨om black hole solution endowed with two different non-Abelian effective charges that depend on the specific combination, χ = 2χ1 + χ2, of the respective coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' After studying the spacetime structure of the black hole, which allows to characterize the parameter space that preserves the weak cosmic censorship conjecture, some astrophysical implica- tions of the black hole solutions are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' First, joint analysis of observations of the EHT’s first images of Sagittarius A⋆ of our Galaxy and the Keck telescope, set the first serious constraint on the free parameters of the theory beyond the theoretical bounds found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Second, we investigate the accretion properties of spherical steady flows around this class of non-Abelian Reissner-Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Specifically, we examine the general conditions under which transonic flow is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Analytical solution for critical accretion is found in terms of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In addition, we explore numerically the effect of changing χ on the radial velocity and mass density, and show how the extremal Reissner-Nordstr¨om and the standard Schwarzschild solutions as limit cases are achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Lastly, working in the fully relativistic regime, an analytical expression for the critical mass accretion rate of a polytropic fluid onto a black hole is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As a main result, we find that the critical accretion rate efficiency can be noticeably improved compared to the Schwarzschild case for a specific region of the parameter space where the non-Abelian charge becomes imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' INTRODUCTION Black holes (BHs) are one of the most fascinating ob- jects in the Universe that arise as a result of gravitational collapse of massive objects, as predicted by general rel- ativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Apart from the fundamental conceptions and bizarre properties they harbor, BHs are ideal labo- ratories, due to their intense gravitational fields, to study high-energy astrophysical processes that take place in neighborhoods around them and, more importantly, they offer a direct route to study the spacetime structure it- self which is a challenge of alternatives theories of gravity beyond general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Motivated by these concerns, BHs have been the central target of current astrophys- ical experiments such as the Event Horizon Telescope (EHT) and the Very Large Telescope global networks [1, 2], GRAVITY collaboration [3] and the LIGO and VIRGO collaborations [4, 5] among other important sci- entific projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Although the results derived from these observations are well consistent with GR, they are not conclusive in the sense that they can be reproduced by ∗ gabriel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='gomez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='d@usach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='cl † jose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='rodriguez2@correo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='co non-trivial spacetime metrics (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [6–9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Such ob- servations however do provide strong evidences about the existence of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Beyond the simple Schwarzschild BH realization, BHs can be endowed with an electric and magnetic charges when, for instance, the Maxwell theory is coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Nevertheless, it is believed that astrophysical BHs are electrically neutral due to charge neutralization by astrophysical plasma, among other suitable physical mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Another less conservative alternative is to consider BHs carrying, for instance, U(1) charge instead of electromagnetic charge due to early universe mecha- nisms within the dark (hidden) sector with no coupling to Standard-Model particles (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [10, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' See also [12] for a discussion from astrophysical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' No matter the underling physical process behind the charge mechanism, this is still an open issue that have brought recently much attention after the measurements made by the EHT of the supermassive BH M87⋆ shadow size and the detection of gravitational waves of compact object binaries, which demand careful examination on the BH charge far beyond academic considerations [13–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' See also [19, 20] for observational limits on the charge of the Galactic center BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' On the other hand, it has been confirmed by several observations that BHs must rotate to account for some arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05222v1 [gr-qc] 12 Jan 2023 2 astrophysical phenomena observed as X-rays streaming off material near BHs as a result of the formation of ac- cretion disk (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Due to the technical complications behind the Kerr solution that accounts for the BHs angular momentum, an useful first approach to study realistic and complex phenomena is to consider the Reissner Nordstr¨om (RN) solution whose charge plays a similar role as the spin does in the event horizon struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Indeed, the effects of the magnetic and electric charges of the RN BH can mimic the spin parameter of a rotating Kerr black hole in observations of the mag- netar J1745-2900 orbiting around the supermassive BH Sagittarius A⋆ (Sgr A⋆) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In addition, a recent anal- ysis of the motion of S-stars has constrained the spin of the object at the center of the Milky Way to be rather small, a/M ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This implies, that even though it is expected that BHs exhibit considerable spin, there are objects whose angular momentum is small and can be described by a stationary and spherically symmetric spacetime [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' After the finding of (purely magnetic) static spheri- cally symmetric non-Abelian BH solutions in the Einstein SU(2) Yang-Mills (EYM) model [25–27], it was shown soon after that they are perturbatively unstable [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Higher order curvature terms of the gauge field [29, 30], as well as non-trivial combinations with other theories have been included to the gravitational sector aiming at resolving this impasse (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [31] for former pro- posals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Some BHs solutions with non-Abelian hairs have been also found in theories beyond the canonical Yang- Mills theory but still in the context of GR [29, 30, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Higher curvature terms of the metric tensor such as f(R) gravity coupled to the Yang-Mills field admit also BH so- lutions with single or double horizons [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Interestingly, in the EYM case, there exists a RN so- lution [34, 35], but unlike the Einstein-Maxwell case, this is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This shows that although both models share the same spacetime configuration, they are pertur- batively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The same happens in the Einstein- Yang-Mills-Higgs, where a RN solution exists, but the Higgs mechanism can stabilize the solution [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In this work, we found an exact solution in the context of a mod- ified gravity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This solution is also of RN spacetime type, but as mentioned before, it is not the same as the EYM RN solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' On the hand, it is possible to build a healthy theory including higher derivative self-interactions of the SU(2) gauge field but still propagating the correct number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The result is the Generalized SU(2) Proca (GSU2P) theory [37, 38], which is the non-Abelian version of the Generalized Proca theory [39–41], and it belongs to a class of vector-tensor theories that lies in the spirit of Horndeski’s theory [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Considering particularly some Lagrangian pieces that involves four gauge fields minimally coupled to gravity, which arise from a system- atic construction in the full theory, gives place to BH so- lutions with two different non-Abelian effective charges that depend on the coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It is worthwhile mentioning that our findings are rooted to modified theo- ries of gravity despite they were particularly derived from these Lagrangian pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' From the astrophysical side, accretion processes of a ideal and polytropic fluids onto black holes has been a theme of intense study (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='g [43–46]), as a probe of concept, either in the heart of GR or in most general frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In particular, accretion flows in an arbitrary spacetime have been extensively investigated as astro- physical probes to reveal any deviation from GR [47–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' After a concise introduction, in Section II we describe the model and derive in detail an exact non-Abelian RN BH solution in terms of the coupling constants of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Some properties of the BH solutions such as the event hori- zon, photon sphere and shadow are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In partic- ular, using observational data of the EHT’s first images of Sagittarius A⋆, places first constraints on the effective coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We also investigate some astrophysi- cal implications of the BH solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Firstly, in Section III, a general description of the hydrodynamics equations of accretion flow is presented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' and secondly, in Section IV the critical accretion rate for both isothermal and poly- tropic fluids are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is done by implementing both analytical and numerical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We finish discussing the main findings of this work as well as pos- sible extensions of it in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Further observational constraints on the theory are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Throughout this paper we use geometrized units with c = G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' REISSNER NORDSTROM BLACK HOLE WITH NON-ABELIAN CHARGE The action of the model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' which corresponds to some Lagrangian pieces of the GSU2P theory [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' includes quartic order self-interactions of the vector field1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' S = 1 16π � √−g d4x[R − FaµνF aµν + χ1BaµBaµBbνBbν + χ2BaµBa νBb µBbν],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (1) where R is the Ricci scalar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Baµ represents the vector fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Faµν = ∂µBaν − ∂νBaµ + ˜gϵabcBbµBcν is the field strength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' ˜g is the gauge coupling constant and ϵabc is the structure constant tensor of the SU(2) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In geometrized units ˜g has units of inverse length, and the free parameters χ1 and χ2 have units of inverse square length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 1 As the inclusion of a mass term µ2AaαAaα spoils the existence of the solution, it has been taken away from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This result is similar to the classical massive vector field, where the mass needs to vanish to guarantee regularity of the solution and to allow therefore a vector hair to exist [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 3 The line element in a stationary and spherical sym- metric spacetime has the following form, ds2 = gtt(r)dt2 + grr(r)dr2 + r2dΩ2 = −e−2δNdt2 + N −1dr2 + r2dΩ2, (2) where N = 1 − 2m/r, δ and m are functions of the co- ordinate r, and dΩ is the solid angle element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Regarding the vector fields we chose the Wu-Yang monopole, A0 = Ar = 0 (3) Aθ = (w/v + 1) tφ (4) Aφ = (v − w) sin θ tθ, (5) where, tθ = cos θ cos φ t1 + cos θ sin φ t2 − sin θ t3, (6) tφ = − sin φ t1 + cos φ t2, (7) ti = −iσi/2 correspond to the vector basis of the SU(2) algebra with σi being the Pauli matrices, w is constant and v is an integer denoting the azimuthal winding num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We use the coupling constant ˜g to define the normal- ized variables, ˆr = r˜g, ˆm = m˜g, ˆχ1 = χ1/˜g2 ˆχ2 = χ2/˜g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The form of the equations in the normalized variables can be obtained effectively by setting ˜g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hereafter, all the equations are normalized, but we drop the hat to ease the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The field equation obtained after varying the action with respect to Baµ is, (v + w) �� v2χ1 + v2χ2 + χ1 � (v + w)2 + w(v − w) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (8) The solutions of this last equation are, wschw = −v, (9) wI,II = v + 2vχ1 + 2v3 (χ1 + χ2) ± � v2 [8v2χ2 + 8 (v2 + 1) χ1 + 1] 2 (1 − v2χ1 − v2χ2 − χ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (10) The first solution (9) is the trivial solution with vanish- ing vector field which corresponds to the Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Instead, the latter solutions (10), with two branches I and II, allow the existence of a non-trivial vector field we shall focus on, and constitutes, therefore, an important outcome of this work, as will be described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The field equations obtained by varying the action (1) with respect to metric are given by, m′ − � v3 − vw2�2 2r2v4 + (v + w)4 � v4χ2 + � v2 + 1 �2 χ1 + χ2 � 4r2v4 = 0 (11) δ′ = 0 (12) m′′ + v2(v − w)(v + 3w)(v + w)2 r3v4 + �� v4 + 5 � χ2 + � v4 + 6v2 + 5 � χ1 � (v + w)4 2r3v4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (13) We look for asymptotically flat solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' when r → ∞ the functions δ → 0, m → M ≡ finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Under these conditions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (12) is easily solved as δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' On the other hand, the solution of mass function m for the cases given by (10) has the following Reissner-Nordstr¨om solution, m = M − Q2 NA 2r , (14) where M is the total gravitational mass and QNA is a constant representing the effective charge and depends on the free parameters of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (11) and (13) must be consistent, which implies an additional constraint between χ1, χ2 and v given by, 4 3 � v4 − 1 � (χ1 + χ2) � 4v3χ2 + 3 � v2 [8v2χ2 + 8 (v2 + 1) χ1 + 1] + 4 � v3 + v � χ1 + 5v �2 [v2χ2 + (v2 + 1) χ1 − 1] 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (15) One solution fixes the winding number as, v = ±1, (16) with the parameters the parameters χ1 and χ2 indepen- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The other solution gives a relation between the free parameters of the action model, χ2 = −χ1, (17) with the winding number now unconstrained2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Despite the solution set by (16) gives two possible solu- tions, it represents only one since changing the sign of the winding number interchanges the solutions wI and wII as can be verified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In the solution given by (16) the value of w depends on the combination χ = 2χ1 +χ2, wI,II = 1 + 2χ ± √1 + 8χ 2 − 2χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (18) In the other solution corresponding to (17), the value of w/v has the same functional form of (18) after making χ �→ χ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Therefore, the physical behavior of both cases can be analyzed by means of (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hence, all subsequent analysis will be carried out in terms of the new effective coupling constant χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Consequently, the value of the effective charge Q2 NA is given by, Q2 NA,I,II = 1 − 4χ(5 + 2χ) ∓ (1 + 8χ)3/2 2(1 − χ)3 , (19) where the sign − corresponds to the branch I, and the sign + to the branch II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The dependence of both quanti- ties on the coupling constant is displayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' There exist solutions for the interval −1/8 < χ < 1 ∪ χ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' When χ = 0 the branch I solution corresponds to the Schwarzschild solution, and the branch II solution corresponds to the EYM charged solution with Q2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' When χ → 1 the branch I has a divergence, in contrast, in the same limit, the branch II is finite and tends to wII → −1/3, Q2 NA,II → 16/27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' On the hand, if we assume that the total mass of the black hole in normalized units is M = 1, we can find regions where Q2 NA > 1, corresponding to naked sin- gularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' These regions are given by −1/8 < χ < (11 − 5 √ 5)/2 ∪ 1 < χ < (11 + 5 √ 5)/2, for the branch 2 It seems that there is another solution given by the vanishing of the expression between braces in the numerator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Nevertheless, this does not constitute a solution because it makes also the denominator to vanish, inducing a divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' I, and −1/8 < χ < 0, for the branch II (see the gray regions in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Finally, it is worthwhile to mention that the en- ergy density associated with the vector fields is ρ = Q2 NA/(8πr4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Thus, when the non-Abelian charge is imaginary the energy density is negative, and this hap- pens only in the branch I in the interval 0 < χ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Event horizon In the stationary and spherically symmetric case, the vanishing of the metric function gtt defines unequivocally the horizon (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' When multiple solutions exist the greatest positive solution is identified with the event horizon of the black hole r+ = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In particular, the met- ric function associated to the Reissner-Nordstr¨om (RN) solution has two distinct roots r± = M ± � M 2 − Q2 NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (20) The internal solution r− is an apparent horizon and the external solution corresponds to the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hence, any observer outside the black hole (in asymp- totically flat spacetimes), or on the event horizon itself, cannot see any singularity because they are protected by an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Otherwise it is said to posses a naked singularity at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We do not mention here all the mi- nor details and conditions about the precise formulation of what is called the weak cosmic censorship conjecture3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The structure of the RN solution has been studied ex- tensively whereby we do not pretend to make here a detailed examination on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Nevertheless, an intriguing query arises when one asks about the impli- cations of the coupling constant χ on the charge, and therefore, on the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In particular, we are in- terested in finding which values of the coupling constant account for the convergence to both the Schwarzschild and the extremal RN black holes, as limit cases of our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' When performing numerical analysis and presenting the corresponding general discussion, we shall work with dimensionless variables by normalizing all physical quan- tities by the black hole mass M, unless otherwise said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Accordingly, we introduce, as usual, the charge to mass ratio qNA = QNA/M and the dimensionless radial coor- dinate x = r/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 3 We refer reader to [63] for a general and robust formulation of the cosmic censor conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 5 10 2 100 102 +1/8 3 2 1 0 1 2 3 Q 2 NA, I wI naked singularity 10 2 100 102 +1/8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 Q 2 NA, II wII naked singularity FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Values of the vector field and effective charge as a function of χ for the branch I (left panel) and for the branch II (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In both cases when χ → ∞ the charge tends to zero, thus the solution becomes the Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' If the mass of the black hole in normalized units is M = 1, values of Q2 NA > 1 represent a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' These last cases are shown as gray regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In the branch II the charge is always real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In the branch I for 0 < χ < 1 the charge is imaginary, which implies that the energy density is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Notice also that the shift χ + 1/8 in the abscissa has been done for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' q NA,I q NA,II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='10 1 10 100 1000 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 χ+1/8 xH q NA,I q NA,II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='50 1 5 10 0 2 4 6 8 10 χ+1/8 xISCO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Left panel: position of the event horizon xH as a function of the coupling constant for both charges as specified by the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Right panel: innermost stable circular orbit xISCO as a function of the coupling parameter χ as well for both non-Abelian charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In both plots red points indicate the match with the extremal RN solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It is evident here the two-fold degeneracy of qNA,I and the convergence to the Schwarzschild solutions when χ → ∞ for both charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The event horizon is depicted in left panel of Figure 2 as a function of the coupling constant for both charges4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For qNA,II (purple solid line) the event horizon is a well- behaved function of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Indeed, it covers continually the full range χ ∈ (0, ∞) where the fi- nite extreme value corresponds to the extremal RN case, in which case both solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (20) meet at xH = 1 (red point on the purple curve);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' while large values lead to the uncharged solution where xH = 2 and the appar- ent horizon (dashed curve) coincides with the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is hence a quite normal behavior that reproduces plainly the standard RN solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 4 When plotting, the shift χ + 1/8 in the abscissa is done for con- venience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' On the contrary, the event horizon for the case qNA,I (blue solid curves) exhibits a peculiar structure: there is a two-fold degeneracy with respect to the constant coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It means that the limit cases, that is, the Schwarzschild solution and the extremal RN black hole, can be described in two distinct regions of the param- eter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The same degeneracy is also presented for the apparent horizon (blue dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is clearly appreciated in the regions defined by the ranges χ ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901, 0) and χ ∈ (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0902, ∞) of left panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' There is also a special region of the parameter space χ ∈ (0, 1), where the square of the effective charge becomes negative and gravity repulsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Out of the men- tioned ranges results in q2 NA > 1, case that describes a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Inner most stable circular orbit Next, we turn our attention to the motion of massive test particles in the equatorial plane, as usual, to find the position of the so-called innermost stable circular orbits (ISCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Following the standard procedure (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [64]), one finds that the geodesic equation can be written as �dr ds �2 = � E2 − N � 1 + L2 r2 �� , (21) where E and L are identified, respectively, as the energy and angular momentum which correspond to conserved quantities as in the classical Keplerian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' From the above equation, one can also identify the effective potential Veff = N � 1 + L2 r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (22) We remind that N = 1 − 2m/r, the mass function m(r) has the form given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (14) and and the (squared) angular momentum is defined as L2 = N ′(r)r3 2N(r) − rN ′(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (23) In marginally stable circular orbits, the local extremum of the effective potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' V ′′ eff = 0, determines the position of the ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It yields rISCO = 2M + 4M 4 − 3M 2Q2 NA + � 8M 6 − 9M 4Q2 NA + 2M 2Q4 NA + � 5M 8Q4 NA − 9M 6Q6 NA + 4M 4Q8 NA �2/3 M � 8M 6 − 9M 4Q2 NA + 2M 2Q4 NA + � 5M 8Q4 NA − 9M 6Q6 NA + 4M 4Q8 NA �1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (24) This expression of course equals the corresponding stan- dard RN case [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' At this point we are able to compute the location of the ISCO in terms of the coupling con- stant χ with the aid of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The result is shown in right panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Clearly the effect of the cou- pling constant is replicated on the ISCO structure in the same fashion as it does on the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hence the same parameter space region establishes the correspond- ing Schwarzschild solution xISCO = 6, and the extremal RN case xISCO = 4 (red point on all curves) for both positive square of the non-Abelian charges as can be read from the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Thus, all the discussed properties for the event horizon are preserved for the ISCO location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It is interesting to notice, however, that for the charge qNA,I, there appears another region where gravity is repulsive in the interval χ ∈ (1, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='092), in comparison to the ones seen for the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For this charge it is admissi- ble then to have stable circular orbits for almost all the parameter space with the exception of its extreme values even though there does not exist event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This par- ticular behavior is reminiscent to the fact of having stable accretion flows onto a RN black hole even thought there is not a physical event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hence, in a naked sin- gularity situation it is (mathematically) possible to have stable circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We will not discuss however this point in detail since it is not of physical interest for the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Photon sphere and shadow A black hole has a central dark area called the shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This shadow is not delimited by the event horizon, but by the photon sphere, which is made of circular photon orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The radius of the photon sphere rph is given by solving, rphg′ tt(rph) − 2gtt(rph) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (25) However, the observed shadow radius, rsh is given by the lensed image of this surface [65], rsh = rph � gtt(rph) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (26) For the Reissner-Nordstr¨om solution the shadow radius is, rsh M = xsh = √ 2 �� 9 − 8q2 NA + 3 � � 4q2 NA+√ 9−8q2 NA−3 q2 NA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (27) This last equation can be inverted such that, for a given observed shadow radius, we can constrain the charge qNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' From this value we can constrain in turn the free param- eters of the theory ˜g and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The observations of Sagittar- ius A∗ made by the EHT collaboration give the following constrains on the size of the shadow, which depend on the mass-to-distance ratio [66], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 ≲ xsh ≲ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5, (28) with Keck and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3 ≲ xsh ≲ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3, (29) with VLTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' With this information, we can place con- straints on the parameters ˜g and χ, shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 7 Since (19) is normalized but depends physically on the ˜g and χ, only in this part of the analysis we put back the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The first conclusion is that an extremal RN black hole is not consistent with the current observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' If the non-Abelian charge is real, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=', Q2 NA > 0, the maximum value of the shadow is 3 √ 3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This value cor- responds to the Schwarzschild case χ → ∞ for both the branches I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The shadow of a Schwarzschild black hole is inside (28) and (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Thus, for the branch II we can give lower limits on χ (see the blue and orange curves in the lower plot of Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In contrast, the branch I exhibits imaginary values of the charge, allowing a greater shadow than the one of a Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Also, in this branch the al- lowed region of values for ˜g and χ is greater than in the branch II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For instance, when ˜gM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='9 the mini- mum value of χM 2 is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0326136 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0414562, for mass-to-distance ratio with Keck and VLTI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This implies a less stringent constraint in the parame- ters than in the branch discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For example, if ˜gM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01, the constraints are −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0545704 ≲ χM 2 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0223263 ∪ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3271 ≲ χM 2, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0708403 ≲ χM 2 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='00774184 ∪ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6237 ≲ χM 2, for Keck and VLTI, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' More cases can be inferred from the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The main purpose now is to figure out how the struc- ture of the non-Abelian RN black hole can impact the transonic properties of accretion flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' After formulat- ing the basic hydrodynamics equations, this subject will be investigated by employing both numerical and analyt- ical treatments in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' SPHERICAL STEADY ACCRETION FLOWS IN A SPHERICALLY SYMMETRIC SPACETIME Bondi accretion processes in a spherically symmetric spacetime is briefly described in this section following the general prescription presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Accordingly, we consider a spacetime with line element given by (2), with m and Q2 NA given, respectively, by (14) and (19), de- scribing a black hole of mass M and non-Abelian charge Q2 NA in Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Although the non- Abelian RN solution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (14) is, in principle, globally in- distinguishable for the standard RN BH solution, it was found, formally speaking, in the framework of modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' So, we keep as much as possible the generality in the description5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' On the other hand, we consider a steady fluid with total density ρ, mass density ρ0 and internal energy density ϵ, such that ρ = ρ0 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For isentropic fluids the pressure can be defined as P = kργ where k is a constant and γ is the adiabatic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For perfect 5 This general treatment serves also as a starting point to other (non-analytical) BH solutions found in the GSU2P theory [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' fluids, the stress energy momentum tensor is given by T µν = (ρ + P)uµuν + Pgµν, (30) where uµ = (ut, ur, 0, 0) is the four velocity of the fluid characterized by infall radial flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The normalization condition allows to obtain the relation between the com- ponents ut = � grr(u2+gtt) gtt , where we have defined for abbreviation u ≡ ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' From the baryon conservation and energy momentum conservation ∇µ(ρ0uµ) = 0, (31) ∇µT µν = 0, (32) one obtains two master equations, respectively ρ′ 0 ρ0 + u′ u +Σ = 0, (33) uu′ + gtt′ 2grrgtt (1 + grru2) + g′ rr 2grr u2 + c2 s grr (1 + grru2)ρ′ 0 ρ0 = 0, (34) where prime denotes radial derivative and the quantity Σ ≡ (√−g)′ √−g has been introduced for brevity as [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' These equations reduce to the Schwarzschild case when −gtt = 1 grr = 1 − 2M r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In finding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (34), we have used the definition of the sound speed of a medium at constant entropy c2 s ≡ dP dρ , and the useful relation P ′ = (ρ+P ) ρ0 c2 s ρ′ 0 derived from the first law of thermodynamics in the form dρ dρ0 = ρ+P ρ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Integration of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (33) and (34) gives, re- spectively, the mass accretion rate ˙M = 4πr2uρ0, (35) and, after some algebraic manipulations, the relativistic version of the Bernoulli equation (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' for more details) gtt(1 + grru2) �ρ + P ρ0 � = C, (36) where C is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Once C is defined by the boundary conditions at infinity for instance, and the metric functions are specified, the inward radial velocity can be computed for a given equation of state P = P(ρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Our primary concern is to solve this equation to deter- mine, in turn, the accretion rate given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Before computing this, we find critical values at which accretion flow is regular and causality is guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' ACCRETION IN A NON-ABELIAN REISSNER-NORDSTR¨OM BLACK HOLE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Critical accretion We study in this part general conditions under which transonic flow can take place in the vicinity of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 gM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 log10( M 2) Branch I xsh =4 (extremal) xsh =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3 xsh =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 xsh =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='19 Keck VLTI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 gM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='10 M 2 Branch I xsh =4 (extremal) xsh =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3 xsh =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3 xsh =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 xsh =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 xsh =3 3 (Schw) Keck VLTI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 gM 1 0 1 2 3 log10( M 2 +1/8) Branch II xsh =4 (extremal) xsh =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3 xsh =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 xsh =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='19 Keck VLTI FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Constraints on the parameters ˜g, and χ obtained from the observation of the shadow of the object located at the Galactic center of the Milky Way, Sagittarius A⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The upper plots correspond to the branch I which has been split into two ranges of χ for better representation, and the lower plot corresponds to the branch II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The different colored curves show the possible values of gc and χ for the lower and upper limits on xsh given by (28) and (29) as indicated by the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For comparison, we have added the curves for xsh = 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='19, 3 √ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The regions with slanted orange lines represent the constraints obtained from the mass-to-distance ratio with Keck, and the blue regions represent the constraints with VLTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It can be seen that a naked singularity is not consistent with the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Next, we shall describe both the spacetime geometry and the fluid nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Let us first write Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (33) and (34) in the more convenient form u′ u = gtt(2c2 s(1 + u2grr)Σ − u2g′ rr) − (1 + u2grr)g′ tt 2grrgtt(u2 − c2s(grru2)) , (37) ρ′ 0 ρ0 = −u2gtt(−2grrΣ + g′ rr) + (1 + u2grr)g′ tt 2grrgtt(u2 − c2s(grru2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (38) Imposing regular condition in both equations, implies that both numerators must vanish simultaneously at some critical point rc, resulting in u2 c = − g′ tt gttg′rr + grr(−2gttΣ + g′ tt), (39) c2 s,c = grrg′ tt 2grrgttΣ − gttg′rr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (40) Causality constraint c2 s < 1 in the flow sets a special point (rc) by which the flow must pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This physical requirement leads to the relation u2 c = − c2 s,c grr −1 + c2s,c , (41) between the radial velocity and the sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' At large radius, the flow is in the subsonic regime u2 < c2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' So far these results are general in the sense that can be applied to any spherically symmetric black hole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' At this point we must specify necessarily the metric functions to obtain the exact forms for the critical velocity and sound speed Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (39) and (40), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Thus, considering the non-Abelian Reissner Norstr¨om BH solution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (14), leads to the critical values u2 c = −Q2 NA − Mr 2r2 , c2 s,c = −Q2 NA + Mr Q2 NA + r(−3M + 2r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (42) 9 From the transonic condition that u2 c = c2 s,c at the critical radius, and considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (41), one obtains un- equivocally the critical radius rc = M + 3c2 s,cM ± � (M + 3c2s,cM)2 − 8c2s,c(1 + c2s,c)Q2 NA 4c2s,c , (43) where the non-trivial contribution of the non-Abelian charge to the Schwarzschild solution, rc,Sch = M(1 + 3c2 s,c)/2c2 s,c and u2 c,Sch = M/2rc, is clearly manifested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (43) we can see that there exist, mathemati- cally speaking, two distinct critical points but the pos- itive branch corresponds only to the physical solution which resides outside the event horizon, and therefore, it is the one we shall pay our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Of course, the negative branch is in a region observationally inaccessible since it is delimited by the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Existence of the critical radius demands Q2 NA M 2 < 1 + 6c2 s,c + 9c4 s,c 8c2s,c + 8c4s,c , (44) which in turn puts constraints on the coupling parameter for a given non-Abelian charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Nevertheless, the result- ing expressions are very lengthy (and not illuminating) to be reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We shall illustrate below this aspect numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In order to understand the striking structure of the critical radius we must necessarily specify the non- Abelian charge and the nature of the fluid which charac- terizes the sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For simplicity in the former anal- ysis, we assume an isothermal fluid6 so that the sound speed equals its equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This simple choice will give us, however, a profound insight about the behavior of the critical radius in terms on the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' So the present case suffices to prove the rich structure of the critical radius due to the non-Abelian charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It should be noticed, on the other hand, that for the (phys- ical) critical radius there are two solutions due to the existence of the non-Abelian charges QNA,I,II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Let us first explore the effect of changing the sound speed on the critical radius for some specific values of the coupling constant that cover mostly the (physical) BH solutions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The behavior of the critical ra- dius depends, however, on the non-Abelian charge cho- sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Here left and right pan- els correspond, respectively, to the non-Abelian charges qNA,I and qNA,II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In particular, for qNA,I and χ = 0, the critical radius matches exactly the Schwarzschild solu- tion, whereas for qNA,II the convergence is possible pro- vided that χ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is the reason why we have 6 Another possibility is to take a polytropic fluid but it introduces an extra parameter that overclouds the present analysis about the structure of the critical radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Nevertheless, it will be suc- cessfully addressed in the next part for the sake of completeness when looking for the critical mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' displayed the exact Schwarzschild solution (green curve) in right panel as an asymptotic limit of large coupling constant values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Accordingly, for the larger case shown, χ = 20, the solution is barely distinguishable from the uncharged case but physically different since the former is endowed with a charge qNA,I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='113438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hence, de- spite taking the same values of χ for both non-Abelian charges, the physical implications do not hold for the crit- ical radius as it also happens for the event horizon and ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' To better illustrate the role of χ in the critical radius, we take the same values of the coupling constant, as de- scribed below, for both charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For instance, taking qNA,I (see left panel) and a given coupling parameter lead to the following physical situations: the extremal case χ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901, the Schwarzschild solution χ = 0, the charged solution χ = 20 (qNA,I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4112) and a naked singularity χ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0222, that affect in a way differ- ent the behavior of the critical radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Now, taking the same values for χ as before but for qNA,II, the previous physical meaning is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Naked singularities can take place, for instance, for χ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901 which in the qNA,I case occurs for χ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0222 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Notice that for the for- mer case accretion is possible only for (subsonic) sound speeds c2 s,c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='3478 while for the latter it occurs out of the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6930 < c2 s,c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This explains why the discontinuity of those curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hence accretion may be possible even though the event horizon vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' An in- teresting discussion about how to distinguish a BH from a naked singularity spacetime by using the image of thin accretion disks is addressed in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As a general trend, for subsonic sound speeds the crit- ical radius can be located far away from the event hori- zon, and for supersonic sound speeds it can be accommo- dated, on the contrary, between the event horizon and the apparent horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For c2 s = 1 all critical points coin- cide with their respective position of the event horizon as can be easily verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Another interesting feature is that for the Schwarzschild case and subsonic speeds c2 s,c < 1 the critical radius is outside the event horizon while for c2 s,c > 1 the critical radius is located behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' See purple and green curves in the left and right panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This general discussion is in agreement with former stud- ies about the accretion of perfect fluids in the RN metric [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In contrast, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 5, the coupling parameter is fixed while the sound speed varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Notice that for the case qNA,I the critical radius exhibits a discontinuity as it was also perceived for the event horizon structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It should be noticed that as c2 s,c decreases (see for instance c2 s,c=1/4) the critical radius can be in a region where naked singularity takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As to the case qNA,II, the critical radius is also a monotonically increasing function of χ, but it is well-behaved until the condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (44) is broken for small values of χ and given squared sound speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This discussion provides, in complement to the one made around Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 4, a full picture on the general conditions under which the critical radius exists in terms 10 of the coupling constant through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Now, we are in a more grounded position to investigate the effect of changing the coupling parameter on the ac- cretion properties, concretely on the radial velocity and on the mass density around this class of black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This will be carried out first numerically for an isothermal fluid, and later with the aid of some analytical treat- ments for polytropic fluids, to allow a more robust and complete exploration of the involved parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' These two cases will be then treated separately in the next sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Isothermal test fluid Before describing accretion flows for a more general fluid, let us first consider a simplistic but useful isother- mal test fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It will provide us some physical insights on how the coupling constant influences the behavior of the infall radial velocity and mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This inquiry is complementary to the discussion on the critical point realized previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We focus for comparison reasons in a range of the coupling constant that resembles the RN BH and the Schwarzschild solution and leave the (allowed) range that provides q2 NA,I < 0 out of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It implies that the corresponding parameter space of χ for a given non- Abelian charge, provide the same physics whereby we focus on qNA,I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' To illustrate this point we show only a limit case for qNA,II to see the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The full range of χ will be, however, considered in the calculation of the critical accretion rate for a polytropic fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The above are also advantageous for numerical facilities since we have many variables involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Accordingly, the equation of state is of the form P = κρ, with κ being a constant, from which the simple rela- tion for the sound speed c2 s,c = κ is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Let us start our analysis by considering a stiff fluid κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As we already discussed, for this case (c2 s,c = 1) critical points coincide with the event horizons no mat- ter the value of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The latter spans the allowed region of the parameter space χ ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09, 0) for the charge qNA,I, as can be seen in the bar legend of Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We are not considering the other possible range of values χ ∈ (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0902, ∞) because the same phys- ical properties are replicated in the already shown range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For the other cases describing an ultra-relativistic fluid κ = 1/2, a radiation fluid κ = 1/3 and a sub-relativistic fluid κ = 1/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' all critical points move out the BH as κ decreases according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As a general trend, all transonic solutions are delimited from below to the ex- tremal RN solution χ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901 and from above to the Schwarzschild case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This latter case is attained whenever the coupling constant increases until it reaches the max- imum value showed here (χ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We have also included the solution χ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 (dashed magenta curve) for the charge qNA,II which matches the RN solution of the qNA,I case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' What is of physical interest here is that all infall radial velocities pass through the corresponding critical points marked as points on the curves and computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (43), guaranteeing thus the transonic flow of all solu- tions, otherwise a stellar wind is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This selects an unique solution with constant inward mass flux that connects the subsonic with the supersonic regimes, as ex- pected in Bondi-type accretion [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Far beyond the BH influence, that is, in the non-relativistic regime, the radial velocities of the particles are too low such that the inflow rate is decreased compared with particles with high-speed velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For the mass density distribution due to the BH grav- itational potential, the RN solution now delimits all so- lutions from above, as seen in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As in the radial velocity case, χ covers the same range of values for each value of the constant κ considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We can observe that as κ reduces, the mass density is more spread out along the radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' So far we have obtained the expected behavior for the radial velocity and mass density within the steady-state, spherical accretion scenario for the discussed range of χ- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As in previous sections, this has been very useful to understand the role of the coupling constant on the transonic flows and how our solutions approach to the ex- tremal RN and Schwarzschild solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Making a more robust description of the fluid, in next section we will perform analytical calculations of the critical accretion rate in the fully relativistic regime along with numerical computations for the entire range of values of χ and both non-Abelian charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Polytropic fluid Once the properties of the steady flows are known at the sonic point, this is, radial velocity, sound speed and critical radius, we can proceed to express such quantities in terms of the boundary conditions, with the help of the Bernoulli equation, as commonly done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The purpose of doing so is to calculate the accretion rate explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It is necessary also to adopt an equation of state for the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' So we study a non-relativistic baryonic gas with a polytropic equation P = Knγ, (45) where γ is the adiabatic index and K is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' With this and from the first of law of thermodynamic one can get [69] ρ = mn + K γ − 1nγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (46) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Relativistic regime Surprisingly, the fully relativistic accretion rate for the Reissner Nordstr¨om solution has not been thoroughly treated in the literature, where most of existing works 11 χ=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901 χ=0 χ=20 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0222 qNA,I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0 1 2 3 4 5 6 cs 2 xc Sch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' χ=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901 χ=0 χ=20 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0222 qNA,II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0 1 2 3 4 5 6 cs 2 xc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Left panel: critical radius xc ≡ rc/M (43) for the non-abelian charge qNA,I for a given coupling parameters that correspond to the extremal case χ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901, Schwarzschild solution χ = 0, charged solution χ = 20 (qNA,I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4112) and a naked singularity χ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0222 (qNA,I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Right panel: critical radius for the non-abelian charge qNA,II for the same values of parameters as left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The exact Schwarzschild solution (green curve) has been included here for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Notice that, on the contrary, the solutions χ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0901, χ = 0, χ = 20 and χ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0222 correspond, respectively, to a naked singularity (qNA,II = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='11352), a extremal case and charge BH cases qNA,II = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='113438 and qNA,II = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='17345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' cs 2=1/4 cs 2=1/3 cs 2=1/2 cs 2=1 qNA,I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0 2 4 6 8 χ+1/8 xc cs 2=1/4 cs 2=1/3 cs 2=1/2 cs 2=1 qNA,II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0 1 2 3 4 5 6 χ+1/8 xc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Left panel: critical radius for the non-abelian charge qNA,I for certain critical sound speeds, as described in the legend, as a function of the coupling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Right panel: critical radius for the non-abelian charge qNA,II for the same parameters as left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' have focused on the non-relativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' It leaves un- doubtedly an incomplete comprehension of the full pic- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We first derive some analytical expressions and show some numerical examples to illustrate better the role of the coupling constant on the mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We follow closely Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [45], where Bondi accretion of steady spherical gas flow onto a Schwarzschild black hole has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' We extend this work to the charged case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' With the aid of the polytropic equation, it is possible to relate the sound speed with the mass density c2 s = γkργ−1 0 1 + γkργ−1 0 /(γ − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (47) This expression can be evaluated at the critical point and in the asymptotic region to provide the useful relation ρ0,s = ρ0,∞ � c2 s,c c2s,∞ � 1 γ−1 � γ − 1 − c2 s,∞ γ − 1 − c2s,c � 1 γ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (48) This closed expression requires the knowledge of the crit- ical sound speed that can be extracted from the relativis- tic Bernoulli equation (1 + 3c2 s,c) � 1 − c2 s,c γ − 1 �2 = � 1 − c2 s,∞ γ − 1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (49) So, once the sound speed at infinity is specified, the sound speed and the mass density at the critical point are uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The Bernoulli equation is ac- tually a cubic equation for c2 s,c with one real solution for 12 χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 κ = 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 x u r χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 κ = 1/2 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 x u r χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 κ = 1/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 x u r χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 κ = 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 x u r FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Infall radial velocity for some specific values of κ describing a stiff fluid (κ = 1), ultra-relativistic fluid (κ = 1/2), radiation fluid (κ = 1/3) and sub-relativistic fluid (κ = 1/4) while the coupling parameter χ spans the allowed range as depicted by the bar legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The dotted magenta curve that delimits all the possible transonic solutions from below corresponds to the extremal case χ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 for qNA,I (or χ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 for qNA,II), while χ → 0 resembles the Schwarzschild solution from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' the range 1 < γ < 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' There are some procedures to solve analytically this equations as, for instance, a stan- dard root-finding schema which we implement to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Having expressed all quantities at the critical radius in terms of the boundary conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' the critical accretion rate ˙M = 4πρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='susr2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (50) can be computed easily ˙M = 4π � M c2s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='∞ �2 c2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='∞ ρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='∞ λNA RN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (51) with the accretion rate eigenvalue λNA RN ≡ � c2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='c c2s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='∞ � 5−3γ γ−1 � γ − 1 − c2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='∞ γ − 1 − c2s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='c � 1 γ−1 (1 + 3c2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='c)3/2 4 β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (52) and the β factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' containing information of the non- Abelian charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' is β = 1 4 � 1 + � 1 − 8c2s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='c(1 + c2s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='c)q2 NA (1 + 3c2s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='c)2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (53) This quantity clearly accounts for the deviation from the Schwarzschild case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In what follows, we quantify such a deviation by computing the ratio of both accretion rates ˙M NA RN ˙MSch = λNA RN λSch = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (54) As it is known, the electric charge of the RN black hole reduces the accretion rate compared to the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Our case may be, however, different in the sense that if the imaginary charge of the black hole is allowed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' when q2 NA = q2 NA,I < 0 and χ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Un- der this choice, ˙M NA RN > ˙MSch as can be verified in all left panels of Figure 8 where the accretion rate has been plotted as a function of the coupling constant χ for differ- ent adiabatic indices as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Notice that all curves meet in the corresponding position of the event horizon χ → 0, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Out of the mentioned range, the expected be- havior of the Reissner Nordstr¨om black hole is displayed just as the case qNA = qNA,II (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Even though the non-Abelian charge case is effectively distinguishable from its electric counterpart in the small range χ ∈ (0, 1) for qNA,I, these numerical computations allow, besides, to understand better the multiplicity of the non-Abelian 13 χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 κ = 1 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 1 2 3 4 0 1 2 3 4 5 6 7 x ρ χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 κ = 1/2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 1 2 3 4 0 1 2 3 4 5 x ρ χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 κ = 1/3 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 0 1 2 3 4 5 0 1 2 3 4 5 x ρ χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='01 κ = 1/4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 χ=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 0 1 2 3 4 5 0 1 2 3 4 5 x ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Mass density distribution for some specific values of κ describing a stiff fluid (κ = 1), ultra-relativistic fluid (κ = 1/2), radiation fluid (κ = 1/3) and sub-relativistic fluid (κ = 1/4) while the coupling parameter χ spans the allowed range as depicted by the bar legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The dotted magenta curve that delimits all the possible transonic solutions from below corresponds to the extremal case χ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 for qNA,I (or χ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='09 for qNA,II), while χ → 0 (or χ → ∞) resembles the Schwarzschild solution from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Reissner Nordstr¨om black solution and its implications in the accretion rate, in particular for the range of val- ues of χ which leads to a significant enhancement of the accretion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As a last remark about the boundary conditions, the larger the boundary sound speed, the shorter the accre- tion rate variations are among the polytropic fluid con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The stiff case γ = 1 is more sensible to the increase of the boundary sound speed: notice how the dashed black curves in right panels meet the other curves for low χ or, which is equivalent, in the non-Abelian ex- tremal case qNA → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Asymptotic limit We do not describe here in detail all derivations con- cerning the asymptotic limit of the mass accretion rate since it can be found, for instance, in reference [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The purpose of this part is to check the consistency with the well-known Newtonian limit and the corresponding de- pendence on the non-Abelian charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' From the Bernoulli equation, one can derive the useful relation c2 s,c ≈ 2c2 s,∞ (5 − 3γ), (55) under the non-relativistic condition cs,c ≪ 1 which holds for reasonable large radius, r ≫ rc, far away from the BH gravitational influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' The same condition leads to the simple relation c2 s,c ≈ Kγ ργ−1 0,c , (56) between the sound speed and the mass density at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This implies that the mass density can be expressed in terms of the sound speed at the infinity in view of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (55) to yield ρ0,c ≈ ρ0,∞ � c2 s,c c2s,∞ � 1 γ−1 ≈ � 2 5 − 3γ � 1 γ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (57) As in the uncharged case, physical solutions require γ < 5/3 in this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' At this point all above is standard and 14 γ=4/3 γ=5/3 γ=1 qNA,I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1 5 10 χ+1/8 M \uf110 RN NA M \uf110 Sch γ=4/3 γ=5/3 γ=1 qNA,II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 χ+1/8 M \uf110 RN NA M \uf110 Sch γ=4/3 γ=5/3 γ=1 qNA,I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1 5 10 χ+1/8 M \uf110 RN NA M \uf110 Sch γ=4/3 γ=5/3 γ=1 qNA,II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 χ+1/8 M \uf110 RN NA M \uf110 Sch γ=4/3 γ=5/3 γ=1 qNA,I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 1 5 10 χ+1/8 M \uf110 RN NA M \uf110 Sch γ=4/3 γ=5/3 γ=1 qNA,II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='2 χ+1/8 M \uf110 RN NA M \uf110 Sch FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Ratio of mass accretion rate in the non-Abelian RN BH to the accretion rate in the Schwarzschild BH for different adiabatic indices as indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Left and right panels correspond to the cases qNA = qNA,I and qNA = qNA,II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In tops panels the boundary condition approaching to the non-relativistic regime c2 s,∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='001 has been taken, whereas in middle and bottom panels a relativistic boundary sound speed c2 s,∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='1 and c2 s,∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='5 have been chosen, respectively, in contrasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In all cases the condition uc > cs,∞ is guaranteed, ensuring thus the transonic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' these quantities do not receive contributions from the ef- fective charge QNA at lowest order in cs,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is not the case however for the critical radius Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (43) where, at leading order, QNA already appear explicitly as second order power rc ≈ M 2c2s,c +3M 2 − 2Q2 NA 2M +2(M 2Q2 NA − Q4 NA) M 3 c2 s,c+O(c4 s,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (58) Keeping only higher order contributions in cs,c and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (55), the critical radius can be approximated to rc ≈ (5 − 3γ) 4c2s,∞ Mη(QNA), (59) where the dimensionless correction factor due to the ef- fective charge has been defined as η(QNA) = 1 + c2 s,∞ (5 − 3γ) � 6 − 4Q2 NA M 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (60) 15 With all this, the critical accretion mass rate can be writ- ten in the familiar form ˙M ≈ 4πρ0,∞M 2c−3 s,∞ �1 2 � γ+1 2(γ−1) �5 − 3γ 4 � 3γ−5 2(γ−1) η(QNA)2, (61) which agrees with [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Finally, the uncharged Newtonian limit is straightforward achieved in the limit QNA → 0 and, appropriately, c2 c,∞ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS Within the framework of modified theories of gravity, in particular in the context of vector-tensor theories that follow the same spirit of Horndeski’s theory, a kind of RN black hole solution with two different non-Abelian effective charges have been found in terms of the cou- pling constants of the involved Lagrangian pieces of the generalized SU(2) Proca theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Such new solutions cor- respond to genuine non-Abelian RN BH solutions in the sense they were derived from a theory where the vec- tor fields belong to the Lie algebra of the SU(2) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' These objects then carry a dark charge because the mag- netic charge is not of electromagnetic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' There do exist other solutions coming from other more involved Lagrangians of the theory [37], but they do not posse ef- fective charges in the asymptotic limit, which means that they converge formally speaking to the Schwarzschild so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This will be reported in a separated paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Even though these solutions retain the main global properties of the standard RN BH derived from the Einstein-Maxwell theory, the solutions found exhibit an appealing structure due to the non-trivial dependence on the coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Interestingly, the solution with qNA,I is characterized by having a negative square of the effective charge for a certain range of values of the cou- pling constant χ, similar to the tidal charge of brane BHs [71–73] and BH solutions in Horndeski theory [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Pre- venting the naked singularity from forming in both non- Abelian RN BHs leads to discard a small region of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This happens particularly for the solu- tions with qNA,I since the solution with qNA,II is always real and well-behaved in the sense that no divergences are present in the event horizon and ISCO structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Some phenomenological implications of the BH solu- tions were also investigated in the astrophysical setting in order to constrain the parameter space in a joint way with the aforementioned theoretical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' They are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Observations of the EHT’s first images of Sagittar- ius A⋆ of our Galaxy, along with Keck telescope results, set the first serious constraint on the free parameters of the theory (˜g, χ), leaving almost all the available parameter space of the non-Abelian RN BHs basically unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' For a given ˜g, a lower limit on χ is determined as can be inferred from the parameter space Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As in the electric RN BH case, these observational constraints also rule out regions of the parameter space of the BH solutions associated with naked singularities and with the extremal BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' On the contrary, the corre- sponding region of the parameter space for which q2 NA,I < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=', imaginary non-Abelian charge, is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As a first step towards a more realistic and elabo- rated description of accretion processes, a fully rel- ativistic treatment of spherical accretion of isother- mal and polytropic fluids onto this class of BHs has been performed to quantify the effect of the non- Abelian charge and, therefore, of the coupling con- stant, on the critical accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Interestingly, we have found some dissimilarities in the accretion process with respect to the standard electric RN case that can serve as a potential observational sig- nature to test the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Concretely, the critical accretion rate efficiency can be noticeably improved compared to the Schwarzschild case (and also to the electric RN case), that is, ˙M NA RN > ˙MSch, provided that χ ∈ (0, 1) and qNA = q2 NA,I < 0 which is, as discussed above, admitted form the observational side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In this regard, we have examined carefully, with the aid of numerical computations for differ- ent adiabatic indices of a polytropic fluid, the role of the coupling parameter on the transonic proper- ties of steady flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' As a way of probing the consistency of the non- Abelian BH solutions, the Schwarzschild solution and the extremal RN BH solution are recovered in our solutions, as limit cases of the theory, for cer- tain values of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is a probe of concept of how the BH solutions found behave in extreme regimes of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' An immediate theoretical extension of this work is to implement the Newman-Janis algorithm to find ro- tating non-Abelian charge BH solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Effectively, a Kerr-(non-Abelian) Newman BH solution is naturally ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Although the applicability of this algorithm must be taken with great care [75], the absence of direct cou- plings of the gauge fields to curvature terms guarantee the viability of this future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Then, we plan to study the main properties of the image of the resulting BHs, such as the shadows and photon rings, surrounded by an optically and geometrically thin accretion disk and the subsequent comparison with current observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' In this regard, it is imperative to use observational constraints from the shadow of the supermassive BH galaxy M87⋆, as was recently done for the tidal charge [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Gravita- tional and electromagnetic waveforms for charged black hole binaries can be used to estimate the charges of BHs in current and future gravitational wave experiments as has been discussed recently [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' This is another in- teresting way to assess the effect of the coupling constants 16 in the strong-field regime in the vicinity of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Hence, gravitational wave observations have also the potential of putting constraints on the coupling constants of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' ACKNOWLEDGMENTS G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' acknowledges financial support from Agencia Nacional de Investigaci´on y Desarrollo (ANID) through the FONDECYT postdoctoral Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 3210417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' thanks financial support from the Patrimonio Aut´onomo - Fondo Nacional de Financiamiento para la Ciencia, la Tecnolog´ıa y la Innovaci´on Francisco Jos´e de Caldas (MINCIENCIAS - COLOMBIA) under the grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 110685269447 RC-80740–465–2020, project 69553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (Event Horizon Telescope), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 875, L1 (2019), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='11238 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (Event Horizon Telescope), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 930, L12 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Abuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' (GRAVITY), Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 636, L5 (2020), arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='07187 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} 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LWA, HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, AT- LAS Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT), Astro- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 848, L12 (2017), arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='05833 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='HE].' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Shapiro, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 502, 3003 (2021), [Er- ratum: Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 506, 3935 (2021)], arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='08797 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Gam- mie, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' Yunes, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' 925, 119 (2022), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content='02178 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf'} +page_content=' [48] K.' metadata={'source': 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0000000000000000000000000000000000000000..cc27c0e502119c353a0c90f60275dee8a66be5bf --- /dev/null +++ b/k9E4T4oBgHgl3EQftQ0b/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:01ec05f0f6160140ffddc1f2641267ad7849ad5a4bd400fbc78da74360da8cb8 +size 190986 diff --git a/mtE3T4oBgHgl3EQf6gu4/content/tmp_files/2301.04791v1.pdf.txt b/mtE3T4oBgHgl3EQf6gu4/content/tmp_files/2301.04791v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ee43cf85b67e145659c4fbc68adc9cc7891455a --- /dev/null +++ b/mtE3T4oBgHgl3EQf6gu4/content/tmp_files/2301.04791v1.pdf.txt @@ -0,0 +1,1917 @@ +Self-Attention Amortized Distributional Projection +Optimization for Sliced Wasserstein Point-Cloud +Reconstruction +Khai Nguyen†,∗ +Dang Nguyen⋄,∗ +Nhat Ho† +University of Texas, Austin†, VinAI Research⋄ +January 13, 2023 +Abstract +Max sliced Wasserstein (Max-SW) distance has been widely known as a solution for redundant +projections of sliced Wasserstein (SW) distance. In applications that have various independent +pairs of probability measures, amortized projection optimization is utilized to predict the “max" +projecting directions given two input measures instead of using projected gradient ascent multiple +times. +Despite being efficient, the first issue of the current framework is the violation of +permutation invariance property and symmetry property. To address the issue, we propose +to design amortized models based on self-attention architecture. Moreover, we adopt efficient +self-attention architectures to make the computation linear in the number of supports. Secondly, +Max-SW and its amortized version cannot guarantee metricity property due to the sub-optimality +of the projected gradient ascent and the amortization gap. Therefore, we propose to replace +Max-SW with distributional sliced Wasserstein distance with von Mises-Fisher (vMF) projecting +distribution (v-DSW). Since v-DSW is a metric with any non-degenerate vMF distribution, its +amortized version can guarantee the metricity when predicting the best discriminate projecting +distribution. With the two improvements, we derive self-attention amortized distributional +projection optimization and show its appealing performance in point-cloud reconstruction and +its downstream applications. +1 +Introduction +Wasserstein distance [49, 38] has been widely recognized in the community of machine learning as +an effective tool. For example, Wasserstein distance is used to explore clusters inside data [21], to +transfer knowledge between different domains [11, 13], to learn generative models [4, 47], to extract +features from graphs [50], to compare datasets [2], and many other applications. Despite being +effective, Wasserstein distance is extremely expensive to compute. In particular, the computational +complexity and memory complexity of Wasserstein distance in the discrete case is O(m3 log m) and +O(m2) respectively with m is the number of supports. The computational problem becomes more +severe for applications that require computing the Wasserstein distance multiple times on different +pairs of measures. Some examples can be named: deep generative modeling [20, 32], deep domain +adaptation [6], comparing datasets [2], topic modeling [22], point-cloud reconstruction [1], and so on. +By adding entropic regularization [12], an ε-approximation of Wasserstein distance can be obtained +in O(m2/ε2). However, this approach cannot reduce the memory complexity of O(m2) due to +the storage of the cost matrix. A more efficient approach is based on the closed-form solution of +Wasserstein distance in one dimension which is known as sliced Wasserstein distance [7]. Sliced +*Khai Nguyen and Dang Nguyen contributed equally to this work +1 +arXiv:2301.04791v1 [stat.ML] 12 Jan 2023 + +Wasserstein (SW) distance is defined as the expectation of the Wasserstein distance between random +one-dimensional push-forward measures from the two original measures. Thanks to the closed- +form solution, SW can be solved in O(m log2 m) while having a linear memory complexity O(m). +Moreover, SW is also better than Wasserstein distance in high-dimensional statistical inference. +Namely, the sample complexity (statistical estimation rate) of SW is O(n−1/2) compared to O(n−1/d) +of Wasserstein distance with d is the number dimension and n is the number of data samples. +Due to appealing properties, SW is utilized successfully in various applications e.g., generative +modeling [17, 35, 32], domain adaptation [26], Bayesian inference [30, 58], point-cloud representation +learning [36, 29], and so on. +The downside of SW is that it treats all projections the same due to the usage of a uniform +distribution over projecting directions. This choice is inappropriate in practice since there exist +projecting directions that cannot discriminate two interested measures [25]. As a solution, max sliced +Wasserstein distance (Max-SW) [16] is introduced by searching for the best projecting direction that +can maximize the projected Wasserstein distance. Max-SW needs to use a projected sub-gradient +ascent algorithm to find the “max" slice. Therefore, in applications that need to evaluate Max-SW +multiple times on different pairs of measures, the repeated optimization procedure is costly. For +example, this paper focuses on point-cloud reconstruction applications where Max-SW needs to be +computed between various pairs of empirical measures over a point-cloud and its reconstruction. +To address the problem, amortized projection optimization is proposed in [31]. As in other amortized +optimization [43, 3] (learning to learn), an amortized model is estimated to predict the best projecting +direction given the two input empirical measures. The authors in [31] propose three types of amortized +models including linear model, generalized linear model, and non-linear model. The linear model +assumes that the “max" projecting direction is a linear combination of supports of two measures. +The generalized linear model injects the linearity through a link function on the supports of two +measures while the non-linear model uses multilayer perceptions to have more expressiveness. +Despite performing well in practice, the previous work has not explored the full potential of +amortized optimization in the sliced Wasserstein setting. +There are two issues in the current +amortized optimization framework. Firstly, the sub-optimality of amortized optimization leads to +losing the metricity of the projected distance from the predicted projecting direction. In particular, +the metricity of Max-SW is only obtained at the global optimum. Therefore, using an amortized +model with sub-optimal solutions cannot achieve the metricity for all pairs of measures. Losing +metricity property could hurt the performance of downstream applications. Secondly, the current +amortized models are not permutation invariant to the supports of two input measures and are +not symmetric. The permutation-invariant and symmetry properties are vital since the “max" +projecting direction is also not changed when permuting supports of two input empirical measures +and exchanging two input empirical measures. By inducing the permutation-invariance and symmetry +to the amortized model, it could learn a better amortized model and reduce the amortization gap +In this paper, we focus on overcoming the two issues of the current amortized projection optimization +framework. For metricity preservation, we propose amortized distributional projection optimization +framework which predicts the best distribution over projecting directions. In particular, we do +amortized optimization for distributional sliced Wasserstein (DSW) distance [33] with von Mises +Fisher (vMF) slicing distribution [23] instead of Max-SW. Thanks to the smoothness of vMF, the +metricity can be preserved even without a zero amortization gap. For the permutation-invariance and +symmetry properties, we propose to use the self-attention mechanism [48] to design the amortized +2 + +model. Moreover, we utilize efficient self-attention approaches that have the computational complexity +scales linearly in the number of supports including efficient attention [42] and linear attention [51]. +Contribution: In summary, our contribution is two-fold: +1. First, we introduce amortized distributional projection amortization framework which predicts the +best location parameter for von Mises-Fisher (vMF) distribution in distributional sliced Wasserstein +(DSW) distance. Due to the smoothness of vMF, the metricity is guaranteed for all pairs of measures. +Moreover, we enhance amortized models by inducing inductive biases which are permutation +invariance and symmetry. To improve the efficiency, we leverage two linear-complexity attention +mechanisms including efficient attention [42] and linear attention [51] to parametrize the amortized +model. Combining the above two improvements, we obtain self-attention amortized distributional +projection amortization framework +2. Second, we adapt the new framework to the point-clouds reconstruction problem. In particular, we +want to learn an autoencoder that can reconstruct (encode and decode) all point-clouds through their +latent representations. The main idea is to treat a point-cloud as an empirical measure and use sliced +Wasserstein distances as the reconstruction losses. Here, amortized optimization serves as a fast way +to yield informative projecting directions for sliced Wasserstein distance to discriminative all pairs +of original point-cloud and reconstructed point-cloud. Empirically, we show that the self-attention +amortized distributional projection amortization provides better reconstructed point-clouds on the +ModelNet40 dataset [54] than the amortized projection optimization framework and widely used +distances. Moreover, on downstream tasks, the new framework also leads to higher classification +accuracy on ModelNet40 and generates ShapeNet chairs with better quality. +Organization: The remainder of the paper is organized as follows. In Section 2, we provide +backgrounds for point-cloud reconstruction and popular distances. In Section 3, we define the +new amortized distributional projection optimization framework for the point-cloud reconstruction +problem. Section 4 benchmarks the proposed method by extensive experiments on point-cloud +reconstruction, transfer learning, and point-cloud generation. Finally, proofs of key results and extra +materials are in the supplementary. +Notation. For any d ≥ 2, we denote U(Sd−1) is the uniform measure over the unit hyper-sphere +Sd−1 := {θ ∈ Rd | ||θ||2 +2 = 1}. For p ≥ 1, Pp(Rd) is the set of all probability measures on Rd that +have finite p-moments. For any two sequences an and bn, the notation an = O(bn) means that +an ≤ Cbn for all n ≥ 1, where C is some universal constant. We denote θ♯µ is the push-forward +measures of µ through the function f : Rd → R that is f(x) = θ⊤x. +2 +Preliminaries +We first review the point-cloud reconstruction framework in Section 2.1. After that, we discuss +famous choices of metrics between two point-clouds in Section 2.2. Finally, we present an adapted +definition of the amortized projection optimization framework in the point-cloud reconstruction +setting in Section 2.3. +3 + +Encoder +Decoder +Figure 1: The reconstruction of a point-cloud X (a plane). +2.1 +Point-Cloud Reconstruction +We denote a point-cloud of m points x1, . . . , xm ∈ Rd (d ≥ 1) as X = (x1, . . . , xm) ∈ Rdm which +is a vector of a concatenation of all points in the point-cloud. We denote the set of all possible +point-clouds as X ⊂ Rdm. +Permutation invariant metric space: Given a permutation one-to-one mapping function σ : +[m] → [m], we have σ(X) ∈ X for all X ∈ X. Moreover, we need a metric D : X × X → R+ such +that D(X, σ(X)) = 0 for all X ∈ X where σ(X) = (xσ(1), . . . , xσ(m)). Here, D is a metric, namely, +it needs to satisfy the non-negativity, symmetry, triangle inequality, and identity property. The pair +(X, D) forms a point-cloud metric space. +Learning representation via reconstruction: The raw representation of point-clouds is hard +to work with in applications due to the complicated metric space. Therefore, a famous approach is +to map point-clouds to points in a different space e.g., Euclidean, which is easier to apply machine +learning algorithms. In more detail, we want to estimate a function fφ : X → Z (φ ∈ Φ) where +Z is a set that belongs to another metric space. Then, we can apply machine learning algorithms +on Z instead of X. The most well-known and effective way to estimate the function fφ is through +reconstruction loss. Namely, we estimate fφ jointly with a function gγ : Z → X (γ ∈ Γ) given a +point-cloud dataset p(X) (distribution over X) by minimizing the objective: +min +φ∈Φ,γ∈Γ EX∼p(X)D(X, gγ(fφ(X))). +(1) +The loss EX∼p(X)D(X, gγ(fφ(X))) is known as the reconstruction loss. If the reconstruction loss is 0, +we have gγ = f−1 +φ +p-almost surely. Therefore, we can move from X to Z and move back from Z to +X without losing information through the functions fφ (referred as the encoder) and gγ (referred as +the decoder). We show an illustration of the framework [1] in Figure 1. After learning how to do the +reconstruction well, other point-cloud tasks can be done using the autoencoder (the pair (fφ, gγ)) +e.g., shape interpolation, shape editing, shape analogy, shape completion, point-cloud classification, +and point-cloud generation [1]. +4 + +2.2 +Metric Spaces for Point-Clouds +We now review some famous choices of the metric D which are Chamfer distance [5], Wasserstein +distance [49], sliced Wasserstein (SW) distance [7], and max sliced Wasserstein (Max-SW) [16] +distance. +Chamfer distance: +For any two point-clouds X and Y , the Chamfer distance is defined as follows: +CD(X, Y ) = +1 +|X| +� +x∈X +min +y∈Y ∥x − y∥2 +2 + 1 +|Y | +� +y∈Y +min +x∈X ∥x − y∥2 +2, +(2) +where |X| denotes the number of points in X. +Wasserstein distance: Given two probability measures µ ∈ Pp(Rd) and ν ∈ Pp(Rd), the Wasser- +stein distance between µ and ν is defined as follows: +Wp(µ, ν) = +� +inf +π∈Π(µ,ν) +� +Rd×Rd ∥x − y∥p +pdπ(x, y) +� 1 +p +(3) +where Π(µ, ν) is set of all couplings whose marginals are µ and ν respectively. Since the Wasserstein +distance is originally defined on probability measures space, we need to convert a point-cloud +X = (x1, . . . , xm) ∈ X to the corresponding empirical probability measure PX = 1 +m +�m +i=1 δxi ∈ P(Rd). +Therefore, we can use D(X, Y ) = Wp(PX, PY ) for X, Y ∈ X. +Sliced Wasserstein distance: As discussed, the Wasserstein distance is expensive to compute +with the time complexity O(m3 log m) and the memory complexity O(m2). Therefore, an alternative +choice is sliced Wasserstein (SW) distance between two probability measures µ ∈ Pp(Rd) and +ν ∈ Pp(Rd) is: +SWp(µ, ν) = +� +Eθ∼U(Sd−1)Wp +p(θ♯µ, θ♯ν) +� 1 +p , +(4) +The benefit of SW is that Wp(θ♯µ, θ♯ν) has a closed-form solution which is +�� 1 +0 |F −1 +θ♯µ(z) − F −1 +θ♯ν(z)|pdz +� 1 +p +with F −1 denotes the inverse CDF function. The expectation is often approximated by Monte Carlo +sampling, namely, it is replaced by the average from θ1, . . . , θL that are drawn i.i.d from U(Sd−1). +The computational complexity and memory complexity of SW becomes O(Lm log2 m) and O(Lm). +Max sliced Wasserstein distance: +It is well-known that SW has a lot of redundant projections +due to the uniform sampling. Therefore, max sliced Wasserstein distance is proposed to use the +most discriminative projecting direction. Max sliced Wasserstein (Max-SW) distance [16] between +µ ∈ Pp(Rd) and ν ∈ Pp(Rd) is introduced as follows: +Max-SWp(µ, ν) = max +θ∈Sd−1 Wp(θ♯µ, θ♯ν), +(5) +Max-SW is often computed by a projected sub-gradient ascent algorithm. When the projected +sub-gradient ascent algorithm has T ≥ 1 iterations, the computation complexity of Max-SW is +O(Tm log2 m) and the memory complexity is O(m). Both SW and Max-SW are applied successfully +in point-cloud reconstruction [36]. +5 + +2.3 +Amortized Projection Optimization +We first revisit the point-cloud reconstruction objective with D(X, Y ) = Max-SWp(PX, PY ): +min +φ∈Φ,γ∈Γ E +� +max +θ∈Sd−1 Wp(θ♯PX, θ♯Pgγ(fφ(X))) +� +, +(6) +where the expectation is with respect to X ∼ p(X). For each point-cloud X ∈ X, we need to +compute a Max-SW distance with an iterative optimization procedure, which is computationally +expensive. +Amortized projection optimization: Authors in [31] propose to use amortized optimization [43, +3] to speed up the problem. Instead of solving all optimization problems independently, an amortized +model is trained to predict optimal solutions to all problems. In greater detail, given a parametric +function aψ : X × X → Sd−1 (ψ ∈ Ψ), the amortized objective is: +min +φ∈Φ,γ∈Γ max +ψ∈Ψ EWp(θψ,γ,φ♯PX, θψ,γ,φ♯Pgγ(fφ(X))), +(7) +where the expectation is with respect to X ∼ p(X), and θψ,γ,φ = aψ(X, gγ(fφ(X))). The above +optimization is solved by an alternative stochastic (projected)-gradient descent-ascent algorithm. +Therefore, it is faster to compute in each update iteration of φ and γ. It is worth noting that +the previous work [31] considers the generative model application which is hard and unstable to +understand. Here, we adapt the framework to the point-cloud reconstruction application which is +easier to explore the behavior of amortized optimization. We refer the reader to Appendix A.1 for +more information about amortized optimization and Algorithms 2-3 in Appendix A.4 for algorithms +on training an autoencoder with Max-SW and amortized projection optimization. +Amortized models: +Authors in [31] propose three types of amortized models that are based on +the literature on linear models [10]. In particular, the linear amortized model is defined as: +Definition 1. Given X, Y ∈ Rdm, the linear amortized model is defined as: +aψ(X, Y ) := +w0 + X′w1 + Y ′w2 +||w0 + X′w1 + Y ′w2||2 +, +where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X and +Y of size dm, ψ = (w0, w1, w2) with w1, w2 ∈ Rm, and w0 ∈ Rd . +Similarly, the generalized linear amortized model and the non-linear amortized model are defined +by injecting non-linearity into the linear model. We review the definitions of the generalized linear +amortized model and non-linear amortized model in Definitions 4-5 in Appendix A.2. +Sub-optimality: Despite being faster, amortized optimization often cannot recover the global +optimum of optimization problems. Namely, we denote θ⋆(X) = argmaxθ∈Sd−1Wp(θ♯PX, θ♯Pgγ(fφ(X))) +and ψ⋆ = arg maxψ∈Ψ EX∈p(X) +� +Wp(θψ,γ,φ♯PX, θψ,γ,φ♯Pgγ(fφ(X))) +� +. Then, it is well-known that the +amortization gap EX∼p(X)[c(θ⋆(X), aψ⋆(X, gγ(fφ(X))))] > 0 for a metric c : Sd−1 × Sd−1 → R+. A +great amortized model is one that can minimize the amortization gap. However, in the amortized +projection optimization setting, we cannot obtain θ⋆(X) since the projected gradient ascent algorithm +can only yield the local optimum. Therefore, the investigation of the amortization gap is intractable. +6 + +Amortized projection +Amortized distributional projection +Figure 2: +The difference between amortized projection optimization and amortized distributional projection +optimization. +3 +Self-Attention Amortized Distributional Projection Optimization +In this section, we propose the self-attention amortized distributional projection optimization +framework. First, we present amortized distributional projection optimization to maintain the +metricity property in Section 3.1. We then introduce self-attention amortized models which are +symmetric and permutation invariant in Section 3.2. +3.1 +Amortized Distributional Projection Optimization +The current amortized projection optimization framework is for predicting the “max" projecting +direction in Max-SW. However, the projected one-dimensional Wasserstein is only a metric on space +of probability measure at the global optimum of Max-SW. Therefore, the local optimum from the +projected sub-gradient ascent algorithm [37] and the prediction from the amortized model only yield +pseudo-metricity for the projected Wasserstein. +Proposition 1. Let the projected one-dimensional Wasserstein be PWp(µ, ν; ˆθ) = Wp(ˆθ♯µ, ˆθ♯ν)) +for any µ, ν ∈ Pp(Rd) (p ≥ 1, d ≥ 1) and ˆθ ∈ Sd−1 such that ˆθ ̸= arg maxθ∈Sd−1 Wp(θ♯µ, θ♯ν) +, PWp(µ, ν; ˆθ) is a pseudo metric on Pp(Rd) since it satisfies symmetry, non-negativity, triangle +inequality, µ = ν implies PWp(µ, ν; ˆθ) = 0, however, PWp(µ, ν; ˆθ) = 0 does not imply µ = ν. +The proof for Proposition 1 is given in Appendix B.1. This result implies that the if reconstruction +loss EX∼p(X)[PWp(PX, Pgγ(fφ(X)); ˆθ(X)) = 0, it does not imply X = gγ(fφ(X)) for p-almost surely +X ∈ X. Therefore, a local maximum for maxθ∈Sd−1 in Max-SW reconstruction (Equation 6) and the +7 + +global maximum for maxψ∈Ψ in amortized Max-SW reconstruction (Equation 7 with a misspecified +amortized model) cannot guarantee perfect reconstruction even when their objectives obtain 0 values. +Amortized Distributional Projection Optimization: To overcome the issue, we propose to +replace Max-SW in Equation 6 with the von Mises Fisher distributional sliced Wasserstein (v-DSW) +distance [34]: +min +φ∈Φ,γ∈Γ EX∼p(X) +� +max +ϵ∈Sd−1 +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯PX, θ♯Pgγ(fφ(X))) +� 1 +p � +, +(8) +where vMF(ϵ, κ) is the von Mises Fisher distribution with the mean location parameter ϵ ∈ Sd−1 and +the concentration parameter κ > 0, and v-DSWp(µ, ν; κ) = maxϵ∈Sd−1 +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯µ, θ♯ν) +� 1 +p +is von Mises Fisher distributional sliced Wasserstein distance. The optimization can be solved by a +stochastic projected gradient ascent algorithm with the vMF reparameterization trick. In particular, +θ1, . . . , θL (L ≥ 1) is sampled i.i.d from vMF(ϵ, κ) via the reparameterized acceptance-rejection +sampling [14] to approximate ∇ϵEvMF(ϵ,κ)[Wp +p(θ♯µ, θ♯ν)] via Monte Carlo integration. We refer +the reader to Section A.3 for more detail about the vMF distribution, its sampling algorithm, its +reparameterization trick, and the stochastic gradient estimators. We present a visualization of the +difference between the new amortized distributional projection optimization framework and the +conventional amortized projection optimization framework in Figure 2. The corresponding amortized +objective is: +min +φ∈Φ,γ∈Γ max +ψ∈Ψ EX∼p(X) +� +Eθ∼vMF(ϵψ,γ,φ,κ)Wp +p(θ♯PX, θ♯Pgγ(fφ(X))) +� 1 +p , +(9) +where ϵψ,γ,φ = aψ(X, gγ(fφ(X))). The optimization is solved by an alternative stochastic (projected)- +gradient descent-ascent algorithm with the vMF reparameterization. +Theorem 1. For any ϵ ∈ Sd−1 and 0 ≤ κ < ∞, if EX∼p(X) +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯PX, θ♯Pgγ(fφ(X))) +� 1 +p = +0, X = gγ(fφ(X)) for p-almost surely X ∈ X. +The proof of Theorem 1 is given in Appendix B.2. The proof is based on proving the metricity +of the non-optimal von Mises Fisher distributional sliced Wasserstein distance (v-DSW) with the +smoothness condition of the vMF distribution. It is worth noting that the proof of metricity of +von Mises Fisher distributional sliced Wasserstein distance is new since the original work [34] +only shows the pseudo-metricity with the global optimality condition. Theorem 1 indicates that a +perfect reconstruction can be obtained with a local optimum for maxϵ∈Sd−1 in v-DSW reconstruction +(Equation 8) and a local optimum for maxψ∈Ψ in amortized v-DSW reconstruction (Equation 9). +Comparison to SW and Max-SW: When κ → 0, the vMF distribution converges weakly to the +uniform distribution over the unit hypersphere. Hence, we can get back the conventional sliced +Wasserstein reconstruction in both Equation 8 and Equation 9. When κ → ∞, vMF distribution +converges weakly to the Dirac delta at the location parameter. Therefore, we obtain Max-SW +reconstruction and amortized Max-SW reconstruction in Equation 8 and Equation 9, respectively. +However, when 0 < κ < ∞, v-DSW reconstruction and amortized v-DSW reconstruction can find a +region of discriminative projecting directions while preserving the metricity for perfect reconstruction. +8 + +NOT SYMMETRIC +NOT PERMUTATION INVARIANT +Figure 3: Visualization of an amortized model that is not symmetric and permutation invariant in two dimensions. +3.2 +Self-Attention Amortized Models +Permutation Invariance and Symmetry: +Let X and Y be two point-clouds, the optimal slicing +distribution vMF(ϵ⋆, κ) of v-DSW between PX and PY is invariant to the permutation of the supports +since Pσ(X) = PX and Pσ(Y ) = PY for a permutation function σ. Moreover, the optimal slicing +distribution vMF(ϵ⋆, κ) is also unchanged when we exchange PX and PY since v-DSW is symmetric. +However, the current amortized models (see Definition 1, Definitions 4-5 in Appendix A.2) are not +permutation invariant and symmetric, namely, aψ(X, Y ) ̸= aψ(X, σ(Y )) and aψ(X, Y ) ̸= aψ(Y, X) . +Therefore, the current amortized models could be strongly misspecified. We show a visualization +of an amortized model that is not symmetric and permutation invariant in Figure 3. To address +the issue, we propose amortized models that are symmetric and permutation invariant based on the +self-attention mechanism. +Self-Attention Mechanism: +Attention is well-known for its effectiveness in learning long-range +dependencies when data are sequences. For example, the attention mechanism has been successfully +utilized to tackle many natural language processing [18, 8, 28, 41], speech recognition [52, 27, 53], +and computer vision tasks [19, 59, 57, 55, 45]. We now revisit the attention mechanism [48]. Given +Q, K ∈ Rm×dk, V ∈ Rm×dv, the scaled dot-product attention operator is defined as: +Att(Q, K, V ) = softmaxrow +�QKT +√dk +� +V +(10) +where softmaxrow denotes the row-wise softmax function. In the self-attention mechanism, the query +matrix Q, the key matrix K, and the value matrix V are usually computed by projecting the input +sequence X into different subspaces. Thus, the self-attention mechanism is given as follows. Given +9 + +X ∈ Rm×d, the self-attention operator is: +Aζ(X) = Att(XWq, XWk, XWv) +(11) +where Wq, Wk ∈ Rd×dk, Wv ∈ Rd×dv and ζ = (Wq, Wk, Wv). The self-attention operator is infamous +for its quadratic memory and computational costs. In particular, given an input sequence of length +m, both the time and space complexity are O(m2). Since we focus on the sliced Wasserstein setting +where the computational complexity should be at most O(m log m), the conventional self-attention +is not appropriate. Several works have been proposed to reduce the overall complexity from O(m2) +to O(m). In this paper, we utilize two linear complexity variants of attention which are efficient +attention [42] and linear attention [51]. Given X ∈ Rm×d, the efficient self-attention is defined as: +EAζ(X) = +softmaxrow(XWq) +� +softmaxcol(XWk)T (XWv) +� +(12) +where Wq, Wk ∈ Rd×dk, Wv ∈ Rd×dv, ζ = (Wq, Wk, Wv), and softmaxcol denotes applying the softmax +function column-wise. The linear self-attention is defined as: +LAζ(X) = Att(XWq, Wk1XWk2, Wv1XWv2) +(13) +where Wq, Wk2 ∈ Rd×dk, Wv2 ∈ Rd×dv, Wk1, Wv1 ∈ Rk×n, and ζ = (Wq, Wk1, Wk2, Wv1, Wv2). The +projected dimension k is chosen such that m ≫ k to reduce the memory and space consumption +significantly. +Self-Attention Amortized Models: Based on the self-attention mechanism, we introduce the self- +attention amortized model which is permutation invariant and symmetric. Formally, the self-attention +amortized model is defined as: +Definition 2. Given X, Y ∈ Rdm, the self-attention amortized model is defined as: aψ(X, Y ) = +w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m +||w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m||2 +, +(14) +where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X and +Y of size dm, 1m is the m-dimensional vector whose all entries are 1, w0 ∈ Rd and ψ = (w0, ζ). +By replacing the conventional self-attention with the linear self-attention and the efficient self- +attention, we obtain the linear self-attention amortized model and the efficient self-attention amortized +model. +Proposition 2. Self-attention amortized models are symmetric and permutation invariant. +The proof of Proposition 2 is given in Appendix B.3. The symmetry follows directly from the +definition of the self-attention amortized models. The permutation invariance is proved by showing +that the self-attention functions Aζ(X), EAζ(X), and LAζ(X) are permutation invariant. +10 + +Table 1: Reconstruction and transfer learning performance on the ModelNet40 dataset. All losses +are multiplied by 100. +Method +CD(↓) +SW(↓) +EMD(↓) +Acc(↑) +Time (↓) +CD +1.23 +660.37 +31.25 +86.63 +95 +EMD +1.17 +205.67 +13.70 +86.43 +207 +SW +0.69 +90.80 +9.45 +87.84 +103 +Max-SW +0.68 +87.13 +9.31 +88.33 +92 +ASW +0.69 +84.10 +9.24 +87.72 +112 +v-DSW +0.68 +85.93 +9.21 +88.29 +202 +L-Max-SW +1.09 +123.97 +11.64 +87.48 +93 +G-Max-SW +11.80 +850.76 +40.86 +86.99 +95 +N-Max-SW +11.97 +836.34 +39.73 +87.64 +94 +Lv-DSW +0.68 +85.85 +9.18 +88.17 +113 +Gv-DSW +0.68 +82.27 +9.11 +87.48 +116 +Nv-DSW +0.67 +82.89 +9.11 +87.93 +115 +Av-DSW +0.69 +82.15 +9.11 +87.84 +255 +EAv-DSW +0.69 +81.51 +9.09 +88.37 +127 +LAv-DSW +0.68 +81.21 +9.07 +88.45 +123 +4 +Experiments +To verify the effectiveness of our proposal, we evaluate our methods on the point-cloud reconstruction +task and its two downstream tasks including transfer learning and point-cloud generation. Three +important questions we want to answer are: 1. Does the sub-optimality issue of amortized Max- +SW occur when working with point-clouds and does replacing Max-SW with v-DSW alleviate the +problem? 2. Does the proposed amortized distribution projection optimization framework improve +the performance over the conventional amortized projection optimization framework and commonly +used distances e.g., Chamfer distance, Earth Mover Distance (Wasserstein distance), SW, Max-SW, +adaptive SW (ASW) [36], and v-DSW? 3. Are self-attention amortized models better than the +previous misspecified amortized models in [31]? +Experiment settings: Our settings, which can be found in Appendix C.1, are identical to the +setting in the paper of ASW [36]. For amortized models, we consider 6 different ones. The prefix L, G, +and N denote the linear, generalized linear, and non-linear amortized models in [31], respectively. +A, EA, and LA are used to represent self-attention, efficient self-attention, and linear self-attention, +respectively. Implementation details for baseline distances and amortized models are given in +Appendices C.2 and C.3, respectively. +Point-cloud reconstruction: Following ASW [36], we measure the reconstruction performance of +different autoencoders on the ModelNet40 dataset [54] using three discrepancies: Chamfer discrepancy +(CD), sliced Wasserstein distance (SW), and EMD. The quantitative results are summarized in +Table 1. For each method, we only report the best performing (based on SW and EMD losses) +model among all choices of hyper-parameters. Full quantitative results can be found in Table 4. +Our method achieves the best performance in all three discrepancies. In contrast, autoencoders +11 + +Figure 4: Qualitative results of reconstructing point-clouds in the ShapeNet Core-55 dataset. From top to bottom: +input, SW, Max-SW, v-DSW, and LAv-DSW. +with amortized Max-SW losses fail in this scenario due to the sub-optimality and losing metricity +issues that we discussed in Appendix 2.3. In addition, using amortized optimization reduces the +training time compared to the conventional computation using the projected sub-gradient ascent +algorithm (e.g. Max-SW and v-DSW). For example, training one iteration of autoencoder using +LAv-DSW only takes 123 seconds while using v-DSW costs 202 seconds. In terms of amortized +models, attention-based amortized models lead to lower distances between reconstructed and input +point-clouds. Qualitative results are given in Figure 4, showing the success of our methods in +reconstructing 3D point-clouds. Full qualitative results can be found in Figure 6. +Amortization Gaps: To validate the advantage of self-attention amortized models over the previous +misspecified amortized models, we compare their effectiveness in approximating v-DSW. We create +a dataset by sampling 1000 pairs of point-clouds from the ShapeNet Core-55 dataset. Due to the +memory constraint when solving amortized optimization, the dataset is divided into 10 batches of size +100. We compute v-DSW and its amortized versions between all pairs of point-clouds and report their +average loss values in Table 2. Compared to previous misspecified amortized models, attention-based +amortized models produce higher losses which are closer to the conventional computation of v-DSW +(T = 100). To achieve the same level as efficient/linear self-attention amortized models, one needs to +run more than 50 sub-gradient iterations, which is more than 10 times slower. +Transfer learning: We further feed the latent vectors learned by the above autoencoders into a +12 + +Table 2: Comparison between amortized models when approximating von Mises Fisher distributional +sliced Wasserstein (v-DSW). T denotes the number of projected sub-gradient ascent iterations. +Method +T +Distance (↑) +Time (↓) +Lv-DSW +1 +52.73 +0.06 +Gv-DSW +1 +50.73 +0.07 +Nv-DSW +1 +51.89 +0.07 +Av-DSW +1 +53.07 +1.00 +EAv-DSW +1 +53.17 +0.17 +LAv-DSW +1 +53.83 +0.14 +v-DSW +1 +51.87 +0.1 +v-DSW +5 +51.90 +0.33 +v-DSW +10 +52.65 +0.5 +v-DSW +50 +53.16 +2.00 +v-DSW +100 +54.39 +4.00 +Table 3: Performance comparison of point-cloud generation on the chair category of ShapeNet. JSD, +MMD-CD, and MMD-EMD are multiplied by 100. +Method +JSD (↓) +MMD (↓) +COV (%, ↑) +1-NNA (%, ↓) +CD +EMD +CD +EMD +CD +EMD +CD +16.62 +1.09 +16.72 +24.37 +11.67 +98.30 +100.00 +EMD +5.25 +0.99 +13.08 +29.69 +26.14 +97.27 +99.48 +SW +1.64 +0.70 +10.69 +39.00 +44.76 +90.25 +88.70 +Max-SW +2.04 +0.71 +10.62 +41.80 +47.86 +91.14 +89.59 +ASW +1.85 +0.71 +10.79 +40.47 +47.27 +90.77 +88.92 +v-DSW +1.37 +0.75 +11.03 +36.78 +43.57 +89.51 +87.74 +Lv-DSW +1.60 +0.71 +10.65 +39.88 +48.89 +89.88 +88.55 +Gv-DSW +1.85 +0.72 +10.84 +40.03 +46.09 +90.47 +88.55 +Nv-DSW +1.33 +0.77 +10.98 +38.70 +49.19 +91.21 +89.73 +EAv-DSW +1.93 +0.66 +10.37 +41.80 +49.48 +89.29 +88.70 +LAv-DSW +1.60 +0.68 +10.49 +42.39 +47.56 +89.51 +88.18 +classifier. Following the settings in ASW’s paper, we train our classifier for 500 epochs with a batch +size of 256. The optimizer is the same as that in the reconstruction experiment. Table 1 illustrates +the classification result. Again, we see a boost in accuracy when using self-attention amortized +v-DSW. +Point-cloud generation: We also evaluate our methods on the 3D point-cloud generation task. +Following [1], the chair category of ShapeNet is divided into train/valid/test sets in an 85/5/10 +ratio. We train each autoencoder on the train set for 100 epochs and evaluate on the valid set. The +generator is then trained to generate latent codes learned by the autoencoder, same as [1]. For +13 + +evaluation, the same set of metrics in [56] is used. The quantitative results of the test set are given in +Table 3. Our methods yield the best performance in 6 out of 7 metrics. In addition, attention-based +amortized models lead to higher performance than previous amortized models in all metrics except +for JSD. Full quantitative results are reported in Table 5. +5 +Conclusion +We have proposed a self-attention amortized distributional projection optimization framework +which uses a self-attention amortized model to predict the best discriminative distribution over +projecting direction for each pair of probability measures. The efficient self-attention mechanism +helps to inject the geometric inductive biases which are permutation invariance and symmetry into +the amortized model while remaining fast computation. Furthermore, the amortized distribution +projection optimization framework guarantees the metricity for all pairs of probability measures +while the amortization gap still exists. On the experimental side, we compare the new proposed +framework to the conventional amortized projection optimization framework and other widely-used +distances in the point-cloud reconstruction application and its two downstream tasks including +transfer learning and point-cloud generation. Overall, the self-attention amortized distributional +projection optimization framework performs the best in all tasks. +Supplement to “Self-Attention Amortized Distributional Projection +Optimization for Sliced Wasserstein Point-Clouds Reconstruction" +In this supplementary, we first provide some additional materials in Appendix A including the detail +of amortized optimization in Appendix A.1, definitions of generalized linear amortized models and +non-linear amortized models in Appendix A.2, the detail of computing von Mises-Fisher distributional +sliced Wasserstein in Appendix A.3, and training algorithms for autoencoders in Appendix A.4. Next, +we collect skipped proofs in the main text in Appendix B. After that, we discuss the experimental +settings of our experiments in Appendix C. Finally, we present additional experimental results in +Appendix D. +A +Additional Materials +A.1 +Amortized Optimization +We start with the definition of amortized optimization. +Definition 3. For each context variable x in the context space X, θ⋆(x) is the solution of the +optimization problem θ⋆(x) = arg minθ∈Θ L(θ, x), where Θ is the solution space. A parametric +function fψ : X → Θ, where ψ ∈ Ψ, is called an amortized model if +fψ(x) ≈ θ⋆(x), +∀x ∈ X. +(15) +The amortized model is trained by the amortized optimization objective which is defined as: +min +ψ∈Ψ Ex∼p(x)L(fψ(x), x), +(16) +where p(x) is a probability measure on X which measures the “importance" of optimization problems. +14 + +The amortized model in Definition 3 is sometimes called a fully amortized model for a distinction +with the other concept of semi amortized model [3]. The gap between the predicted solution and the +optimal solution Ex∼p(x)||fψ(x) − θ⋆(x)||2 is called the amortization gap. However, understanding +this gap depends on specific configurations of the objective L(·, x), such as convexity and smoothness, +which are often non-trivial to obtain in practice. +A.2 +Amortized models +We now review the generalized linear amortized model and the non-linear amortized model [31]. +Definition 4. Given X, Y ∈ Rdm, the generalized linear amortized model is defined as: +fψ(X, Y ) := +w0 + gψ1(X)′w1 + gψ1(Y )′w2 +||w0 + gψ1(X)′w1 + gψ1(Y )′w2||2 +2 +, +(17) +where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X +and Y of size dm, w1, w2 ∈ Rm, w0 ∈ Rd, ψ1 ∈ Ψ1, gψ1 : Rdm → Rdm, ψ = (w0, w1, w2, ψ1), and +gψ1(X) = (x′ +1, . . . , x′ +m) and gψ1(Y ) = (y′ +1, . . . , y′ +m). To specify, we let gψ1(X) = (W2σ(W1x1) + +b0, . . . , W2σ(W1xm) + b0), where σ(·) is the Sigmoid function, W1 ∈ Rd×d, W2 ∈ Rd×d, and b0 ∈ Rd. +Definition 5. Given X, Y ∈ Rdm, the non-linear amortized model is defined as: +fψ(X, Y ) := +hψ2(w0 + X′w1 + Y ′w2) +||hψ2(w0 + X′w1 + Y ′w2)||2 +2 +, +(18) +where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X +and Y of size dm, w1, w2 ∈ Rm, w0 ∈ Rd, ψ2 ∈ Ψ2, hψ2 : Rd → Rd, ψ = (w0, w1, w2, ψ2), and +hψ2(x) = W4σ(W3x)) + b0 where σ(·) is the Sigmoid function. +A.3 +Von Mises-Fisher distributional sliced Wasserstein distance +We first start with the definition of von Mises Fisher (vMF) distribution. The von Mises–Fisher +distribution (vMF)[23] is a probability distribution on the unit hypersphere Sd−1 with the density +function is : +f(x|ϵ, κ) := Cd(κ) exp(κϵ⊤x), +(19) +where ϵ ∈ Sd−1 is the location vector, κ ≥ 0 is the concentration parameter, and Cd(κ) := +κd/2−1 +(2π)d/2Id/2−1(κ) is the normalization constant. Here, Iv is the modified Bessel function of the first +kind at order v [46]. +The vMF distribution is a continuous distribution, its mass concentrates around the mean ϵ, and its +density decrease when x goes away from ϵ. When κ → 0, vMF converges in distribution to U(Sd−1), +and when κ → ∞, vMF converges in distribution to the Dirac distribution centered at ϵ [44]. +Reparameterized Rejection Sampling: +The sampling process of vMF distribution is based +on the rejection sampling procedure. We review the sampling process in Algorithm 1 [14, 34]. +The algorithm performs the reparameterization for the proposal distribution. We now derive the +gradient estimator for ∇ϵEvMF(θ|ϵ,κ) +� +f(θ) +� +for a general function f(θ) to find the maxima ϵ∗ in the +optimization problem maxϵ∈Sd−1 EvMF(θ|ϵ,κ) +� +f(θ) +� +. +15 + +In d > 0 dimension, let (ϵ, κ) be the parameters of vMF distribution. We denotes b = −2κ+√ +4κ2+(d−1)2 +d−1 +, +two conditional distributions: g(ω | κ) = +2(πd/2) +Γ(d/2) Cd(κ) +exp(ωκ)(1−ω2) +1 +2 (d−3) +Beta( 1 +2 , 1 +2 (d−1)) +and +r(ω|κ) = +2b1/2(d−1) +Beta( 1 +2 (d−1), 1 +2 (d−1)) +(1−ω2) +1/2(d−3) +[(1+b)−(1−b)ω]d−1 , distribution s(ψ) := Beta +� 1 +2(d − 1), 1 +2(d − 1) +� +, func- +tion h(ψ, κ) = 1−(1+b)ψ +1−(1−b)ψ, distributions π1(ψ|κ) = s(ψ) g(h(ψ,κ)|κ) +r(h(ψ,κ)|κ), π2(v) := U(Sd−2), and function +T(ω, v, ϵ) = +� +I − 2 +e1−ϵ +||e1−ϵ||2 +e1−ϵ +||e1−ϵ||2 +⊤�� +ω, +√ +1 − ω2v⊤�⊤ := θ. +We can obtain the gradient estimator by following Lemma 2 in [15],: +∇ϵEvMF(θ|ϵ,κ) +� +f(θ) +� += ∇ϵE(ψ,v)∼π1(ψ|κ)π2(v) +� +f +� +T(h(ψ, κ), v, ϵ) +�� += E(ψ,v)∼π1(ψ|κ)π2(v) +� +∇ϵf +� +T(h(ψ, κ), v, ϵ) +�� +. +In v-DSW case, we have f(θ) = Wp +p(θ♯µ, θ♯ν). Therefore, we have: +∇ϵEvMF(θ|ϵ,κ) +� +Wp +p(θ♯µ, θ♯ν) +� += E(ψ,v)∼π1(ψ|κ)π2(v) +� +∇ϵWp +p(f +� +T(h(ψ, κ), v, ϵ)♯µ, f +� +T(h(ψ, κ), v, ϵ)♯ν) +�� +. +Then we can get a gradient estimator by using Monte-Carlo estimation scheme: +∇ϵEvMF(θ|ϵ,κ) +� +Wp +p(θ♯µ, θ♯ν) +� +≈ 1 +L +L +� +i=1 +� +∇ϵWp +p(f +� +T(h(ψi, κ), vi, ϵ)♯µ, f +� +T(h(ψi, κ), vi, ϵ)♯ν) +�� +, +where {ψi}L +i=1 ∼ π1(ψ|κ) i.i.d, {vi}L +i=1 ∼ π2(v) i.i.d, and L is the number of projections. Sampling +from π1(ψ|κ) is equivalent to the acceptance-rejection scheme in vMF sampling procedure, sampling +π2(v) is directly from U(Sd−2). It is worth noting that the gradient estimator for ∇κEvMF(θ|ϵ,κ) +� +f(θ) +� +can be derived by using the log-derivative trick, however, we do not need it here since we do not +optimize for κ in v-DSW. +A.4 +Training algorithms +Training point-cloud autoencoder with Max-SW: We present the algorithm of training au- +toencoder with Max-SW in Algorithm 2. The algorithm contains a nested loop: one is for training +the autoencoder, one is for finding the max projecting direction for Max-SW. +Training point-cloud autoencoder with amortized projection optimization: We present +the training algorithm for point-cloud autoencoder with amortized projection optimization in +Algorithm 3. With amortized optimization, the inner loop for finding the max projecting direction +is removed. +Training point-cloud autoencoder with v-DSW: We present the algorithm of training autoen- +coder with v-DSW in Algorithm 4. The algorithm contains a nested loop: one is for training the +autoencoder, one is for finding the best distribution over projecting directions for v-DSW. +Training point-cloud autoencoder with amortized distributonal projection optimiza- +tion: We present the training algorithm for point-cloud autoencoder with amortized distributional +projection optimization in Algorithm 5. With amortized distributional optimization, the inner loop +for finding the best distribution over projecting directions is removed. +16 + +Algorithm 1 Sampling from vMF distribution +Input: location ϵ, concentration κ, dimension d, unit vector e1 = (1, 0, .., 0) +Draw v ∼ U(Sd−2) +b ← −2κ+√ +4κ2+(d−1)2 +d−1 +, a ← (d−1)+2κ+√ +4κ2+(d−1)2 +4 +, m ← +4ab +(1+b) − (d − 1) log(d − 1) +repeat +Draw ψ ∼ Beta +� 1 +2(d − 1), 1 +2(d − 1) +� +ω ← h(ψ, κ) = 1−(1+b)ψ +1−(1−b)ψ +t ← +2ab +1−(1−b)ψ +Draw u ∼ U([0, 1]) +until (d − 1) log(t) − t + m ≥ log(u) +h1 ← (ω, +√ +1 − ω2v⊤)⊤ +ϵ′ ← e1 − ϵ +u = +ϵ′ +||ϵ′||2 +U = I − 2uu⊤ +Output: Uh1 +Algorithm 2 Training point-cloud autoencoder with max sliced Wasserstein distance +Input: Point-cloud distribution p(X), learning rate η, slice learning rate ηs, model maximum +number of iterations T , slice maximum number of iterations T, mini-batch size k. +Initialization: Initialize the encoder fφ and the decoder gγ +while φ, γ not converge or reach T do +Sample a mini-batch X1, . . . , Xk i.i.d from p(X) +∇φ = 0, ∇γ = 0 +for i = 1 to k do +Initialize θ +while θ not converge or reach T do +θ = θ + ηs · ∇θWp(θ♯PXi, θ♯Pgγ(fφ(Xi))) # Other update rules can be used +θ = +θ +||θ||2 #Project back to the unit-hypersphere Sd−1 +end while +∇φ = ∇φ + 1 +k∇φWp(θ♯PXi, θ♯Pgγ(fφ(Xi))) +∇γ = ∇γ + 1 +k∇γWp(θ♯PXi, θ♯Pgγ(fφ(Xi))) +end for +φ = φ − η · ∇φ # Other update rules can be used +γ = γ − η · ∇γ # Other update rules can be used +end while +Return: φ, γ +B +Proofs +B.1 +Proof for Proposition 1 +We first recall the definition of the projected one-dimensional Wasserstein between two probability +measures µ and ν: PWp(µ, ν; ˆθ) = Wp(ˆθ♯µ, ˆθ♯ν) for ˆθ ̸= argmaxθ∈Sd−1Wp(θ♯µ, θ♯ν). +17 + +Algorithm 3 Training point-cloud autoencoder with amortized projection optimization +Input: Point-cloud distribution p(X), learning rate η, slice learning rate ηs, model maximum +number of iterations T , mini-batch size k. +Initialization: Initialize the encoder fφ, the decoder gγ, and the amortized model aψ +while φ, γ, ψ not converge or reach T do +Sample a mini-batch X1, . . . , Xk i.i.d from p(X) +∇φ = 0, ∇γ = 0, ∇ψ = 0 +for i = 1 to k do +θψ,γ,φ = aψ(Xi, gγ(fφ(Xi))) +∇ψ = ∇ψ + 1 +k∇ψWp(θψ,γ,φ♯PXi, θψ,γ,φ♯Pgγ(fφ(Xi))) +∇φ = ∇φ + 1 +k∇φWp(θψ,γ,φ♯PXi, θψ,γ,φ♯Pgγ(fφ(Xi))) +∇γ = ∇γ + 1 +k∇γWp(θψ,γ,φ♯PXi, θψ,γ,φ♯Pgγ(fφ(Xi))) +end for +ψ = ψ + ηs · ∇ψ # Other update rules can be used +φ = φ − η · ∇φ # Other update rules can be used +γ = γ − η · ∇γ # Other update rules can be used +end while +Return: φ, γ +Non-negativity and Symmetry: Due to the non-negativity and symmetry of the Wasserstein +distance, the non-negativity and symmetry of the projected Wasserstein follow directly from its +definition. +Triangle inequality: +For any three probability measures µ1, µ2, µ3 ∈ Pp(Rd), we have: +PWp(µ1, µ3; ˆθ) = Wp(ˆθ♯µ1, ˆθ♯µ3) +≤ Wp(ˆθ♯µ1, ˆθ♯µ2) + Wp(ˆθ♯µ2, ˆθ♯µ3) += PWp(µ1, µ2; ˆθ) + PWp(µ2, µ3; ˆθ), +where the first inequality is due to the triangle inequality of the Wasserstein distance. +Identity: +If µ = ν, we have PWp(µ, ν; ˆθ) = 0 due to the identity of the Wasserstein distance. +However, if PWp(µ, ν; ˆθ) = 0, there exists θ′ ∈ Sd−1 such that 0 = PWp(µ, ν; ˆθ) < PWp(µ, ν; θ′). Let +F[γ](w) = +� +Rd′ e−i⟨w,x⟩dγ(x) be the Fourier transform of γ ∈ P(Rd′), for any t ∈ R, we have +F[µ](tθ′) = +� +Rd e−it⟨θ′,x⟩dµ(x) = +� +R +e−itzdθ′♯µ(z) = F[θ′♯µ](t) +̸= F[θ′♯ν](t) = +� +R +e−itzdθ′♯ν(z) = +� +Rd e−it⟨θ′,x⟩dν(x) = F[ν](tθ′). +Therefore, we have µ ̸= ν. We complete the proof. +B.2 +Proof for Theorem 1 +We first start with proving the metricity of the non-optimal von Mises Fisher distributional sliced +Wasserstein distance (v-DSW). For any two probability measures µ, ν ∈ Pp(Rd), the non-optimal +18 + +Algorithm 4 Training point-cloud autoencoder with von-Mises Fisher distributional sliced Wasser- +stein distance +Input: Point-cloud distribution p(X), learning rate η, slice learning rate ηs, model maximum +number of iterations T , slice maximum number of iterations T, mini-batch size k, the number of +projections L, and the concentration hyperparameter κ. +Initialization: Initialize the encoder fφ and the decoder gγ +while φ, γ not converge or reach T do +Sample a mini-batch X1, . . . , Xk i.i.d from p(X) +∇φ = 0, ∇γ = 0 +for i = 1 to k do +Initialize ϵ +while ϵ not converge or reach T do +Sample θϵ +1, . . . , θϵ +L i.i.d from vMF(ϵ, κ) via the reparameterized acceptance-rejection sam- +pling in Algorithm 1 +ϵ = ϵ + ηs · 1 +L +�L +l=1 ∇ϵWp(θϵ +l ♯PXi, θϵ +l ♯Pgγ(fφ(Xi))) # Other update rules can be used +ϵ = +ϵ +||ϵ||2 #Project back to the unit-hypersphere Sd−1 +end while +Sample θϵ +1, . . . , θϵ +L i.i.d from vMF(ϵ, κ) via the reparameterized acceptance-rejection sampling +in Algorithm 1. +∇φ = ∇φ + 1 +k +1 +L +�L +i=l ∇φWp(θϵ +l ♯PXi, θϵ +l ♯Pgγ(fφ(Xi))) +∇γ = ∇γ + 1 +k +1 +L +�L +i=l ∇γWp(θϵ +l ♯PXi, θϵ +l ♯Pgγ(fφ(Xi))) +end for +φ = φ − η · ∇φ # Other update rules can be used +γ = γ − η · ∇γ # Other update rules can be used +end while +Return: φ, γ +von Mises Fisher distributional sliced Wasserstein distance (v-DSW) is defined as follow: +v-DSWp(µ, ν; ϵ, κ) = +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯µ, θ♯ν) +� 1 +p , +where ϵ ∈ Sd−1 and 0 < κ < ∞. +Lemma 1. For any ϵ ∈ Sd−1 and κ < ∞, v-DSWp(·, ·; ϵ, κ) is a valid metric on the space of +probability measures. +Proof. We now prove that v-DSW satisfies non-negativity, symmetry, triangle inequality, and identity. +Non-negativity and Symmetry: The non-negativity and symmetry of v-DSW follow directly +the non-negativity and symmetry of the Wasserstein distance. +19 + +Algorithm 5 Training point-cloud autoencoder with amortized projection optimization +Input: Point-cloud distribution p(X), learning rate η, slice learning rate ηs, model maximum +number of iterations T , mini-batch size k. +Initialization: Initialize the encoder fφ, the decoder gγ, and the amortized model aψ +while φ, γ, ψ not converge or reach T do +Sample a mini-batch X1, . . . , Xk i.i.d from p(X) +∇φ = 0, ∇γ = 0, ∇ψ = 0 +for i = 1 to k do +ϵψ,γ,φ = aψ(Xi, gγ(fφ(Xi))) +Sample θψ,γ,φ +1 +, . . . , θψ,γ,φ +L +i.i.d from vMF(ϵψ,γ,φ, κ) via the reparameterized acceptance-rejection +sampling in Algorithm 1 +∇ψ = ∇ψ + 1 +k +1 +L +�L +i=l ∇ψWp(θψ,γ,φ +l +♯PXi, θψ,γ,φ +l +♯Pgγ(fφ(Xi))) +∇φ = ∇φ + 1 +k +1 +L +�L +i=l ∇φWp(θψ,γ,φ +l +♯PXi, θψ,γ,φ +l +♯Pgγ(fφ(Xi))) +∇γ = ∇γ + 1 +k +1 +L +�L +i=l ∇γWp(θψ,γ,φ +l +♯PXi, θψ,γ,φ +l +♯Pgγ(fφ(Xi))) +end for +ψ = ψ + ηs · ∇ψ # Other update rules can be used +φ = φ − η · ∇φ # Other update rules can be used +γ = γ − η · ∇γ # Other update rules can be used +end while +Return: φ, γ +Triangle inequality: For any three probability measures µ1, µ2, µ3 ∈ Pp(Rd), we have +v-DSWp(µ1, µ3; ϵ, κ) = +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯µ1, θ♯µ3) +� 1 +p +≤ +� +Eθ∼vMF(ϵ,κ) [Wp(θ♯µ1, θ♯µ2) + Wp(θ♯µ2, θ♯µ3)]p� 1 +p +≤ +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯µ1, θ♯µ2) +� 1 +p + +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯µ2, θ♯µ3) +� 1 +p += v-DSWp(µ1, µ2; ϵ, κ) + v-DSWp(µ2, µ3; ϵ, κ) +Identity: From the definition, if µ = ν, we obtain v-DSWp(µ, ν; ϵ, κ) = 0. Now, we need to show +that if v-DSWp(µ, ν; ϵ, κ) = 0, then µ = ν. +If v-DSWp(µ, ν; ϵ, κ) = 0, we have +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯µ, θ♯ν) +� 1 +p = 0 which implies Eθ∼vMF(ϵ,κ)Wp +p(θ♯µ, θ♯ν) = +0. Therefore, Wp(θ♯µ, θ♯ν) = 0 for vMF(ϵ, κ) almost surely θ ∈ Sd−1. Using the identity property of +the Wasserstein distance, we obtain θ♯µ = θ♯ν for vMF(ϵ, κ) almost surely θ ∈ Sd−1. Since vMF(ϵ, κ) +with 0 < κ < ∞ has the supports on all Sd−1, for any t ∈ R and θ ∈ Sd−1, we have: +F[µ](tθ) = +� +Rd e−it⟨θ,x⟩dµ(x) = +� +R +e−itzdθ♯µ(z) = F[θ♯µ](t) += F[θ♯ν](t) = +� +R +e−itzdθ♯ν(z) = +� +Rd e−it⟨θ,x⟩dν(x) = F[ν](tθ), +where F[γ](w) = +� +Rd′ e−i⟨w,x⟩dγ(x) denotes the Fourier transform of γ ∈ P(Rd′). We then obtain +µ = ν by the injectivity of the Fourier transform. We complete the proof. +20 + +By abuse of notation, we denote v-DSW(X, Y ; ϵ, κ) = v-DSW(PX, PY ; ϵ, κ) for X, Y ∈ X are two +point-clouds, PX = 1 +m +�m +i=1 δxi, and PY = 1 +m +�m +i=1 δyi. We cast the v-DSW from a metric on the +space of probability measures to the space of point-clouds X. +Corollary 1. For any ϵ ∈ Sd−1 and κ < ∞, v-DSWp(·, ; ϵ, κ) is a valid metric on the space of +point-clouds X. +Proof. Since PX, PY ∈ Pp(Rd), the non-negativity, symmetry, triangle inequality, and identity +properties follow directly from Lemma 1. We now only need to show that v-DSW is invariant to +permutation. This property is straightforward from the definition of empirical probability measures. +For any permutation function σ, we have PX = 1 +m +�m +i=1 δxi = 1 +m +�m +i=1 δxσ(i) = Pσ(X) which completes +the proof. +We now continue the proof of Theorem 1. If EX∼p(X) +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯PX, θ♯Pgγ(fφ(X))) +� 1 +p = 0, we +obtain +� +Eθ∼vMF(ϵ,κ)Wp +p(θ♯PX, θ♯Pgγ(fφ(X))) +� 1 +p = v-DSW(X, gγ(fφ(X)); ϵ, κ) = 0 for p-almost surely +X ∈ X. By Collorary 1, we obtain X = gγ(fφ(X)) for p-almost surely X ∈ X. We complete the +proof. +B.3 +Proof for Proposition 2 +We first recall the definition of the self-attention amortized model in Definition 2: +aψ(X, Y ) = +w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m +||w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m||2 +, +Symmetry: Since the self-attention amortized model use the same attention weight ζ for both X +and Y , exchanging X and Y yields the same results aψ(X, Y ) = aψ(Y, X). +Permutation invariance: We first show that self-attention is permutation invariant. In particular, +we have: +Aζ(X′⊤)⊤1m = Att(X′⊤Wq, X′⊤Wk, X′⊤Wv)⊤1m += +� +softmaxrow +�X′⊤WqW ⊤ +k X′ +√dk +� +X′⊤Wv +�⊤ +1m += +� +softmaxrow +�σ(X)′⊤WqW ⊤ +k σ(X)′ +√dk +� +σ(X)′⊤Wv +�⊤ +1m += Aζ(σ(X)′⊤)⊤1m. +Similarly, the proof holds for both linear self-attention and efficient self-attention. +C +Experiment settings +In this section, we first provide the details of the training process and the architecture for point-cloud +reconstruction, transfer learning, and point-cloud generation. Then, we present the implementation +detail and hyper-parameters settings for different distances used in our experiments. +21 + +conv1d(64, 1) +conv1d(64, 1) +conv1d(64, 1) +conv1d(128, 1) +conv1d(1024, 1) +maxpool +fc 256 +z +fc 1024 +fc 1024 +fc Nx3 +tanh +fc 512 +fc 256 +fc 40 +fc 256 +fc 1024 +fc 1024 +encoder +decoder +classifier +Figure 5: The architecture of the Point-Net variant in our experiments. For transfer learning, we use a simple +classifier with 3 fully-connected layers. All layers are followed by ReLU activation and batch normalization by default, +except for the final layers. +C.1 +Details of point-cloud reconstruction and downstream applications +Point-cloud reconstruction: We use the same settings in ASW [36] to train autoencoders. We +utilize a variant of Point-Net [40] with an embedding size of 256 proposed in [39]. The architecture +of the autoencoder and classifier are shown in Figure 5. Our autoencoder is trained on the ShapeNet +Core-55 dataset [9] with a batch size of 128 and a point-cloud size of 2048. We train it for 300 +epochs using an SGD optimizer with an initial learning rate of 1e-3, a momentum of 0.9, and a +weight decay of 5e-4. All experiments are run on NVIDIA V100 GPUs. +Next, we detail the process of conducting two downstream applications of point-cloud reconstruction. +Transfer learning: A classifier is trained on the latent space of the autoencoder. Particularly, we +extract a 256-dimension latent vector of an input 3D point-cloud via the pre-trained encoder. Then, +this vector is fed into a multi-layer perceptron with hidden layers of size 512 and 256. The last layer +outputs a 40-dimension vector representing the prediction of 40 classes of the ModelNet40 dataset. +Point-cloud generation: Our generative model is trained on the latent space of the autoencoder +as follows. First, we extract a 256-dimension latent vector of an input 3D point-cloud via the +pre-trained encoder. Then a 64-dimensional vector is drawn from a normal distribution N(0, I64), +where I64 is the 64x64 identity matrix, and fed into a generator which also outputs a 256-dimension +vector. Finally, the generator learns by minimizing the optimal transport distance between the +generated and ground truth latent codes. +22 + +Figure 6: Qualitative results of reconstructing point-clouds in the ShapeNet Core-55 dataset. From top to bottom: +input, CD, EMD, SW, Max-SW, ASW, v-DSW, Nv-DSW, EAv-DSW, and LAv-DSW. +23 + +Table 4: Reconstruction and transfer learning performance of different autoencoders on the Model- +Net40 dataset. For v-DSW and Max-SW, T denotes the number of projected sub-gradient ascent +iterations. +In Table 1, v-DSW and Max-SW have T = 10 and 1 iterations, respectively. +All +reconstructed losses are multiplied by 100. +Method +CD (↓) +SW (↓) +EMD (↓) +Acc (↑) +Time (↓) +CD +1.23 +660.37 +31.25 +86.63 +95 +EMD +1.17 +205.67 +13.70 +86.43 +207 +SW +0.69 +90.80 +9.45 +87.84 +103 +Max-SW (T = 1) +0.68 +87.13 +9.31 +88.33 +92 +Max-SW (T = 10) +0.68 +90.90 +9.37 +87.76 +107 +Max-SW (T = 50) +0.68 +90.12 +9.30 +88.17 +124 +ASW +0.69 +84.10 +9.24 +87.72 +112 +v-DSW (T = 1) +0.68 +89.34 +9.30 +88.09 +115 +v-DSW (T = 10) +0.68 +85.93 +9.21 +88.29 +202 +v-DSW (T = 50) +0.67 +88.55 +9.26 +88.37 +638 +L-Max-SW +1.09 +123.97 +11.64 +87.48 +93 +G-Max-SW +11.80 +850.76 +40.86 +86.99 +95 +N-Max-SW +11.97 +836.34 +39.73 +87.64 +94 +Lv-DSW (ours) +0.68 +85.85 +9.18 +88.17 +113 +Gv-DSW (ours) +0.68 +82.27 +9.11 +87.48 +116 +Nv-DSW (ours) +0.67 +82.89 +9.11 +87.93 +115 +Av-DSW (ours) +0.69 +82.15 +9.11 +87.84 +255 +EAv-DSW (ours) +0.69 +81.51 +9.09 +88.37 +127 +LAv-DSW (ours) +0.68 +81.21 +9.07 +88.45 +123 +C.2 +Details of baseline distances +We want to emphasize that we use the same set of hyper-parameters reported in [36] for Chamfer, +EMD, SW, and Max-SW. +Chamfer and EMD: We use the CUDA implementation from [56]. +SW: We use the Monte Carlo estimation with 100 slices. +Max-SW: We use the projected sub-gradient ascent algorithm to optimize the projection. It is +trained with an Adam optimizer with an initial learning rate of 1e-4. The number of iterations T is +chosen from {1, 10, 50}. +Adaptive SW: We use Algorithm 1 in [36] with the same set of parameters as follows: N0 = 2, s = +1, ϵ = 0.5, and M = 500. +v-DSW: We use stochastic projected gradient ascent algorithm to optimize the location vector ϵ in +Equation 19 while we fix the concentration parameter κ to 1 for both v-DSW and all of its amortized +versions. Similar to Max-SW, it is trained with an Adam optimizer with an initial learning rate +of 1e-4. The number of iterations T is selected from {1, 10, 50} based on the task performance. +Intuitively, increasing the number of iterations leads to a better approximation that is closer to +24 + +Table 5: Performance comparison of point-cloud generation on the chair category of ShapeNet. For +v-DSW and Max-SW, T denotes the number of projected sub-gradient ascent iterations. In Table 3, +v-DSW and Max-SW have T = 10 and 1 iterations, respectively. JSD, MMD-CD, and MMD-EMD +are multiplied by 100. +Method +JSD (↓) +MMD (↓) +COV (%, ↑) +1-NNA (%, ↓) +CD +EMD +CD +EMD +CD +EMD +CD +16.62 +1.09 +16.72 +24.37 +11.67 +98.30 +100.00 +EMD +5.25 +0.99 +13.08 +29.69 +26.14 +97.27 +99.48 +SW +1.64 +0.70 +10.69 +39.00 +44.76 +90.25 +88.70 +Max-SW (T = 1) +2.04 +0.71 +10.62 +41.80 +47.86 +91.14 +89.59 +Max-SW (T = 10) +1.64 +0.73 +10.76 +41.21 +46.82 +90.84 +88.77 +Max-SW (T = 50) +1.50 +0.73 +10.82 +40.18 +47.12 +91.88 +90.55 +ASW +1.85 +0.71 +10.79 +40.47 +47.27 +90.77 +88.92 +v-DSW (T = 1) +1.85 +0.77 +11.14 +37.96 +45.64 +90.69 +88.04 +v-DSW (T = 10) +1.37 +0.75 +11.03 +36.78 +43.57 +89.51 +87.74 +v-DSW (T = 50) +1.55 +0.74 +10.98 +38.40 +46.23 +90.18 +89.36 +Lv-DSW (ours) +1.60 +0.71 +10.65 +39.88 +48.89 +89.88 +88.55 +Gv-DSW (ours) +1.85 +0.72 +10.84 +40.03 +46.09 +90.47 +88.55 +Nv-DSW (ours) +1.33 +0.77 +10.98 +38.70 +49.19 +91.21 +89.73 +EAv-DSW (ours) +1.93 +0.66 +10.37 +41.80 +49.48 +89.29 +88.70 +LAv-DSW (ours) +1.60 +0.68 +10.49 +42.39 +47.56 +89.51 +88.18 +the optimal value but comes with an expensive computational cost. We also use the Monte Carlo +estimation with 100 slices as in SW. +C.3 +Details of amortized sliced Wasserstein distances +Linear, generalized linear, and non-linear models: We adopt the official implementations +in [31]. +Self-attention-based models: We adapt the official implementations from their corresponding +papers in our experiments. For all variants, dv is set to 3, which equals the dimension of point-clouds +while dk is chosen from {32, 64, 128}. In Equation 13, the projected dimension k is also selected +from {32, 64, 128}. +Training amortized models: The learning rate is set to 1e-3 and the optimizer is set to Adam [24] +with (β1, β2) = (0, 0.9). +D +Additional experimental results +Point-cloud reconstruction: Table 4 illustrates the full quantitative results of the point-cloud +reconstruction experiment. For Max-SW and v-DSW, we vary the number of gradient iterations T in +{1, 10, 50}. Because CD is not a proper distance so we choose the best number of iterations based on +25 + +SW and EMD losses. As can be seen from the table, with all choices of T, the performance of both +Max-SW and v-DSW are worse than our amortized methods except for Av-DSW which is slower. +The qualitative results are given in Figure 6. As can be seen, CD and EMD fail to reconstruct some +point-clouds while all SW variants can generate good quality point-clouds. +Point-cloud generation: We summarize the full quantitative results for point-cloud generation +in Table 5. For Max-SW and v-DSW, we again change the number of gradient iterations T in +{1, 10, 50}. Note that Av-DSW cannot be used in this experiment due to being out of memory while +the performance of amortized Max-SW is too bad. Therefore, their results are not reported in this +experiment. +26 + +References +[1] P. Achlioptas, O. Diamanti, I. Mitliagkas, and L. Guibas. 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(Cited on page 9.) +31 + diff --git a/mtE3T4oBgHgl3EQf6gu4/content/tmp_files/load_file.txt b/mtE3T4oBgHgl3EQf6gu4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f35b2f25f19fe6a83747157ef43b352f1ab73a66 --- /dev/null +++ b/mtE3T4oBgHgl3EQf6gu4/content/tmp_files/load_file.txt @@ -0,0 +1,1523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf,len=1522 +page_content='Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstruction Khai Nguyen†,∗ Dang Nguyen⋄,∗ Nhat Ho† University of Texas, Austin†, VinAI Research⋄ January 13, 2023 Abstract Max sliced Wasserstein (Max-SW) distance has been widely known as a solution for redundant projections of sliced Wasserstein (SW) distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In applications that have various independent pairs of probability measures, amortized projection optimization is utilized to predict the “max" projecting directions given two input measures instead of using projected gradient ascent multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Despite being efficient, the first issue of the current framework is the violation of permutation invariance property and symmetry property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' To address the issue, we propose to design amortized models based on self-attention architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Moreover, we adopt efficient self-attention architectures to make the computation linear in the number of supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Secondly, Max-SW and its amortized version cannot guarantee metricity property due to the sub-optimality of the projected gradient ascent and the amortization gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, we propose to replace Max-SW with distributional sliced Wasserstein distance with von Mises-Fisher (vMF) projecting distribution (v-DSW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Since v-DSW is a metric with any non-degenerate vMF distribution, its amortized version can guarantee the metricity when predicting the best discriminate projecting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' With the two improvements, we derive self-attention amortized distributional projection optimization and show its appealing performance in point-cloud reconstruction and its downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 1 Introduction Wasserstein distance [49, 38] has been widely recognized in the community of machine learning as an effective tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For example, Wasserstein distance is used to explore clusters inside data [21], to transfer knowledge between different domains [11, 13], to learn generative models [4, 47], to extract features from graphs [50], to compare datasets [2], and many other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Despite being effective, Wasserstein distance is extremely expensive to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, the computational complexity and memory complexity of Wasserstein distance in the discrete case is O(m3 log m) and O(m2) respectively with m is the number of supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The computational problem becomes more severe for applications that require computing the Wasserstein distance multiple times on different pairs of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Some examples can be named: deep generative modeling [20, 32], deep domain adaptation [6], comparing datasets [2], topic modeling [22], point-cloud reconstruction [1], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' By adding entropic regularization [12], an ε-approximation of Wasserstein distance can be obtained in O(m2/ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' However, this approach cannot reduce the memory complexity of O(m2) due to the storage of the cost matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A more efficient approach is based on the closed-form solution of Wasserstein distance in one dimension which is known as sliced Wasserstein distance [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Sliced Khai Nguyen and Dang Nguyen contributed equally to this work 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='04791v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='ML] 12 Jan 2023 Wasserstein (SW) distance is defined as the expectation of the Wasserstein distance between random one-dimensional push-forward measures from the two original measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Thanks to the closed- form solution, SW can be solved in O(m log2 m) while having a linear memory complexity O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Moreover, SW is also better than Wasserstein distance in high-dimensional statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Namely, the sample complexity (statistical estimation rate) of SW is O(n−1/2) compared to O(n−1/d) of Wasserstein distance with d is the number dimension and n is the number of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Due to appealing properties, SW is utilized successfully in various applications e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=', generative modeling [17, 35, 32], domain adaptation [26], Bayesian inference [30, 58], point-cloud representation learning [36, 29], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The downside of SW is that it treats all projections the same due to the usage of a uniform distribution over projecting directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' This choice is inappropriate in practice since there exist projecting directions that cannot discriminate two interested measures [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' As a solution, max sliced Wasserstein distance (Max-SW) [16] is introduced by searching for the best projecting direction that can maximize the projected Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Max-SW needs to use a projected sub-gradient ascent algorithm to find the “max" slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, in applications that need to evaluate Max-SW multiple times on different pairs of measures, the repeated optimization procedure is costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For example, this paper focuses on point-cloud reconstruction applications where Max-SW needs to be computed between various pairs of empirical measures over a point-cloud and its reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' To address the problem, amortized projection optimization is proposed in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' As in other amortized optimization [43, 3] (learning to learn), an amortized model is estimated to predict the best projecting direction given the two input empirical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The authors in [31] propose three types of amortized models including linear model, generalized linear model, and non-linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The linear model assumes that the “max" projecting direction is a linear combination of supports of two measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The generalized linear model injects the linearity through a link function on the supports of two measures while the non-linear model uses multilayer perceptions to have more expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Despite performing well in practice, the previous work has not explored the full potential of amortized optimization in the sliced Wasserstein setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' There are two issues in the current amortized optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Firstly, the sub-optimality of amortized optimization leads to losing the metricity of the projected distance from the predicted projecting direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, the metricity of Max-SW is only obtained at the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, using an amortized model with sub-optimal solutions cannot achieve the metricity for all pairs of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Losing metricity property could hurt the performance of downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Secondly, the current amortized models are not permutation invariant to the supports of two input measures and are not symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The permutation-invariant and symmetry properties are vital since the “max" projecting direction is also not changed when permuting supports of two input empirical measures and exchanging two input empirical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' By inducing the permutation-invariance and symmetry to the amortized model, it could learn a better amortized model and reduce the amortization gap In this paper, we focus on overcoming the two issues of the current amortized projection optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For metricity preservation, we propose amortized distributional projection optimization framework which predicts the best distribution over projecting directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, we do amortized optimization for distributional sliced Wasserstein (DSW) distance [33] with von Mises Fisher (vMF) slicing distribution [23] instead of Max-SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Thanks to the smoothness of vMF, the metricity can be preserved even without a zero amortization gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For the permutation-invariance and symmetry properties, we propose to use the self-attention mechanism [48] to design the amortized 2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Moreover, we utilize efficient self-attention approaches that have the computational complexity scales linearly in the number of supports including efficient attention [42] and linear attention [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Contribution: In summary, our contribution is two-fold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' First, we introduce amortized distributional projection amortization framework which predicts the best location parameter for von Mises-Fisher (vMF) distribution in distributional sliced Wasserstein (DSW) distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Due to the smoothness of vMF, the metricity is guaranteed for all pairs of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Moreover, we enhance amortized models by inducing inductive biases which are permutation invariance and symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' To improve the efficiency, we leverage two linear-complexity attention mechanisms including efficient attention [42] and linear attention [51] to parametrize the amortized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Combining the above two improvements, we obtain self-attention amortized distributional projection amortization framework 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Second, we adapt the new framework to the point-clouds reconstruction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, we want to learn an autoencoder that can reconstruct (encode and decode) all point-clouds through their latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The main idea is to treat a point-cloud as an empirical measure and use sliced Wasserstein distances as the reconstruction losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Here, amortized optimization serves as a fast way to yield informative projecting directions for sliced Wasserstein distance to discriminative all pairs of original point-cloud and reconstructed point-cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Empirically, we show that the self-attention amortized distributional projection amortization provides better reconstructed point-clouds on the ModelNet40 dataset [54] than the amortized projection optimization framework and widely used distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Moreover, on downstream tasks, the new framework also leads to higher classification accuracy on ModelNet40 and generates ShapeNet chairs with better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Organization: The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In Section 2, we provide backgrounds for point-cloud reconstruction and popular distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In Section 3, we define the new amortized distributional projection optimization framework for the point-cloud reconstruction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Section 4 benchmarks the proposed method by extensive experiments on point-cloud reconstruction, transfer learning, and point-cloud generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Finally, proofs of key results and extra materials are in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For any d ≥ 2, we denote U(Sd−1) is the uniform measure over the unit hyper-sphere Sd−1 := {θ ∈ Rd | ||θ||2 2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For p ≥ 1, Pp(Rd) is the set of all probability measures on Rd that have finite p-moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For any two sequences an and bn, the notation an = O(bn) means that an ≤ Cbn for all n ≥ 1, where C is some universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We denote θ♯µ is the push-forward measures of µ through the function f : Rd → R that is f(x) = θ⊤x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 2 Preliminaries We first review the point-cloud reconstruction framework in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' After that, we discuss famous choices of metrics between two point-clouds in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Finally, we present an adapted definition of the amortized projection optimization framework in the point-cloud reconstruction setting in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 3 Encoder Decoder Figure 1: The reconstruction of a point-cloud X (a plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1 Point-Cloud Reconstruction We denote a point-cloud of m points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , xm ∈ Rd (d ≥ 1) as X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , xm) ∈ Rdm which is a vector of a concatenation of all points in the point-cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We denote the set of all possible point-clouds as X ⊂ Rdm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Permutation invariant metric space: Given a permutation one-to-one mapping function σ : [m] → [m], we have σ(X) ∈ X for all X ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Moreover, we need a metric D : X × X → R+ such that D(X, σ(X)) = 0 for all X ∈ X where σ(X) = (xσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , xσ(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Here, D is a metric, namely, it needs to satisfy the non-negativity, symmetry, triangle inequality, and identity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The pair (X, D) forms a point-cloud metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Learning representation via reconstruction: The raw representation of point-clouds is hard to work with in applications due to the complicated metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, a famous approach is to map point-clouds to points in a different space e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=', Euclidean, which is easier to apply machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In more detail, we want to estimate a function fφ : X → Z (φ ∈ Φ) where Z is a set that belongs to another metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Then, we can apply machine learning algorithms on Z instead of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The most well-known and effective way to estimate the function fφ is through reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Namely, we estimate fφ jointly with a function gγ : Z → X (γ ∈ Γ) given a point-cloud dataset p(X) (distribution over X) by minimizing the objective: min φ∈Φ,γ∈Γ EX∼p(X)D(X, gγ(fφ(X))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' (1) The loss EX∼p(X)D(X, gγ(fφ(X))) is known as the reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' If the reconstruction loss is 0, we have gγ = f−1 φ p-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, we can move from X to Z and move back from Z to X without losing information through the functions fφ (referred as the encoder) and gγ (referred as the decoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We show an illustration of the framework [1] in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' After learning how to do the reconstruction well, other point-cloud tasks can be done using the autoencoder (the pair (fφ, gγ)) e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=', shape interpolation, shape editing, shape analogy, shape completion, point-cloud classification, and point-cloud generation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2 Metric Spaces for Point-Clouds We now review some famous choices of the metric D which are Chamfer distance [5], Wasserstein distance [49], sliced Wasserstein (SW) distance [7], and max sliced Wasserstein (Max-SW) [16] distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Chamfer distance: For any two point-clouds X and Y , the Chamfer distance is defined as follows: CD(X, Y ) = 1 |X| � x∈X min y∈Y ∥x − y∥2 2 + 1 |Y | � y∈Y min x∈X ∥x − y∥2 2, (2) where |X| denotes the number of points in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Wasserstein distance: Given two probability measures µ ∈ Pp(Rd) and ν ∈ Pp(Rd), the Wasser- stein distance between µ and ν is defined as follows: Wp(µ, ν) = � inf π∈Π(µ,ν) � Rd×Rd ∥x − y∥p pdπ(x, y) � 1 p (3) where Π(µ, ν) is set of all couplings whose marginals are µ and ν respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Since the Wasserstein distance is originally defined on probability measures space, we need to convert a point-cloud X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , xm) ∈ X to the corresponding empirical probability measure PX = 1 m �m i=1 δxi ∈ P(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, we can use D(X, Y ) = Wp(PX, PY ) for X, Y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Sliced Wasserstein distance: As discussed, the Wasserstein distance is expensive to compute with the time complexity O(m3 log m) and the memory complexity O(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, an alternative choice is sliced Wasserstein (SW) distance between two probability measures µ ∈ Pp(Rd) and ν ∈ Pp(Rd) is: SWp(µ, ν) = � Eθ∼U(Sd−1)Wp p(θ♯µ, θ♯ν) � 1 p , (4) The benefit of SW is that Wp(θ♯µ, θ♯ν) has a closed-form solution which is �� 1 0 |F −1 θ♯µ(z) − F −1 θ♯ν(z)|pdz � 1 p with F −1 denotes the inverse CDF function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The expectation is often approximated by Monte Carlo sampling, namely, it is replaced by the average from θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , θL that are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from U(Sd−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The computational complexity and memory complexity of SW becomes O(Lm log2 m) and O(Lm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Max sliced Wasserstein distance: It is well-known that SW has a lot of redundant projections due to the uniform sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, max sliced Wasserstein distance is proposed to use the most discriminative projecting direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Max sliced Wasserstein (Max-SW) distance [16] between µ ∈ Pp(Rd) and ν ∈ Pp(Rd) is introduced as follows: Max-SWp(µ, ν) = max θ∈Sd−1 Wp(θ♯µ, θ♯ν), (5) Max-SW is often computed by a projected sub-gradient ascent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' When the projected sub-gradient ascent algorithm has T ≥ 1 iterations, the computation complexity of Max-SW is O(Tm log2 m) and the memory complexity is O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Both SW and Max-SW are applied successfully in point-cloud reconstruction [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3 Amortized Projection Optimization We first revisit the point-cloud reconstruction objective with D(X, Y ) = Max-SWp(PX, PY ): min φ∈Φ,γ∈Γ E � max θ∈Sd−1 Wp(θ♯PX, θ♯Pgγ(fφ(X))) � , (6) where the expectation is with respect to X ∼ p(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For each point-cloud X ∈ X, we need to compute a Max-SW distance with an iterative optimization procedure, which is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Amortized projection optimization: Authors in [31] propose to use amortized optimization [43, 3] to speed up the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Instead of solving all optimization problems independently, an amortized model is trained to predict optimal solutions to all problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In greater detail, given a parametric function aψ : X × X → Sd−1 (ψ ∈ Ψ), the amortized objective is: min φ∈Φ,γ∈Γ max ψ∈Ψ EWp(θψ,γ,φ♯PX, θψ,γ,φ♯Pgγ(fφ(X))), (7) where the expectation is with respect to X ∼ p(X), and θψ,γ,φ = aψ(X, gγ(fφ(X))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The above optimization is solved by an alternative stochastic (projected)-gradient descent-ascent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, it is faster to compute in each update iteration of φ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' It is worth noting that the previous work [31] considers the generative model application which is hard and unstable to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Here, we adapt the framework to the point-cloud reconstruction application which is easier to explore the behavior of amortized optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We refer the reader to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1 for more information about amortized optimization and Algorithms 2-3 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='4 for algorithms on training an autoencoder with Max-SW and amortized projection optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Amortized models: Authors in [31] propose three types of amortized models that are based on the literature on linear models [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, the linear amortized model is defined as: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Given X, Y ∈ Rdm, the linear amortized model is defined as: aψ(X, Y ) := w0 + X′w1 + Y ′w2 ||w0 + X′w1 + Y ′w2||2 , where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X and Y of size dm, ψ = (w0, w1, w2) with w1, w2 ∈ Rm, and w0 ∈ Rd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Similarly, the generalized linear amortized model and the non-linear amortized model are defined by injecting non-linearity into the linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We review the definitions of the generalized linear amortized model and non-linear amortized model in Definitions 4-5 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Sub-optimality: Despite being faster, amortized optimization often cannot recover the global optimum of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Namely, we denote θ⋆(X) = argmaxθ∈Sd−1Wp(θ♯PX, θ♯Pgγ(fφ(X))) and ψ⋆ = arg maxψ∈Ψ EX∈p(X) � Wp(θψ,γ,φ♯PX, θψ,γ,φ♯Pgγ(fφ(X))) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Then, it is well-known that the amortization gap EX∼p(X)[c(θ⋆(X), aψ⋆(X, gγ(fφ(X))))] > 0 for a metric c : Sd−1 × Sd−1 → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A great amortized model is one that can minimize the amortization gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' However, in the amortized projection optimization setting, we cannot obtain θ⋆(X) since the projected gradient ascent algorithm can only yield the local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, the investigation of the amortization gap is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 6 Amortized projection Amortized distributional projection Figure 2: The difference between amortized projection optimization and amortized distributional projection optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 3 Self-Attention Amortized Distributional Projection Optimization In this section, we propose the self-attention amortized distributional projection optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' First, we present amortized distributional projection optimization to maintain the metricity property in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We then introduce self-attention amortized models which are symmetric and permutation invariant in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1 Amortized Distributional Projection Optimization The current amortized projection optimization framework is for predicting the “max" projecting direction in Max-SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' However, the projected one-dimensional Wasserstein is only a metric on space of probability measure at the global optimum of Max-SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, the local optimum from the projected sub-gradient ascent algorithm [37] and the prediction from the amortized model only yield pseudo-metricity for the projected Wasserstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Let the projected one-dimensional Wasserstein be PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) = Wp(ˆθ♯µ, ˆθ♯ν)) for any µ, ν ∈ Pp(Rd) (p ≥ 1, d ≥ 1) and ˆθ ∈ Sd−1 such that ˆθ ̸= arg maxθ∈Sd−1 Wp(θ♯µ, θ♯ν) , PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) is a pseudo metric on Pp(Rd) since it satisfies symmetry, non-negativity, triangle inequality, µ = ν implies PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) = 0, however, PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) = 0 does not imply µ = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The proof for Proposition 1 is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' This result implies that the if reconstruction loss EX∼p(X)[PWp(PX, Pgγ(fφ(X));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ(X)) = 0, it does not imply X = gγ(fφ(X)) for p-almost surely X ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, a local maximum for maxθ∈Sd−1 in Max-SW reconstruction (Equation 6) and the 7 global maximum for maxψ∈Ψ in amortized Max-SW reconstruction (Equation 7 with a misspecified amortized model) cannot guarantee perfect reconstruction even when their objectives obtain 0 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Amortized Distributional Projection Optimization: To overcome the issue, we propose to replace Max-SW in Equation 6 with the von Mises Fisher distributional sliced Wasserstein (v-DSW) distance [34]: min φ∈Φ,γ∈Γ EX∼p(X) � max ϵ∈Sd−1 � Eθ∼vMF(ϵ,κ)Wp p(θ♯PX, θ♯Pgγ(fφ(X))) � 1 p � , (8) where vMF(ϵ, κ) is the von Mises Fisher distribution with the mean location parameter ϵ ∈ Sd−1 and the concentration parameter κ > 0, and v-DSWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' κ) = maxϵ∈Sd−1 � Eθ∼vMF(ϵ,κ)Wp p(θ♯µ, θ♯ν) � 1 p is von Mises Fisher distributional sliced Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The optimization can be solved by a stochastic projected gradient ascent algorithm with the vMF reparameterization trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , θL (L ≥ 1) is sampled i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from vMF(ϵ, κ) via the reparameterized acceptance-rejection sampling [14] to approximate ∇ϵEvMF(ϵ,κ)[Wp p(θ♯µ, θ♯ν)] via Monte Carlo integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We refer the reader to Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3 for more detail about the vMF distribution, its sampling algorithm, its reparameterization trick, and the stochastic gradient estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We present a visualization of the difference between the new amortized distributional projection optimization framework and the conventional amortized projection optimization framework in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The corresponding amortized objective is: min φ∈Φ,γ∈Γ max ψ∈Ψ EX∼p(X) � Eθ∼vMF(ϵψ,γ,φ,κ)Wp p(θ♯PX, θ♯Pgγ(fφ(X))) � 1 p , (9) where ϵψ,γ,φ = aψ(X, gγ(fφ(X))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The optimization is solved by an alternative stochastic (projected)- gradient descent-ascent algorithm with the vMF reparameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For any ϵ ∈ Sd−1 and 0 ≤ κ < ∞, if EX∼p(X) � Eθ∼vMF(ϵ,κ)Wp p(θ♯PX, θ♯Pgγ(fφ(X))) � 1 p = 0, X = gγ(fφ(X)) for p-almost surely X ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The proof of Theorem 1 is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The proof is based on proving the metricity of the non-optimal von Mises Fisher distributional sliced Wasserstein distance (v-DSW) with the smoothness condition of the vMF distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' It is worth noting that the proof of metricity of von Mises Fisher distributional sliced Wasserstein distance is new since the original work [34] only shows the pseudo-metricity with the global optimality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Theorem 1 indicates that a perfect reconstruction can be obtained with a local optimum for maxϵ∈Sd−1 in v-DSW reconstruction (Equation 8) and a local optimum for maxψ∈Ψ in amortized v-DSW reconstruction (Equation 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Comparison to SW and Max-SW: When κ → 0, the vMF distribution converges weakly to the uniform distribution over the unit hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Hence, we can get back the conventional sliced Wasserstein reconstruction in both Equation 8 and Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' When κ → ∞, vMF distribution converges weakly to the Dirac delta at the location parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, we obtain Max-SW reconstruction and amortized Max-SW reconstruction in Equation 8 and Equation 9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' However, when 0 < κ < ∞, v-DSW reconstruction and amortized v-DSW reconstruction can find a region of discriminative projecting directions while preserving the metricity for perfect reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 8 NOT SYMMETRIC NOT PERMUTATION INVARIANT Figure 3: Visualization of an amortized model that is not symmetric and permutation invariant in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2 Self-Attention Amortized Models Permutation Invariance and Symmetry: Let X and Y be two point-clouds, the optimal slicing distribution vMF(ϵ⋆, κ) of v-DSW between PX and PY is invariant to the permutation of the supports since Pσ(X) = PX and Pσ(Y ) = PY for a permutation function σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Moreover, the optimal slicing distribution vMF(ϵ⋆, κ) is also unchanged when we exchange PX and PY since v-DSW is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' However, the current amortized models (see Definition 1, Definitions 4-5 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2) are not permutation invariant and symmetric, namely, aψ(X, Y ) ̸= aψ(X, σ(Y )) and aψ(X, Y ) ̸= aψ(Y, X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, the current amortized models could be strongly misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We show a visualization of an amortized model that is not symmetric and permutation invariant in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' To address the issue, we propose amortized models that are symmetric and permutation invariant based on the self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Self-Attention Mechanism: Attention is well-known for its effectiveness in learning long-range dependencies when data are sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For example, the attention mechanism has been successfully utilized to tackle many natural language processing [18, 8, 28, 41], speech recognition [52, 27, 53], and computer vision tasks [19, 59, 57, 55, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We now revisit the attention mechanism [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Given Q, K ∈ Rm×dk, V ∈ Rm×dv, the scaled dot-product attention operator is defined as: Att(Q, K, V ) = softmaxrow �QKT √dk � V (10) where softmaxrow denotes the row-wise softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In the self-attention mechanism, the query matrix Q, the key matrix K, and the value matrix V are usually computed by projecting the input sequence X into different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Thus, the self-attention mechanism is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Given 9 X ∈ Rm×d, the self-attention operator is: Aζ(X) = Att(XWq, XWk, XWv) (11) where Wq, Wk ∈ Rd×dk, Wv ∈ Rd×dv and ζ = (Wq, Wk, Wv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The self-attention operator is infamous for its quadratic memory and computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, given an input sequence of length m, both the time and space complexity are O(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Since we focus on the sliced Wasserstein setting where the computational complexity should be at most O(m log m), the conventional self-attention is not appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Several works have been proposed to reduce the overall complexity from O(m2) to O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In this paper, we utilize two linear complexity variants of attention which are efficient attention [42] and linear attention [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Given X ∈ Rm×d, the efficient self-attention is defined as: EAζ(X) = softmaxrow(XWq) � softmaxcol(XWk)T (XWv) � (12) where Wq, Wk ∈ Rd×dk, Wv ∈ Rd×dv, ζ = (Wq, Wk, Wv), and softmaxcol denotes applying the softmax function column-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The linear self-attention is defined as: LAζ(X) = Att(XWq, Wk1XWk2, Wv1XWv2) (13) where Wq, Wk2 ∈ Rd×dk, Wv2 ∈ Rd×dv, Wk1, Wv1 ∈ Rk×n, and ζ = (Wq, Wk1, Wk2, Wv1, Wv2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The projected dimension k is chosen such that m ≫ k to reduce the memory and space consumption significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Self-Attention Amortized Models: Based on the self-attention mechanism, we introduce the self- attention amortized model which is permutation invariant and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Formally, the self-attention amortized model is defined as: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Given X, Y ∈ Rdm, the self-attention amortized model is defined as: aψ(X, Y ) = w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m ||w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m||2 , (14) where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X and Y of size dm, 1m is the m-dimensional vector whose all entries are 1, w0 ∈ Rd and ψ = (w0, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' By replacing the conventional self-attention with the linear self-attention and the efficient self- attention, we obtain the linear self-attention amortized model and the efficient self-attention amortized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Self-attention amortized models are symmetric and permutation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The proof of Proposition 2 is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The symmetry follows directly from the definition of the self-attention amortized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The permutation invariance is proved by showing that the self-attention functions Aζ(X), EAζ(X), and LAζ(X) are permutation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 10 Table 1: Reconstruction and transfer learning performance on the ModelNet40 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' All losses are multiplied by 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Method CD(↓) SW(↓) EMD(↓) Acc(↑) Time (↓) CD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='23 660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='25 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='63 95 EMD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='17 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='67 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='70 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='43 207 SW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='80 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='45 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='84 103 Max-SW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='31 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='33 92 ASW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='24 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='72 112 v-DSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='93 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='21 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='29 202 L-Max-SW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='97 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='64 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='48 93 G-Max-SW 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='80 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='76 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='86 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='99 95 N-Max-SW 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='97 836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='34 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='73 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='64 94 Lv-DSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='85 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='18 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='17 113 Gv-DSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='48 116 Nv-DSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='67 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='89 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='93 115 Av-DSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='84 255 EAv-DSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 127 LAv-DSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='21 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='07 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='45 123 4 Experiments To verify the effectiveness of our proposal, we evaluate our methods on the point-cloud reconstruction task and its two downstream tasks including transfer learning and point-cloud generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Three important questions we want to answer are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Does the sub-optimality issue of amortized Max- SW occur when working with point-clouds and does replacing Max-SW with v-DSW alleviate the problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Does the proposed amortized distribution projection optimization framework improve the performance over the conventional amortized projection optimization framework and commonly used distances e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=', Chamfer distance, Earth Mover Distance (Wasserstein distance), SW, Max-SW, adaptive SW (ASW) [36], and v-DSW?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Are self-attention amortized models better than the previous misspecified amortized models in [31]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Experiment settings: Our settings, which can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1, are identical to the setting in the paper of ASW [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For amortized models, we consider 6 different ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The prefix L, G, and N denote the linear, generalized linear, and non-linear amortized models in [31], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A, EA, and LA are used to represent self-attention, efficient self-attention, and linear self-attention, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Implementation details for baseline distances and amortized models are given in Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Point-cloud reconstruction: Following ASW [36], we measure the reconstruction performance of different autoencoders on the ModelNet40 dataset [54] using three discrepancies: Chamfer discrepancy (CD), sliced Wasserstein distance (SW), and EMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The quantitative results are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For each method, we only report the best performing (based on SW and EMD losses) model among all choices of hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Full quantitative results can be found in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Our method achieves the best performance in all three discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In contrast, autoencoders 11 Figure 4: Qualitative results of reconstructing point-clouds in the ShapeNet Core-55 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' From top to bottom: input, SW, Max-SW, v-DSW, and LAv-DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' with amortized Max-SW losses fail in this scenario due to the sub-optimality and losing metricity issues that we discussed in Appendix 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In addition, using amortized optimization reduces the training time compared to the conventional computation using the projected sub-gradient ascent algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Max-SW and v-DSW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For example, training one iteration of autoencoder using LAv-DSW only takes 123 seconds while using v-DSW costs 202 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In terms of amortized models, attention-based amortized models lead to lower distances between reconstructed and input point-clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Qualitative results are given in Figure 4, showing the success of our methods in reconstructing 3D point-clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Full qualitative results can be found in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Amortization Gaps: To validate the advantage of self-attention amortized models over the previous misspecified amortized models, we compare their effectiveness in approximating v-DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We create a dataset by sampling 1000 pairs of point-clouds from the ShapeNet Core-55 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Due to the memory constraint when solving amortized optimization, the dataset is divided into 10 batches of size 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We compute v-DSW and its amortized versions between all pairs of point-clouds and report their average loss values in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Compared to previous misspecified amortized models, attention-based amortized models produce higher losses which are closer to the conventional computation of v-DSW (T = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' To achieve the same level as efficient/linear self-attention amortized models, one needs to run more than 50 sub-gradient iterations, which is more than 10 times slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Transfer learning: We further feed the latent vectors learned by the above autoencoders into a 12 Table 2: Comparison between amortized models when approximating von Mises Fisher distributional sliced Wasserstein (v-DSW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' T denotes the number of projected sub-gradient ascent iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Method T Distance (↑) Time (↓) Lv-DSW 1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='06 Gv-DSW 1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='07 Nv-DSW 1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='07 Av-DSW 1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='00 EAv-DSW 1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='17 LAv-DSW 1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='14 v-DSW 1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1 v-DSW 5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='33 v-DSW 10 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='5 v-DSW 50 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='00 v-DSW 100 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='00 Table 3: Performance comparison of point-cloud generation on the chair category of ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' JSD, MMD-CD, and MMD-EMD are multiplied by 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Method JSD (↓) MMD (↓) COV (%, ↑) 1-NNA (%, ↓) CD EMD CD EMD CD EMD CD 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='72 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 11.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='57 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='51 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='74 Lv-DSW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='71 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='65 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='88 48.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='48 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='29 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='70 LAv-DSW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='49 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='39 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='56 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='51 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='18 classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Following the settings in ASW’s paper, we train our classifier for 500 epochs with a batch size of 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The optimizer is the same as that in the reconstruction experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Table 1 illustrates the classification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Again, we see a boost in accuracy when using self-attention amortized v-DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Point-cloud generation: We also evaluate our methods on the 3D point-cloud generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Following [1], the chair category of ShapeNet is divided into train/valid/test sets in an 85/5/10 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We train each autoencoder on the train set for 100 epochs and evaluate on the valid set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The generator is then trained to generate latent codes learned by the autoencoder, same as [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For 13 evaluation, the same set of metrics in [56] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The quantitative results of the test set are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Our methods yield the best performance in 6 out of 7 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In addition, attention-based amortized models lead to higher performance than previous amortized models in all metrics except for JSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Full quantitative results are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 5 Conclusion We have proposed a self-attention amortized distributional projection optimization framework which uses a self-attention amortized model to predict the best discriminative distribution over projecting direction for each pair of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The efficient self-attention mechanism helps to inject the geometric inductive biases which are permutation invariance and symmetry into the amortized model while remaining fast computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Furthermore, the amortized distribution projection optimization framework guarantees the metricity for all pairs of probability measures while the amortization gap still exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' On the experimental side, we compare the new proposed framework to the conventional amortized projection optimization framework and other widely-used distances in the point-cloud reconstruction application and its two downstream tasks including transfer learning and point-cloud generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Overall, the self-attention amortized distributional projection optimization framework performs the best in all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Supplement to “Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Clouds Reconstruction" In this supplementary, we first provide some additional materials in Appendix A including the detail of amortized optimization in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1, definitions of generalized linear amortized models and non-linear amortized models in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2, the detail of computing von Mises-Fisher distributional sliced Wasserstein in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3, and training algorithms for autoencoders in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Next, we collect skipped proofs in the main text in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' After that, we discuss the experimental settings of our experiments in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Finally, we present additional experimental results in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A Additional Materials A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1 Amortized Optimization We start with the definition of amortized optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For each context variable x in the context space X, θ⋆(x) is the solution of the optimization problem θ⋆(x) = arg minθ∈Θ L(θ, x), where Θ is the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A parametric function fψ : X → Θ, where ψ ∈ Ψ, is called an amortized model if fψ(x) ≈ θ⋆(x), ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' (15) The amortized model is trained by the amortized optimization objective which is defined as: min ψ∈Ψ Ex∼p(x)L(fψ(x), x), (16) where p(x) is a probability measure on X which measures the “importance" of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 14 The amortized model in Definition 3 is sometimes called a fully amortized model for a distinction with the other concept of semi amortized model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The gap between the predicted solution and the optimal solution Ex∼p(x)||fψ(x) − θ⋆(x)||2 is called the amortization gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' However, understanding this gap depends on specific configurations of the objective L(·, x), such as convexity and smoothness, which are often non-trivial to obtain in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2 Amortized models We now review the generalized linear amortized model and the non-linear amortized model [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Given X, Y ∈ Rdm, the generalized linear amortized model is defined as: fψ(X, Y ) := w0 + gψ1(X)′w1 + gψ1(Y )′w2 ||w0 + gψ1(X)′w1 + gψ1(Y )′w2||2 2 , (17) where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X and Y of size dm, w1, w2 ∈ Rm, w0 ∈ Rd, ψ1 ∈ Ψ1, gψ1 : Rdm → Rdm, ψ = (w0, w1, w2, ψ1), and gψ1(X) = (x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , x′ m) and gψ1(Y ) = (y′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , y′ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' To specify, we let gψ1(X) = (W2σ(W1x1) + b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , W2σ(W1xm) + b0), where σ(·) is the Sigmoid function, W1 ∈ Rd×d, W2 ∈ Rd×d, and b0 ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Given X, Y ∈ Rdm, the non-linear amortized model is defined as: fψ(X, Y ) := hψ2(w0 + X′w1 + Y ′w2) ||hψ2(w0 + X′w1 + Y ′w2)||2 2 , (18) where X′ and Y ′ are matrices of size d × m that are reshaped from the concatenated vectors X and Y of size dm, w1, w2 ∈ Rm, w0 ∈ Rd, ψ2 ∈ Ψ2, hψ2 : Rd → Rd, ψ = (w0, w1, w2, ψ2), and hψ2(x) = W4σ(W3x)) + b0 where σ(·) is the Sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3 Von Mises-Fisher distributional sliced Wasserstein distance We first start with the definition of von Mises Fisher (vMF) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The von Mises–Fisher distribution (vMF)[23] is a probability distribution on the unit hypersphere Sd−1 with the density function is : f(x|ϵ, κ) := Cd(κ) exp(κϵ⊤x), (19) where ϵ ∈ Sd−1 is the location vector, κ ≥ 0 is the concentration parameter, and Cd(κ) := κd/2−1 (2π)d/2Id/2−1(κ) is the normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Here, Iv is the modified Bessel function of the first kind at order v [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The vMF distribution is a continuous distribution, its mass concentrates around the mean ϵ, and its density decrease when x goes away from ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' When κ → 0, vMF converges in distribution to U(Sd−1), and when κ → ∞, vMF converges in distribution to the Dirac distribution centered at ϵ [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Reparameterized Rejection Sampling: The sampling process of vMF distribution is based on the rejection sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We review the sampling process in Algorithm 1 [14, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The algorithm performs the reparameterization for the proposal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We now derive the gradient estimator for ∇ϵEvMF(θ|ϵ,κ) � f(θ) � for a general function f(θ) to find the maxima ϵ∗ in the optimization problem maxϵ∈Sd−1 EvMF(θ|ϵ,κ) � f(θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 15 In d > 0 dimension, let (ϵ, κ) be the parameters of vMF distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We denotes b = −2κ+√ 4κ2+(d−1)2 d−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' two conditional distributions: g(ω | κ) = 2(πd/2) Γ(d/2) Cd(κ) exp(ωκ)(1−ω2) 1 2 (d−3) Beta( 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 1 2 (d−1)) and r(ω|κ) = 2b1/2(d−1) Beta( 1 2 (d−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 1 2 (d−1)) (1−ω2) 1/2(d−3) [(1+b)−(1−b)ω]d−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' distribution s(ψ) := Beta � 1 2(d − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 1 2(d − 1) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' func- tion h(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' κ) = 1−(1+b)ψ 1−(1−b)ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' distributions π1(ψ|κ) = s(ψ) g(h(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='κ)|κ) r(h(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='κ)|κ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' π2(v) := U(Sd−2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' and function T(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ) = � I − 2 e1−ϵ ||e1−ϵ||2 e1−ϵ ||e1−ϵ||2 ⊤�� ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' √ 1 − ω2v⊤�⊤ := θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We can obtain the gradient estimator by following Lemma 2 in [15],: ∇ϵEvMF(θ|ϵ,κ) � f(θ) � = ∇ϵE(ψ,v)∼π1(ψ|κ)π2(v) � f � T(h(ψ, κ), v, ϵ) �� = E(ψ,v)∼π1(ψ|κ)π2(v) � ∇ϵf � T(h(ψ, κ), v, ϵ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In v-DSW case, we have f(θ) = Wp p(θ♯µ, θ♯ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, we have: ∇ϵEvMF(θ|ϵ,κ) � Wp p(θ♯µ, θ♯ν) � = E(ψ,v)∼π1(ψ|κ)π2(v) � ∇ϵWp p(f � T(h(ψ, κ), v, ϵ)♯µ, f � T(h(ψ, κ), v, ϵ)♯ν) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Then we can get a gradient estimator by using Monte-Carlo estimation scheme: ∇ϵEvMF(θ|ϵ,κ) � Wp p(θ♯µ, θ♯ν) � ≈ 1 L L � i=1 � ∇ϵWp p(f � T(h(ψi, κ), vi, ϵ)♯µ, f � T(h(ψi, κ), vi, ϵ)♯ν) �� , where {ψi}L i=1 ∼ π1(ψ|κ) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d, {vi}L i=1 ∼ π2(v) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d, and L is the number of projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Sampling from π1(ψ|κ) is equivalent to the acceptance-rejection scheme in vMF sampling procedure, sampling π2(v) is directly from U(Sd−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' It is worth noting that the gradient estimator for ∇κEvMF(θ|ϵ,κ) � f(θ) � can be derived by using the log-derivative trick, however, we do not need it here since we do not optimize for κ in v-DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='4 Training algorithms Training point-cloud autoencoder with Max-SW: We present the algorithm of training au- toencoder with Max-SW in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The algorithm contains a nested loop: one is for training the autoencoder, one is for finding the max projecting direction for Max-SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Training point-cloud autoencoder with amortized projection optimization: We present the training algorithm for point-cloud autoencoder with amortized projection optimization in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' With amortized optimization, the inner loop for finding the max projecting direction is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Training point-cloud autoencoder with v-DSW: We present the algorithm of training autoen- coder with v-DSW in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The algorithm contains a nested loop: one is for training the autoencoder, one is for finding the best distribution over projecting directions for v-DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Training point-cloud autoencoder with amortized distributonal projection optimiza- tion: We present the training algorithm for point-cloud autoencoder with amortized distributional projection optimization in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' With amortized distributional optimization, the inner loop for finding the best distribution over projecting directions is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 16 Algorithm 1 Sampling from vMF distribution Input: location ϵ, concentration κ, dimension d, unit vector e1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='.,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 0) Draw v ∼ U(Sd−2) b ← −2κ+√ 4κ2+(d−1)2 d−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' a ← (d−1)+2κ+√ 4κ2+(d−1)2 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' m ← 4ab (1+b) − (d − 1) log(d − 1) repeat Draw ψ ∼ Beta � 1 2(d − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 1 2(d − 1) � ω ← h(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' κ) = 1−(1+b)ψ 1−(1−b)ψ t ← 2ab 1−(1−b)ψ Draw u ∼ U([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 1]) until (d − 1) log(t) − t + m ≥ log(u) h1 ← (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' √ 1 − ω2v⊤)⊤ ϵ′ ← e1 − ϵ u = ϵ′ ||ϵ′||2 U = I − 2uu⊤ Output: Uh1 Algorithm 2 Training point-cloud autoencoder with max sliced Wasserstein distance Input: Point-cloud distribution p(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' learning rate η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' slice learning rate ηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' model maximum number of iterations T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' slice maximum number of iterations T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' mini-batch size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Initialization: Initialize the encoder fφ and the decoder gγ while φ, γ not converge or reach T do Sample a mini-batch X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , Xk i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from p(X) ∇φ = 0, ∇γ = 0 for i = 1 to k do Initialize θ while θ not converge or reach T do θ = θ + ηs · ∇θWp(θ♯PXi, θ♯Pgγ(fφ(Xi))) # Other update rules can be used θ = θ ||θ||2 #Project back to the unit-hypersphere Sd−1 end while ∇φ = ∇φ + 1 k∇φWp(θ♯PXi, θ♯Pgγ(fφ(Xi))) ∇γ = ∇γ + 1 k∇γWp(θ♯PXi, θ♯Pgγ(fφ(Xi))) end for φ = φ − η · ∇φ # Other update rules can be used γ = γ − η · ∇γ # Other update rules can be used end while Return: φ, γ B Proofs B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1 Proof for Proposition 1 We first recall the definition of the projected one-dimensional Wasserstein between two probability measures µ and ν: PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) = Wp(ˆθ♯µ, ˆθ♯ν) for ˆθ ̸= argmaxθ∈Sd−1Wp(θ♯µ, θ♯ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 17 Algorithm 3 Training point-cloud autoencoder with amortized projection optimization Input: Point-cloud distribution p(X), learning rate η, slice learning rate ηs, model maximum number of iterations T , mini-batch size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Initialization: Initialize the encoder fφ, the decoder gγ, and the amortized model aψ while φ, γ, ψ not converge or reach T do Sample a mini-batch X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , Xk i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from p(X) ∇φ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ∇γ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ∇ψ = 0 for i = 1 to k do θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ = aψ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' gγ(fφ(Xi))) ∇ψ = ∇ψ + 1 k∇ψWp(θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ♯PXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ♯Pgγ(fφ(Xi))) ∇φ = ∇φ + 1 k∇φWp(θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ♯PXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ♯Pgγ(fφ(Xi))) ∇γ = ∇γ + 1 k∇γWp(θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ♯PXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ♯Pgγ(fφ(Xi))) end for ψ = ψ + ηs · ∇ψ # Other update rules can be used φ = φ − η · ∇φ # Other update rules can be used γ = γ − η · ∇γ # Other update rules can be used end while Return: φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' γ Non-negativity and Symmetry: Due to the non-negativity and symmetry of the Wasserstein distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' the non-negativity and symmetry of the projected Wasserstein follow directly from its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Triangle inequality: For any three probability measures µ1, µ2, µ3 ∈ Pp(Rd), we have: PWp(µ1, µ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) = Wp(ˆθ♯µ1, ˆθ♯µ3) ≤ Wp(ˆθ♯µ1, ˆθ♯µ2) + Wp(ˆθ♯µ2, ˆθ♯µ3) = PWp(µ1, µ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) + PWp(µ2, µ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ), where the first inequality is due to the triangle inequality of the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Identity: If µ = ν, we have PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) = 0 due to the identity of the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' However, if PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) = 0, there exists θ′ ∈ Sd−1 such that 0 = PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ˆθ) < PWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Let F[γ](w) = � Rd′ e−i⟨w,x⟩dγ(x) be the Fourier transform of γ ∈ P(Rd′), for any t ∈ R, we have F[µ](tθ′) = � Rd e−it⟨θ′,x⟩dµ(x) = � R e−itzdθ′♯µ(z) = F[θ′♯µ](t) ̸= F[θ′♯ν](t) = � R e−itzdθ′♯ν(z) = � Rd e−it⟨θ′,x⟩dν(x) = F[ν](tθ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, we have µ ̸= ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2 Proof for Theorem 1 We first start with proving the metricity of the non-optimal von Mises Fisher distributional sliced Wasserstein distance (v-DSW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For any two probability measures µ, ν ∈ Pp(Rd), the non-optimal 18 Algorithm 4 Training point-cloud autoencoder with von-Mises Fisher distributional sliced Wasser- stein distance Input: Point-cloud distribution p(X), learning rate η, slice learning rate ηs, model maximum number of iterations T , slice maximum number of iterations T, mini-batch size k, the number of projections L, and the concentration hyperparameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Initialization: Initialize the encoder fφ and the decoder gγ while φ, γ not converge or reach T do Sample a mini-batch X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , Xk i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from p(X) ∇φ = 0, ∇γ = 0 for i = 1 to k do Initialize ϵ while ϵ not converge or reach T do Sample θϵ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , θϵ L i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from vMF(ϵ, κ) via the reparameterized acceptance-rejection sam- pling in Algorithm 1 ϵ = ϵ + ηs · 1 L �L l=1 ∇ϵWp(θϵ l ♯PXi, θϵ l ♯Pgγ(fφ(Xi))) # Other update rules can be used ϵ = ϵ ||ϵ||2 #Project back to the unit-hypersphere Sd−1 end while Sample θϵ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , θϵ L i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from vMF(ϵ, κ) via the reparameterized acceptance-rejection sampling in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ∇φ = ∇φ + 1 k 1 L �L i=l ∇φWp(θϵ l ♯PXi, θϵ l ♯Pgγ(fφ(Xi))) ∇γ = ∇γ + 1 k 1 L �L i=l ∇γWp(θϵ l ♯PXi, θϵ l ♯Pgγ(fφ(Xi))) end for φ = φ − η · ∇φ # Other update rules can be used γ = γ − η · ∇γ # Other update rules can be used end while Return: φ, γ von Mises Fisher distributional sliced Wasserstein distance (v-DSW) is defined as follow: v-DSWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) = � Eθ∼vMF(ϵ,κ)Wp p(θ♯µ, θ♯ν) � 1 p , where ϵ ∈ Sd−1 and 0 < κ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For any ϵ ∈ Sd−1 and κ < ∞, v-DSWp(·, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) is a valid metric on the space of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We now prove that v-DSW satisfies non-negativity, symmetry, triangle inequality, and identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Non-negativity and Symmetry: The non-negativity and symmetry of v-DSW follow directly the non-negativity and symmetry of the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 19 Algorithm 5 Training point-cloud autoencoder with amortized projection optimization Input: Point-cloud distribution p(X), learning rate η, slice learning rate ηs, model maximum number of iterations T , mini-batch size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Initialization: Initialize the encoder fφ, the decoder gγ, and the amortized model aψ while φ, γ, ψ not converge or reach T do Sample a mini-batch X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , Xk i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from p(X) ∇φ = 0, ∇γ = 0, ∇ψ = 0 for i = 1 to k do ϵψ,γ,φ = aψ(Xi, gγ(fφ(Xi))) Sample θψ,γ,φ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' , θψ,γ,φ L i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='d from vMF(ϵψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' κ) via the reparameterized acceptance-rejection sampling in Algorithm 1 ∇ψ = ∇ψ + 1 k 1 L �L i=l ∇ψWp(θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ l ♯PXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ l ♯Pgγ(fφ(Xi))) ∇φ = ∇φ + 1 k 1 L �L i=l ∇φWp(θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ l ♯PXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ l ♯Pgγ(fφ(Xi))) ∇γ = ∇γ + 1 k 1 L �L i=l ∇γWp(θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ l ♯PXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' θψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='φ l ♯Pgγ(fφ(Xi))) end for ψ = ψ + ηs · ∇ψ # Other update rules can be used φ = φ − η · ∇φ # Other update rules can be used γ = γ − η · ∇γ # Other update rules can be used end while Return: φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' γ Triangle inequality: For any three probability measures µ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' µ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' µ3 ∈ Pp(Rd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' we have v-DSWp(µ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' µ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) = � Eθ∼vMF(ϵ,κ)Wp p(θ♯µ1, θ♯µ3) � 1 p ≤ � Eθ∼vMF(ϵ,κ) [Wp(θ♯µ1, θ♯µ2) + Wp(θ♯µ2, θ♯µ3)]p� 1 p ≤ � Eθ∼vMF(ϵ,κ)Wp p(θ♯µ1, θ♯µ2) � 1 p + � Eθ∼vMF(ϵ,κ)Wp p(θ♯µ2, θ♯µ3) � 1 p = v-DSWp(µ1, µ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) + v-DSWp(µ2, µ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) Identity: From the definition, if µ = ν, we obtain v-DSWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Now, we need to show that if v-DSWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) = 0, then µ = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' If v-DSWp(µ, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) = 0, we have � Eθ∼vMF(ϵ,κ)Wp p(θ♯µ, θ♯ν) � 1 p = 0 which implies Eθ∼vMF(ϵ,κ)Wp p(θ♯µ, θ♯ν) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, Wp(θ♯µ, θ♯ν) = 0 for vMF(ϵ, κ) almost surely θ ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Using the identity property of the Wasserstein distance, we obtain θ♯µ = θ♯ν for vMF(ϵ, κ) almost surely θ ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Since vMF(ϵ, κ) with 0 < κ < ∞ has the supports on all Sd−1, for any t ∈ R and θ ∈ Sd−1, we have: F[µ](tθ) = � Rd e−it⟨θ,x⟩dµ(x) = � R e−itzdθ♯µ(z) = F[θ♯µ](t) = F[θ♯ν](t) = � R e−itzdθ♯ν(z) = � Rd e−it⟨θ,x⟩dν(x) = F[ν](tθ), where F[γ](w) = � Rd′ e−i⟨w,x⟩dγ(x) denotes the Fourier transform of γ ∈ P(Rd′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We then obtain µ = ν by the injectivity of the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 20 By abuse of notation, we denote v-DSW(X, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) = v-DSW(PX, PY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) for X, Y ∈ X are two point-clouds, PX = 1 m �m i=1 δxi, and PY = 1 m �m i=1 δyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We cast the v-DSW from a metric on the space of probability measures to the space of point-clouds X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For any ϵ ∈ Sd−1 and κ < ∞, v-DSWp(·, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) is a valid metric on the space of point-clouds X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Since PX, PY ∈ Pp(Rd), the non-negativity, symmetry, triangle inequality, and identity properties follow directly from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We now only need to show that v-DSW is invariant to permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' This property is straightforward from the definition of empirical probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For any permutation function σ, we have PX = 1 m �m i=1 δxi = 1 m �m i=1 δxσ(i) = Pσ(X) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We now continue the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' If EX∼p(X) � Eθ∼vMF(ϵ,κ)Wp p(θ♯PX, θ♯Pgγ(fφ(X))) � 1 p = 0, we obtain � Eθ∼vMF(ϵ,κ)Wp p(θ♯PX, θ♯Pgγ(fφ(X))) � 1 p = v-DSW(X, gγ(fφ(X));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' ϵ, κ) = 0 for p-almost surely X ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' By Collorary 1, we obtain X = gγ(fφ(X)) for p-almost surely X ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3 Proof for Proposition 2 We first recall the definition of the self-attention amortized model in Definition 2: aψ(X, Y ) = w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m ||w0 + Aζ(X′⊤)⊤1m + Aζ(Y ′⊤)⊤1m||2 , Symmetry: Since the self-attention amortized model use the same attention weight ζ for both X and Y , exchanging X and Y yields the same results aψ(X, Y ) = aψ(Y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Permutation invariance: We first show that self-attention is permutation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In particular, we have: Aζ(X′⊤)⊤1m = Att(X′⊤Wq, X′⊤Wk, X′⊤Wv)⊤1m = � softmaxrow �X′⊤WqW ⊤ k X′ √dk � X′⊤Wv �⊤ 1m = � softmaxrow �σ(X)′⊤WqW ⊤ k σ(X)′ √dk � σ(X)′⊤Wv �⊤ 1m = Aζ(σ(X)′⊤)⊤1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Similarly, the proof holds for both linear self-attention and efficient self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' C Experiment settings In this section, we first provide the details of the training process and the architecture for point-cloud reconstruction, transfer learning, and point-cloud generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Then, we present the implementation detail and hyper-parameters settings for different distances used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 21 conv1d(64, 1) conv1d(64, 1) conv1d(64, 1) conv1d(128, 1) conv1d(1024, 1) maxpool fc 256 z fc 1024 fc 1024 fc Nx3 tanh fc 512 fc 256 fc 40 fc 256 fc 1024 fc 1024 encoder decoder classifier Figure 5: The architecture of the Point-Net variant in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For transfer learning, we use a simple classifier with 3 fully-connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' All layers are followed by ReLU activation and batch normalization by default, except for the final layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='1 Details of point-cloud reconstruction and downstream applications Point-cloud reconstruction: We use the same settings in ASW [36] to train autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We utilize a variant of Point-Net [40] with an embedding size of 256 proposed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The architecture of the autoencoder and classifier are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Our autoencoder is trained on the ShapeNet Core-55 dataset [9] with a batch size of 128 and a point-cloud size of 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We train it for 300 epochs using an SGD optimizer with an initial learning rate of 1e-3, a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='9, and a weight decay of 5e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' All experiments are run on NVIDIA V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Next, we detail the process of conducting two downstream applications of point-cloud reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Transfer learning: A classifier is trained on the latent space of the autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Particularly, we extract a 256-dimension latent vector of an input 3D point-cloud via the pre-trained encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Then, this vector is fed into a multi-layer perceptron with hidden layers of size 512 and 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The last layer outputs a 40-dimension vector representing the prediction of 40 classes of the ModelNet40 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Point-cloud generation: Our generative model is trained on the latent space of the autoencoder as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' First, we extract a 256-dimension latent vector of an input 3D point-cloud via the pre-trained encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Then a 64-dimensional vector is drawn from a normal distribution N(0, I64), where I64 is the 64x64 identity matrix, and fed into a generator which also outputs a 256-dimension vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Finally, the generator learns by minimizing the optimal transport distance between the generated and ground truth latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 22 Figure 6: Qualitative results of reconstructing point-clouds in the ShapeNet Core-55 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' From top to bottom: input, CD, EMD, SW, Max-SW, ASW, v-DSW, Nv-DSW, EAv-DSW, and LAv-DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 23 Table 4: Reconstruction and transfer learning performance of different autoencoders on the Model- Net40 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For v-DSW and Max-SW, T denotes the number of projected sub-gradient ascent iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In Table 1, v-DSW and Max-SW have T = 10 and 1 iterations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' All reconstructed losses are multiplied by 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Method CD (↓) SW (↓) EMD (↓) Acc (↑) Time (↓) CD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='23 660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='25 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='63 95 EMD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='17 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='67 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='70 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='43 207 SW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='80 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='45 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='84 103 Max-SW (T = 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='31 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='33 92 Max-SW (T = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='90 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='76 107 Max-SW (T = 50) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='30 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='17 124 ASW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='24 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='72 112 v-DSW (T = 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='34 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='30 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 115 v-DSW (T = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='93 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='21 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='29 202 v-DSW (T = 50) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='67 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='55 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='26 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 638 L-Max-SW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='97 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='64 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='48 93 G-Max-SW 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='80 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='76 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='86 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='99 95 N-Max-SW 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='97 836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='34 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='73 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='64 94 Lv-DSW (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='85 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='18 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='17 113 Gv-DSW (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='48 116 Nv-DSW (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='67 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='89 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='93 115 Av-DSW (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='84 255 EAv-DSW (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 127 LAv-DSW (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='21 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='07 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='45 123 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='2 Details of baseline distances We want to emphasize that we use the same set of hyper-parameters reported in [36] for Chamfer, EMD, SW, and Max-SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Chamfer and EMD: We use the CUDA implementation from [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' SW: We use the Monte Carlo estimation with 100 slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Max-SW: We use the projected sub-gradient ascent algorithm to optimize the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' It is trained with an Adam optimizer with an initial learning rate of 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The number of iterations T is chosen from {1, 10, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Adaptive SW: We use Algorithm 1 in [36] with the same set of parameters as follows: N0 = 2, s = 1, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='5, and M = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' v-DSW: We use stochastic projected gradient ascent algorithm to optimize the location vector ϵ in Equation 19 while we fix the concentration parameter κ to 1 for both v-DSW and all of its amortized versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Similar to Max-SW, it is trained with an Adam optimizer with an initial learning rate of 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The number of iterations T is selected from {1, 10, 50} based on the task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Intuitively, increasing the number of iterations leads to a better approximation that is closer to 24 Table 5: Performance comparison of point-cloud generation on the chair category of ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For v-DSW and Max-SW, T denotes the number of projected sub-gradient ascent iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In Table 3, v-DSW and Max-SW have T = 10 and 1 iterations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' JSD, MMD-CD, and MMD-EMD are multiplied by 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Method JSD (↓) MMD (↓) COV (%, ↑) 1-NNA (%, ↓) CD EMD CD EMD CD EMD CD 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='72 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='67 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='30 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='00 EMD 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='99 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='08 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='14 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='27 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='48 SW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='70 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='00 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='76 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='25 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='70 Max-SW (T = 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='71 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='62 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='80 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='86 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='14 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='59 Max-SW (T = 10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='73 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='76 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='21 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='82 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='84 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='77 Max-SW (T = 50) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='73 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='82 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='18 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='12 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='88 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='55 ASW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='71 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='79 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='47 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='27 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='77 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='92 v-DSW (T = 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='77 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='14 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='96 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='64 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='69 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='04 v-DSW (T = 10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='03 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='78 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='57 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='51 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='74 v-DSW (T = 50) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='74 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='98 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='40 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='23 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='18 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='36 Lv-DSW (ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='71 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='65 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='88 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='89 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='88 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='55 Gv-DSW (ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='72 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='84 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='03 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='09 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='47 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='55 Nv-DSW (ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='77 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='98 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='70 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='19 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='21 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='73 EAv-DSW (ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='37 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='80 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='48 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='29 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='70 LAv-DSW (ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='68 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='49 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='39 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='56 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='51 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='18 the optimal value but comes with an expensive computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' We also use the Monte Carlo estimation with 100 slices as in SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='3 Details of amortized sliced Wasserstein distances Linear, generalized linear, and non-linear models: We adopt the official implementations in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Self-attention-based models: We adapt the official implementations from their corresponding papers in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For all variants, dv is set to 3, which equals the dimension of point-clouds while dk is chosen from {32, 64, 128}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In Equation 13, the projected dimension k is also selected from {32, 64, 128}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Training amortized models: The learning rate is set to 1e-3 and the optimizer is set to Adam [24] with (β1, β2) = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' D Additional experimental results Point-cloud reconstruction: Table 4 illustrates the full quantitative results of the point-cloud reconstruction experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For Max-SW and v-DSW, we vary the number of gradient iterations T in {1, 10, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Because CD is not a proper distance so we choose the best number of iterations based on 25 SW and EMD losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' As can be seen from the table, with all choices of T, the performance of both Max-SW and v-DSW are worse than our amortized methods except for Av-DSW which is slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' The qualitative results are given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' As can be seen, CD and EMD fail to reconstruct some point-clouds while all SW variants can generate good quality point-clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Point-cloud generation: We summarize the full quantitative results for point-cloud generation in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' For Max-SW and v-DSW, we again change the number of gradient iterations T in {1, 10, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Note that Av-DSW cannot be used in this experiment due to being out of memory while the performance of amortized Max-SW is too bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Therefore, their results are not reported in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' 26 References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Achlioptas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Diamanti, 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[58] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Yi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' Sliced Wasserstein variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' In Fourth Symposium on Advances in Approximate Bayesian Inference, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' (Cited on page 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=') [59] X.' metadata={'source': 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transformers for end-to-end object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content='04159, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=' (Cited on page 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} +page_content=') 31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf'} diff --git a/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf b/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8f706c01edbde83a2114fc5b360e279071ad9755 --- /dev/null +++ b/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a44e70e155795ad56754718d5ac32c684b8ba0d613cbaada2c411263801ea62a +size 5031757 diff --git a/pNE1T4oBgHgl3EQf2AWA/content/tmp_files/2301.03474v1.pdf.txt b/pNE1T4oBgHgl3EQf2AWA/content/tmp_files/2301.03474v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..269be9d5e0c0a952b7af5b656dca0eca529fa39d --- /dev/null +++ b/pNE1T4oBgHgl3EQf2AWA/content/tmp_files/2301.03474v1.pdf.txt @@ -0,0 +1,2358 @@ +Ab initio derivation of lattice gauge theory dynamics for cold gases in optical lattices +Federica Maria Surace,1, ∗ Pierre Fromholz,2, 3, † Nelson Darkwah Oppong,4, 5, ‡ +Marcello Dalmonte,2, 3 and Monika Aidelsburger4, 5, § +1Department of Physics and Institute for Quantum Information and Matter, +California Institute of Technology, Pasadena, California 91125, USA +2The Abdus Salam International Centre for Theoretical Physics (ICTP), strada Costiera 11, 34151 Trieste, Italy +3International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy +4Faculty of Physics, Ludwig-Maximilians-Universit¨at M¨unchen, Schellingstr. 4, D-80799 Munich, Germany +5Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 Munich, Germany +(Dated: January 10, 2023) +We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, +utilizing alkaline-earth(-like) atoms in state-dependent optical lattices. The proposal enables the +study of both gauge and fermionic-matter fields without integrating out one of them in one and +two dimensions. +We focus on a realistic and robust implementation that utilizes the long-lived +metastable clock state available in alkaline-earth(-like) atomic species. Starting from an ab initio +modelling of the experimental setting, we systematically carry out a derivation of the target U(1) +gauge theory. This approach allows us to identify and address conceptual and practical challenges +for the implementation of lattice gauge theories that – while pivotal for a successful implementation +– have never been rigorously addressed in the literature: those include the specific engineering of +lattice potentials to achieve the desired structure of Wannier functions, and the subtleties involved +in realizing the proper separation of energy scales to enable gauge-invariant dynamics. We discuss +realistic experiments that can be carried out within such a platform using the fermionic isotope +173Yb, addressing via simulations all key sources of imperfections, and provide concrete parameter +estimates for relevant energy scales in both one- and two-dimensional settings. +I. +INTRODUCTION +In the last decade, the rapid development of quantum +simulators has motivated an increasingly large interest +in possible applications to nuclear and particle physics. +The study of lattice gauge theories (LGTs) [1–3], one of +the most successful theoretical frameworks to regularize +strong interacting field theories, could take great advan- +tage of the use of quantum devices. +First formulated +in the 1970s [1], classical simulations of LGTs based on +Monte Carlo sampling soon became a pillar of our under- +standing of quantum chromodynamics (QCD) [3], with +applications as diverse as low-energy spectra [4, 5], phase +diagrams [6–8], and even precision measurements in the +context of the recently puzzling muon magnetic moment +results [9]. Quantum simulators promise to extend our +understanding of LGTs to regimes that are presently in- +accessible to Monte Carlo methods, including real-time +dynamics, or the physics of the early universe and neu- +tron stars [10–15]. +Starting from early theoretical proposals, the field of +quantum simulation of LGTs has rapidly evolved driven +by two factors: the development of quantum simulation +tools and schemes tailored to the specificity of gauge the- +ories (in particular, gauge invariance), and a series of +∗ fsurace@caltech.edu +† pierre.fromholz@unibas.ch +‡ Current address: JILA, University of Colorado and National In- +stitute of Standards and Technology, and Department of Physics, +University of Colorado, Boulder, Colorado 80309, USA +§ monika.aidelsburger@physik.uni-muenchen.de +first experimental steps that have been taken to demon- +strate the feasibility of the proposed schemes. The for- +mer has been pioneered by the trapped ion experiment +reported in Ref. [16] (later extended in Ref. [17]), where +the dynamics of a few-site Schwinger model [i.e., quan- +tum electrodynamics in (1+1)-dimension (d)] with up to +six sites was demonstrated. Moreover, building blocks of +matter-gauge interactions have been successfully demon- +strated in cold atom settings, including both Z2 [18] and +U(1) [19] gauge theories in 1d, utilizing the quantum link +formulation (QLM) of LGTs, where the dimension of the +local Hilbert space of the gauge link is truncated and +therefore finite. More recently, large-scale quantum sim- +ulations of Abelian LGTs have been reported in Ryd- +berg atom arrays (Schwinger model [20, 21], as well as +a (2+1)-d Ising-Higgs gauge theory [22]) and with ultra- +cold bosonic atoms in tilted optical superlattices [23–25]. +In the continuum, quantum simulation of a topological +gauge theory was realized in an optically-dressed Bose- +Einstein condensate by realising a one-dimensional re- +duction of the Chern-Simons theory, the so-called chiral +BF theory [26]. +The first generation of experimental realizations has +already proven that quantum simulators of LGTs can +reach system sizes and timescales at the boundaries of +the capabilities of classical numerical simulations [20, 21]. +Nevertheless, many challenges still have to be overcome +before we can utilize quantum devices for making accu- +rate predictions on complex gauge theories like QCD. De- +spite the recent experimental progress mentioned above, +there is no direct path at this point towards quantum +simulation of LGTs with fermionic matter in more than +arXiv:2301.03474v1 [cond-mat.quant-gas] 9 Jan 2023 + +2 +one dimension, where both gauge and matter fields are +simulated [27]. +In the pioneering experiments imple- +mented with trapped ions [16, 17] gauge fields are elim- +inated by a Jordan-Wigner transformation, which maps +the original Schwinger model to a spin model with ex- +otic long-range interactions. A related approach is fol- +lowed for implementations of QLMs in Rydberg atom +arrays, where the matter fields are integrated out [21]. +Keeping both matter and gauge fields is more challeng- +ing and most proposals require implementations based +on ultracold mixtures [19], which significantly increases +the experimental complexity. QLMs on the other hand +are characterized by a finite-dimensional Hilbert space +for the gauge degrees of freedom, offering the possibil- +ity of implementing matter and gauge degrees of freedom +with a single atomic species. This has been demonstrated +with ultracold bosons in tilted optical superlattice poten- +tials [23–25] and a possible extension of this scheme to +higher dimensions is currently explored theoretically [28]. +Moreover, schemes based on Floquet engineering appear +challenging due to the presence of higher-order terms +that need to be suppressed in order to respect gauge in- +variance [18], for instance by implementations of addi- +tional stabilizers [29]. In order to overcome these limita- +tions, we have developed a new scheme for the realization +of U(1) QLMs with ultracold alkaline-earth(-like) atoms +(AELA) that offers a direct implementation of fermionic +matter and gauge fields, as well as a straightforward ex- +tension to two dimensions. +To ensure a robust experimental implementation, it +is crucial to further bridge the gap between theoreti- +cal proposals, which focus on conceptual developments +and novel implementation schemes, and experimental re- +alizations, which require microscopic derivations of the +gauge-theory dynamics. The last step is vital in order +to understand at a qualitative and quantitative level the +impact of often neglected challenges or even roadblocks— +including particle losses, limited coherence time (e.g., due +to spontaneous emission), and practical difficulties (such +as challenges in realizing the required optical potentials). +In this work, we introduce a novel scheme and present +an ab initio derivation of a U(1) lattice gauge theory in +both one and two spatial dimensions using AELA in op- +tical lattices. The backbone framework is an implemen- +tation that relies on protecting gauge invariance utilizing +a combination of energy penalty and locality, through +a specific design of state-dependent optical potentials, +that are particularly well suited for atomic species with +a long-lived electronically excited state [30, 31]. We carry +out a thorough numerical study of the experimental pa- +rameters of the optical lattice needed to obtain a regime +for our quantum simulation that features the best ratio +between coherent and incoherent dynamics. +We com- +pute the parameters of the lattice model, and simulate +the corresponding microscopic dynamics, showing that a +moderate amount of on-site disorder would only mildly +affect the observed time evolution. +δe +∆ g +∆ e +δg +g +e +2j+1/2 +2j+3/2 +2j+1 +2j +x/a +... +... +g +e +g atom +detuned +lattice site +empty +lattice site +correlated hopping +e atom +electric field +(negative) +(positive) +charge (+) +(−) +empty site +− +pair creation/annihilation +− ++ ++ +(a) +(c) +(b) +FIG. 1. +Illustration of the mapping between atomic +states in the optical lattice potential and the U(1) +quantum link model considered in our proposal. +(a) Schematic optical lattice potentials for the g (blue line) +and e atoms (orange line) together with the relevant energy +scales δg,e and ∆g,e. +Circles indicate an exemplary initial +state of g (blue circles) and e atoms (orange circles) in the +optical lattice. +The black arrows on the right indicate the +correlated hopping of atoms between adjacent lattice sites. +(b) Simplified schematic of the optical lattice potential in +panel (a) showing the bonds across which hopping is energet- +ically allowed (thick gray lines) and forbidden (dashed gray +lines). +(c) State in the quantum link model corresponding +to the atomic configuration in panels (a) and (b). +In the +mapping, every lattice site that can only be occupied by e +or g atoms is interpreted as a matter site, shown as circles +with the + (−) labels indicating filled sites with charge + +(−). The lattice sites that can be occupied by both e and g +are interpreted as link (l) or gauge-field sites, where triangles +pointing right (left) indicate positive (negative) electric field. +Colors indicate the corresponding internal state of the atoms +[see panels (a) and (b)] in the proposed experimental imple- +mentation and gray indicates an empty site. Note that the +correlated hopping of atoms is mapped to a gauge-invariant +pair creation/annihilation process, as shown on the right. +II. +QUANTUM LINK MODEL +In this work we focus on the realization of a U(1) lattice +gauge theory with fermionic matter, the lattice version +of the Schwinger model, i.e., quantum electrodynamics +in one spatial dimension [32]. Despite its simplicity, the +Schwinger model displays several salient features in com- +mon with more complicated ones, including confinement, +chiral symmetry breaking, and non-trivial real-time dy- +namics [33]. + +3 +In one spatial dimension, the Hamiltonian has the +form [2]: +HLGT = −w +� +j +� +ψ† +jUj,j+1ψj+1 + H.c. +� ++ m +� +j +(−1)jψ† +jψj + g +� +j +(Ej,j+1 + E0)2, +(1) +where ψ† +j, ψj are fermionic creation and annihilation op- +erators on site j of a 1d lattice. The operators Uj,j+1 +and Ej,j+1 are respectively the parallel transporter and +the electric field operators, with commutation relation +[Ei,i+1, Uj,j+1] = δijUj,j+1: these operators represent a +U(1) gauge field on the link connecting the sites j and +j + 1. The nearest-neighbor hopping term, of amplitude +w, is made gauge invariant by the parallel transporter +Uj,j+1. The fermionic mass is staggered, according to the +Kogut-Susskind formulation [34]: on even sites, an occu- +pied fermionic site represents a “positron” with charge ++1, while “electrons”, of charge −1, are represented by +holes on odd sites [Fig. 1(c)]. We can therefore define the +local charge as +qj = ψ† +jψj − 1 − (−1)j +2 +. +(2) +The mass term in Eq. (1) assigns the mass m to both +electrons and positrons. Finally, the term proportional +to g is the energy of the electric field, and E0 represents +a static background electric field. The Hamiltonian in +Eq. (1) has a gauge symmetry generated by the local +operators Gj, defined as +Gj = Ej,j+1 − Ej−1,j − qj. +(3) +The physical states for the LGT are the ones that sat- +isfy the local constraint (Gauss’ law) Gj |Ψ⟩ = 0 for every +site j. +In the following, we will consider the quantum link +formulation of the model [35–38]: in this formulation, all +the gauge fields are represented by a finite d-dimensional +Hilbert space (we choose d = 2), and the operators Ej,j+1 +and Uj,j+1 have the form of Sz and S+ operators respec- +tively. Compared to the usual Wilsonian lattice gauge +theories [1, 2], this formulation is particularly suitable for +quantum simulations, because it exploits discrete quan- +tum degrees of freedom. +Here we focus on the spin-1/2 representation [39]. In +this case, the electric field has the two possible values +Ej,j+1 = ±1/2 [Fig. 1(c)], that have the same energy +for E0 = 0. This choice, with half-integer values of the +electric field, is generally denoted as having a topological +angle θ = π (in contrast to the case of integer electric field +values, having θ = 0) [21]. The topological angle can be +tuned by changing the static background field E0, whose +effect is to split the degeneracy between the two electric +field states [40, 41]. For the spin-1/2 representation it +is useful to define τ = 2gE0. Then, the Hamiltonian (1) +becomes (up to an additive constant) +HQLM = −w +� +j +� +ψ† +jUj,j+1ψj+1 + H.c. +� ++ m +� +j +(−1)jψ† +jψj + τ +� +j +Ej,j+1. +(4) +With this notation, choosing τ ̸= 0 effectively changes +the topological angle to θ ̸= π. We note that the model +above can be exactly mapped to a spin chain via direct +integration of Gauss’ law [21]. +III. +QUANTUM SIMULATION +A. +Optical lattice +In the proposed experimental setup, we consider cold +fermionic atoms in two different electronic states α = +{g, e}, realized by the ground and meta-stable excited +clock states g ≡ 1S0 and e ≡ 3P0 of AELA. The atoms +are considered to be spin polarized in a given nuclear +Zeeman state mF , so that the corresponding Hamiltonian +is given by H = Hnon-int + Hint, with [42] +Hnon-int = +� +α +� +d3rΨ† +α (r) +� +− ℏ2 +2M ∇2 + Vα (r) +� +Ψα (r) , +Hint =g− +eg +� +d3rρe (r) ρg (r) . +(5) +Here Ψα(r) denotes the fermion field operator for atoms +in the internal state |α mF ⟩. The density operators are +defined as ρα (r) = Ψ† +α(r)Ψα(r). +Since the atoms are +polarized in the same nuclear Zeeman state, the inter- +action strength g− +eg = 4πℏ2a− +eg/M (atomic mass M) is +associated with the scattering length a− +eg of the anti- +symmetric electronic state [43, 44]. The term Vα (r) de- +notes a 3d lattice potential Vα(r) = V x +α (x) + V y +α (y) + +V z +α (z), where V x +α (x) is the state-dependent potential de- +picted in Fig. 1(a) and V y +α (y) and V z +α (z) are deep state- +independent optical lattices with amplitude Fg = Fe and +lattice spacing dy = dz that isolate individual 1d chains +and provide strong radial confinement. For simplicity, we +choose equal amplitudes for the transverse lattices along +y and z. The state-dependent lattice along x is defined +as +V x +α (x) = − Aα sin2 � π +2ax + ϕ +� +− Bα sin2 �π +a x +� +− Cα sin2 +�2π +a x + π +2 +� +. +(6) +It has a unit cell of length 2a with three “low”-energy +lattice sites and one “high”-energy site, which suppresses + +4 +30 +20 +10 +Vx +g(x) [kHz] +(a) +0 +1 +2 +3 +4 +5 +x/a +20 +0 +Vx +e(x) [kHz] +(b) +0 +2 +wx +g, s(x +3a) +s = +s = 0 +s = + +0 +2 +wx +e, s(x +2a) +FIG. 2. +Optical lattice potential and Wannier func- +tions. The x-component of the optical lattice potential de- +fined in Eq. (6) for the (a) g and (b) e atoms is plotted in +gray for ϕ = 0. The parameters Aα, Bα, Cα are reported in +Table I. In red, the x-components of the Wannier functions +centered on site x = 3a and x = 2a are shown. The parame- +ter s = {−, 0, +} labels the three different orbitals in the unit +cell. +tunneling to that site as shown schematically in Fig. 1(b). +The triple wells of the g and e lattices are shifted relative +to each other by a distance a. +The optical potential along x can be realized by super- +imposing three different optical lattices. Each of them +is generated from a pair of monochromatic laser beams +at either the magic wavelength λm [45, 46], which cor- +responds to a state-independent potential, or the anti- +magic wavelength λam [47], where the potentials for +atoms in the g and e state are equal in magnitude but +have opposite signs. Moreover, the lattice spacing can +be set by tuning the intersection angle θ between the in- +terfering pair of laser beams according to λ/ [2 sin(θ/2)]. +The two shorter-spacing lattices in Eq. (6) are operated +at the magic wavelength λm (Be = Bg and Ce = Cg) and +intersection angles θC = 180◦ and θB = 60◦. The corre- +sponding lattice spacings are a = λm/2 and 2a, such that +their combination yields a symmetric double-well poten- +tial [48, 49]. The third long-lattice at lattice spacing 4a, +which can be generated at a smaller intersection angle, +is operated at λam with Ag = −Ae generating a triple +well potential that is shifted for g and e atoms as shown +in Fig. 1(a) for ϕ = 0. +Note that the required opti- +cal potentials could also be generated using a hybrid ap- +proach using a combination of optical lattices and tweez- +ers, which have recently been employed for Hubbard-type +physics [50, 51]. +B. +Lattice Hamiltonian +To obtain a lattice Hamiltonian for the model, we as- +sume that only the three lowest Bloch bands are occupied +both for the g and e states, and we express the field oper- +ator Ψα(r) in terms of the Wannier functions wα,s, where +s = {−, 0, +} labels the three Wannier centers in a unit +cell (Fig. 2): +Ψg(r) = +� +j odd +� +wg,+(r − rj)cj+1/2 ++wg,0(r − rj)cj + wg,−(r − rj)cj−1/2 +� +, +(7) +Ψe(r) = +� +j even +� +we,+(r − rj)dj+1/2 ++we,0(r − rj)dj + we,−(r − rj)dj−1/2 +� +, +(8) +where rj = jaˆx, ˆx is the unit vector and cj (dj) is the lat- +tice fermionic annihilation operator of a g atom (e atom) +on lattice site j. +Substituting the expressions for the field operators +in Eq. (5), we obtain the lattice Hamiltonian (see Ap- +pendix A) +Hlatt = Hg + He + HU + HD + Hlr + const, +(9) +where Hg and He denote the terms containing hopping +and chemical potentials of the g and e atoms within a +single triple well respectively +Hg = +� +j odd +� +−tg +� +c† +jcj+1/2 + c† +jcj−1/2 + H.c. +� ++ δgc† +jcj +� +, +(10) +He = +� +j even +� +−te +� +d† +jdj+1/2 + d† +jdj−1/2 + H.c. +� ++ δed† +jdj +� +. +(11) +We assumed, for the moment, that ϕ = 0, so the model +is symmetric under reflections centered on the matter +sites: this implies that the chemical potentials of the sites +s = + and s = − are the same (and can be chosen as a +reference level and set to zero). +The terms HU and HD are obtained from the interact- +ing term in Eq. (5), and read (see Fig. 3) +HU = U +� +j +d† +j+1/2dj+1/2c† +j+1/2cj+1/2, +(12) +HD = Dg +� +j odd +� +d† +j+1/2dj+1/2c† +jcj+1/2 +(13) ++d† +j−1/2dj−1/2c† +jcj−1/2 + H.c. +� ++ De +� +j even +� +c† +j+1/2cj+1/2d† +jdj+1/2 ++c† +j−1/2cj−1/2d† +jdj−1/2 + H.c. +� +. +Finally, Hlr contains all the additional terms, of the form +of longer-range hoppings and interactions, that have very + +5 +g +e +onsite +interaction U +−te +−te+ De +(a) +(b) +(c) +FIG. 3. +Illustration of the interacting terms in HU +and HD. (a) On-site interaction U between a single g (blue +circle) and e atom (orange circle). (b) Hopping of a single e +atom to an empty lattice site. (c) In the presence of interac- +tions, tunneling is additionally modified by a density-assisted +tunneling with amplitude De. +small amplitudes and can be neglected (we explicitly ver- +ify that these terms are negligible for the parameters re- +ported in Section IV A). +It is useful to define ϵ = (δg − δe)/2, δ = (δg + δe)/2, +and the total number of atoms on each site j+1/2, which +corresponds to a link in the QLM (Fig. 1), +n(l) +j+1/2 = d† +j+1/2dj+1/2 + c† +j+1/2cj+1/2. +(14) +Here (l) is a redundant superscript to indicate we are on a +link. We now assume ϵ, tα, Dα ≪ δ, U − δ: in this regime +it is convenient to split the Hamiltonian into three parts +Hlatt = H0 + H1 + Hlr with different energy scales, i.e., +H0 = (Ng+Ne−Nl)δ+ +� +j +� +−δ + U +2 n(l) +j+1/2 +� +(n(l) +j+1/2−1), +(15) +where Ng and Ne are the total numbers of atoms in the +g and e states respectively and Nl is the total number +of links. From Eq. (15) it is immediate to see that for +δ, U − δ > 0 the lowest energy states of H0 have exactly +one atom (either g or e) on each half-integer site, i.e., +n(l) +j+1/2 = 1 for every j: a double occupancy n(l) +j+1/2 = 2 +costs energy U − δ, while having a hole n(l) +j+1/2 = 0 costs +energy δ. The term H1 has the form +H1 = +� +j odd +� +−tg +� +c† +jcj+1/2 + c† +jcj−1/2 + H.c. +� ++ ϵ c† +jcj +� ++ +� +j even +� +−te +� +d† +jdj+1/2 + d† +jdj−1/2 + H.c. +� +− ϵ d† +jdj +� ++ HD, +(16) +where HD is the Hamiltonian in Eq. (13). +We initialize the system with two g atoms for every g +triple well and one e atom for every e triple well (all these +quantities are locally conserved, if we neglect Hlr). The +effective Hamiltonian describing the resonant dynamics is +obtained using perturbation theory: we neglect Hlr, and +we treat H1 as a perturbation to H0. To second order, +the effective Hamiltonian has the form (see Appendix B) +H(2) +eff = −w +� +j odd +(c† +jcj+1/2d† +j+1/2dj+1 + H.c) +− w +� +j even +(d† +jdj+1/2c† +j+1/2cj+1 + H.c) ++ m +� +j odd +c† +jcj − m +� +j even +d† +jdj, +(17) +with +w = +tgteU +δ(δ − U) + −Detg − Dgte + DeDg +δ − U +, +(18) +m = ϵ + 2t2 +g − t2 +e +2δ +− (tg − Dg)2 − 2(te − De)2 +2(U − δ) +. +(19) +C. +Mapping to the quantum link model +We now prove that there is an exact mapping between +the effective Hamiltonian H(2) +eff +and the quantum link +Hamiltonian in Eq. (4). +The fermionic operator ψj for the matter is defined as +ψj = +� +cj +j odd, +dj +j even, +(20) +(and an analogous definition is used for ψ† +j). The electric +field Ej,j+1 and the parallel transporter Uj,j+1 on the +link are represented by +Ej,j+1 = (−1)j +2 +(c† +j+1/2cj+1/2 − d† +j+1/2dj+1/2), +(21) +Uj,j+1 = +� +cj+1/2d† +j+1/2 +j odd, +dj+1/2c† +j+1/2 +j even, +(22) +and +satisfy +the +desired +commutation +relation +[Ei,i+1, Uj,j+1] += +δi,jUj,j+1. +With the definitions +in Eqs. (20) and (21), the operator Gj takes the form +Gj = 1 +2(n(l) +j+1/2 + n(l) +j−1/2) − n(b) +j ++ (1 − (−1)j) +2 +, +(23) +with n(b) +j +being the number of atoms in the j-th triple- +well (or “block”), i.e., +n(b) +j += +� +c† +jcj + c† +j+1/2cj+1/2 + c† +j−1/2cj−1/2 j odd, +d† +jdj + d† +j+1/2dj+1/2 + d† +j−1/2dj−1/2 j even. +(24) +With this mapping, which is schematically shown in +Fig. 1, we obtain that H(2) +eff is equivalent to the Hamil- +tonian HQLM with τ = 0. +The gauge-invariant sub- +space corresponds to the sector with nj+1/2 = 1 and + +6 +n(b) +j += [3−(−1)j]/2 for every j. Some examples of gauge- +invariant states are shown in Fig. 4. +In Fig. 4(a), all g atoms sit on the links, and all e +atoms are on the matter sites: the corresponding electric +field takes values Ej,j+1 = (−1)j/2, while ψ† +jψj = 1, 0 +for even and odd sites respectively, leading to alternating +positive and negative charges on matter sites. This state +is the ground state of the model in the limit m → −∞. +Similarly, it is easy to show that the states represented in +Fig. 4(b) and 4(c) have no charges on matter sites, and +have uniform (negative or positive) electric field. These +states (vacua) are degenerate ground states in the limit +m ≫ |w| with τ = 0, while the degeneracy is split for +τ ̸= 0. +D. +Theta term +We now show how to tune the parameters of the op- +tical lattice to obtain τ ̸= 0. The lattice Hamiltonian +Eq. (9) was derived with the assumption that ϕ = 0. We +now slightly perturb this model, by introducing a small +shift ϕ ≪ 1. To first order in ϕ, the shift produces an +additional potential along x +V x +α → V x +α − Aα sin +�π +a x +� +ϕ. +(25) +The main effect of this additional term is to change the +chemical potential at half-integer positions x = (j+1/2)a +by a quantity −Aαϕ(−1)j. We obtain +Hlatt → Hlatt − +� +j +(−1)jϕ(Agc† +j+1/2cj+1/2 ++ Aed† +j+1/2dj+1/2) += Hlatt − +� +j +(−1)jϕ +�Ag + Ae +2 +n(l) +j+1/2 ++Ag − Ae +2 +(c† +j+1/2cj+1/2 − d† +j+1/2dj+1/2) +� +. +(26) +The term � +j(−1)jn(l) +j+1/2 cancels in the resonant sector, +and the remaining term is mapped to � +j τEj,j+1, with +τ = (Ae − Ag)ϕ. +(27) +IV. +EXPERIMENTAL IMPLEMENTATION +The ab-initio calculation of the band structure and +Wannier functions allows us to estimate the energy scales +involved in the quantum simulation. +These estima- +tions explicitly verify that the desired parameter range is +achievable in present-day experiments. The quantitative +estimation of the parameters is also crucial to understand +the limitations of our proposal such as the amplitude and +duration of the signal and to identify the main sources of ++ +− ++ +(a) +(b) +(c) +vacuum with +negative electric field +neighboring pairs of ++ and − charges +vacuum with +positive electric field +e atom +detuned site +empty site +g atom +FIG. 4. +Examples of the mapping between atomic +configurations +in +the +optical +lattice +and +gauge- +invariant states in the U(1) QLM. In each panel, the +atomic states in the optical lattice are shown in the top row +and the states in the QLM are shown in the bottom row. +The panels show (a) the state with neighboring pairs of + +and − charges, (b) the vacuum state with homogeneous neg- +ative electric field E = −1/2, and (c) the vacuum state with +homogeneous positive electric field E = +1/2. +error such as the population of higher bands, higher-order +perturbative processes, longer-range terms, and dissipa- +tion. +To set values, we choose the fermionic isotope 173Yb +with mass M ≈ 173u and the interorbital scattering +length a− +eg = 219.7 a0 [52]; here u denotes the atomic +mass unit and a0 the Bohr radius. However, we note that +our proposal can be similarly applied to other fermionic +AELA species such as 171Yb and 87Sr. The experimental +parameters require scaling to account for the modified +atomic mass and scattering length [53, 54] with no con- +ceptual change in the design of the experiment. +A. +Realistic parameters +We define ∆g/e as the gap between the third and fourth +energy band in the lattice for the g/e atoms (see Section +A). The parameters used here are chosen to satisfy the +hierarchy of energy scales +∆g/e ≫ δg/e, U − δg/e ≫ tg/e, Dg/e ≫ terms in Hlr. +(28) +We note that for the 3d lattice potential Vα(r) +the Wannier functions obtained by solving the non- +interacting Hamiltonian Hnon-int can be factorized in the +three directions. The hoppings tα, the chemical poten- +tials δα, and the gaps ∆α do not depend on the y and +z components φy +α(y) and φz +α(z) of the Wannier functions +(see Appendix A). The interactions U and Dα, on the +other hand, are proportional to the quantity +Jyz = +� +dy dz |φy +g(y)|2|φy +e(y)|2|φz +g(z)|2|φz +e(z)|2. +(29) +We can therefore tune Fα and dy/z to change the value of +Jyz and thus enhance or suppress the interaction terms +U and Dg/e independently from the other parameters. +In Table I, we report a possible choice for the parame- +ters of the optical lattice, and the corresponding param- + +7 +TABLE I. Experimental parameters. All values are given +in units of h·kHz. +Ag = −Ae +Bg = Be +Cg = Ce +9.827 +6.343 +15.832 +∆g = ∆e +δg = δe +U +tg = te +Dg = De +7.22 +1.02 +2.03 +0.085 +0.023 +eters of the lattice Hamiltonian. We choose a = λm = +0.7594 µm [55] and Jyz = 48.566 µm−2. +With these parameters, from Eqs. (18) and (19) we +obtain m/h = 9 Hz and w/h = −18 Hz; where h denotes +Planck’s constant. The value of Jyz reported here can be +obtained with a transverse confinement Fg/h = Fe/h = +48.9 kHz and dy/z = λm/2. For this choice of transverse +potential, the hopping in the y and z direction is ty/z +α /h = +2 Hz, much smaller than the relevant scales m and w. +The transverse hopping can be made even smaller by +using two beams intersecting at a shallow angle instead +of using retro-reflected beams: the lattice spacing dy/z +is increased, and the same value of Jyz is obtained for a +larger Fα, thus suppressing the transverse hopping. +We explicitly check that all the terms included in Hlr, +which were neglected in the derivation of HQLM, are small +with respect to w. The nearest-neighbor density-density +interaction is of the order of ∼ 0.6 h·Hz, while the hop- +ping between the sites s = + and s = − in a triple well +is ∼ 2 h·Hz, and is negligible because it is suppressed by +the on-site interaction U. +The parameters in Table I can be readily generalized +to other atomic species: we can define the adimensional +lattice constant ˜a = a/λm and the adimensional param- +eter ˜Aα = Aα2M/ℏ2λ2 +m. Similarly we can define dimen- +sionless parameters for the other energy scales Bα, Cα, +∆α, δα, U, tα, Dα, m, w, and for the quantity Jyz. Im- +plementations with different atomic species but same adi- +mensional parameters of the optical lattice yield the same +adimensional values for the terms in the lattice Hamilto- +nian. +B. +Initial state preparation +We propose two distinct ways of a two-step prepa- +ration of the system in gauge-invariant initial states as +shown in Fig. 4, which require specific atom configura- +tions. First, the correct atom-number distribution needs +to be prepared (independent of the internal state). One +option is to prepare it starting from a sample with one +g atom per lattice site followed by the removal of atoms +on selected lattice sites yielding the desired occupation. +This is a standard technique in quantum gas micro- +scopes [56, 57]. Alternatively, the atoms could directly +be placed at their desired location with moving optical +tweezer potentials [51]. In the second step, g atoms can +be converted to e atoms on selected lattice sites using +a global clock laser excitation pulse exploiting the local +differential light shifts δα and ∆α [Fig. 1(a)]. The conver- +sion could also be performed locally by using clock laser +light focused onto single lattice sites. +V. +REAL-TIME DYNAMICS +Using numerical simulations, we study here the real- +time dynamics of the model of Hamiltonian (9) that we +compare with the dynamics of the quantum link model +in Eq. (4). In both cases we start with the initial state +shown in Fig. 4(a). The time evolution of the model is +simulated exactly for a system of length 4a with periodic +boundary conditions, with the parameters reported in +Section IV A. Longer-range terms from Hlr in Eq. (9) are +included in the numerical simulation: the dynamics is +exact as long as higher bands are not occupied. +A. +Gauge invariance and gauge field +The system is effectively gauge invariant as long as +Gauss’ law applies. To quantify the violation of Gauss’ +law in Eq. (9), we examine the time evolution of n(l) +j+1/2, +and n(b) +j . The simulation results are shown in Fig. 5(a)- +(c). +We find that the conditions n(b) +j += [3 + (−1)j]/2 +and n(l) +j+1/2 = 1 are preserved up to 5 · 10−4 and 10−1 +respectively within the first 70 ms after initialization, +which should be compared to the characteristic interac- +tion timescale ℏ/|w| ≃ 8.8 ms. +In Fig. 5(d) we examine the evolution of the electric +field with (i) the Hamiltonian Hlatt, (ii) the second-order +effective Hamiltonian H(2) +eff , equivalent to HQLM, and (iii) +the fourth-order effective Hamiltonian H(4) +eff +(from the +Schrieffer Wolff procedure). The differences between the +electric field value obtained in case (i) and with the ap- +proximate Hamiltonians are plotted in Fig. 5(e). +For +times up to ∼ 40 ms, the results obtained in the three +cases are in good agreement: this shows that the second- +order effective Hamiltonian HQLM captures the main fea- +tures of the time evolution, at least at short and inter- +mediate time scales, and that longer-range terms and +higher-order perturbative processes are minor sources of +error. For times of the order of ≳ 50 ms, we find that +fourth-order corrections have to be considered in order +to obtain a good prediction of the time evolution in the +experiment. We remark that the fourth-order corrections +do not violate Gauss’ law, but correspond to additional +gauge-invariant terms. A feature of the evolution induced +by Hlatt that is not observed in the effective Hamiltonians +is the presence of fast oscillations with small amplitude. +These oscillations have a frequency compatible with the + +8 +FIG. 5. +Dynamics in the U(1) quantum link model. +Time evolution of (a) the number of atoms per link, (b,c) the +number of atoms on each odd/even block. For each observ- +able O, the shaded area indicates the interval ⟨O⟩± +� +Var(O). +The initial state is depicted in Fig. 4(a) and is evolved under +Hlatt. The system has periodic boundary conditions and fi- +nite size 4a. The top-left schematic in each panel illustrates +the observable. +(d) Time evolution of the electric field on +an odd-even link. +The exact dynamics given by Hlatt are +compared with the second-order effective Hamiltonian [using +Eqs. (18,19)] and with the fourth-order effective Hamiltonian +H(4) +eff . The latter two are both equivalent to HQLM. (e) Dif- +ference ∆Eodd,even = ⟨Eodd,even⟩Hlatt − ⟨Eodd,even⟩H (the sub- +script denotes the Hamiltonian that generates the time evo- +lution) for the two cases H = HQLM and H = H(4) +eff . +energy scale of H0 and are averaged out in the perturba- +tive approach. +To make this separation of energy scales more evi- +dent, we examine the Fourier transform of the signal +E(ω) = +� tmax +0 +E(t)e−iωtdt in Fig. 6. +As expected, the +evolution under Hlatt shows peaks at frequency ω ∼ +δ/ℏ = (U − δ)/ℏ = 6.4 kHz [Fig. 6(a)], that are not +observed for the effective Hamiltonians. Zooming in at +smaller frequencies [Fig. 6(b)], we see that the agree- +ment in the Fourier transforms is good between Hlatt +and HQLM and is excellent between Hlatt and H(4) +eff . +0 +2 +4 +6 +8 + [kHz] +10 +1 +101 +|E( +)| +Hlatt +HQLM +H(4) +eff +0.0 +0.2 +0.4 +0.6 +0.8 + [kHz] +10 +1 +100 +101 +|E( +)| +Hlatt +HQLM +H(4) +eff +(a) +(b) +FIG. 6. +Fourier transform of the time evolution of +the electric field. (a) Spectrum corresponding to the time +evolution shown in Fig. 5(d). Panel (b) shows a zoom-in of +panel (a) revealing details at small frequencies. +B. +Dissipation +Finite dissipation can become a crucial issue in the ex- +periment when it occurs on time scales comparable to the +relevant dynamic evolution of the system. Therefore, we +focus on identifying the fastest dissipation process. This +allows us to estimate for how long the experimental sys- +tem closely follows the coherent dynamics of our model. +For the AELA in an optical lattice considered here, the +typically dominant dissipation channels are lossy colli- +sions between pairs of atoms and off-resonant scattering +of optical lattice photons. +In the following, we evalu- +ate the relevance of both for our proposed experimental +implementation. +Lossy collisions between pairs of atoms with one or +both of the two atoms in the e state can lead to a particu- +larly fast atom loss [42, 43]. For our proposed implemen- +tation, however, double occupancies of lattice sites are +(purposefully) strongly suppressed, either by fermionic +quantum statistics (ee-pairs) or a large on-site interac- +tion energy (eg-pairs). As a consequence, lossy collisions +between pairs are not expected to be a limiting factor. +In contrast, off-resonant photon scattering from lattice +photons was identified as the dominant limiting factor in +our proposed implementation. While off-resonant photon +scattering eventually leads to heating and atom loss, an- +other effect could become relevant at earlier times. When +an atom in the e state scatters an optical lattice photon, +short-lived intermediate states can be populated and de- +cay back to the g state with a finite probability. Conver- +sion of e atoms to g atoms due to this optical pumping +process has already been observed and characterized in +optical lattice experiments with AELA [58, 59]. Here, we +employ these results to estimate the expected time scale + +9 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 + [kHz] +10 +1 +100 +101 +|E( +)| +0 +0.005 +0.01 +0.02 +FIG. 7. +Influence of disorder on the dynamics of the +electric field. Fourier transform of the time evolution of the +electric field under HW for different values of W. The legend +indicates the values of W/h in kHz. The results are averaged +over 100 disorder realizations. +for our parameters (see Table I). Focusing on the pro- +posed implementation in Section III A for 173Yb and the +off-resonant scattering of magic-wavelength lattice light, +we estimate a repumping rate Γ ≈ 111 mHz. Comparison +of this estimate to the quantity |w|/ℏ = 113 Hz suggests +that the dynamics of our model can be faithfully observed +for many characteristic time scales. +C. +Disorder +An experimental implementation could also exhibit fi- +nite disorder. Based on previous experimental work [60], +disorder occurs in particular when utilizing hybrid po- +tentials generated with both optical lattices and optical +tweezers. To estimate how much disorder affects the dy- +namic evolution of the system, we consider a quenched +disorder in the chemical potentials of the atoms of the +form +HW =Hlatt + +� +j odd +� +Wg,j+1/2c† +j+1/2cj+1/2 +(30) ++Wg,j−1/2c† +j−1/2cj−1/2 + Wj c† +jcj +� ++ +� +j even +� +We,j+1/2d† +j+1/2dj+1/2 ++We,j−1/2d† +j−1/2dj−1/2 + Wj d† +jdj +� +, +where Wα,j+1/2, Wj are taken randomly from a uniform +distribution in the interval [0, W). +In Fig. 7 we plot the Fourier transform E(ω) averaged +over 100 disorder realizations for different values of W. +We find that disorder has the effect of smearing out the +peaks, which nevertheless remain visible up to W/h ∼ +0.01 kHz. +VI. +THE 2D MODEL +We now generalize the implementation examined in the +previous sections to the quantum link model in two spa- +tial dimensions, and we show how this can be simulated +with realistic experimental setups. +A. +Quantum link model +The two-dimensional quantum link model is described +by the Hamiltonian [61] +HQLM = − w +� +r +� +k=ˆx,ˆy +� +ψ† +rUr,r+kψr+k + H.c. +� ++ m +� +r +srψ† +rψr + τ +� +r +� +k=ˆx,ˆy +Er,r+k, +(31) +where the sums run over the points r = (i, j) of a +two-dimensional square lattice (i, j are integers), and +sr = (−1)i+j; here ˆx and ˆy denote unit vectors along the +respective direction. The gauge fields sit on the links and, +similarly to the one-dimensional case, are represented by +spin variables with finite d-dimensional Hilbert spaces +(here, d = 2). The generators of the gauge symmetry +read +Gr = +� +k=ˆx,ˆy +(Er,r+k − Er,r−k) − ψ† +rψr + 1 − sr +2 +. +(32) +A state |Ψ⟩ is gauge-invariant if it satisfies Gauss’ law +Gr |Ψ⟩ = 0 for every lattice site r. +An example of a +gauge-invariant state is shown in Fig. 8(d). +B. +Quantum simulation +Our desired optical lattice in two dimensions consists +of cross-shaped “blocks” of g and e sites [see Fig. 8(a-c)]. +While in the one-dimensional case each block consisted +of a triple well, here a block contains five sites: a central +matter site at r = (x/a, y/a) = (i, j), with i, j integers, +and four gauge sites around it at positions r ± (1/2, 0) +and r ± (0, 1/2). Blocks of g and e sites alternate in a +checkerboard pattern, as shown in Fig. 8(c), with over- +lapping g and e gauge sites. A lattice of this type can be +realized with the potential +V x,y +α +(x, y) = +− Aα sin2 � π +2a(x + y) + ϕ +� +− Aα sin2 � π +2a(x − y) +� +− Bα sin2 �π +a (x + y) +� +− Bα sin2 �π +a (x − y) +� +− Cα sin2 +�2π +a x + π +2 +� +− Cα sin2 +�2π +a y + π +2 +� +. +(33) +Fig. 8(a,b) depicts the profiles of V x,y +g +and V x,y +e +for the +values of Ag, Ae, Bg, Be, Cg, Ce reported in Table II. + +10 +(a) +(b) +(c) +(d) ++ +− +electric field +(negative) +electric field +(positve) +charge (−) +charge (+) +e atom +g atom +detuned +lattice site +hopping +bond +empty +lattice site +0 +2 +4 +x/a +0 +1 +2 +3 +4 +y/a +165 +150 +135 +120 +105 +90 +75 +60 +45 +0 +2 +4 +x/a +0 +1 +2 +3 +4 +75 +60 +45 +30 +15 +0 +15 +30 +45 +60 +FIG. 8. +Implementation of the quantum link model +in two dimensions. Optical lattice for the (a) g and (b) +e atoms. (c) Example of a gauge-invariant state belonging +to the resonant subspace. +Blue and orange circles repre- +sent g and e atoms, respectively. (d) Corresponding gauge- +invariant state in the QLM. Dark/light gray circles indicate +the occupied/empty matter sites (with charge +, −, or no +charge). Red (blue) arrows represent a link with electric field +Er,r+k = +1/2 (Er,r+k = −1/2). +The steps for deriving the lattice Hamiltonian and +mapping it to the QLM are analogous to the one- +dimensional case. +We report here the mapping of the +operators: +Er,r+k = sr +2 (c† +r+k/2cr+k/2 − d† +r+k/2dr+k/2), +(34) +ψr = +� +cr if sr = −1, +dr if sr = +1, +(35) +Ur,r+k = +� +cr+k/2d† +r+k/2 +if sr = +1, +dr+k/2c† +r+k/2 +if sr = −1. +(36) +The system is initialized with three g atoms on each odd +block, and two e atoms on each even block. To illustrate +the mapping, we show in Fig. 8(c) and (d) an example +of a resonant state in the local occupation basis and the +corresponding gauge-invariant state in the quantum link +model. +Using the same derivation as the one-dimensional case, +we obtain the effective Hamiltonian Eq. (31). In Table II +we report the parameters obtained for the lattice poten- +tial depicted in Fig. 8(a,b). We set a = λm = 0.7594 µm +and Jz ≡ +� +dz|φz +g(z)|2|φz +e(z)|2 = 19.235 µm−1. +For +this choice of parameters we obtain m = 10 h·Hz and +w = 20 h·Hz for the QLM Hamiltonian Eq. (31). +VII. +CONCLUSIONS +We have presented a proposal for the scalable quan- +tum simulation of lattice gauge theories coupling (stag- +gered) fermions to U(1) gauge fields utilizing a mixture +of alkaline-earth(-like) atoms in both a ground and a +metastable state in optical potentials. The key element +of our proposal is a careful treatment of the full system +dynamics, that are derived ab initio from microscopic +interactions between atoms and light, and atoms them- +selves. While the proposal can be applied to a variety +of atomic species, we have drawn a complete blueprint +utilizing concrete estimates based on 173Yb atoms. +Our treatment highlights concrete challenges in the +quantum simulation of lattice gauge theories that have +so far mostly been overlooked. In particular, it makes +clear that the superposition of lattice potentials required +for such simulations, while certainly realistic experiment- +wise, gives rise to complicated band structures that must +be quantitatively understood to access the reliability and +feasibility of any quantum simulation. The reason for this +is twofold: band separation can become much smaller +than what is naively expected, making protection of +gauge invariance very challenging; in parallel, intrinsic +energy scales of desired processes can be considerably re- +duced with respect to simplistic deep lattice estimates +based on highly localized Wannier functions. These lim- +itations are particularly pernicious for single-body terms +in the lattice potential, whose estimate crucially requires +a quantitative approach as the one carried out here. +Within the context of our proposal, we have shown that +optimal parameter regimes can still be found for observ- +ing the correct and expected LGT dynamics. This can be +achieved thanks to the detailed microscopic understand- +ing our treatment leads to. We have demonstrated this +conclusion by comparing numerical simulations of both +ideal and effective dynamics of string relaxation, includ- +ing also effects of inhomogeneities. +Based on our findings, we believe that the ab initio +approach we propose will be, in the long term, the one +needed to fully determine the capabilities of quantum +simulators of lattice gauge theories. We have taken the +first step beyond Abelian 1D models, by extending them +to 2D geometries. +Future works will be fundamental +TABLE II. Experimental +parameters +for +the +two- +dimensional quantum link model. All values are given +in units of h·kHz. +Ag = −Ae +Bg = Be +Cg = Ce +50.029 +25.915 +27.516 +∆ +δg = δe +U +tg = te +Dg = De +9.63 +1.08 +2.16 +0.087 +0.033 + +11 +to address the experimental capabilities to realize non- +Abelian lattice gauge theories, which to date have been +proposed only in very few settings [62–70]. +ACKNOWLEDGMENTS +We thank M. Burrello, G. Pagano and E. Rico for in- +sightful discussions, and F. Scazza for collaboration on +a related work. +The work of M. D., P. F. and F. S. +was partly supported by the ERC under grant num- +ber 758329 (AGEnTh), and by the MIUR Programme +FARE (MEPH). M. A. and N. D. 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We diagonalize the +single-particle Hamiltonian hα(r) +hα(r) = − ℏ2 +2M ∇2 + Vα (r) . +(A1) +with α = g, e. For the quantum simulation of the one- +dimensional model discussed in Section III, the potential +is +Vα(r) = V x +α (x) + V y +α (y) + V z +α (z). +(A2) +We then look for a factorized and localized three- +dimensional complete basis of wavefunctions for all states +involved in the dynamic evolution of the system. The po- +tential is periodic in space for each component x, y, z such +that the Bloch theorem applies to Eq. (A1). In Fig. 9, +we plot the potential V x +g +(with the parameters of Ta- +ble I) and the lowest bands obtained by solving the cor- +responding periodic Hamiltonian. Using a unitary trans- +formation (similar to the reverse Fourier transformation) +of the Bloch eigenfunctions, we obtain a set of localized +orthonormal wavefunctions wα,s called the Wannier func- +tions. Because of the form of the potential Eq. (A2), the +Bloch functions are factorizable. Likewise, the Wannier +functions factorize along x, y, z: +wα,s(r − rj) = φx +α,s(x − ja)φy +α(y)φz +α(z), +(A3) +where w is the three-dimensional Wannier function, and +the φi are the one dimensional Wannier functions for each +direction i = x, y, z. +α = g, e is the electronic state, +s = {0, +, −} denotes the three functions corresponding +to the three sites of a triple well, and rj = jaˆx is the +Wannier center. As we target a one dimensional system, +we consider only Wannier functions in the transverse di- +rection y, z centered around y = z = 0 by convention. +They involve only the lowest Bloch band. The x com- +ponent is instead obtained from the three lowest bands +such that we have three Wannier centers per unit cell. +To derive an expression for the localized Wannier func- +tions useful to estimate the overlap, we compute the +eigenstates of the projection of the position operator onto +a given set of Bloch states [71, 72]. In one dimension, +this method always gives the maximally localized Wan- +nier functions [73, 74]. In many other cases, this deriva- +tion still gives a good approximation of the maximally +localized Wannier functions. +We can then define discrete operators for the discrete +Hamiltonian. We expand the fermionic operators in the +basis of the Wannier functions +Ψg(r) = +� +j odd +� +wg,+(r − rj)cj+1/2 ++wg,0(r − rj)cj + wg,−(r − rj)cj−1/2 +� +, +(A4) +Ψe(r) = +� +j even +� +we,+(r − rj)dj+1/2 ++we,0(r − rj)dj + we,−(r − rj)dj−1/2 +� +. +(A5) +We stress that the only approximations performed so far +are (i) neglecting the higher bands and (ii) only consid- +ering the chain localized at y, z = 0. (i) is justified when +the gaps ∆g, ∆e between the third and the closest higher +bands (see Fig. 9) are much larger than the energy scales + +14 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +x/a +30 +25 +20 +15 +10 +5 +Vx +g(x)/h [kHz] +/2 +0 +/2 +k +30 +25 +20 +15 +10 +5 +/h [kHz] +/2 +0 +/2 +21.835 +21.830 +21.825 +/h [kHz] +FIG. 9. Lattice potential and band structure along x. Left panel: x component of the optical lattice potential V x +g (x). +The defining parameters are reported in Table I. Right panel: corresponding band structure using the same scale as the panel +on the left. The two lowest bands are almost degenerate (they correspond to symmetric and antisymmetric superpositions of +the s = {+, −} sites in the triple well), and are separated from the third one (corresponding to the central site 0 of the triple +well) by an energy ∼ δg. Higher bands are separated from the first three by an energy gap ∆g. +of the dynamics that we are interested in. (ii) is justi- +fied if the transverse hoppings ty +α, tz +α are small compared +to the energy scales of our interest. For the gaps and +the transverse hoppings of Section IV A, both approxi- +mations are appropriate. +The discrete parameters emerge when substituting +Eqs. (A4) and (A5) in the Hamiltonian H = Hnon-int + +Hint in Eq. (5). +These parameters include the chem- +ical potentials, the hoppings (both from Hnon-int), the +on-site and off-site density-density interactions, the den- +sity mediated hoppings, and the correlated hoppings be- +tween g and e atoms. We included all these terms in the +numerical simulations of the real-time dynamics in Sec- +tion V. For clarity, we give the lattice Hamiltonian with +the terms of highest amplitude. We define the chemical +potentials such that +µα,s = +� +d3r w∗ +α,s(r − rα)hα(r)wα,s(r − rα) += µx +α,s + µy +α + µz +α +(A6) +where α = g, e, s = {0, +, −}, and the Wannier centers +are rg = aˆx, re = 0. We further define the difference +between the chemical potentials in each triple well +δα,± = µα,0 − µα,± = µx +α,0 − µx +α,±, +(A7) +and the nearest-neighbour hoppings within a triple well +tα,± = − +� +d3r w∗ +α,0(r − rα)hα(r)wα,±(r − rα) += − +� +dx [φx +α,0(x − ja)]∗hx +α(x)φx +α,±(x − ja). +(A8) +For ϕ = 0 the triple well is designed to be symmetrical, +such that δα,+ = δα,− = δα and tα,+ = tα,− = tα. These +two terms result in the Hamiltonian terms Hg and He in +Eq. (9). +The term with largest amplitude obtained from Hint is +the on-site interaction on the sites where g and e triple +wells overlap. This amplitude is given by +U = g− +eg +� +d3r |wg,−(r − aˆx)|2|we,+(r)|2 += g− +egJyz +� +dx |φx +g,−(x − ja)|2|φx +e,+(x)|2, +(A9) +with +Jyz = +� +dy dz |φy +g(y)|2|φy +e(y)|2|φz +g(z)|2|φz +e(z)|2, +(A10) +and yields the Hamiltonian term HU in Eq. (9). +The +terms with the next largest amplitude with our choice +of parameters are density-assisted hoppings. Specifically, +where a g or e atom hops between two sites of a triple +well, provided that an atom of the opposite electronic +state sits in either the initial or the final site (see Fig. 3). +The amplitude has the form +Dg = g− +eg +� +d3r w∗ +g,0(r − aˆx)wg,−(r − a)|we,+(r)|2 += g− +egJyz +� +dx [φx +g,0(x − ja)]∗φx +g,−(x − ja)|φx +e,+(x)|2, +(A11) +De = g− +eg +� +d3r |wg,−(r − aˆx)|2w∗ +e,0(r)we,+(r), += g− +egJyz +� +dx |φx +g,−(x − ja)|2[φx +e,0(x)]∗φx +e,+(x), +(A12) +and results in the Hamiltonian HD in Eq. (9). For our +choice of parameters, all the other terms (that we gener- +ically include in Hlr) have sufficiently small amplitudes +to be negligible according to Sec. V A. +2. +Two-dimensional case +The steps of the derivation of the lattice Hamiltonian +in the two-dimensional system are very similar to the + +15 +2.5 +0.0 +2.5 +kx +2 +0 +2 +ky +n = 0 +2.5 +0.0 +2.5 +kx +2 +0 +2 +n = 1 +41.00335455 +41.00335450 +41.00335445 +2.5 +0.0 +2.5 +2 +0 +2 +ky +n = 2 +2.5 +0.0 +2.5 +2 +0 +2 +n = 3 +2.5 +0.0 +2.5 +kx +2 +0 +2 +ky +n = 4 +2.5 +0.0 +2.5 +kx +2 +0 +2 +n = 5 +40.98013355 +40.98013350 +2.5 +0.0 +2.5 +kx +2 +0 +2 +ky +n = 6 +2.5 +0.0 +2.5 +kx +2 +0 +2 +n = 7 +40.97734702 +40.97734700 +40.97734698 +40.97734696 +40.97734694 +2.5 +0.0 +2.5 +kx +2 +0 +2 +ky +n = 8 +2.5 +0.0 +2.5 +kx +2 +0 +2 +n = 9 +39.868653180 +39.868653178 +39.868653176 +FIG. 10. Bandstructure for the 2d lattice. The 10 lowest +bands corresponding the two-dimensional lattice V x,y +e +(x, y) in +Fig. 8. +one-dimensional case. +The main difference lies in the +potential that is now +Vα(r) = V x,y +α +(x, y) + V z +α (z). +(A13) +As a consequence, only the z component of the Bloch +(and Wannier) functions can be factorized out, while for +the x − y plane we have to solve a two-dimensional sin- +gle particle Hamiltonian. The first 10 two-dimensional +Bloch bands for the parameters in Table II are plotted in +Fig. 10. +The unit cell we consider is defined by the lattice vec- +2.5 +3.0 +x/a +2.2 +2.4 +2.6 +2.8 +3.0 +3.2 +3.4 +y/a +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +FIG. 11. Wannier functions for the 2d lattice. Two- +dimensional Wannier functions of two g and two e blocks. In +orange/light blue, we plot the 4 Wannier functions localized +on the links of each block for the e/g state. In red/dark blue, +we plot the Wannier function localized on the center of each +block for the e/g state. +tors (2a, 0) and (0, 2a) and contains two blocks for each +electronic state. Because each block contains five sites, +we need 10 Wannier functions per unit cell for each state. +To find the Wannier functions we first find the eigen- +states of the projection of the x position operator on +the 10 lowest bands: we collect the groups of eigenstates +with (almost) degenerate eigenvalue and diagonalize the +y position operator projected on each group. The Wan- +nier functions obtained with this procedure are plotted +in Fig. 11. The coefficients of the lattice Hamiltonian are +then obtained as in the one-dimensional case. +Appendix B: Perturbative theory of the lattice +Hamiltonian +The Hamiltonian Eq. (5) describing the cold atoms +in the optical lattice and its lattice formulation Eq. (9) +can be mapped to the QLM Hamiltonian Eq. (4), when +considering the coupling between the targeted gauge- +invariant Hilbert subspace and the rest of the Hilbert +space as a perturbation. +Such a regime occurs when +ϵ, tα, Dα ≪ δ, U − δ and Hlr is negligible. +To second +order in perturbation, we obtain the correction Eq. (17). +We present here the computation in more detail. +The resonant targeted subspace verifies: +∀j even, +ng +j−1/2 + ng +j + ng +j+1/2 = 2, +(B1) +∀j odd, +ne +j−1/2 + ne +j + ne +j+1/2 = 1, +(B2) +∀j, +ne +j+1/2 + ng +j+1/2 = 1, +(B3) +which satisfy the gauge-invariant condition Gi|ψ⟩ = 0 +for all sites i and |ψ⟩ in the targeted subspace. All states +within this targeted subspace have the same energy rela- +tively to the Hamiltonian H0 in Eq. (15), although they +are not its ground states. This subspace is separated from +all other orthogonal states coupled by H1 in Eq. (16) by +an energy proportional to δ and U. It is thus possible to + +16 +apply standard quantum perturbation theory by treating +H1 as a perturbation to H0 with ratios of tα and Dα with +1/δ or 1/(U − δ) as the small parameters. To first order, +the terms in ϵ of H1 generate a contribution in the stag- +gered mass m. Further corrections due to these terms +are negligible and neglected. The eigenfunctions of the +resonant subspace in the Fock basis are not modified to +first order. +To find Eq. (17), we continue the perturbation to sec- +ond order. We use the perturbation formula: +H(2) +eff = +� +i,j +� +φ +|ψi⟩⟨ψi|H1|φ⟩⟨φ|H1|ψj⟩⟨ψj| +Eφ − Eψ +, +(B4) +with H0|ψi⟩ = Eψ|ψi⟩ for all i where the ψi generate the +resonant subspace, and the φ are all states orthogonal to +the ψi such that ⟨ψi|H1|φ⟩ ̸= 0. By definition, we take +H0|φ⟩ = Eφ|φ⟩. Both set of states {|ψi⟩} and {|φ⟩} are +separable in the local Fock basis and H1 is short-ranged +such that, in Eq. (B4), we may only consider a couple of +processes (i.e., matrix elements of ⟨ψi|H1|φ⟩⟨φ|H1|ψj⟩) +involving one link or one site. +All of these processes, +their amplitude, and the amplitude they contribute to +are listed in Table III. +Processes +Amplitude +Add to +− tgteU +δ(U−δ) − Detg +U−δ +− Dgte +U−δ − DeDg +U−δ +w +− +t2 +e +U−δ +−2 teDe +U−δ − +D2 +e +U−δ +δe (x2) +− +t2 +e +δ +δe +− +t2 +g +δ +δg (x2) +− +t2 +g +U−δ +−2 tgDg +U−δ − +D2 +g +U−δ +δg +TABLE III: A couple of processes (arrows) starting and ending +with a gauge-invariant state (full color) are illustrated. Trans- +parent dots correspond to the intermediate state. First order +corrections to the energy are neglected in the amplitude of each +processes. m = (δe − δg)/2. + diff --git a/pNE1T4oBgHgl3EQf2AWA/content/tmp_files/load_file.txt b/pNE1T4oBgHgl3EQf2AWA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..284c35095cd12d57ca073887a816385b57ce2de5 --- /dev/null +++ b/pNE1T4oBgHgl3EQf2AWA/content/tmp_files/load_file.txt @@ -0,0 +1,1397 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf,len=1396 +page_content='Ab initio derivation of lattice gauge theory dynamics for cold gases in optical lattices Federica Maria Surace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' ∗ Pierre Fromholz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' † Nelson Darkwah Oppong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' ‡ Marcello Dalmonte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 3 and Monika Aidelsburger4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' § 1Department of Physics and Institute for Quantum Information and Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' California 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' USA 2The Abdus Salam International Centre for Theoretical Physics (ICTP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' strada Costiera 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 34151 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Italy 3International School for Advanced Studies (SISSA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' via Bonomea 265,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 34136 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Italy 4Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Ludwig-Maximilians-Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Schellingstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4, D-80799 Munich, Germany 5Munich Center for Quantum Science and Technology (MCQST), Schellingstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4, D-80799 Munich, Germany (Dated: January 10, 2023) We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent optical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The proposal enables the study of both gauge and fermionic-matter fields without integrating out one of them in one and two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Starting from an ab initio modelling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theories that – while pivotal for a successful implementation – have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions, and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope 173Yb, addressing via simulations all key sources of imperfections, and provide concrete parameter estimates for relevant energy scales in both one- and two-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' INTRODUCTION In the last decade, the rapid development of quantum simulators has motivated an increasingly large interest in possible applications to nuclear and particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The study of lattice gauge theories (LGTs) [1–3], one of the most successful theoretical frameworks to regularize strong interacting field theories, could take great advan- tage of the use of quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' First formulated in the 1970s [1], classical simulations of LGTs based on Monte Carlo sampling soon became a pillar of our under- standing of quantum chromodynamics (QCD) [3], with applications as diverse as low-energy spectra [4, 5], phase diagrams [6–8], and even precision measurements in the context of the recently puzzling muon magnetic moment results [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Quantum simulators promise to extend our understanding of LGTs to regimes that are presently in- accessible to Monte Carlo methods, including real-time dynamics, or the physics of the early universe and neu- tron stars [10–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Starting from early theoretical proposals, the field of quantum simulation of LGTs has rapidly evolved driven by two factors: the development of quantum simulation tools and schemes tailored to the specificity of gauge the- ories (in particular, gauge invariance), and a series of ∗ fsurace@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='edu † pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='fromholz@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='ch ‡ Current address: JILA, University of Colorado and National In- stitute of Standards and Technology, and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA § monika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='aidelsburger@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='uni-muenchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='de first experimental steps that have been taken to demon- strate the feasibility of the proposed schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The for- mer has been pioneered by the trapped ion experiment reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' [16] (later extended in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' [17]), where the dynamics of a few-site Schwinger model [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', quan- tum electrodynamics in (1+1)-dimension (d)] with up to six sites was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Moreover, building blocks of matter-gauge interactions have been successfully demon- strated in cold atom settings, including both Z2 [18] and U(1) [19] gauge theories in 1d, utilizing the quantum link formulation (QLM) of LGTs, where the dimension of the local Hilbert space of the gauge link is truncated and therefore finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' More recently, large-scale quantum sim- ulations of Abelian LGTs have been reported in Ryd- berg atom arrays (Schwinger model [20, 21], as well as a (2+1)-d Ising-Higgs gauge theory [22]) and with ultra- cold bosonic atoms in tilted optical superlattices [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In the continuum, quantum simulation of a topological gauge theory was realized in an optically-dressed Bose- Einstein condensate by realising a one-dimensional re- duction of the Chern-Simons theory, the so-called chiral BF theory [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The first generation of experimental realizations has already proven that quantum simulators of LGTs can reach system sizes and timescales at the boundaries of the capabilities of classical numerical simulations [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Nevertheless, many challenges still have to be overcome before we can utilize quantum devices for making accu- rate predictions on complex gauge theories like QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' De- spite the recent experimental progress mentioned above, there is no direct path at this point towards quantum simulation of LGTs with fermionic matter in more than arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='03474v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='quant-gas] 9 Jan 2023 2 one dimension, where both gauge and matter fields are simulated [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In the pioneering experiments imple- mented with trapped ions [16, 17] gauge fields are elim- inated by a Jordan-Wigner transformation, which maps the original Schwinger model to a spin model with ex- otic long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A related approach is fol- lowed for implementations of QLMs in Rydberg atom arrays, where the matter fields are integrated out [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Keeping both matter and gauge fields is more challeng- ing and most proposals require implementations based on ultracold mixtures [19], which significantly increases the experimental complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' QLMs on the other hand are characterized by a finite-dimensional Hilbert space for the gauge degrees of freedom, offering the possibil- ity of implementing matter and gauge degrees of freedom with a single atomic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This has been demonstrated with ultracold bosons in tilted optical superlattice poten- tials [23–25] and a possible extension of this scheme to higher dimensions is currently explored theoretically [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Moreover, schemes based on Floquet engineering appear challenging due to the presence of higher-order terms that need to be suppressed in order to respect gauge in- variance [18], for instance by implementations of addi- tional stabilizers [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In order to overcome these limita- tions, we have developed a new scheme for the realization of U(1) QLMs with ultracold alkaline-earth(-like) atoms (AELA) that offers a direct implementation of fermionic matter and gauge fields, as well as a straightforward ex- tension to two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To ensure a robust experimental implementation, it is crucial to further bridge the gap between theoreti- cal proposals, which focus on conceptual developments and novel implementation schemes, and experimental re- alizations, which require microscopic derivations of the gauge-theory dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The last step is vital in order to understand at a qualitative and quantitative level the impact of often neglected challenges or even roadblocks— including particle losses, limited coherence time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', due to spontaneous emission), and practical difficulties (such as challenges in realizing the required optical potentials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In this work, we introduce a novel scheme and present an ab initio derivation of a U(1) lattice gauge theory in both one and two spatial dimensions using AELA in op- tical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The backbone framework is an implemen- tation that relies on protecting gauge invariance utilizing a combination of energy penalty and locality, through a specific design of state-dependent optical potentials, that are particularly well suited for atomic species with a long-lived electronically excited state [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We carry out a thorough numerical study of the experimental pa- rameters of the optical lattice needed to obtain a regime for our quantum simulation that features the best ratio between coherent and incoherent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We com- pute the parameters of the lattice model, and simulate the corresponding microscopic dynamics, showing that a moderate amount of on-site disorder would only mildly affect the observed time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' δe ∆ g ∆ e δg g e 2j+1/2 2j+3/2 2j+1 2j x/a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' g e g atom detuned lattice site empty lattice site correlated hopping e atom electric field (negative) (positive) charge (+) (−) empty site − pair creation/annihilation − + + (a) (c) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Illustration of the mapping between atomic states in the optical lattice potential and the U(1) quantum link model considered in our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (a) Schematic optical lattice potentials for the g (blue line) and e atoms (orange line) together with the relevant energy scales δg,e and ∆g,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Circles indicate an exemplary initial state of g (blue circles) and e atoms (orange circles) in the optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The black arrows on the right indicate the correlated hopping of atoms between adjacent lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (b) Simplified schematic of the optical lattice potential in panel (a) showing the bonds across which hopping is energet- ically allowed (thick gray lines) and forbidden (dashed gray lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (c) State in the quantum link model corresponding to the atomic configuration in panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In the mapping, every lattice site that can only be occupied by e or g atoms is interpreted as a matter site, shown as circles with the + (−) labels indicating filled sites with charge + (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The lattice sites that can be occupied by both e and g are interpreted as link (l) or gauge-field sites, where triangles pointing right (left) indicate positive (negative) electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Colors indicate the corresponding internal state of the atoms [see panels (a) and (b)] in the proposed experimental imple- mentation and gray indicates an empty site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Note that the correlated hopping of atoms is mapped to a gauge-invariant pair creation/annihilation process, as shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' QUANTUM LINK MODEL In this work we focus on the realization of a U(1) lattice gauge theory with fermionic matter, the lattice version of the Schwinger model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', quantum electrodynamics in one spatial dimension [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Despite its simplicity, the Schwinger model displays several salient features in com- mon with more complicated ones, including confinement, chiral symmetry breaking, and non-trivial real-time dy- namics [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 3 In one spatial dimension, the Hamiltonian has the form [2]: HLGT = −w � j � ψ† jUj,j+1ψj+1 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � + m � j (−1)jψ† jψj + g � j (Ej,j+1 + E0)2, (1) where ψ† j, ψj are fermionic creation and annihilation op- erators on site j of a 1d lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The operators Uj,j+1 and Ej,j+1 are respectively the parallel transporter and the electric field operators, with commutation relation [Ei,i+1, Uj,j+1] = δijUj,j+1: these operators represent a U(1) gauge field on the link connecting the sites j and j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The nearest-neighbor hopping term, of amplitude w, is made gauge invariant by the parallel transporter Uj,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The fermionic mass is staggered, according to the Kogut-Susskind formulation [34]: on even sites, an occu- pied fermionic site represents a “positron” with charge +1, while “electrons”, of charge −1, are represented by holes on odd sites [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We can therefore define the local charge as qj = ψ† jψj − 1 − (−1)j 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (2) The mass term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (1) assigns the mass m to both electrons and positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Finally, the term proportional to g is the energy of the electric field, and E0 represents a static background electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (1) has a gauge symmetry generated by the local operators Gj, defined as Gj = Ej,j+1 − Ej−1,j − qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (3) The physical states for the LGT are the ones that sat- isfy the local constraint (Gauss’ law) Gj |Ψ⟩ = 0 for every site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In the following, we will consider the quantum link formulation of the model [35–38]: in this formulation, all the gauge fields are represented by a finite d-dimensional Hilbert space (we choose d = 2), and the operators Ej,j+1 and Uj,j+1 have the form of Sz and S+ operators respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Compared to the usual Wilsonian lattice gauge theories [1, 2], this formulation is particularly suitable for quantum simulations, because it exploits discrete quan- tum degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Here we focus on the spin-1/2 representation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In this case, the electric field has the two possible values Ej,j+1 = ±1/2 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1(c)], that have the same energy for E0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This choice, with half-integer values of the electric field, is generally denoted as having a topological angle θ = π (in contrast to the case of integer electric field values, having θ = 0) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The topological angle can be tuned by changing the static background field E0, whose effect is to split the degeneracy between the two electric field states [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For the spin-1/2 representation it is useful to define τ = 2gE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Then, the Hamiltonian (1) becomes (up to an additive constant) HQLM = −w � j � ψ† jUj,j+1ψj+1 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � + m � j (−1)jψ† jψj + τ � j Ej,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (4) With this notation, choosing τ ̸= 0 effectively changes the topological angle to θ ̸= π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We note that the model above can be exactly mapped to a spin chain via direct integration of Gauss’ law [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' QUANTUM SIMULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Optical lattice In the proposed experimental setup, we consider cold fermionic atoms in two different electronic states α = {g, e}, realized by the ground and meta-stable excited clock states g ≡ 1S0 and e ≡ 3P0 of AELA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The atoms are considered to be spin polarized in a given nuclear Zeeman state mF , so that the corresponding Hamiltonian is given by H = Hnon-int + Hint, with [42] Hnon-int = � α � d3rΨ† α (r) � − ℏ2 2M ∇2 + Vα (r) � Ψα (r) , Hint =g− eg � d3rρe (r) ρg (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (5) Here Ψα(r) denotes the fermion field operator for atoms in the internal state |α mF ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The density operators are defined as ρα (r) = Ψ† α(r)Ψα(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Since the atoms are polarized in the same nuclear Zeeman state, the inter- action strength g− eg = 4πℏ2a− eg/M (atomic mass M) is associated with the scattering length a− eg of the anti- symmetric electronic state [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The term Vα (r) de- notes a 3d lattice potential Vα(r) = V x α (x) + V y α (y) + V z α (z), where V x α (x) is the state-dependent potential de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1(a) and V y α (y) and V z α (z) are deep state- independent optical lattices with amplitude Fg = Fe and lattice spacing dy = dz that isolate individual 1d chains and provide strong radial confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For simplicity, we choose equal amplitudes for the transverse lattices along y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The state-dependent lattice along x is defined as V x α (x) = − Aα sin2 � π 2ax + ϕ � − Bα sin2 �π a x � − Cα sin2 �2π a x + π 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (6) It has a unit cell of length 2a with three “low”-energy lattice sites and one “high”-energy site, which suppresses 4 30 20 10 Vx g(x) [kHz] (a) 0 1 2 3 4 5 x/a 20 0 Vx e(x) [kHz] (b) 0 2 wx g, s(x 3a) s = s = 0 s = + 0 2 wx e, s(x 2a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Optical lattice potential and Wannier func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The x-component of the optical lattice potential de- fined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (6) for the (a) g and (b) e atoms is plotted in gray for ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The parameters Aα, Bα, Cα are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In red, the x-components of the Wannier functions centered on site x = 3a and x = 2a are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The parame- ter s = {−, 0, +} labels the three different orbitals in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' tunneling to that site as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The triple wells of the g and e lattices are shifted relative to each other by a distance a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The optical potential along x can be realized by super- imposing three different optical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Each of them is generated from a pair of monochromatic laser beams at either the magic wavelength λm [45, 46], which cor- responds to a state-independent potential, or the anti- magic wavelength λam [47], where the potentials for atoms in the g and e state are equal in magnitude but have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Moreover, the lattice spacing can be set by tuning the intersection angle θ between the in- terfering pair of laser beams according to λ/ [2 sin(θ/2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The two shorter-spacing lattices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (6) are operated at the magic wavelength λm (Be = Bg and Ce = Cg) and intersection angles θC = 180◦ and θB = 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The corre- sponding lattice spacings are a = λm/2 and 2a, such that their combination yields a symmetric double-well poten- tial [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The third long-lattice at lattice spacing 4a, which can be generated at a smaller intersection angle, is operated at λam with Ag = −Ae generating a triple well potential that is shifted for g and e atoms as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1(a) for ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Note that the required opti- cal potentials could also be generated using a hybrid ap- proach using a combination of optical lattices and tweez- ers, which have recently been employed for Hubbard-type physics [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Lattice Hamiltonian To obtain a lattice Hamiltonian for the model, we as- sume that only the three lowest Bloch bands are occupied both for the g and e states, and we express the field oper- ator Ψα(r) in terms of the Wannier functions wα,s, where s = {−, 0, +} labels the three Wannier centers in a unit cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 2): Ψg(r) = � j odd � wg,+(r − rj)cj+1/2 +wg,0(r − rj)cj + wg,−(r − rj)cj−1/2 � , (7) Ψe(r) = � j even � we,+(r − rj)dj+1/2 +we,0(r − rj)dj + we,−(r − rj)dj−1/2 � , (8) where rj = jaˆx, ˆx is the unit vector and cj (dj) is the lat- tice fermionic annihilation operator of a g atom (e atom) on lattice site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Substituting the expressions for the field operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (5), we obtain the lattice Hamiltonian (see Ap- pendix A) Hlatt = Hg + He + HU + HD + Hlr + const, (9) where Hg and He denote the terms containing hopping and chemical potentials of the g and e atoms within a single triple well respectively Hg = � j odd � −tg � c† jcj+1/2 + c† jcj−1/2 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � + δgc† jcj � , (10) He = � j even � −te � d† jdj+1/2 + d† jdj−1/2 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � + δed† jdj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (11) We assumed, for the moment, that ϕ = 0, so the model is symmetric under reflections centered on the matter sites: this implies that the chemical potentials of the sites s = + and s = − are the same (and can be chosen as a reference level and set to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The terms HU and HD are obtained from the interact- ing term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (5), and read (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 3) HU = U � j d† j+1/2dj+1/2c† j+1/2cj+1/2, (12) HD = Dg � j odd � d† j+1/2dj+1/2c† jcj+1/2 (13) +d† j−1/2dj−1/2c† jcj−1/2 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � + De � j even � c† j+1/2cj+1/2d† jdj+1/2 +c† j−1/2cj−1/2d† jdj−1/2 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Finally, Hlr contains all the additional terms, of the form of longer-range hoppings and interactions, that have very 5 g e onsite interaction U −te −te+ De (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Illustration of the interacting terms in HU and HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (a) On-site interaction U between a single g (blue circle) and e atom (orange circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (b) Hopping of a single e atom to an empty lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (c) In the presence of interac- tions, tunneling is additionally modified by a density-assisted tunneling with amplitude De.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' small amplitudes and can be neglected (we explicitly ver- ify that these terms are negligible for the parameters re- ported in Section IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' It is useful to define ϵ = (δg − δe)/2, δ = (δg + δe)/2, and the total number of atoms on each site j+1/2, which corresponds to a link in the QLM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1), n(l) j+1/2 = d† j+1/2dj+1/2 + c† j+1/2cj+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (14) Here (l) is a redundant superscript to indicate we are on a link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We now assume ϵ, tα, Dα ≪ δ, U − δ: in this regime it is convenient to split the Hamiltonian into three parts Hlatt = H0 + H1 + Hlr with different energy scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', H0 = (Ng+Ne−Nl)δ+ � j � −δ + U 2 n(l) j+1/2 � (n(l) j+1/2−1), (15) where Ng and Ne are the total numbers of atoms in the g and e states respectively and Nl is the total number of links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (15) it is immediate to see that for δ, U − δ > 0 the lowest energy states of H0 have exactly one atom (either g or e) on each half-integer site, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', n(l) j+1/2 = 1 for every j: a double occupancy n(l) j+1/2 = 2 costs energy U − δ, while having a hole n(l) j+1/2 = 0 costs energy δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The term H1 has the form H1 = � j odd � −tg � c† jcj+1/2 + c† jcj−1/2 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � + ϵ c† jcj � + � j even � −te � d† jdj+1/2 + d† jdj−1/2 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � − ϵ d† jdj � + HD, (16) where HD is the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We initialize the system with two g atoms for every g triple well and one e atom for every e triple well (all these quantities are locally conserved, if we neglect Hlr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The effective Hamiltonian describing the resonant dynamics is obtained using perturbation theory: we neglect Hlr, and we treat H1 as a perturbation to H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To second order, the effective Hamiltonian has the form (see Appendix B) H(2) eff = −w � j odd (c† jcj+1/2d† j+1/2dj+1 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c) − w � j even (d† jdj+1/2c† j+1/2cj+1 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c) + m � j odd c† jcj − m � j even d† jdj, (17) with w = tgteU δ(δ − U) + −Detg − Dgte + DeDg δ − U , (18) m = ϵ + 2t2 g − t2 e 2δ − (tg − Dg)2 − 2(te − De)2 2(U − δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (19) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Mapping to the quantum link model We now prove that there is an exact mapping between the effective Hamiltonian H(2) eff and the quantum link Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The fermionic operator ψj for the matter is defined as ψj = � cj j odd, dj j even, (20) (and an analogous definition is used for ψ† j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The electric field Ej,j+1 and the parallel transporter Uj,j+1 on the link are represented by Ej,j+1 = (−1)j 2 (c† j+1/2cj+1/2 − d† j+1/2dj+1/2), (21) Uj,j+1 = � cj+1/2d† j+1/2 j odd, dj+1/2c† j+1/2 j even, (22) and satisfy the desired commutation relation [Ei,i+1, Uj,j+1] = δi,jUj,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' With the definitions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (20) and (21), the operator Gj takes the form Gj = 1 2(n(l) j+1/2 + n(l) j−1/2) − n(b) j + (1 − (−1)j) 2 , (23) with n(b) j being the number of atoms in the j-th triple- well (or “block”), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', n(b) j = � c† jcj + c† j+1/2cj+1/2 + c† j−1/2cj−1/2 j odd, d† jdj + d† j+1/2dj+1/2 + d† j−1/2dj−1/2 j even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (24) With this mapping, which is schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1, we obtain that H(2) eff is equivalent to the Hamil- tonian HQLM with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The gauge-invariant sub- space corresponds to the sector with nj+1/2 = 1 and 6 n(b) j = [3−(−1)j]/2 for every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Some examples of gauge- invariant states are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4(a), all g atoms sit on the links, and all e atoms are on the matter sites: the corresponding electric field takes values Ej,j+1 = (−1)j/2, while ψ† jψj = 1, 0 for even and odd sites respectively, leading to alternating positive and negative charges on matter sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This state is the ground state of the model in the limit m → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Similarly, it is easy to show that the states represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4(b) and 4(c) have no charges on matter sites, and have uniform (negative or positive) electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' These states (vacua) are degenerate ground states in the limit m ≫ |w| with τ = 0, while the degeneracy is split for τ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Theta term We now show how to tune the parameters of the op- tical lattice to obtain τ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The lattice Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (9) was derived with the assumption that ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We now slightly perturb this model, by introducing a small shift ϕ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To first order in ϕ, the shift produces an additional potential along x V x α → V x α − Aα sin �π a x � ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (25) The main effect of this additional term is to change the chemical potential at half-integer positions x = (j+1/2)a by a quantity −Aαϕ(−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We obtain Hlatt → Hlatt − � j (−1)jϕ(Agc† j+1/2cj+1/2 + Aed† j+1/2dj+1/2) = Hlatt − � j (−1)jϕ �Ag + Ae 2 n(l) j+1/2 +Ag − Ae 2 (c† j+1/2cj+1/2 − d† j+1/2dj+1/2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (26) The term � j(−1)jn(l) j+1/2 cancels in the resonant sector, and the remaining term is mapped to � j τEj,j+1, with τ = (Ae − Ag)ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (27) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' EXPERIMENTAL IMPLEMENTATION The ab-initio calculation of the band structure and Wannier functions allows us to estimate the energy scales involved in the quantum simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' These estima- tions explicitly verify that the desired parameter range is achievable in present-day experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The quantitative estimation of the parameters is also crucial to understand the limitations of our proposal such as the amplitude and duration of the signal and to identify the main sources of + − + (a) (b) (c) vacuum with negative electric field neighboring pairs of + and − charges vacuum with positive electric field e atom detuned site empty site g atom FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Examples of the mapping between atomic configurations in the optical lattice and gauge- invariant states in the U(1) QLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In each panel, the atomic states in the optical lattice are shown in the top row and the states in the QLM are shown in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The panels show (a) the state with neighboring pairs of + and − charges, (b) the vacuum state with homogeneous neg- ative electric field E = −1/2, and (c) the vacuum state with homogeneous positive electric field E = +1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' error such as the population of higher bands, higher-order perturbative processes, longer-range terms, and dissipa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To set values, we choose the fermionic isotope 173Yb with mass M ≈ 173u and the interorbital scattering length a− eg = 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='7 a0 [52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' here u denotes the atomic mass unit and a0 the Bohr radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' However, we note that our proposal can be similarly applied to other fermionic AELA species such as 171Yb and 87Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The experimental parameters require scaling to account for the modified atomic mass and scattering length [53, 54] with no con- ceptual change in the design of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Realistic parameters We define ∆g/e as the gap between the third and fourth energy band in the lattice for the g/e atoms (see Section A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The parameters used here are chosen to satisfy the hierarchy of energy scales ∆g/e ≫ δg/e, U − δg/e ≫ tg/e, Dg/e ≫ terms in Hlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (28) We note that for the 3d lattice potential Vα(r) the Wannier functions obtained by solving the non- interacting Hamiltonian Hnon-int can be factorized in the three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The hoppings tα, the chemical poten- tials δα, and the gaps ∆α do not depend on the y and z components φy α(y) and φz α(z) of the Wannier functions (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The interactions U and Dα, on the other hand, are proportional to the quantity Jyz = � dy dz |φy g(y)|2|φy e(y)|2|φz g(z)|2|φz e(z)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (29) We can therefore tune Fα and dy/z to change the value of Jyz and thus enhance or suppress the interaction terms U and Dg/e independently from the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In Table I, we report a possible choice for the parame- ters of the optical lattice, and the corresponding param- 7 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Experimental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' All values are given in units of h·kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Ag = −Ae Bg = Be Cg = Ce 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='827 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='343 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='832 ∆g = ∆e δg = δe U tg = te Dg = De 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='023 eters of the lattice Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We choose a = λm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='7594 µm [55] and Jyz = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='566 µm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' With these parameters, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (18) and (19) we obtain m/h = 9 Hz and w/h = −18 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' where h denotes Planck’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The value of Jyz reported here can be obtained with a transverse confinement Fg/h = Fe/h = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='9 kHz and dy/z = λm/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For this choice of transverse potential, the hopping in the y and z direction is ty/z α /h = 2 Hz, much smaller than the relevant scales m and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The transverse hopping can be made even smaller by using two beams intersecting at a shallow angle instead of using retro-reflected beams: the lattice spacing dy/z is increased, and the same value of Jyz is obtained for a larger Fα, thus suppressing the transverse hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We explicitly check that all the terms included in Hlr, which were neglected in the derivation of HQLM, are small with respect to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The nearest-neighbor density-density interaction is of the order of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='6 h·Hz, while the hop- ping between the sites s = + and s = − in a triple well is ∼ 2 h·Hz, and is negligible because it is suppressed by the on-site interaction U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The parameters in Table I can be readily generalized to other atomic species: we can define the adimensional lattice constant ˜a = a/λm and the adimensional param- eter ˜Aα = Aα2M/ℏ2λ2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Similarly we can define dimen- sionless parameters for the other energy scales Bα, Cα, ∆α, δα, U, tα, Dα, m, w, and for the quantity Jyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Im- plementations with different atomic species but same adi- mensional parameters of the optical lattice yield the same adimensional values for the terms in the lattice Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Initial state preparation We propose two distinct ways of a two-step prepa- ration of the system in gauge-invariant initial states as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4, which require specific atom configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' First, the correct atom-number distribution needs to be prepared (independent of the internal state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' One option is to prepare it starting from a sample with one g atom per lattice site followed by the removal of atoms on selected lattice sites yielding the desired occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This is a standard technique in quantum gas micro- scopes [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Alternatively, the atoms could directly be placed at their desired location with moving optical tweezer potentials [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In the second step, g atoms can be converted to e atoms on selected lattice sites using a global clock laser excitation pulse exploiting the local differential light shifts δα and ∆α [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The conver- sion could also be performed locally by using clock laser light focused onto single lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' REAL-TIME DYNAMICS Using numerical simulations, we study here the real- time dynamics of the model of Hamiltonian (9) that we compare with the dynamics of the quantum link model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In both cases we start with the initial state shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The time evolution of the model is simulated exactly for a system of length 4a with periodic boundary conditions, with the parameters reported in Section IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Longer-range terms from Hlr in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (9) are included in the numerical simulation: the dynamics is exact as long as higher bands are not occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Gauge invariance and gauge field The system is effectively gauge invariant as long as Gauss’ law applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To quantify the violation of Gauss’ law in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (9), we examine the time evolution of n(l) j+1/2, and n(b) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The simulation results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 5(a)- (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We find that the conditions n(b) j = [3 + (−1)j]/2 and n(l) j+1/2 = 1 are preserved up to 5 · 10−4 and 10−1 respectively within the first 70 ms after initialization, which should be compared to the characteristic interac- tion timescale ℏ/|w| ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='8 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 5(d) we examine the evolution of the electric field with (i) the Hamiltonian Hlatt, (ii) the second-order effective Hamiltonian H(2) eff , equivalent to HQLM, and (iii) the fourth-order effective Hamiltonian H(4) eff (from the Schrieffer Wolff procedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The differences between the electric field value obtained in case (i) and with the ap- proximate Hamiltonians are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 5(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For times up to ∼ 40 ms, the results obtained in the three cases are in good agreement: this shows that the second- order effective Hamiltonian HQLM captures the main fea- tures of the time evolution, at least at short and inter- mediate time scales, and that longer-range terms and higher-order perturbative processes are minor sources of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For times of the order of ≳ 50 ms, we find that fourth-order corrections have to be considered in order to obtain a good prediction of the time evolution in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We remark that the fourth-order corrections do not violate Gauss’ law, but correspond to additional gauge-invariant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A feature of the evolution induced by Hlatt that is not observed in the effective Hamiltonians is the presence of fast oscillations with small amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' These oscillations have a frequency compatible with the 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Dynamics in the U(1) quantum link model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Time evolution of (a) the number of atoms per link, (b,c) the number of atoms on each odd/even block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For each observ- able O, the shaded area indicates the interval ⟨O⟩± � Var(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The initial state is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 4(a) and is evolved under Hlatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The system has periodic boundary conditions and fi- nite size 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The top-left schematic in each panel illustrates the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (d) Time evolution of the electric field on an odd-even link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The exact dynamics given by Hlatt are compared with the second-order effective Hamiltonian [using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (18,19)] and with the fourth-order effective Hamiltonian H(4) eff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The latter two are both equivalent to HQLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (e) Dif- ference ∆Eodd,even = ⟨Eodd,even⟩Hlatt − ⟨Eodd,even⟩H (the sub- script denotes the Hamiltonian that generates the time evo- lution) for the two cases H = HQLM and H = H(4) eff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' energy scale of H0 and are averaged out in the perturba- tive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To make this separation of energy scales more evi- dent, we examine the Fourier transform of the signal E(ω) = � tmax 0 E(t)e−iωtdt in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' As expected, the evolution under Hlatt shows peaks at frequency ω ∼ δ/ℏ = (U − δ)/ℏ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='4 kHz [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 6(a)], that are not observed for the effective Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Zooming in at smaller frequencies [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 6(b)], we see that the agree- ment in the Fourier transforms is good between Hlatt and HQLM and is excellent between Hlatt and H(4) eff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 0 2 4 6 8 [kHz] 10 1 101 |E( )| Hlatt HQLM H(4) eff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='8 [kHz] 10 1 100 101 |E( )| Hlatt HQLM H(4) eff (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Fourier transform of the time evolution of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (a) Spectrum corresponding to the time evolution shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Panel (b) shows a zoom-in of panel (a) revealing details at small frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Dissipation Finite dissipation can become a crucial issue in the ex- periment when it occurs on time scales comparable to the relevant dynamic evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Therefore, we focus on identifying the fastest dissipation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This allows us to estimate for how long the experimental sys- tem closely follows the coherent dynamics of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For the AELA in an optical lattice considered here, the typically dominant dissipation channels are lossy colli- sions between pairs of atoms and off-resonant scattering of optical lattice photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In the following, we evalu- ate the relevance of both for our proposed experimental implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Lossy collisions between pairs of atoms with one or both of the two atoms in the e state can lead to a particu- larly fast atom loss [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For our proposed implemen- tation, however, double occupancies of lattice sites are (purposefully) strongly suppressed, either by fermionic quantum statistics (ee-pairs) or a large on-site interac- tion energy (eg-pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' As a consequence, lossy collisions between pairs are not expected to be a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In contrast, off-resonant photon scattering from lattice photons was identified as the dominant limiting factor in our proposed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' While off-resonant photon scattering eventually leads to heating and atom loss, an- other effect could become relevant at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' When an atom in the e state scatters an optical lattice photon, short-lived intermediate states can be populated and de- cay back to the g state with a finite probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Conver- sion of e atoms to g atoms due to this optical pumping process has already been observed and characterized in optical lattice experiments with AELA [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Here, we employ these results to estimate the expected time scale 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='7 [kHz] 10 1 100 101 |E( )| 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='02 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Influence of disorder on the dynamics of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Fourier transform of the time evolution of the electric field under HW for different values of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The legend indicates the values of W/h in kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The results are averaged over 100 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' for our parameters (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Focusing on the pro- posed implementation in Section III A for 173Yb and the off-resonant scattering of magic-wavelength lattice light, we estimate a repumping rate Γ ≈ 111 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Comparison of this estimate to the quantity |w|/ℏ = 113 Hz suggests that the dynamics of our model can be faithfully observed for many characteristic time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Disorder An experimental implementation could also exhibit fi- nite disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Based on previous experimental work [60], disorder occurs in particular when utilizing hybrid po- tentials generated with both optical lattices and optical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To estimate how much disorder affects the dy- namic evolution of the system, we consider a quenched disorder in the chemical potentials of the atoms of the form HW =Hlatt + � j odd � Wg,j+1/2c† j+1/2cj+1/2 (30) +Wg,j−1/2c† j−1/2cj−1/2 + Wj c† jcj � + � j even � We,j+1/2d† j+1/2dj+1/2 +We,j−1/2d† j−1/2dj−1/2 + Wj d† jdj � , where Wα,j+1/2, Wj are taken randomly from a uniform distribution in the interval [0, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 7 we plot the Fourier transform E(ω) averaged over 100 disorder realizations for different values of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We find that disorder has the effect of smearing out the peaks, which nevertheless remain visible up to W/h ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='01 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' THE 2D MODEL We now generalize the implementation examined in the previous sections to the quantum link model in two spa- tial dimensions, and we show how this can be simulated with realistic experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Quantum link model The two-dimensional quantum link model is described by the Hamiltonian [61] HQLM = − w � r � k=ˆx,ˆy � ψ† rUr,r+kψr+k + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' � + m � r srψ† rψr + τ � r � k=ˆx,ˆy Er,r+k, (31) where the sums run over the points r = (i, j) of a two-dimensional square lattice (i, j are integers), and sr = (−1)i+j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' here ˆx and ˆy denote unit vectors along the respective direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The gauge fields sit on the links and, similarly to the one-dimensional case, are represented by spin variables with finite d-dimensional Hilbert spaces (here, d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The generators of the gauge symmetry read Gr = � k=ˆx,ˆy (Er,r+k − Er,r−k) − ψ† rψr + 1 − sr 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (32) A state |Ψ⟩ is gauge-invariant if it satisfies Gauss’ law Gr |Ψ⟩ = 0 for every lattice site r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' An example of a gauge-invariant state is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Quantum simulation Our desired optical lattice in two dimensions consists of cross-shaped “blocks” of g and e sites [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8(a-c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' While in the one-dimensional case each block consisted of a triple well, here a block contains five sites: a central matter site at r = (x/a, y/a) = (i, j), with i, j integers, and four gauge sites around it at positions r ± (1/2, 0) and r ± (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Blocks of g and e sites alternate in a checkerboard pattern, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8(c), with over- lapping g and e gauge sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A lattice of this type can be realized with the potential V x,y α (x, y) = − Aα sin2 � π 2a(x + y) + ϕ � − Aα sin2 � π 2a(x − y) � − Bα sin2 �π a (x + y) � − Bα sin2 �π a (x − y) � − Cα sin2 �2π a x + π 2 � − Cα sin2 �2π a y + π 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (33) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8(a,b) depicts the profiles of V x,y g and V x,y e for the values of Ag, Ae, Bg, Be, Cg, Ce reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 10 (a) (b) (c) (d) + − electric field (negative) electric field (positve) charge (−) charge (+) e atom g atom detuned lattice site hopping bond empty lattice site 0 2 4 x/a 0 1 2 3 4 y/a 165 150 135 120 105 90 75 60 45 0 2 4 x/a 0 1 2 3 4 75 60 45 30 15 0 15 30 45 60 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Implementation of the quantum link model in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Optical lattice for the (a) g and (b) e atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (c) Example of a gauge-invariant state belonging to the resonant subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Blue and orange circles repre- sent g and e atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (d) Corresponding gauge- invariant state in the QLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Dark/light gray circles indicate the occupied/empty matter sites (with charge +, −, or no charge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Red (blue) arrows represent a link with electric field Er,r+k = +1/2 (Er,r+k = −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The steps for deriving the lattice Hamiltonian and mapping it to the QLM are analogous to the one- dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We report here the mapping of the operators: Er,r+k = sr 2 (c† r+k/2cr+k/2 − d† r+k/2dr+k/2), (34) ψr = � cr if sr = −1, dr if sr = +1, (35) Ur,r+k = � cr+k/2d† r+k/2 if sr = +1, dr+k/2c† r+k/2 if sr = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (36) The system is initialized with three g atoms on each odd block, and two e atoms on each even block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To illustrate the mapping, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8(c) and (d) an example of a resonant state in the local occupation basis and the corresponding gauge-invariant state in the quantum link model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Using the same derivation as the one-dimensional case, we obtain the effective Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In Table II we report the parameters obtained for the lattice poten- tial depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We set a = λm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='7594 µm and Jz ≡ � dz|φz g(z)|2|φz e(z)|2 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='235 µm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For this choice of parameters we obtain m = 10 h·Hz and w = 20 h·Hz for the QLM Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' CONCLUSIONS We have presented a proposal for the scalable quan- tum simulation of lattice gauge theories coupling (stag- gered) fermions to U(1) gauge fields utilizing a mixture of alkaline-earth(-like) atoms in both a ground and a metastable state in optical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The key element of our proposal is a careful treatment of the full system dynamics, that are derived ab initio from microscopic interactions between atoms and light, and atoms them- selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' While the proposal can be applied to a variety of atomic species, we have drawn a complete blueprint utilizing concrete estimates based on 173Yb atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Our treatment highlights concrete challenges in the quantum simulation of lattice gauge theories that have so far mostly been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In particular, it makes clear that the superposition of lattice potentials required for such simulations, while certainly realistic experiment- wise, gives rise to complicated band structures that must be quantitatively understood to access the reliability and feasibility of any quantum simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The reason for this is twofold: band separation can become much smaller than what is naively expected, making protection of gauge invariance very challenging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' in parallel, intrinsic energy scales of desired processes can be considerably re- duced with respect to simplistic deep lattice estimates based on highly localized Wannier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' These lim- itations are particularly pernicious for single-body terms in the lattice potential, whose estimate crucially requires a quantitative approach as the one carried out here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Within the context of our proposal, we have shown that optimal parameter regimes can still be found for observ- ing the correct and expected LGT dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This can be achieved thanks to the detailed microscopic understand- ing our treatment leads to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We have demonstrated this conclusion by comparing numerical simulations of both ideal and effective dynamics of string relaxation, includ- ing also effects of inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Based on our findings, we believe that the ab initio approach we propose will be, in the long term, the one needed to fully determine the capabilities of quantum simulators of lattice gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We have taken the first step beyond Abelian 1D models, by extending them to 2D geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Future works will be fundamental TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Experimental parameters for the two- dimensional quantum link model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' All values are given in units of h·kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Ag = −Ae Bg = Be Cg = Ce 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='029 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='915 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='516 ∆ δg = δe U tg = te Dg = De 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='033 11 to address the experimental capabilities to realize non- Abelian lattice gauge theories, which to date have been proposed only in very few settings [62–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Burrello, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Pagano and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Rico for in- sightful discussions, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Scazza for collaboration on a related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The work of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' was partly supported by the ERC under grant num- ber 758329 (AGEnTh), and by the MIUR Programme FARE (MEPH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, Ger- man Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868, from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 803047) and from the German Federal Ministry of Education and Research via the funding pro- gram quantum technologies – from basic research to mar- ket (contract number 13N15895 FermiQP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' further acknowledge funding within the Quan- tERA II Programme that has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No 101017733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Marzari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Mostofi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Yates, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Souza, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Vanderbilt, Maximally localized wannier functions: Theory and applications, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 84, 1419 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Appendix A: Wannier functions and lattice Hamiltonian 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' One-dimensional case In this appendix, we discuss the ab-initio derivation of the lattice Hamiltonian from the Wannier functions for the case of the one-dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' First, we solve the non interacting Hamiltonian Hnon-int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We diagonalize the single-particle Hamiltonian hα(r) hα(r) = − ℏ2 2M ∇2 + Vα (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A1) with α = g, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For the quantum simulation of the one- dimensional model discussed in Section III, the potential is Vα(r) = V x α (x) + V y α (y) + V z α (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A2) We then look for a factorized and localized three- dimensional complete basis of wavefunctions for all states involved in the dynamic evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The po- tential is periodic in space for each component x, y, z such that the Bloch theorem applies to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 9, we plot the potential V x g (with the parameters of Ta- ble I) and the lowest bands obtained by solving the cor- responding periodic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Using a unitary trans- formation (similar to the reverse Fourier transformation) of the Bloch eigenfunctions, we obtain a set of localized orthonormal wavefunctions wα,s called the Wannier func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Because of the form of the potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A2), the Bloch functions are factorizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Likewise, the Wannier functions factorize along x, y, z: wα,s(r − rj) = φx α,s(x − ja)φy α(y)φz α(z), (A3) where w is the three-dimensional Wannier function, and the φi are the one dimensional Wannier functions for each direction i = x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' α = g, e is the electronic state, s = {0, +, −} denotes the three functions corresponding to the three sites of a triple well, and rj = jaˆx is the Wannier center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' As we target a one dimensional system, we consider only Wannier functions in the transverse di- rection y, z centered around y = z = 0 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' They involve only the lowest Bloch band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The x com- ponent is instead obtained from the three lowest bands such that we have three Wannier centers per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To derive an expression for the localized Wannier func- tions useful to estimate the overlap, we compute the eigenstates of the projection of the position operator onto a given set of Bloch states [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In one dimension, this method always gives the maximally localized Wan- nier functions [73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In many other cases, this deriva- tion still gives a good approximation of the maximally localized Wannier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We can then define discrete operators for the discrete Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We expand the fermionic operators in the basis of the Wannier functions Ψg(r) = � j odd � wg,+(r − rj)cj+1/2 +wg,0(r − rj)cj + wg,−(r − rj)cj−1/2 � , (A4) Ψe(r) = � j even � we,+(r − rj)dj+1/2 +we,0(r − rj)dj + we,−(r − rj)dj−1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A5) We stress that the only approximations performed so far are (i) neglecting the higher bands and (ii) only consid- ering the chain localized at y, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (i) is justified when the gaps ∆g, ∆e between the third and the closest higher bands (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 9) are much larger than the energy scales 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 x/a 30 25 20 15 10 5 Vx g(x)/h [kHz] /2 0 /2 k 30 25 20 15 10 5 /h [kHz] /2 0 /2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='835 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='830 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='825 /h [kHz] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Lattice potential and band structure along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Left panel: x component of the optical lattice potential V x g (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The defining parameters are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Right panel: corresponding band structure using the same scale as the panel on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The two lowest bands are almost degenerate (they correspond to symmetric and antisymmetric superpositions of the s = {+, −} sites in the triple well), and are separated from the third one (corresponding to the central site 0 of the triple well) by an energy ∼ δg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Higher bands are separated from the first three by an energy gap ∆g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' of the dynamics that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (ii) is justi- fied if the transverse hoppings ty α, tz α are small compared to the energy scales of our interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For the gaps and the transverse hoppings of Section IV A, both approxi- mations are appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The discrete parameters emerge when substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A4) and (A5) in the Hamiltonian H = Hnon-int + Hint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' These parameters include the chem- ical potentials, the hoppings (both from Hnon-int), the on-site and off-site density-density interactions, the den- sity mediated hoppings, and the correlated hoppings be- tween g and e atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We included all these terms in the numerical simulations of the real-time dynamics in Sec- tion V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For clarity, we give the lattice Hamiltonian with the terms of highest amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We define the chemical potentials such that µα,s = � d3r w∗ α,s(r − rα)hα(r)wα,s(r − rα) = µx α,s + µy α + µz α (A6) where α = g, e, s = {0, +, −}, and the Wannier centers are rg = aˆx, re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We further define the difference between the chemical potentials in each triple well δα,± = µα,0 − µα,± = µx α,0 − µx α,±, (A7) and the nearest-neighbour hoppings within a triple well tα,± = − � d3r w∗ α,0(r − rα)hα(r)wα,±(r − rα) = − � dx [φx α,0(x − ja)]∗hx α(x)φx α,±(x − ja).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A8) For ϕ = 0 the triple well is designed to be symmetrical, such that δα,+ = δα,− = δα and tα,+ = tα,− = tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' These two terms result in the Hamiltonian terms Hg and He in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The term with largest amplitude obtained from Hint is the on-site interaction on the sites where g and e triple wells overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This amplitude is given by U = g− eg � d3r |wg,−(r − aˆx)|2|we,+(r)|2 = g− egJyz � dx |φx g,−(x − ja)|2|φx e,+(x)|2, (A9) with Jyz = � dy dz |φy g(y)|2|φy e(y)|2|φz g(z)|2|φz e(z)|2, (A10) and yields the Hamiltonian term HU in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The terms with the next largest amplitude with our choice of parameters are density-assisted hoppings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Specifically, where a g or e atom hops between two sites of a triple well, provided that an atom of the opposite electronic state sits in either the initial or the final site (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The amplitude has the form Dg = g− eg � d3r w∗ g,0(r − aˆx)wg,−(r − a)|we,+(r)|2 = g− egJyz � dx [φx g,0(x − ja)]∗φx g,−(x − ja)|φx e,+(x)|2, (A11) De = g− eg � d3r |wg,−(r − aˆx)|2w∗ e,0(r)we,+(r), = g− egJyz � dx |φx g,−(x − ja)|2[φx e,0(x)]∗φx e,+(x), (A12) and results in the Hamiltonian HD in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' For our choice of parameters, all the other terms (that we gener- ically include in Hlr) have sufficiently small amplitudes to be negligible according to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' V A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Two-dimensional case The steps of the derivation of the lattice Hamiltonian in the two-dimensional system are very similar to the 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 ky n = 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 n = 1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='00335455 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='00335450 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='00335445 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 2 0 2 ky n = 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 2 0 2 n = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 ky n = 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 n = 5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='98013355 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='98013350 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 ky n = 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 n = 7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='97734702 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='97734700 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='97734698 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='97734696 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='97734694 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 ky n = 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 kx 2 0 2 n = 9 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='868653180 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='868653178 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='868653176 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Bandstructure for the 2d lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The 10 lowest bands corresponding the two-dimensional lattice V x,y e (x, y) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The main difference lies in the potential that is now Vα(r) = V x,y α (x, y) + V z α (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (A13) As a consequence, only the z component of the Bloch (and Wannier) functions can be factorized out, while for the x − y plane we have to solve a two-dimensional sin- gle particle Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The first 10 two-dimensional Bloch bands for the parameters in Table II are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The unit cell we consider is defined by the lattice vec- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 x/a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='4 y/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Wannier functions for the 2d lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Two- dimensional Wannier functions of two g and two e blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In orange/light blue, we plot the 4 Wannier functions localized on the links of each block for the e/g state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' In red/dark blue, we plot the Wannier function localized on the center of each block for the e/g state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' tors (2a, 0) and (0, 2a) and contains two blocks for each electronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Because each block contains five sites, we need 10 Wannier functions per unit cell for each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To find the Wannier functions we first find the eigen- states of the projection of the x position operator on the 10 lowest bands: we collect the groups of eigenstates with (almost) degenerate eigenvalue and diagonalize the y position operator projected on each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The Wan- nier functions obtained with this procedure are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The coefficients of the lattice Hamiltonian are then obtained as in the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Appendix B: Perturbative theory of the lattice Hamiltonian The Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (5) describing the cold atoms in the optical lattice and its lattice formulation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (9) can be mapped to the QLM Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (4), when considering the coupling between the targeted gauge- invariant Hilbert subspace and the rest of the Hilbert space as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Such a regime occurs when ϵ, tα, Dα ≪ δ, U − δ and Hlr is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To second order in perturbation, we obtain the correction Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We present here the computation in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The resonant targeted subspace verifies: ∀j even, ng j−1/2 + ng j + ng j+1/2 = 2, (B1) ∀j odd, ne j−1/2 + ne j + ne j+1/2 = 1, (B2) ∀j, ne j+1/2 + ng j+1/2 = 1, (B3) which satisfy the gauge-invariant condition Gi|ψ⟩ = 0 for all sites i and |ψ⟩ in the targeted subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' All states within this targeted subspace have the same energy rela- tively to the Hamiltonian H0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (15), although they are not its ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' This subspace is separated from all other orthogonal states coupled by H1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (16) by an energy proportional to δ and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' It is thus possible to 16 apply standard quantum perturbation theory by treating H1 as a perturbation to H0 with ratios of tα and Dα with 1/δ or 1/(U − δ) as the small parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To first order, the terms in ϵ of H1 generate a contribution in the stag- gered mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Further corrections due to these terms are negligible and neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' The eigenfunctions of the resonant subspace in the Fock basis are not modified to first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' To find Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (17), we continue the perturbation to sec- ond order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' We use the perturbation formula: H(2) eff = � i,j � φ |ψi⟩⟨ψi|H1|φ⟩⟨φ|H1|ψj⟩⟨ψj| Eφ − Eψ , (B4) with H0|ψi⟩ = Eψ|ψi⟩ for all i where the ψi generate the resonant subspace, and the φ are all states orthogonal to the ψi such that ⟨ψi|H1|φ⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' By definition, we take H0|φ⟩ = Eφ|φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Both set of states {|ψi⟩} and {|φ⟩} are separable in the local Fock basis and H1 is short-ranged such that, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' (B4), we may only consider a couple of processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=', matrix elements of ⟨ψi|H1|φ⟩⟨φ|H1|ψj⟩) involving one link or one site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' All of these processes, their amplitude, and the amplitude they contribute to are listed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Processes Amplitude Add to − tgteU δ(U−δ) − Detg U−δ − Dgte U−δ − DeDg U−δ w − t2 e U−δ −2 teDe U−δ − D2 e U−δ δe (x2) − t2 e δ δe − t2 g δ δg (x2) − t2 g U−δ −2 tgDg U−δ − D2 g U−δ δg TABLE III: A couple of processes (arrows) starting and ending with a gauge-invariant state (full color) are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' Trans- parent dots correspond to the intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' First order corrections to the energy are neglected in the amplitude of each processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} +page_content=' m = (δe − δg)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf'} diff --git a/pNE1T4oBgHgl3EQf2AWA/vector_store/index.faiss b/pNE1T4oBgHgl3EQf2AWA/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..06f6374cfc76f96b0e47396d1adfecfd7a2d28f5 --- /dev/null +++ b/pNE1T4oBgHgl3EQf2AWA/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:906ea69126d6d9ca8b175e1b1d34d658745b70be4ebae860aa78897db2326cce +size 5701677 diff --git a/pNE1T4oBgHgl3EQf2AWA/vector_store/index.pkl b/pNE1T4oBgHgl3EQf2AWA/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..4b2b5fb1e324d07e6a5275e4bcc289074f662c33 --- 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explaining their predictions has limited +their applicability in certain application areas. Due to the difficulty in iden- +tifying causal relationships between the input and output of such black-box +methods, they rarely have been adopted in domains such as legal and med- +ical fields in which the reliability and interpretability of the results can be +essential. +In this paper, we propose DANLIP, a novel deep learning-based +probabilistic time series forecasting architecture that is intrinsically inter- +pretable. We conduct experiments with multiple datasets and performance +metrics and empirically show that our model is not only interpretable but +also provides comparable performance to state-of-the-art probabilistic time +series forecasting methods. Furthermore, we demonstrate that interpreting +the parameters of the stochastic processes of interest can provide useful in- +sights into several application areas. +Keywords: +interpretability, time series, deep learning, forecasting +1. Introduction +Interpretable machine learning, a subfield of statistical learning, has re- +ceived a lot of attention in the last few years. Despite the significant achieve- +ments of statistical learning techniques, such methods are not fully adopted +in some decision-critical settings such as those frequently seen in legal and +medical domains [15]. It is primarily because, in such cases, decision-makers +are typically required to answer the question “Why did the machine learning +model make such decisions?”, as described by Molnar [16]. +Preprint submitted to Elsevier +January 9, 2023 +arXiv:2301.02332v1 [cs.LG] 5 Jan 2023 + +According to Doshi-Velez and Kim [4], interpretability in the context +of machine learning is defined as “the ability to explain or to present in +understandable terms to a human”. +Under such a criterion, naive neural +network-based learning methodologies fail to achieve interpretability due to +their black-box nature. Specifically, the neuron activation status of the net- +work is likely to fail to provide human-level understanding of how the neural +network makes the prediction. +Linear regression is frequently used as a traditional forecasting algorithm +and it can be considered a typical example of interpretable statistical learning +methodologies. +This method allows users to express the predictions as a +linear combination of covariates. That is, the model assumes the response +variable expressed as +y = +� +f∈F +γfxf + ε +(1) +where y is target, F is set of features, γf ∈ R is linear model coefficient for all +f ∈ F, and ε is a Gaussian noise. Standardized linear regression is regarded +as an interpretable forecasting method, as each linear combination coefficient +γf for feature f ∈ F can be taken as a degree of importance of that feature. +While linear regression has been extensively used, the method is not al- +ways applicable in practice. Its functional form restriction inherently limits +the expressive abilities of the model, as there exist many complex functions +that cannot be expressed as a linear combination of features. In this regard, +to achieve better performance, the use of complex learning methods is usu- +ally inevitable. However, such methodologies, e.g., deep neural networks, are +not necessarily interpretable. +Ensuring the interpretability of the model is extremely important in many +disciplines [15]. Even in some domains where interpretability is not deemed +to be as important, interpretable models can greatly help decision-makers +to make more informed decisions. For instance, in the case of a large retail +company, understanding the key factors that highly affect the sales of flagship +products might be of great interest in designing better production, inventory +management, and marketing policies. +There exist two main branches in the realm of interpretable machine +learning [16]. Post-hoc methods consist of applying techniques that analyze +the model after training, e.g., LIME [22] and SHAP [12]. On the other hand, +intrinsic methods involve building inherently interpretable models. This is +2 + +commonly achieved by restricting the complexity of the machine learning +model itself, e.g., RETAIN [3]. +In general, post-hoc methods such as SHAP are slow and not efficient +for large-scale machine learning problems. These methods usually require +a large number of experiments to calculate interpretability scores. Indeed, +the runtime complexity of the SHAP method is exponential in the num- +ber of features in the dataset [16]. In contrast, intrinsically interpretable +methodologies generally work fast. It has been shown that the performance +of proposed methods can be comparable to the non-interpretable methods [3]. +Accordingly, we concentrate on developing an intrinsically interpretable time +series forecasting method using neural networks. As intrinsic methods typ- +ically have a faster training time, this framework can be more suitable to +develop prediction algorithms over a large amount of training data. +Only a few studies have explored interpretable deep neural network-based +time series forecasting methods. Among these works, Temporal Fusion Trans- +former (TFT) [11] is highly related to our work, as the architecture also +achieves interpretability, while performing time series forecasts. However, +the TFT is fundamentally different from our work as it only computes impor- +tance scores rather than contribution scores of features. Unlike importance +scores, contribution scores allow users to quantify both the positive and neg- +ative impact of features on the prediction. We summarize the contributions +of our study as follows: +• We present DANLIP, a deep learning-based interpretable parametric +probabilistic time series forecasting model. +• We show our model can achieve comparable performance to those +of state-of-the-art probabilistic time series forecasting methods while +maintaining high interpretability. We conduct an extensive numerical +study with three real-world datasets to show the performance of our +proposed method. +• We empirically show how interpreting the predictions of the model +using the parameters of stochastic processes that are fitted to the model +can be used to provide important practical insights. +The rest of this paper is structured as follows. In Section 2, we briefly re- +view the relevant literature with a specific focus on RETAIN [3] and DeepAR [23] +models as two highly relevant approaches for probabilistic forecasting. In Sec- +tion 3, we present our interpretable time series forecasting model. Through- +3 + +out the section, we describe the model architecture and show how the pre- +diction results can be interpreted in terms of both mean and variance. In +Section 4, we show the performance of our model empirically using multiple +datasets evaluated over various performance metrics. Moreover, we discuss +how interpreting both the mean and variance of the probabilistic forecasts +can provide useful insights using real-world datasets. Finally, in Section 5, +we summarize our findings and discuss future work. +2. Background +In this section, we first briefly review the most relevant studies from the +literature and summarize time series forecasting basics. Then, we review the +studies by Salinas et al. [23] and Choi et al. [3] in detail, which provide the +preliminaries for our own methodology. +2.1. Related works +There is a vast literature on time series forecasting. Earlier studies in the +field mainly focused on statistical models such as Moving Average (MA), Au- +toregressive Moving Average (ARMA), and Autoregressive Integrated Mov- +ing Average (ARIMA) [2]. Decision trees and their ensembles, namely gradi- +ent boosted trees and random forests were also frequently used for time series +forecasting as they possess interpretable structures and have the ability to +capture nonlinear relations between various features in the dataset [5, 14]. +While the majority of the previous studies focus on point forecasting, +several recent studies proposed methodologies that are designed to perform +high-quality probabilistic forecasts. A probabilistic forecast typically refers +to the confidence interval around the point forecasts and is specified by the +lower and upper limits. Standard methods (e.g., ARIMA and exponential +smoothing) can generate probabilistic forecasts through closed-form expres- +sions for the target predictive distribution or via simulations [2]. +As more and more data has become available over the recent years, many +studies have focused on deep learning-based approaches for time series fore- +casting. For instance, recent studies by Rangapuram et al. [21] and Salinas +et al. [23] proposed deep learning models for probabilistic forecasting that +can directly predict the parameters (e.g., mean and variance) of the proba- +bility distribution that specifies the probabilistic forecast. These approaches +show substantial performance improvements over standard approaches for +datasets which consist of a large number of time series (e.g., in the order of +4 + +hundreds or thousands). Despite the success of these deep learning methods +for forecasting, they came with an important caveat of having highly com- +plex, black-box architectures, which lack interpretability and explainability. +We refer the reader to recent review articles by Parmezan et al. [20] for a +detailed overview of statistical and machine learning models for time series +forecasting. +In addition, recent forecasting competitions provide valuable +insights on best-performing time series forecasting methods [1, 13, 14]. +Explainable artificial intelligence also gained significant attention in re- +cent years. Post-hoc interpretability methods have been used to interpret +the decisions of time series models. Mujkanovic [17] used SHAP to interpret +time series classifiers, whereas Ozyegen et al. [19] evaluated three post-hoc +interpretability methods, including SHAP, to interpret the time series fore- +casting models. On the other hand, many time series forecasting methods +take into account interpretability considerations in model development [3, 8]. +While intrinsically interpretable methods usually trade off the performance +to provide more interpretability, the level of this trade-off varies according +to the model and the prediction task. It has been shown that some intrin- +sically interpretable models achieve similar performance levels to those of +black-box nature [10, 11]. Our study fundamentally differs from prior work +on intepretable time series forecasting, as the proposed architecture is able +to discern which features positively and negatively affect the prediction out- +come. +2.2. Preliminaries +In our analysis, we mainly rely on the standard neural network method- +ology. For the sake of convenience, we additionally introduce the following +notations: +• [N] = {1, 2, . . . , N} for any N ∈ N. +• Given A ∈ Rm×n, A[i, :] denotes the ith row of matrix A, and A[:, j] +denotes the jth column of matrix A. For one- and multi- dimensional +tensors, the notations are analogously extended. +• Given A ∈ Rm×n, A[i1 : i2, :] denotes (Aij)i∈{i1,...,i2−1},j∈[n]. The defini- +tions for A[:, j1 : j2] and A[i1 : i2, j1 : j2] are analogous. For one- and +multi- dimensional tensors, the notations are analogously extended. +• Icondition is an indicator function which takes value 1 if the condition +provided as the subscript is satisfied, and 0 otherwise. +5 + +• For any a ∈ RA and b ∈ RB, [a, b] refers to concatenation of two vectors, +i.e., [a, b] = [a1, · · · , aA, b1, · · · , bB]. +Following the self-explaining neural network model proposed by Melis and +Jaakkola [15], which is analogous to the RETAIN architecture of Choi et al. +[3], we ensure the interpretability of our forecasting method by restricting +the form of the output of the neural network as +y = +� +f∈F +γΘ(x)fxf. +(2) +Equation (2) is highly similar to Equation (1), however, different from Equa- +tion (1), the term that is multiplied with xf is now a function of input, +γΘ : D → R|F|, where D represents the domain of the input data. We char- +acterize the function γΘ(·) with Θ, which denotes the set of neural network +parameters. +2.3. Review of DeepAR architecture +We consider DeepAR [23], a state-of-the-art probabilistic parametric time +series forecasting method, as a baseline in our numerical study. DeepAR is +an RNN-based parametric probabilistic time series forecasting architecture, +which seeks to solve the log-likelihood maximization problem. Figure 1 pro- +vides an intuitive visual summary of the DeepAR architecture. +Network +Input +Figure 1: Simplified architecture of DeepAR +2.4. Review of RETAIN architecture +RETAIN is a neural network-based interpretable binary classification +method developed by Choi et al. [3]. +The architecture restricts the form +6 + +of prediction to be of Equation (2) and defines contribution to the prediction +of feature f as γ(x)f · xf. The architecture uses two RNNs. The sigmoid +function is applied over the sum of multiplication of the embedded input +vectors and output of RNNs, to predict binary variable y, i.e., variable of +interest. A simplified visualization of the RETAIN architecture is provided +in Figure 2. +Input +Embedding layer +Dense layer +Figure 2: Simplified architecture of RETAIN +More formally, given the input data (xt)t∈[T], where xt ∈ Rp for all t ∈ +[T], p being the number of features, the RETAIN architecture involves the +following set of mathematical operations: +vi = Wembxi +(3) +gi, . . . , g1 = RNNα(vi, . . . , v1) +(4) +ej = w⊤ +α gj + bα +for j ∈ [i] +(5) +α1, . . . , αT = Softmax(e1, . . . , ei) +(6) +hi, . . . , h1 = RNNβ(vi, . . . , v1) +(7) +βj = tanh(w⊤ +β hj + bβ) +for j ∈ [i] +(8) +ct = +� +j∈[i] +αjβj ⊙ vj +(9) +P(yi|(xt)t∈[i]) = Softmax(Wci + b) +(10) +7 + +where α computes time step-wise attention scores and ⊙ corresponds to the +Hadamard product operator. +Let r be the embedding dimension of the features. To define a feature +level contribution for the output, Choi et al. [3] show that Equation (10) can +be expanded as follows: +P(yi|(xt)t∈[i]) = Softmax +Ñ +� +j∈[i] +� +k∈[r] +xj,kαjW (βj ⊙ Wemb[:, k]) + b +é +(11) +Then, based on Equation (11), the contribution of feature k at time step j +for the prediction of yi can be obtained as +ω(yi; xj,k) = αjW(βj ⊙ Wemb)[:, k]xj,j +(12) +Note that the definition of quantity ω(yi; xj,k) is analogous to γΘ(x) compo- +nent of the Equation (2), which can be used to explain why RETAIN is an +interpretable model. +2.5. Probabilistic parametric time series forecasting +In general, time series forecasting methodologies can be classified as point +forecasting and probabilistic forecasting. In point forecasting, the objective +is to directly estimate the value for the subsequent steps of the target time +series. On the other hand, in probabilistic forecasting, the probability of +predicting a particular value is quantified for the target time series. More +specifically, let t be the time step and I ∈ N be the number of time series. Let +(xij, zi,j−1)t∈[I],j∈{t−h,··· ,t−1} such that (xij, zi,j−1) ∈ R|F|+1 is the collection of +time series dataset, where h is the history size. We call (xij)ij as covariates, +and (zij)ij as target series. In addition, assume that for all i ∈ [I], (xij)t∈[t,T]∩N +are given, where t + F > t with F ∈ N. Then, the goal of probabilistic +forecasting is to estimate the quantity given by +P((zi,j)i∈[I],t∈[t,...,t+F]∩N|(zi,j)i∈[I],j∈{t−h,··· ,t−1}, (xij)i∈[I],j∈{t−h,··· ,t+F}) +(13) +where F is the forecasting time steps. +There are two main approaches to estimate the quantity shown in Equa- +tion (13): +• Parametric methods involve estimating (zi,j)i∈[I],j∈{t+1,t+F}∩N by as- +suming (zi,j)i∈[I],j∈{t−h,··· ,t−1} is a stochastic process from some joint +8 + +density π(ς) with known π and unknown ς. As an example of paramet- +ric time series forecasting methodologies, we refer the readers to the +work of Salinas et al. [23]. +• Non-parametric methods involve directly estimating Equation (13) with- +out assuming any knowledge about π. For further details of such a +non-parametric methodology, we refer readers to the works of Koenker +and Bassett Jr [9], and Lim et al. [11]. +In this paper, we only focus on the parametric case. The primary benefit +of proposing interpretable parametric time series forecasting methods is that +it allows interpreting parameters of the stochastic process. For instance, by +interpreting the parameters of a Gaussian process, we can achieve interpre- +tation for both the mean value and standard deviation of the probabilistic +forecasts. +3. DANLIP Architecture +We next describe DANLIP by detailing the model specifications and the ar- +chitecture, and we discuss how the interpretability is achieved by this model. +3.1. Model description +DANLIP is designed to perform joint density parameter estimation via +performing time step-wise parameter forecasts. More precisely, we restrict +the stochastic processes to be Gaussian, as it is a probability distribution +which directly use the mean and standard deviation as its input. However, +we note that DANLIP architecture can be appropriately modified to handle +other types of stochastic processes. For instance, following the steps proposed +by Salinas et al. [23], such modifications can be easily achieved. For the +sake of better readability, we eliminate the time series indices i in below +mathematical equations. +Firstly, let M be number of continuous features, and N be number of +categorical features. We note that the input data can be decomposed as a +vector containing both target series value, continuous features and discrete +features. More precisely, input data can be represented as shown in Equa- +tion (14), that is, for all j ∈ {t − h, · · · , t − 1}, where h is history size, time +series with covariate can be written as +[zj, (xCont +jf +)M +f=1, (xCat +jf )N +f=1] +(14) +9 + +where zj is target series value at time step j, (xCont +jf +)M +f=1 denotes vector of +continuous components of vector x, while (xCat +jf )N +f=1 denotes the categorical +components. For the sake of notational convenience, we make the following +assumptions: +• Without loss of generality, we assume such an index ordering as default. +• We let FCat be set of categorical feature indices of the vector shown in +Equation (14). Analogously, we define FCont for continuous features. +• We assume xCat +jf ∈ N, i.e., categories are represented in terms of natural +numbers. We assume that the values of (xCat +itf )it lie in a subset of ordered +natural numbers without any gap starting from 0, so the values of +categorical features can be one-hot encoded. +The DANLIP is defined as concatenation of three important parts: feature +processing layer, encoding layer and decoding layer. In Figure 3, we present +a simplified overview of the entire neural network architecture for DANLIP. +We start the model description by depicting feature processing layer. For +each time step j ∈ {t − h, t − 1}, +eCat +f += Embedding(xCat +jf ) +∀f ∈ FCat +(15) +vj = (zj, xCont +j,1 +· · · xCont +j,M , ej,1 · · · ej,N) +(16) +where for all f ∈ FCat, eCat +jf +∈ Rξf with ξf representing the embedding size +of categorical feature f ∈ FCat. Using notations above, we define the input +for the encoding layer as +v = [vt−h, · · · , vt−1] +(17) +Then we can subsequently define the encoding layer as follows: +gt−h, . . . , gt−1 = RNNα(vth, . . . , vt1) +(18) +qj = w⊤ +α gj + bα +∀j ∈ {t − h, · · · , t − 1} +(19) +αt−h, . . . , αt−1 = Softmax(qt−h, . . . , qt−1) +(20) +ht−h, . . . , ht−1 = RNNβ(vt−h, . . . , vt−1) +(21) +βj = tanh(w⊤ +β hj + bβ) +∀j ∈ {t − h, · · · , t − 1} +(22) +cj = αjβj ⊙ vj +∀j ∈ {t − h, · · · , t − 1} +(23) +10 + +Identity +Embedding +PREPROCESSING +Concatenation +RNN 𝛼 +(Layer 1) +ENCODER +RNN 𝛼 +(Layer 𝑳𝜶) +RNN 𝛽 +(Layer 1) +RNN 𝛽 +(Layer 𝑳𝜷) +DECODER +[𝑧𝑖,𝑡−ℎ|𝑥𝑖,𝑡−ℎ +𝐶𝑜𝑛𝑡] +[𝑧𝑖,𝑡−1|𝑥𝑖,𝑡−1 +𝐶𝑜𝑛𝑡] +𝑥𝑖,𝑡−ℎ +𝐶𝑎𝑡 +𝑥𝑖,𝑡−1 +𝐶𝑎𝑡 +⋯ +⋯ +𝑣𝑖,𝑡−ℎ +𝑣𝑖,𝑡−1 +⋯ +𝛼𝑖,𝑡−ℎ +𝛼𝑖,𝑡−1 +⋯ +𝛽𝑖,𝑡−1 +𝛽𝑖,𝑡−1 +⋯ +⊙ +⊙ +⋯ +𝑐𝑖,𝑡−ℎ +𝑐𝑖,𝑡−1 +⋯ +⋯ +⋯ +𝑐 = +Dense +Dense +Dense +Dense +𝑣 = +[𝑐|𝑑𝑡] +[𝑐|𝑑𝑡+𝐹] +Ƹ𝜇𝑡 +ො𝜎𝑡 +Ƹ𝜇𝑡+𝐹 +ො𝜎𝑡+𝐹 +⋯ +⋯ +Figure 3: DANLIP model architecture +where both RNNα(·) and RNNβ can be either uni- or bi-directional. +The output of the encoding layer is +c = (ct−h, · · · , ct−1) +(24) +11 + +which can be used to define the input for the decoding layer as +dj = [zj−1, xCont +j,1 +· · · xCont +j,M , ej,1 · · · ej,N] +∀j ∈ {t, · · · , t + F} +(25) +d = (dt, · · · , dt+F) +(26) +Using the sequence of vectors (cj)j∈{t−h,··· ,t−1}, we can finally perform mean +and standard deviation prediction in the decoding layer by using the following +formula: +µj = W ⊤ +µ,j · [c|dj] +∀j ∈ {t, · · · , t + F} +(27) +σj = Softplus(W ⊤ +σ,j · [c|dj]) +∀j ∈ {t, · · · , t + F} +(28) +where Wµ,j, Wσ,j ∈ R1×N×� +f∈F df. We note that both µit and σit are later +used to define the parameters for output conditional univariate Gaussian +distribution. +Given the neural network output, the neural network weights are opti- +mized to solve the likelihood maximization problem given by +max +Θ +� +i∈I +t+F +� +j=t +ℓUVN(zi,j|µij(Θ), σij(Θ)) +(29) +where, instead, we solve the log-likelihood maximization problem described +by Equation (30), that is, +max +Θ +� +i∈I +t+F +� +j=t +log(ℓUVN(zi,j|µij(Θ), σij(Θ))) +(30) +We recover the eliminated index i to describe the time series indices. We use +ℓUVN(·|·) to denote univariate Gaussian likelihood, where Θ is the vector of +all network parameters optimized according to the network inputs. +During the training of the DANLIP, the input for decoding layer is defined +using true target series value zj for each j ∈ {t−h, · · · , t−1}. However, dur- +ing the prediction phase, the future information zj for j ∈ {t, · · · , t + F} are +not accessible. To overcome this issue, we perform autoregressive prediction, +and we replace the decoding layer input vector dj by ˆdj. Specifically, +dt = [zt−1, xCont +t,1 +· · · xCont +t,M , et,1 · · · et,N] +(31) +12 + +ˆdj = [ˆzj−1, xCont +j,1 +· · · xCont +j,M , ej,1 · · · ej,N] +∀j ∈ {t + 1, · · · , t + F} +(32) +ˆd = (dt, ˆdt+1, · · · , ˆdt+F) +(33) +where ˆzj−1 is a sample drawn from π(ˆµj−1, ˆσj−1), which denotes the predicted +target series value for all j ∈ {t+1, · · · , t+F}. During the forecasting phase, +DANLIP performs sequence sampling as shown in Figure 4. +DECODER IN PREDICTION PHASE +Dense +Dense +Dense +Dense +[𝑐|𝑑𝑡] +Ƹ𝜇𝑡 +ො𝜎𝑡 +Ƹ𝜇𝑡+𝐹 +ො𝜎𝑡+𝐹 +⋯ +Dense +Dense +[𝑐| መ𝑑𝑡+1] +Ƹ𝜇𝑡+1 +ො𝜎𝑡+1 +[𝑐| መ𝑑𝑡+𝐹] +Figure 4: Decoder of the DANLIP architecture during forecasting phase +3.2. Model interpretation +We can assess how much the input features contribute to the model output +when the DANLIP makes a prediction. For the sake of brevity, let φ = 1 + +|FCont| + �f +r=0 ξr be the length of the one time step context vector. We +can define the contribution of feature f at time step s for µj prediction as +follows: +ωµ(f, s, j) = αsWµ,j[(s − (t − h)) · φ] · (βs[0] · zs), +f /∈ FCont ∪ FCat +(34) +ωµ(f, s, j) = αsWµ,j[(s − (t − h)) · φ + f] · (βs[f] · xsf), +f ∈ FCont +(35) +ωµ(f, s, j) = αsWµ,j +ï +(s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ +ò +· +� +βs +ï +(s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ +ò +⊙ W Cat,Emb +f +[:, xsf] +� +, +f ∈ FCat +(36) +13 + +Similarly, the contribution for σj prediction is defined as +ωσ(f, s, j) = αsWσ,j[(s − (t − h)) · φ] · (βs[0] · zs), +f /∈ FCont ∪ FCat +(37) +ωσ(f, s, j) = αsWσ,j[(s − (t − h)) · φ + f] · (βs[f] · xsf), +f ∈ FCont +(38) +ωσ(f, s, j) = αsWσ,j +ï +(s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ +ò +· +� +βs +ï +(s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ +ò +⊙ W Cat,Emb +f +[:, xsf] +� +, +f ∈ FCat +(39) +We note that the provided formula is valid, as long as all continuous +features come before categorical features, and target series come before all +continuous features. Without such an assumption, the formula for the con- +tribution score must be appropriately modified. When ω(f, s, j) > 0 the +contribution of feature f at timestep j is positive. This means the provided +component contributes to an increase in the value of the prediction. On the +other hand, a negative value means that the contribution score is negative. +Without the monotonically increasing property of the activation functions of +the dense layers in the decoder, such an interpretation would not be possible. +It is also important to note that the contributions of features on standard +deviation are nonlinear, which is due to the existence of the Softplus opera- +tor. By taking the norm of the contribution scores |ω(f, s, j)|, we can obtain +the importance of the features, instead of contribution scores. These features +can then be averaged over the prediction steps and samples of the dataset to +obtain the average importance of the input features over the dataset, which +enables the global-level contribution score computation for the features. +4. Numerical Study +In this section, we present the results from detailed experiments to eval- +uate the forecasting performance and interpretability of the proposed model, +DANLIP. We first provide the experimental setup and detail the considered +forecasting models, hyperparameters and datasets. Then, we compare the +14 + +forecasting performance of several models using well-known point and prob- +abilistic forecasting evaluation metrics. Next, we focus on the results that +demonstrate the interpretability aspects of DANLIP. Specifically, we first show +the explanations produced by the model, and then discuss these explanations +as they relate to the underlying prediction task. +4.1. Experimental setup +We consider three diverse datasets with different characteristics in terms +of size, and observed seasonality/trends among others. +We use a sliding +window method for framing the datasets, and separate the last two forecast- +ing horizons of each time series in the datasets as validation and test sets. +We perform a detailed hyperparameter tuning for all models using Tree- +structured Parzen Estimator (TPE) algorithm. The parameter ranges for +the hyperparameter tuning are shown in Table 1. For DANLIP and DeepAR, +we experiment with different number of RNN layers, RNN cell types, hidden +units, dropout and learning rates. For the MLP model, we experiment with a +similar range of hidden layers, learning and dropout rates. For hidden units, +we use a wider range, as dense neural networks are usually trained with a +larger number of hidden units to achieve similar performance compared to +the RNN based networks. For the GBR model, we test three important pa- +rameters which are the number of trees in the ensemble, the number of leaves +in each tree (i.e., max depth), and the minimum number of samples required +to split an internal node (i.e., min samples split). +Table 1: The hyperparameter tuning search space. +Model +Search space +DANLIP/ DeepAR +# Hidden units: [16, 128], +Dropout rate: [0, 0.5], +Cell Type: LSTM or GRU, +# RNN layers: [1, 8], +Learning Rate: [1e-4, 1e-1] (log uniform) +MLP +# Hidden units: [50, 500], +Dropout rate: [0, 0.5], +# Hidden layers: [1, 8], +Learning Rate: [1e-4, 1e-1] (log uniform) +GBR +# of trees: [10, 200], +Max depth: [2, 5], +Min samples split: [2, 15] +We run the TPE algorithm for 100 trials, optimizing over the normalized +deviation metric on the validation set. The neural network models are im- +15 + +plemented using Pytorch, the GBR model is implemented using Scikit-learn, +and hyperparameter tuning is performed using the Optuna library. All the +experiments are run on a computing node with RTX2070 Super 8GB GPU, +and 128GB of RAM, running on Debian Linux OS. +We consider two sets of performance metrics to evaluate both point and +probabilistic forecasting performance. +For measuring the quality of point +forecasts, we use Normalized Root Mean Squared Error (NRMSE) and Nor- +malized Deviation (ND) similar to Salinas et al. [23]. NRMSE and ND can +be obtained for ground truth values (y) and the prediction (ˆy) as follows: +NRMSE(y, ˆy) = +» +1 +N +�N +i=1(ˆyi − yi)2 +1 +N +�N +i=1 |yi| +, +ND(y, ˆy) = +�N +i=1 |ˆyi − yi| +�N +i=1 |yi| +(40) +To evaluate the probabilistic forecasting ability of the models, we consider +the ρ-risk, also known as quantile loss. The ρ−risk can be obtained as follows: +ρα(y, ˆyα) = +�N +i=1 max{α(yi − ˆyi +α), (1 − α)(ˆyi +α − yi)} +�N +i=1 |yi| +(41) +where ˆyα represents the prediction at quantile level α. +Note that having +a lower value of ρ-risk is better. Finally, we apply Friedman test, a non- +parametric test, to compare the forecasting models on multiple datasets [6]. +For this comparison, we fill each group with the average error results on the +tested datasets. Then, we apply the Friedman test to find out whether there +is a statically significant difference between the mean of the groups. +We consider three datasets in our experiments, namely, Electricity, Ross- +mann and Walmart datasets, which we briefly describe below. +• Electricity: The Electricity dataset contains the hourly electricity con- +sumption records of 370 households. +The dataset has been exten- +sively used in previous studies for performance benchmarking purposes +[11, 23]. In our experiments, besides the electricity consumption time +series, we use additional covariates such as hour of the day, day of the +week, week of the month, and month. Input window size and forecasting +horizon are taken as 168 and 12, respectively. +• Rossmann: The Rossmann sales dataset was published as a part of +a Kaggle competition in 2015 and it contains a rich set of features +and extensive daily sales histories. The dataset consists of daily sales +16 + +records of multiple Rossmann stores, with various covariates. Among +the available features, we use sales value, store index, store open indi- +cator, promotion indicator, state holiday indicator, school holiday indi- +cator, weekday, month, and week of the month. Input window size and +forecasting horizon are taken as 30 and 12, respectively. +• Walmart: The Walmart store sales forecasting dataset contains weekly +store sales of 77 departments in 45 stores, and it was made available +as part of a Kaggle competition in 2014. The dataset includes multiple +features such as temperature, information on markdowns, and various +economic indicators (e.g., unemployment rates, fuel prices and CPI). +Input window size and forecasting horizon are taken as 30 and 6, re- +spectively. +4.2. Results on model performances +We provide summary statistics on the model performances in Table 2. +These results are obtained after performing rigorous fine-tuning for each +dataset-model pair on the validation set. The Friedman test over the ag- +gregate results (i.e., results for all three datasets combined) returns p-values +of 0.12, 0.08, 0.12, and 0.12 for NRMSE, ND, ρ0.75, and ρ0.90, respectively, +which indicates that there is no statistically significant difference (p-value > +0.05) between DANLIP and DeepAR models. Similarly, we see that DANLIP +and DeepAR achieve similar forecasting performance in terms of average +performance values. DANLIP performs marginally better for the Rossmann +and Electricity datasets, whereas DeepAR performs marginally better for the +Walmart dataset. +Baseline models, MLP and GBR, are significantly less complex compared +to DeepAR and DANLIP. MLP leads to the poorest forecasting performance +among the tested models for all the datasets as shown by various forecasting +performance values. GBR outperforms DANLIP and DeepAR on the Ross- +mann dataset, where the dataset shows a significant amount of seasonality. +For the Electricity dataset, GBR ranks second in terms of the NRMSE met- +ric, and first in terms of the ND metric, with a marginal difference in both +cases. For the Walmart dataset, GBR ranks third after DeepAR and DANLIP +in terms of the NRMSE metric, and second in terms of the ND metric. The +high performance of the GBR can be attributed to the way we train this +model. Specifically, for the GBR model, we train a separate model for each +time series on the dataset, whereas, for the other models, we train a single +17 + +Table 2: Summary statistics of model performances. Mean and standard deviation across +10 randomly seeded runs are reported over the test sets. +Dataset +Model +NRMSE +ND +ρ0.75 +ρ0.90 +mean +std +mean +std +mean +std +mean +std +Electricity DeepAR +0.255 +0.017 +0.077 +0.005 +0.133 +0.026 +0.250 +0.041 +GBR +0.223 +0.002 +0.066 +0.000 +0.116 +0.000 +0.229 +0.002 +MLP +0.359 +0.043 +0.120 +0.010 +0.238 +0.019 +0.451 +0.035 +DANLIP +0.221 +0.020 +0.070 +0.004 +0.130 +0.017 +0.249 +0.027 +Rossmann +DeepAR +0.177 +0.027 +0.119 +0.020 +0.241 +0.070 +0.463 +0.121 +GBR +0.148 +0.000 +0.102 +0.000 +0.195 +0.000 +0.483 +0.001 +MLP +0.179 +0.005 +0.122 +0.003 +0.281 +0.013 +0.596 +0.032 +DANLIP +0.153 +0.005 +0.105 +0.004 +0.214 +0.027 +0.407 +0.056 +Walmart +DeepAR +0.152 +0.013 +0.076 +0.007 +0.136 +0.026 +0.248 +0.043 +GBR +0.198 +0.005 +0.093 +0.001 +0.164 +0.007 +0.363 +0.025 +MLP +0.228 +0.023 +0.120 +0.013 +0.250 +0.037 +0.465 +0.062 +DANLIP +0.186 +0.010 +0.095 +0.006 +0.151 +0.032 +0.307 +0.061 +model for all the time series in the dataset. This training methodology for +GBR is computationally more expensive. However, it can result in a sig- +nificantly improved forecasting performance when the availability of similar +time series does not benefit the prediction. Our preliminary analysis indicates +that training a single GBR model for each dataset (i.e., similar to the other +three models) leads to a significant deterioration in forecasting performance +across the datasets, hence we adopt the above-explained approach. We note +that above results for these datasets are largely inline with the forecasting +performance values reported in previous studies (e.g., see [8, 19, 23]). +Figure 5 shows the visualization of forecasts by DeepAR and DANLIP for +three datasets. We use blue lines to represent the ground truth, and orange +lines to denote the predicted mean values of models. We depict 75%, 90% +and 98% prediction intervals as shaded areas. The forecasting horizons are 28 +days, 24 hours, and 6 weeks for Rossmann, Electricity, and Walmart datasets, +respectively. Overall, we observe that all the models can generate predictions +that capture the general trends in the datasets and probabilistic forecasts are +highly similar for these two models. +18 + +(a) Rossmann: DeepAR +(b) Rossmann: DANLIP +(c) Electricity: DeepAR +(d) Electricity: DANLIP +(e) Walmart: DeepAR +(f) Walmart: DANLIP +Figure 5: Visualization of DeepAR and DANLIP predictions on random time series samples. +Each figure displays forecasts for one prediction window. Point forecasts are shown in the +orange line, 75%, 90% and 98% quantile forecasts are shown as orange regions. +Each +model generates predictions that can capture the trends for the provided sample. +19 + +observed +predicted +45000 +40000 +DOSE +ADODE +25000 +24000 +15000 +-30-25-20- +5 +0 +Time indextobserved +predicted +14000 - +12000 +8400 +DO+ +4000 +2400 +0 +30 +20 +-i0 +2f +Time indeobserved +predicted +12000 +8+00 +00 +4000 +2400 +0 +-30 +20 +-i0 +0 +2 +Time indextobserved +predicted ++00 +5000 +4000 +3000 +2400 +1400 +DSL- +-125 +-100 +75 +50 +-25 +0 +25 +Time indextobserved +predicted +00 +5000 +4000 +3400 +2400 +1000 +DSL- +-125 +-100 +75 +05- +-25 +0 +25 +Time indextobserved +predicted +45000 +40000 +ADOSE +DODE +25000 +15000 +5 +0 +Tirme indet4.3. Results on model interpretability +We next provide results on the interpretability of DANLIP. We focus on the +Rossmann dataset as the representative case, as it has a large, interesting set +of features that correlate well with the target variable (e.g., Promo). For the +Electricity dataset, we see that the target value (i.e., electricity load) has the +most impact on the predictions, and certain time covariates such as “week- +of-month” help reduce the variance of the predictions. +For the Walmart +dataset, we again see that the target value (i.e., weekly sales) has the most +impact on the predictions. The interpretability results for the Walmart and +Electricity datasets are provided in the Appendix (see Appendix A.1). +We discuss the interpretability of DANLIP by analyzing the explanations +obtained from the model weights. +Following the similar work [7, 18], we +visualize the explanations as feature contribution heatmaps. The contribu- +tion score formulas described in the methodology are applied to obtain the +contribution of each input feature to the prediction. Note that we collect a +separate contribution score for each sample, input feature, and forecasting +horizon. +This allows us to achieve local interpretability, which helps un- +derstanding how the features contribute to a single prediction. +It is also +possible to aggregate these scores over forecasting horizons and samples to +achieve global interpretability, which helps understand the features that are +important for the model. +Figure 6 shows the contributions scores obtained from the model for a +single sample. We select the 9th forecasting horizon of the first sample from +the Rossmann test set to generate these visualizations. This sample and the +forecasting horizon provide a clear overview of how the model behaves under +certain events. The forecasted date for this sample is a Monday, the store is +Open, there is a promotion event, and there is no state or school holiday. +Figure 6a shows that ‘DayOfWeek’, ‘Open’, ‘Promo’, and ‘StateHoliday’ +have the biggest positive contribution to the prediction. This is intuitive since +there are no sales on the weekends, and Mondays should expect a higher sale. +The store being open, the promotion events and the state holidays are all +positive contributors to the prediction. Looking at the encoder contribu- +tions, we observe that the encoder features have negligible impact on the +prediction. However, we also note that recent and certain seasonal values +of the ‘DayOfWeek’ feature have higher impacts on predictions compared to +the other encoder features. +Figure 6b shows the contribution of the features to the predicted vari- +ance. Overall, we observe that the ‘StateHoliday‘, ‘Store’, and ‘SchoolHoli- +20 + +day’ decoder features have significant negative contributions, and ‘Open’ and +‘Promo’ have positive contributions to the prediction. The negative values +for the store and holiday features show that they reduce the variance of the +predictions. Interestingly, ‘StateHoliday’ has the largest negative contribu- +tion to the variance. Since there is no date corresponding to a holiday in +this example, the explanation may indicate that state holidays can result +in a higher uncertainty (variance), which requires further investigation. On +the other hand, we observe a positive contribution for ‘Open’, and ‘Promo’ +when the store is open and there is a promotion. This is expected as there is +higher uncertainty in the predicted sales when the store is open as opposed +to the days when the store is closed and there are no sales. The explanation +also suggests that the existence of promotion has a similar impact on the +prediction, and it causes a higher variance. +(a) µ contributions +(b) σ contributions +Figure 6: Contribution scores obtained from DANLIP for the first sample and 9th target +horizon of the Rossmann test set. Figure (a) shows the contributions to the µ and (b) +shows contributions to the σ. +In certain cases, it might be useful to calculate the overall importance +of the features for the model, instead of their importance for a particular +prediction. These feature importance values can then be used for different +purposes including feature selection. We apply the following steps to the +contribution scores to find the overall importance. First, we obtain contri- +bution scores for each sample in the dataset. The resulting array has the +shape: #forecasting horizon × #samples × #timesteps × #features. We +then take the absolute values of this array, as we are interested in the over- +all importance of the features, not their contributions. Finally, we average +21 + +Encoder +30 +24. +0.050 +21 +-18 + 0.025 +15 +0.D0O +12 +.025 +050' +Decoder +xpyaug +Dayorweek +uado +StateHolideyEncoder +30 +24 + 0.05 +21 +18 +15 + 0.00 +12 +50'0- +Decoder +0.5 +0.0 +±.5 +xp +Dayorweek +StateHolidey +Taug +. +4the array over the forecasting horizons, and then over the samples to obtain +the importance scores of the input features. We show the visualizations of +these arrays for µ and σ in Figure 7a and Figure 7b, respectively. These +figures show the overall importance of the features for the model. Thus, all +the values displayed are positive numbers, and they are displayed in green +color. +Looking at the importance scores for µ in Figure 7a, we observe +that the same four decoder features from the sample contributions (‘Day- +OfWeek’, ‘Open’, ‘Promo’, and ‘StateHoliday’) are found to be important +for the model. Additionally, we note that various encoder timesteps of the +‘Sales’, and ‘DayOfWeek’ features are also important, suggesting that the +model attends to different timesteps of these features for different samples of +the dataset. Overall importance scores for σ shown in Figure 7b follow a sim- +ilar pattern compared to the sample contribution scores. The same decoder +features, i.e., ‘Stateholiday’, ‘Store’, ‘Open’, ‘Promo’, and ‘SchoolHoliday’ +are all important for the model. Additionally, the ‘DayOfWeek’ feature of +the decoder is also important for predicting the target value. On the decoder +side, certain past values of ‘Sales’, ‘DayOfWeek’ and ‘StateHoliday’ are all +found to be important to the variance prediction. +(a) µ average importance +(b) σ average importance +Figure 7: Contribution scores obtained from DANLIP for all samples, and averaged over +samples to obtain global importance scores. Figure (a) shows the average importance to +the µ and (b) shows average importance to the σ in Rossmann dataset. +5. Conclusion and Discussions +In this paper, we present a novel deep probabilistic intrinsically inter- +pretable time series forecasting method, DANLIP, which is designed to predict +22 + +Encoder +27 +30 +24 +0.D2 +21 +&T +0.D1 +15 + 0.00 +12 +.0.01 +0.02 +Decoder +paug +aos +Dayorweek +uadb +StateHolideyEncoder +27 +30 + 0.04 +24 +21 +0.D2 +-18 +-15 + 0.00 +-12 +0.02 +0.04 +Decoder +0.0 +0.5 +xp aug +aos +Dayorweek. +uadb +PrDka +StateHolidey +丽both mean and standard deviation of the underlying Gaussian processes. We +show how ordinal and categorical covariates can be appropriately incorpo- +rated into DANLIP. Then, we describe procedures to compute the contribution +of both ordinal and discrete features using DANLIP network parameters. We +also compare DANLIP against various baselines and show that our method has +a competitive forecasting performance to DeepAR, a state-of-the-art prob- +abilistic time series forecasting method. We also analyze the contribution +scores generated for mean and standard deviation. We show how contribu- +tion scores can be used to analyze model predictions for a single sample and +to find the overall importance of the features for the model. +An area that could be further explored is the idea of using the con- +tribution scores for feature selection. Because the contribution scores are +calculated directly from the model weights, they may achieve a higher per- +formance in feature selection compared to the alternative methods. Secondly, +the proposed architecture can be tested with other datasets. While DANLIP +achieves similar performance to DeepAR for the tested datasets, the architec- +ture might require further tuning for other datasets and problems. Finally, +the role of regularization can be analyzed in finding the most useful explana- +tions. Methods such as L1 regularization and Dropout can be used to tweak +the contributions assigned to the correlated features. Depending on the use +case, certain regularization hyperparameters can be selected to obtain the +most useful explanation. +Acknowledgment +This research is in part supported by LG Science Park. +Statements and Declarations +No potential conflict of interest was reported by the authors. +References +[1] C. S. Bojer and J. P. Meldgaard. Kaggle forecasting competitions: An +overlooked learning opportunity. International Journal of Forecasting, +37(2):587–603, 2021. +[2] G. E. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung. Time series +analysis: forecasting and control. John Wiley & Sons, 2015. +23 + +[3] E. Choi, M. T. Bahadori, J. Sun, J. Kulas, A. Schuetz, and W. Stewart. +Retain: An interpretable predictive model for healthcare using reverse +time attention mechanism. In Advances in Neural Information Process- +ing Systems, pages 3504–3512, 2016. +[4] F. Doshi-Velez and B. Kim. Towards a rigorous science of interpretable +machine learning. arXiv preprint arXiv:1702.08608, 2017. +[5] A. Galicia, +R. Talavera-Llames, +A. Troncoso, +I. Koprinska, +and +F. Mart´ınez-´Alvarez. +Multi-step forecasting for big data time series +based on ensemble learning. Knowledge-Based Systems, 163:830–841, +2019. +[6] S. Garc´ıa, A. Fern´andez, J. Luengo, and F. Herrera. Advanced non- +parametric tests for multiple comparisons in the design of experiments +in computational intelligence and data mining: Experimental analysis +of power. Information sciences, 180(10):2044–2064, 2010. +[7] T. Guo, T. Lin, and N. 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A unified approach to interpreting model +predictions. arXiv preprint arXiv:1705.07874, 2017. +[13] S. Makridakis, E. Spiliotis, and V. Assimakopoulos. The M4 compe- +tition: 100,000 time series and 61 forecasting methods. International +Journal of Forecasting, 36(1):54–74, 2020. +[14] S. Makridakis, E. Spiliotis, and V. Assimakopoulos. The M5 accuracy +competition: Results, findings and conclusions. International Journal +of Forecasting, 2020. +24 + +[15] D. A. Melis and T. Jaakkola. Towards robust interpretability with self- +explaining neural networks. In Advances in Neural Information Process- +ing Systems, pages 7775–7784, 2018. +[16] C. Molnar. Interpretable Machine Learning. Lulu. com, 2020. +[17] F. Mujkanovic. Explaining the predictions of any time series classifier. +Master’s thesis, Hasso Plattner Institut, 7 2019. +[18] B. Norgeot, D. Lituiev, B. S. Glicksberg, and A. J. Butte. Time ag- +gregation and model interpretation for deep multivariate longitudinal +patient outcome forecasting systems in chronic ambulatory care. arXiv +preprint arXiv:1811.12589, 2018. +[19] O. Ozyegen, I. Ilic, and M. Cevik. +Evaluation of local explana- +tion methods for multivariate time series forecasting. +arXiv preprint +arXiv:2009.09092, 2020. +[20] A. R. S. Parmezan, V. M. Souza, and G. E. Batista. Evaluation of statis- +tical and machine learning models for time series prediction: Identifying +the state-of-the-art and the best conditions for the use of each model. +Information Sciences, 484:302–337, 2019. +[21] S. S. Rangapuram, M. W. Seeger, J. Gasthaus, L. Stella, Y. Wang, and +T. Januschowski. Deep state space models for time series forecasting. +In Advances in neural information processing systems, pages 7785–7794, +2018. +[22] M. T. Ribeiro, S. Singh, and C. Guestrin. ” why should i trust you?” ex- +plaining the predictions of any classifier. In Proceedings of the 22nd ACM +SIGKDD international conference on knowledge discovery and data min- +ing, pages 1135–1144, 2016. +[23] D. Salinas, V. Flunkert, J. Gasthaus, and T. Januschowski. Deepar: +Probabilistic forecasting with autoregressive recurrent networks. Inter- +national Journal of Forecasting, 36(3):1181–1191, 2020. +25 + +Appendix A. +Appendix A.1. Additional Results on DANLIP Interpretability +Figure A.1 shows the contribution scores obtained from the model for a +single sample, and Figure A.2 shows the overall importance of the features +for the model. Overall, we find that the target feature (Weekly Sales) has the +most significant impact, particularly at the decoder step, and at the recent +timesteps of the encoder. Looking closely at the contribution scores, we also +observe that various timesteps of “Dept”, “year”, “month”, ”weekofmonth”, +and “day” features contribute to the predictions. However, looking at the +importance scores in Figure A.2, we observe that the overall impact of these +features is insignificant for the predicted µ, but significant for the predicted +σ. +(a) µ contributions +(b) σ contributions +Figure A.1: Contribution scores obtained from DANLIP for the first sample and 4th target +horizon of the Walmart test set. Figure (a) shows the contributions to the µ and (b) shows +contributions to the σ. +Figure A.3 shows the contribution scores obtained from the model for a +single sample, and Figure A.4 shows the overall importance of the features +for the model. Similar to the Walmart dataset, we observe that the target +feature (series) has the most significant impact, particularly at the decoder +timestep. Comparing contribution scores to the overall importance scores, +we find that in different samples, the model tends to attend different past +encoder timesteps of the “series” feature. This is unlike the Walmart dataset, +in which the model attends more to the recent timesteps of the encoder. +This suggests that, for the Walmart dataset, the recent timesteps are the +26 + +Encoder +o- +0.02 +0.00 +: +0.04 +Decoder + 0.5 +0.0 +0.5 +idx +ko +Unemployinent . +Weekly_ Sakes +aios +a +derak +weekofinonith +Aep +Taugmost informative, whereas, for the Electricity, the seasonal values of the +target feature (t − 24, t − 48 · · · ) are the most informative. Looking at the +contribution scores for σ in Figure A.3b, we find that the “house” feature, +which indicates from which house the electricity load information is collected, +leads to an increase in the predicted variance. On the other hand, decoder +timesteps of the time covariates (“hour”, weekday”, and “weekofmonth”) +reduce the predicted variance. +(a) µ average importance +(b) σ average importance +Figure A.2: Contribution scores obtained from DANLIP for all samples, and averaged over +samples to obtain global importance scores. Figure (a) shows the average importance to +the µ and (b) shows average importance to the σ in Walmart dataset. +(a) µ contributions +(b) σ contributions +Figure A.3: Contribution scores obtained from DANLIP for the first sample and 4th target +horizon of the Electricity test set. Figure (a) shows the contributions to the µ and (b) +shows contributions to the σ. +27 + +Encoder + 0.04 +0.D2 +0.D0 +0.02 +0.04 +Decoder +azs +Emperature +ko +Unemplpyinent . +Weekly_ Sales +Type. +JepEncoder +0.D2 +0.00 +±.02 +Decoder +0.5 +xpy awug +Emperature. +kD +Unemplpyinent +Weekly_Sales +aos +a +derak +Lauouyeyaam +Aep +IsHolidByEncoder +168 +144 + 0.D4 +120 + 0.02 +-72 + 0.00 +48 +0.02 +24 +Decoder +上8.8 +4.5 +xpl aug +weekofinorrth +menth +国Encoder +168 +-144 +120 + 0.02 +-72 +0.00 +48 +0.02 +24 +Decoder +E3. +xp! +BB +weekofinorith +menth +Faug(a) µ average importance +(b) σ average importance +Figure A.4: Contribution scores obtained from DANLIP for all samples, and averaged over +samples to obtain global importance scores. Figure (a) shows the average importance to +the µ and (b) shows average importance to the σ in Electricity dataset. +28 + +Encoder +168 +-144 +0.0050 +120 +0.D025 +-72 +0.0000 +48 +0.0025 +24 +Decoder +xp! +SBHU +asn +weekofinorith +FaugEncoder +-168 +-144 +0.005 +120 +-72 +0.000 +48 +24 +0.005 +Decoder +L-2 +xpI +weekofinonith \ No newline at end of file diff --git a/qNE0T4oBgHgl3EQfaQBn/content/tmp_files/load_file.txt b/qNE0T4oBgHgl3EQfaQBn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5933923cb2f616998c54ca3eaa30f8464d3aba99 --- /dev/null +++ b/qNE0T4oBgHgl3EQfaQBn/content/tmp_files/load_file.txt @@ -0,0 +1,937 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf,len=936 +page_content='DANLIP: Deep Autoregressive Networks for Locally Interpretable Probabilistic Forecasting Ozan Ozyegena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Juyoung Wangb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Mucahit Cevika aDepartment of Mechanical and Industrial Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Toronto Metropolitan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Canada bDepartment of Mechanical and Industrial Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' University of Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Canada Abstract Despite the high performance of neural network-based time series forecasting methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' the inherent challenge in explaining their predictions has limited their applicability in certain application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Due to the difficulty in iden- tifying causal relationships between the input and output of such black-box methods, they rarely have been adopted in domains such as legal and med- ical fields in which the reliability and interpretability of the results can be essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In this paper, we propose DANLIP, a novel deep learning-based probabilistic time series forecasting architecture that is intrinsically inter- pretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We conduct experiments with multiple datasets and performance metrics and empirically show that our model is not only interpretable but also provides comparable performance to state-of-the-art probabilistic time series forecasting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Furthermore, we demonstrate that interpreting the parameters of the stochastic processes of interest can provide useful in- sights into several application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Keywords: interpretability, time series, deep learning, forecasting 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Introduction Interpretable machine learning, a subfield of statistical learning, has re- ceived a lot of attention in the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Despite the significant achieve- ments of statistical learning techniques, such methods are not fully adopted in some decision-critical settings such as those frequently seen in legal and medical domains [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' It is primarily because, in such cases, decision-makers are typically required to answer the question “Why did the machine learning model make such decisions?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', as described by Molnar [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Preprint submitted to Elsevier January 9, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='02332v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='LG] 5 Jan 2023 According to Doshi-Velez and Kim [4], interpretability in the context of machine learning is defined as “the ability to explain or to present in understandable terms to a human”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Under such a criterion, naive neural network-based learning methodologies fail to achieve interpretability due to their black-box nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Specifically, the neuron activation status of the net- work is likely to fail to provide human-level understanding of how the neural network makes the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Linear regression is frequently used as a traditional forecasting algorithm and it can be considered a typical example of interpretable statistical learning methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This method allows users to express the predictions as a linear combination of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' That is, the model assumes the response variable expressed as y = � f∈F γfxf + ε (1) where y is target, F is set of features, γf ∈ R is linear model coefficient for all f ∈ F, and ε is a Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Standardized linear regression is regarded as an interpretable forecasting method, as each linear combination coefficient γf for feature f ∈ F can be taken as a degree of importance of that feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' While linear regression has been extensively used, the method is not al- ways applicable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Its functional form restriction inherently limits the expressive abilities of the model, as there exist many complex functions that cannot be expressed as a linear combination of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In this regard, to achieve better performance, the use of complex learning methods is usu- ally inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' However, such methodologies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', deep neural networks, are not necessarily interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Ensuring the interpretability of the model is extremely important in many disciplines [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Even in some domains where interpretability is not deemed to be as important, interpretable models can greatly help decision-makers to make more informed decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For instance, in the case of a large retail company, understanding the key factors that highly affect the sales of flagship products might be of great interest in designing better production, inventory management, and marketing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' There exist two main branches in the realm of interpretable machine learning [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Post-hoc methods consist of applying techniques that analyze the model after training, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', LIME [22] and SHAP [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' On the other hand, intrinsic methods involve building inherently interpretable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This is 2 commonly achieved by restricting the complexity of the machine learning model itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', RETAIN [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In general, post-hoc methods such as SHAP are slow and not efficient for large-scale machine learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' These methods usually require a large number of experiments to calculate interpretability scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Indeed, the runtime complexity of the SHAP method is exponential in the num- ber of features in the dataset [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In contrast, intrinsically interpretable methodologies generally work fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' It has been shown that the performance of proposed methods can be comparable to the non-interpretable methods [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Accordingly, we concentrate on developing an intrinsically interpretable time series forecasting method using neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' As intrinsic methods typ- ically have a faster training time, this framework can be more suitable to develop prediction algorithms over a large amount of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Only a few studies have explored interpretable deep neural network-based time series forecasting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Among these works, Temporal Fusion Trans- former (TFT) [11] is highly related to our work, as the architecture also achieves interpretability, while performing time series forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' However, the TFT is fundamentally different from our work as it only computes impor- tance scores rather than contribution scores of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Unlike importance scores, contribution scores allow users to quantify both the positive and neg- ative impact of features on the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We summarize the contributions of our study as follows: We present DANLIP, a deep learning-based interpretable parametric probabilistic time series forecasting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We show our model can achieve comparable performance to those of state-of-the-art probabilistic time series forecasting methods while maintaining high interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We conduct an extensive numerical study with three real-world datasets to show the performance of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We empirically show how interpreting the predictions of the model using the parameters of stochastic processes that are fitted to the model can be used to provide important practical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In Section 2, we briefly re- view the relevant literature with a specific focus on RETAIN [3] and DeepAR [23] models as two highly relevant approaches for probabilistic forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In Sec- tion 3, we present our interpretable time series forecasting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Through- 3 out the section, we describe the model architecture and show how the pre- diction results can be interpreted in terms of both mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In Section 4, we show the performance of our model empirically using multiple datasets evaluated over various performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Moreover, we discuss how interpreting both the mean and variance of the probabilistic forecasts can provide useful insights using real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Finally, in Section 5, we summarize our findings and discuss future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Background In this section, we first briefly review the most relevant studies from the literature and summarize time series forecasting basics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Then, we review the studies by Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [23] and Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [3] in detail, which provide the preliminaries for our own methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Related works There is a vast literature on time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Earlier studies in the field mainly focused on statistical models such as Moving Average (MA), Au- toregressive Moving Average (ARMA), and Autoregressive Integrated Mov- ing Average (ARIMA) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Decision trees and their ensembles, namely gradi- ent boosted trees and random forests were also frequently used for time series forecasting as they possess interpretable structures and have the ability to capture nonlinear relations between various features in the dataset [5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' While the majority of the previous studies focus on point forecasting, several recent studies proposed methodologies that are designed to perform high-quality probabilistic forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' A probabilistic forecast typically refers to the confidence interval around the point forecasts and is specified by the lower and upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Standard methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', ARIMA and exponential smoothing) can generate probabilistic forecasts through closed-form expres- sions for the target predictive distribution or via simulations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' As more and more data has become available over the recent years, many studies have focused on deep learning-based approaches for time series fore- casting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For instance, recent studies by Rangapuram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [21] and Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [23] proposed deep learning models for probabilistic forecasting that can directly predict the parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', mean and variance) of the proba- bility distribution that specifies the probabilistic forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' These approaches show substantial performance improvements over standard approaches for datasets which consist of a large number of time series (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', in the order of 4 hundreds or thousands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Despite the success of these deep learning methods for forecasting, they came with an important caveat of having highly com- plex, black-box architectures, which lack interpretability and explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We refer the reader to recent review articles by Parmezan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [20] for a detailed overview of statistical and machine learning models for time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In addition, recent forecasting competitions provide valuable insights on best-performing time series forecasting methods [1, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Explainable artificial intelligence also gained significant attention in re- cent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Post-hoc interpretability methods have been used to interpret the decisions of time series models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Mujkanovic [17] used SHAP to interpret time series classifiers, whereas Ozyegen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [19] evaluated three post-hoc interpretability methods, including SHAP, to interpret the time series fore- casting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' On the other hand, many time series forecasting methods take into account interpretability considerations in model development [3, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' While intrinsically interpretable methods usually trade off the performance to provide more interpretability, the level of this trade-off varies according to the model and the prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' It has been shown that some intrin- sically interpretable models achieve similar performance levels to those of black-box nature [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Our study fundamentally differs from prior work on intepretable time series forecasting, as the proposed architecture is able to discern which features positively and negatively affect the prediction out- come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Preliminaries In our analysis, we mainly rely on the standard neural network method- ology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the sake of convenience, we additionally introduce the following notations: [N] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , N} for any N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Given A ∈ Rm×n, A[i, :] denotes the ith row of matrix A, and A[:, j] denotes the jth column of matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For one- and multi- dimensional tensors, the notations are analogously extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Given A ∈ Rm×n, A[i1 : i2, :] denotes (Aij)i∈{i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=',i2−1},j∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The defini- tions for A[:, j1 : j2] and A[i1 : i2, j1 : j2] are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For one- and multi- dimensional tensors, the notations are analogously extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Icondition is an indicator function which takes value 1 if the condition provided as the subscript is satisfied, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 5 For any a ∈ RA and b ∈ RB, [a, b] refers to concatenation of two vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', [a, b] = [a1, · · · , aA, b1, · · · , bB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Following the self-explaining neural network model proposed by Melis and Jaakkola [15], which is analogous to the RETAIN architecture of Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [3], we ensure the interpretability of our forecasting method by restricting the form of the output of the neural network as y = � f∈F γΘ(x)fxf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' (2) Equation (2) is highly similar to Equation (1), however, different from Equa- tion (1), the term that is multiplied with xf is now a function of input, γΘ : D → R|F|, where D represents the domain of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We char- acterize the function γΘ(·) with Θ, which denotes the set of neural network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Review of DeepAR architecture We consider DeepAR [23], a state-of-the-art probabilistic parametric time series forecasting method, as a baseline in our numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' DeepAR is an RNN-based parametric probabilistic time series forecasting architecture, which seeks to solve the log-likelihood maximization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure 1 pro- vides an intuitive visual summary of the DeepAR architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Network Input Figure 1: Simplified architecture of DeepAR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Review of RETAIN architecture RETAIN is a neural network-based interpretable binary classification method developed by Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The architecture restricts the form 6 of prediction to be of Equation (2) and defines contribution to the prediction of feature f as γ(x)f · xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The architecture uses two RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The sigmoid function is applied over the sum of multiplication of the embedded input vectors and output of RNNs, to predict binary variable y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', variable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' A simplified visualization of the RETAIN architecture is provided in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Input Embedding layer Dense layer Figure 2: Simplified architecture of RETAIN More formally, given the input data (xt)t∈[T], where xt ∈ Rp for all t ∈ [T], p being the number of features, the RETAIN architecture involves the following set of mathematical operations: vi = Wembxi (3) gi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , g1 = RNNα(vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , v1) (4) ej = w⊤ α gj + bα for j ∈ [i] (5) α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , αT = Softmax(e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , ei) (6) hi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , h1 = RNNβ(vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , v1) (7) βj = tanh(w⊤ β hj + bβ) for j ∈ [i] (8) ct = � j∈[i] αjβj ⊙ vj (9) P(yi|(xt)t∈[i]) = Softmax(Wci + b) (10) 7 where α computes time step-wise attention scores and ⊙ corresponds to the Hadamard product operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Let r be the embedding dimension of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' To define a feature level contribution for the output, Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [3] show that Equation (10) can be expanded as follows: P(yi|(xt)t∈[i]) = Softmax Ñ � j∈[i] � k∈[r] xj,kαjW (βj ⊙ Wemb[:, k]) + b é (11) Then, based on Equation (11), the contribution of feature k at time step j for the prediction of yi can be obtained as ω(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' xj,k) = αjW(βj ⊙ Wemb)[:, k]xj,j (12) Note that the definition of quantity ω(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' xj,k) is analogous to γΘ(x) compo- nent of the Equation (2), which can be used to explain why RETAIN is an interpretable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Probabilistic parametric time series forecasting In general, time series forecasting methodologies can be classified as point forecasting and probabilistic forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In point forecasting, the objective is to directly estimate the value for the subsequent steps of the target time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' On the other hand, in probabilistic forecasting, the probability of predicting a particular value is quantified for the target time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' More specifically, let t be the time step and I ∈ N be the number of time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Let (xij, zi,j−1)t∈[I],j∈{t−h,··· ,t−1} such that (xij, zi,j−1) ∈ R|F|+1 is the collection of time series dataset, where h is the history size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We call (xij)ij as covariates, and (zij)ij as target series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In addition, assume that for all i ∈ [I], (xij)t∈[t,T]∩N are given, where t + F > t with F ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Then, the goal of probabilistic forecasting is to estimate the quantity given by P((zi,j)i∈[I],t∈[t,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=',t+F]∩N|(zi,j)i∈[I],j∈{t−h,··· ,t−1}, (xij)i∈[I],j∈{t−h,··· ,t+F}) (13) where F is the forecasting time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' There are two main approaches to estimate the quantity shown in Equa- tion (13): Parametric methods involve estimating (zi,j)i∈[I],j∈{t+1,t+F}∩N by as- suming (zi,j)i∈[I],j∈{t−h,··· ,t−1} is a stochastic process from some joint 8 density π(ς) with known π and unknown ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' As an example of paramet- ric time series forecasting methodologies, we refer the readers to the work of Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Non-parametric methods involve directly estimating Equation (13) with- out assuming any knowledge about π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For further details of such a non-parametric methodology, we refer readers to the works of Koenker and Bassett Jr [9], and Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In this paper, we only focus on the parametric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The primary benefit of proposing interpretable parametric time series forecasting methods is that it allows interpreting parameters of the stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For instance, by interpreting the parameters of a Gaussian process, we can achieve interpre- tation for both the mean value and standard deviation of the probabilistic forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' DANLIP Architecture We next describe DANLIP by detailing the model specifications and the ar- chitecture, and we discuss how the interpretability is achieved by this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Model description DANLIP is designed to perform joint density parameter estimation via performing time step-wise parameter forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' More precisely, we restrict the stochastic processes to be Gaussian, as it is a probability distribution which directly use the mean and standard deviation as its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' However, we note that DANLIP architecture can be appropriately modified to handle other types of stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For instance, following the steps proposed by Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [23], such modifications can be easily achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the sake of better readability, we eliminate the time series indices i in below mathematical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Firstly, let M be number of continuous features, and N be number of categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We note that the input data can be decomposed as a vector containing both target series value, continuous features and discrete features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' More precisely, input data can be represented as shown in Equa- tion (14), that is, for all j ∈ {t − h, · · · , t − 1}, where h is history size, time series with covariate can be written as [zj, (xCont jf )M f=1, (xCat jf )N f=1] (14) 9 where zj is target series value at time step j, (xCont jf )M f=1 denotes vector of continuous components of vector x, while (xCat jf )N f=1 denotes the categorical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the sake of notational convenience, we make the following assumptions: Without loss of generality, we assume such an index ordering as default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We let FCat be set of categorical feature indices of the vector shown in Equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Analogously, we define FCont for continuous features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We assume xCat jf ∈ N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', categories are represented in terms of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We assume that the values of (xCat itf )it lie in a subset of ordered natural numbers without any gap starting from 0, so the values of categorical features can be one-hot encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The DANLIP is defined as concatenation of three important parts: feature processing layer, encoding layer and decoding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In Figure 3, we present a simplified overview of the entire neural network architecture for DANLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We start the model description by depicting feature processing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For each time step j ∈ {t − h, t − 1}, eCat f = Embedding(xCat jf ) ∀f ∈ FCat (15) vj = (zj, xCont j,1 · · xCont j,M , ej,1 · · · ej,N) (16) where for all f ∈ FCat, eCat jf ∈ Rξf with ξf representing the embedding size of categorical feature f ∈ FCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Using notations above, we define the input for the encoding layer as v = [vt−h, · · · , vt−1] (17) Then we can subsequently define the encoding layer as follows: gt−h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , gt−1 = RNNα(vth, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , vt1) (18) qj = w⊤ α gj + bα ∀j ∈ {t − h, · · · , t − 1} (19) αt−h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , αt−1 = Softmax(qt−h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , qt−1) (20) ht−h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' , ht−1 = RNNβ(vt−h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' vt−1) (21) βj = tanh(w⊤ β hj + bβ) ∀j ∈ {t − h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' t − 1} (22) cj = αjβj ⊙ vj ∀j ∈ {t − h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' t − 1} (23) 10 Identity Embedding PREPROCESSING Concatenation RNN 𝛼 (Layer 1) ENCODER RNN 𝛼 (Layer 𝑳𝜶) RNN 𝛽 (Layer 1) RNN 𝛽 (Layer 𝑳𝜷) DECODER [𝑧𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−ℎ|𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−ℎ 𝐶𝑜𝑛𝑡] [𝑧𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1|𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1 𝐶𝑜𝑛𝑡] 𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−ℎ 𝐶𝑎𝑡 𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1 𝐶𝑎𝑡 ⋯ ⋯ 𝑣𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−ℎ 𝑣𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1 ⋯ 𝛼𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−ℎ 𝛼𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1 ⋯ 𝛽𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1 𝛽𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1 ⋯ ⊙ ⊙ ⋯ 𝑐𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−ℎ 𝑐𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='𝑡−1 ⋯ ⋯ ⋯ 𝑐 = Dense Dense Dense Dense 𝑣 = [𝑐|𝑑𝑡] [𝑐|𝑑𝑡+𝐹] Ƹ𝜇𝑡 ො𝜎𝑡 Ƹ𝜇𝑡+𝐹 ො𝜎𝑡+𝐹 ⋯ ⋯ Figure 3: DANLIP model architecture where both RNNα(·) and RNNβ can be either uni- or bi-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The output of the encoding layer is c = (ct−h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' ct−1) (24) 11 which can be used to define the input for the decoding layer as dj = [zj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' xCont j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1 · · xCont j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1 · · · ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='N] ∀j ∈ {t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' t + F} (25) d = (dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' dt+F) (26) Using the sequence of vectors (cj)j∈{t−h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='t−1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' we can finally perform mean and standard deviation prediction in the decoding layer by using the following formula: µj = W ⊤ µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j · [c|dj] ∀j ∈ {t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' t + F} (27) σj = Softplus(W ⊤ σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j · [c|dj]) ∀j ∈ {t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' t + F} (28) where Wµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j ∈ R1×N×� f∈F df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We note that both µit and σit are later used to define the parameters for output conditional univariate Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Given the neural network output, the neural network weights are opti- mized to solve the likelihood maximization problem given by max Θ � i∈I t+F � j=t ℓUVN(zi,j|µij(Θ), σij(Θ)) (29) where, instead, we solve the log-likelihood maximization problem described by Equation (30), that is, max Θ � i∈I t+F � j=t log(ℓUVN(zi,j|µij(Θ), σij(Θ))) (30) We recover the eliminated index i to describe the time series indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We use ℓUVN(·|·) to denote univariate Gaussian likelihood, where Θ is the vector of all network parameters optimized according to the network inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' During the training of the DANLIP, the input for decoding layer is defined using true target series value zj for each j ∈ {t−h, · · · , t−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' However, dur- ing the prediction phase, the future information zj for j ∈ {t, · · · , t + F} are not accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' To overcome this issue, we perform autoregressive prediction, and we replace the decoding layer input vector dj by ˆdj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Specifically, dt = [zt−1, xCont t,1 · · xCont t,M , et,1 · · · et,N] (31) 12 ˆdj = [ˆzj−1, xCont j,1 · · xCont j,M , ej,1 · · · ej,N] ∀j ∈ {t + 1, · · · , t + F} (32) ˆd = (dt, ˆdt+1, · · · , ˆdt+F) (33) where ˆzj−1 is a sample drawn from π(ˆµj−1, ˆσj−1), which denotes the predicted target series value for all j ∈ {t+1, · · · , t+F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' During the forecasting phase, DANLIP performs sequence sampling as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' DECODER IN PREDICTION PHASE Dense Dense Dense Dense [𝑐|𝑑𝑡] Ƹ𝜇𝑡 ො𝜎𝑡 Ƹ𝜇𝑡+𝐹 ො𝜎𝑡+𝐹 ⋯ Dense Dense [𝑐| መ𝑑𝑡+1] Ƹ𝜇𝑡+1 ො𝜎𝑡+1 [𝑐| መ𝑑𝑡+𝐹] Figure 4: Decoder of the DANLIP architecture during forecasting phase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Model interpretation We can assess how much the input features contribute to the model output when the DANLIP makes a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the sake of brevity, let φ = 1 + |FCont| + �f r=0 ξr be the length of the one time step context vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We can define the contribution of feature f at time step s for µj prediction as follows: ωµ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' j) = αsWµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j[(s − (t − h)) · φ] · (βs[0] · zs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' f /∈ FCont ∪ FCat (34) ωµ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' j) = αsWµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j[(s − (t − h)) · φ + f] · (βs[f] · xsf),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' f ∈ FCont (35) ωµ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' j) = αsWµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j ï (s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ ò � βs ï (s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ ò ⊙ W Cat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='Emb f [:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' xsf] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' f ∈ FCat (36) 13 Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' the contribution for σj prediction is defined as ωσ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' j) = αsWσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j[(s − (t − h)) · φ] · (βs[0] · zs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' f /∈ FCont ∪ FCat (37) ωσ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' j) = αsWσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j[(s − (t − h)) · φ + f] · (βs[f] · xsf),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' f ∈ FCont (38) ωσ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' j) = αsWσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='j ï (s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ ò � βs ï (s − (t − h)) · φ + (φ − ξf) : (s − (t − h)) · φ + φ ò ⊙ W Cat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='Emb f [:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' xsf] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' f ∈ FCat (39) We note that the provided formula is valid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' as long as all continuous features come before categorical features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' and target series come before all continuous features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Without such an assumption, the formula for the con- tribution score must be appropriately modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' When ω(f, s, j) > 0 the contribution of feature f at timestep j is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This means the provided component contributes to an increase in the value of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' On the other hand, a negative value means that the contribution score is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Without the monotonically increasing property of the activation functions of the dense layers in the decoder, such an interpretation would not be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' It is also important to note that the contributions of features on standard deviation are nonlinear, which is due to the existence of the Softplus opera- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' By taking the norm of the contribution scores |ω(f, s, j)|, we can obtain the importance of the features, instead of contribution scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' These features can then be averaged over the prediction steps and samples of the dataset to obtain the average importance of the input features over the dataset, which enables the global-level contribution score computation for the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Numerical Study In this section, we present the results from detailed experiments to eval- uate the forecasting performance and interpretability of the proposed model, DANLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We first provide the experimental setup and detail the considered forecasting models, hyperparameters and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Then, we compare the 14 forecasting performance of several models using well-known point and prob- abilistic forecasting evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Next, we focus on the results that demonstrate the interpretability aspects of DANLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Specifically, we first show the explanations produced by the model, and then discuss these explanations as they relate to the underlying prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Experimental setup We consider three diverse datasets with different characteristics in terms of size, and observed seasonality/trends among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We use a sliding window method for framing the datasets, and separate the last two forecast- ing horizons of each time series in the datasets as validation and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We perform a detailed hyperparameter tuning for all models using Tree- structured Parzen Estimator (TPE) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The parameter ranges for the hyperparameter tuning are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For DANLIP and DeepAR, we experiment with different number of RNN layers, RNN cell types, hidden units, dropout and learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the MLP model, we experiment with a similar range of hidden layers, learning and dropout rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For hidden units, we use a wider range, as dense neural networks are usually trained with a larger number of hidden units to achieve similar performance compared to the RNN based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the GBR model, we test three important pa- rameters which are the number of trees in the ensemble, the number of leaves in each tree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', max depth), and the minimum number of samples required to split an internal node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', min samples split).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Table 1: The hyperparameter tuning search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Model Search space DANLIP/ DeepAR # Hidden units: [16, 128], Dropout rate: [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5], Cell Type: LSTM or GRU, # RNN layers: [1, 8], Learning Rate: [1e-4, 1e-1] (log uniform) MLP # Hidden units: [50, 500], Dropout rate: [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5], # Hidden layers: [1, 8], Learning Rate: [1e-4, 1e-1] (log uniform) GBR # of trees: [10, 200], Max depth: [2, 5], Min samples split: [2, 15] We run the TPE algorithm for 100 trials, optimizing over the normalized deviation metric on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The neural network models are im- 15 plemented using Pytorch, the GBR model is implemented using Scikit-learn, and hyperparameter tuning is performed using the Optuna library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' All the experiments are run on a computing node with RTX2070 Super 8GB GPU, and 128GB of RAM, running on Debian Linux OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We consider two sets of performance metrics to evaluate both point and probabilistic forecasting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For measuring the quality of point forecasts, we use Normalized Root Mean Squared Error (NRMSE) and Nor- malized Deviation (ND) similar to Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' NRMSE and ND can be obtained for ground truth values (y) and the prediction (ˆy) as follows: NRMSE(y, ˆy) = » 1 N �N i=1(ˆyi − yi)2 1 N �N i=1 |yi| , ND(y, ˆy) = �N i=1 |ˆyi − yi| �N i=1 |yi| (40) To evaluate the probabilistic forecasting ability of the models, we consider the ρ-risk, also known as quantile loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The ρ−risk can be obtained as follows: ρα(y, ˆyα) = �N i=1 max{α(yi − ˆyi α), (1 − α)(ˆyi α − yi)} �N i=1 |yi| (41) where ˆyα represents the prediction at quantile level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Note that having a lower value of ρ-risk is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Finally, we apply Friedman test, a non- parametric test, to compare the forecasting models on multiple datasets [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For this comparison, we fill each group with the average error results on the tested datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Then, we apply the Friedman test to find out whether there is a statically significant difference between the mean of the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We consider three datasets in our experiments, namely, Electricity, Ross- mann and Walmart datasets, which we briefly describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Electricity: The Electricity dataset contains the hourly electricity con- sumption records of 370 households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The dataset has been exten- sively used in previous studies for performance benchmarking purposes [11, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In our experiments, besides the electricity consumption time series, we use additional covariates such as hour of the day, day of the week, week of the month, and month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Input window size and forecasting horizon are taken as 168 and 12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Rossmann: The Rossmann sales dataset was published as a part of a Kaggle competition in 2015 and it contains a rich set of features and extensive daily sales histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The dataset consists of daily sales 16 records of multiple Rossmann stores, with various covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Among the available features, we use sales value, store index, store open indi- cator, promotion indicator, state holiday indicator, school holiday indi- cator, weekday, month, and week of the month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Input window size and forecasting horizon are taken as 30 and 12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Walmart: The Walmart store sales forecasting dataset contains weekly store sales of 77 departments in 45 stores, and it was made available as part of a Kaggle competition in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The dataset includes multiple features such as temperature, information on markdowns, and various economic indicators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', unemployment rates, fuel prices and CPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Input window size and forecasting horizon are taken as 30 and 6, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Results on model performances We provide summary statistics on the model performances in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' These results are obtained after performing rigorous fine-tuning for each dataset-model pair on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The Friedman test over the ag- gregate results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', results for all three datasets combined) returns p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='12, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='12 for NRMSE, ND, ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='75, and ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='90, respectively, which indicates that there is no statistically significant difference (p-value > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='05) between DANLIP and DeepAR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Similarly, we see that DANLIP and DeepAR achieve similar forecasting performance in terms of average performance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' DANLIP performs marginally better for the Rossmann and Electricity datasets, whereas DeepAR performs marginally better for the Walmart dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Baseline models, MLP and GBR, are significantly less complex compared to DeepAR and DANLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' MLP leads to the poorest forecasting performance among the tested models for all the datasets as shown by various forecasting performance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' GBR outperforms DANLIP and DeepAR on the Ross- mann dataset, where the dataset shows a significant amount of seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the Electricity dataset, GBR ranks second in terms of the NRMSE met- ric, and first in terms of the ND metric, with a marginal difference in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the Walmart dataset, GBR ranks third after DeepAR and DANLIP in terms of the NRMSE metric, and second in terms of the ND metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The high performance of the GBR can be attributed to the way we train this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Specifically, for the GBR model, we train a separate model for each time series on the dataset, whereas, for the other models, we train a single 17 Table 2: Summary statistics of model performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Mean and standard deviation across 10 randomly seeded runs are reported over the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Dataset Model NRMSE ND ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='75 ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='90 mean std mean std mean std mean std Electricity DeepAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='041 GBR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='002 MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='035 DANLIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='027 Rossmann DeepAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='121 GBR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='001 MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='032 DANLIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='056 Walmart DeepAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='043 GBR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='025 MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='062 DANLIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='061 model for all the time series in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This training methodology for GBR is computationally more expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' However, it can result in a sig- nificantly improved forecasting performance when the availability of similar time series does not benefit the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Our preliminary analysis indicates that training a single GBR model for each dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', similar to the other three models) leads to a significant deterioration in forecasting performance across the datasets, hence we adopt the above-explained approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We note that above results for these datasets are largely inline with the forecasting performance values reported in previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', see [8, 19, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure 5 shows the visualization of forecasts by DeepAR and DANLIP for three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We use blue lines to represent the ground truth, and orange lines to denote the predicted mean values of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We depict 75%, 90% and 98% prediction intervals as shaded areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The forecasting horizons are 28 days, 24 hours, and 6 weeks for Rossmann, Electricity, and Walmart datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Overall, we observe that all the models can generate predictions that capture the general trends in the datasets and probabilistic forecasts are highly similar for these two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 18 (a) Rossmann: DeepAR (b) Rossmann: DANLIP (c) Electricity: DeepAR (d) Electricity: DANLIP (e) Walmart: DeepAR (f) Walmart: DANLIP Figure 5: Visualization of DeepAR and DANLIP predictions on random time series samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Each figure displays forecasts for one prediction window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Point forecasts are shown in the orange line, 75%, 90% and 98% quantile forecasts are shown as orange regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Each model generates predictions that can capture the trends for the provided sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='observed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='predicted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='Tirme indet4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Results on model interpretability We next provide results on the interpretability of DANLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We focus on the Rossmann dataset as the representative case, as it has a large, interesting set of features that correlate well with the target variable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', Promo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the Electricity dataset, we see that the target value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', electricity load) has the most impact on the predictions, and certain time covariates such as “week- of-month” help reduce the variance of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' For the Walmart dataset, we again see that the target value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', weekly sales) has the most impact on the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The interpretability results for the Walmart and Electricity datasets are provided in the Appendix (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We discuss the interpretability of DANLIP by analyzing the explanations obtained from the model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Following the similar work [7, 18], we visualize the explanations as feature contribution heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The contribu- tion score formulas described in the methodology are applied to obtain the contribution of each input feature to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Note that we collect a separate contribution score for each sample, input feature, and forecasting horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This allows us to achieve local interpretability, which helps un- derstanding how the features contribute to a single prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' It is also possible to aggregate these scores over forecasting horizons and samples to achieve global interpretability, which helps understand the features that are important for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure 6 shows the contributions scores obtained from the model for a single sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We select the 9th forecasting horizon of the first sample from the Rossmann test set to generate these visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This sample and the forecasting horizon provide a clear overview of how the model behaves under certain events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The forecasted date for this sample is a Monday, the store is Open, there is a promotion event, and there is no state or school holiday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure 6a shows that ‘DayOfWeek’, ‘Open’, ‘Promo’, and ‘StateHoliday’ have the biggest positive contribution to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This is intuitive since there are no sales on the weekends, and Mondays should expect a higher sale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The store being open, the promotion events and the state holidays are all positive contributors to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Looking at the encoder contribu- tions, we observe that the encoder features have negligible impact on the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' However, we also note that recent and certain seasonal values of the ‘DayOfWeek’ feature have higher impacts on predictions compared to the other encoder features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure 6b shows the contribution of the features to the predicted vari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Overall, we observe that the ‘StateHoliday‘, ‘Store’, and ‘SchoolHoli- 20 day’ decoder features have significant negative contributions, and ‘Open’ and ‘Promo’ have positive contributions to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The negative values for the store and holiday features show that they reduce the variance of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Interestingly, ‘StateHoliday’ has the largest negative contribu- tion to the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Since there is no date corresponding to a holiday in this example, the explanation may indicate that state holidays can result in a higher uncertainty (variance), which requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' On the other hand, we observe a positive contribution for ‘Open’, and ‘Promo’ when the store is open and there is a promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This is expected as there is higher uncertainty in the predicted sales when the store is open as opposed to the days when the store is closed and there are no sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The explanation also suggests that the existence of promotion has a similar impact on the prediction, and it causes a higher variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' (a) µ contributions (b) σ contributions Figure 6: Contribution scores obtained from DANLIP for the first sample and 9th target horizon of the Rossmann test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure (a) shows the contributions to the µ and (b) shows contributions to the σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In certain cases, it might be useful to calculate the overall importance of the features for the model, instead of their importance for a particular prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' These feature importance values can then be used for different purposes including feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We apply the following steps to the contribution scores to find the overall importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' First, we obtain contri- bution scores for each sample in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The resulting array has the shape: #forecasting horizon × #samples × #timesteps × #features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We then take the absolute values of this array, as we are interested in the over- all importance of the features, not their contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Finally, we average 21 Encoder 30 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='050 21 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='025 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='D0O 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content="025 050' Decoder xpyaug Dayorweek uado StateHolideyEncoder 30 24 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='05 21 18 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content="00 12 50'0- Decoder 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='0 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5 xp Dayorweek StateHolidey Taug .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 4the array over the forecasting horizons, and then over the samples to obtain the importance scores of the input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We show the visualizations of these arrays for µ and σ in Figure 7a and Figure 7b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' These figures show the overall importance of the features for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Thus, all the values displayed are positive numbers, and they are displayed in green color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Looking at the importance scores for µ in Figure 7a, we observe that the same four decoder features from the sample contributions (‘Day- OfWeek’, ‘Open’, ‘Promo’, and ‘StateHoliday’) are found to be important for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Additionally, we note that various encoder timesteps of the ‘Sales’, and ‘DayOfWeek’ features are also important, suggesting that the model attends to different timesteps of these features for different samples of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Overall importance scores for σ shown in Figure 7b follow a sim- ilar pattern compared to the sample contribution scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' The same decoder features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=', ‘Stateholiday’, ‘Store’, ‘Open’, ‘Promo’, and ‘SchoolHoliday’ are all important for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Additionally, the ‘DayOfWeek’ feature of the decoder is also important for predicting the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' On the decoder side, certain past values of ‘Sales’, ‘DayOfWeek’ and ‘StateHoliday’ are all found to be important to the variance prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' (a) µ average importance (b) σ average importance Figure 7: Contribution scores obtained from DANLIP for all samples, and averaged over samples to obtain global importance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure (a) shows the average importance to the µ and (b) shows average importance to the σ in Rossmann dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Conclusion and Discussions In this paper, we present a novel deep probabilistic intrinsically inter- pretable time series forecasting method, DANLIP, which is designed to predict 22 Encoder 27 30 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='D2 21 &T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='D1 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='00 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='02 Decoder paug aos Dayorweek uadb StateHolideyEncoder 27 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='04 24 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='D2 18 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='00 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='04 Decoder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5 xp aug aos Dayorweek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' uadb PrDka StateHolidey 丽both mean and standard deviation of the underlying Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We show how ordinal and categorical covariates can be appropriately incorpo- rated into DANLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Then, we describe procedures to compute the contribution of both ordinal and discrete features using DANLIP network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We also compare DANLIP against various baselines and show that our method has a competitive forecasting performance to DeepAR, a state-of-the-art prob- abilistic time series forecasting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We also analyze the contribution scores generated for mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' We show how contribu- tion scores can be used to analyze model predictions for a single sample and to find the overall importance of the features for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' An area that could be further explored is the idea of using the con- tribution scores for feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Because the contribution scores are calculated directly from the model weights, they may achieve a higher per- formance in feature selection compared to the alternative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Secondly, the proposed architecture can be tested with other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' While DANLIP achieves similar performance to DeepAR for the tested datasets, the architec- ture might require further tuning for other datasets and problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Finally, the role of regularization can be analyzed in finding the most useful explana- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Methods such as L1 regularization and Dropout can be used to tweak the contributions assigned to the correlated features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Depending on the use case, certain regularization hyperparameters can be selected to obtain the most useful explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Acknowledgment This research is in part supported by LG Science Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Statements and Declarations No potential conflict of interest was reported by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Bojer and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' P.' 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7785–7794, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Ribeiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Singh, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Guestrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' ” why should i trust you?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' ex- plaining the predictions of any classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data min- ing, pages 1135–1144, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Salinas, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Flunkert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Gasthaus, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Januschowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Deepar: Probabilistic forecasting with autoregressive recurrent networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Inter- national Journal of Forecasting, 36(3):1181–1191, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 25 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Additional Results on DANLIP Interpretability Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1 shows the contribution scores obtained from the model for a single sample, and Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='2 shows the overall importance of the features for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Overall, we find that the target feature (Weekly Sales) has the most significant impact, particularly at the decoder step, and at the recent timesteps of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Looking closely at the contribution scores, we also observe that various timesteps of “Dept”, “year”, “month”, ”weekofmonth”, and “day” features contribute to the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' However, looking at the importance scores in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='2, we observe that the overall impact of these features is insignificant for the predicted µ, but significant for the predicted σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' (a) µ contributions (b) σ contributions Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='1: Contribution scores obtained from DANLIP for the first sample and 4th target horizon of the Walmart test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure (a) shows the contributions to the µ and (b) shows contributions to the σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='3 shows the contribution scores obtained from the model for a single sample, and Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='4 shows the overall importance of the features for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Similar to the Walmart dataset, we observe that the target feature (series) has the most significant impact, particularly at the decoder timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Comparing contribution scores to the overall importance scores, we find that in different samples, the model tends to attend different past encoder timesteps of the “series” feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This is unlike the Walmart dataset, in which the model attends more to the recent timesteps of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' This suggests that, for the Walmart dataset, the recent timesteps are the 26 Encoder o- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='00 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='04 Decoder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='5 idx ko Unemployinent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Weekly_ Sakes aios a derak weekofinonith Aep Taugmost informative, whereas, for the Electricity, the seasonal values of the target feature (t − 24, t − 48 · · · ) are the most informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Looking at the contribution scores for σ in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='3b, we find that the “house” feature, which indicates from which house the electricity load information is collected, leads to an increase in the predicted variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' On the other hand, decoder timesteps of the time covariates (“hour”, weekday”, and “weekofmonth”) reduce the predicted variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' (a) µ average importance (b) σ average importance Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='2: Contribution scores obtained from DANLIP for all samples, and averaged over samples to obtain global importance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure (a) shows the average importance to the µ and (b) shows average importance to the σ in Walmart dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' (a) µ contributions (b) σ contributions Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='3: Contribution scores obtained from DANLIP for the first sample and 4th target horizon of the Electricity test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Figure (a) shows the contributions to the µ and (b) shows contributions to the σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' 27 Encoder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='D2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='D0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='04 Decoder azs Emperature ko Unemplpyinent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' Weekly_ Sales Type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content=' JepEncoder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='D2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf'} +page_content='00 ±.' metadata={'source': 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Bouchet2,3 +1Research Institute for Applied Mechanics, Kyushu University, +Kasuga, Fukuoka, Japan. +2Univ. Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, +Lyon, France. +3Laboratoire de Météorologie Dynamique, CNRS, Ecole Normale +Supérieure, Institut Pierre-Simon Laplace, Paris Sciences Lettres +Université, Paris, France. +*Corresponding author(s). E-mail(s): onuki@riam.kyushu-u.ac.jp; +Abstract +The wave kinetic equation predicts the averaged temporal evolution of +a continuous spectral density of waves either randomly interacting or +scattered by the fine structure of a medium. In a wide range of sys- +tems, the wave kinetic equation is derived from a fundamental equation +of wave motion, which is symmetric through time-reversal. By contrast, +the corresponding wave kinetic equations is time-irreversible: its solu- +tions monotonically increase an entropy-like quantity. A similar paradox +appears whenever one make a mesoscopic description of the evolution +of a very large number of microscopic degrees of freedom, the paradig- +matic example being the kinetic theory of dilute gas molecules leading +to the Boltzmann equation. Since Boltzmann, it has been understood +that a probabilistic understanding solves the apparent paradox. More +recently, it has been understood that the kinetic description itself, at +a mesoscopic level, should not break time reversal symmetry [1]. The +time reversal symmetry remains a fundamental property of the meso- +scopic stochastic process: without external forcing the path probabilities +obey a detailed balance relation with respect to an equilibrium quasipo- +tential. The proper theoretical or mathematical tool to derive fully this +mesoscopic time reversal stochastic process is large deviation theory: a +1 + +Springer Nature 2021 LATEX template +2 +Large deviations for linear wave kinetic equation +large deviation principle uncovers a time reversible field theory, char- +acterized by a large deviation Hamiltonian, for which the deterministic +wave kinetic equation appears as the most probable evolution. Its irre- +versibility appears as a consequence of an incomplete description, rather +than as a consequence of the kinetic limit itself, or some related chaotic +hypothesis. This paper follows [1] and a series of other works that derive +the large deviation Hamiltonians of the main classical kinetic theories, +for instance [2] for homogeneous wave kinetics. We propose here a deriva- +tion of the large deviation principle in an inhomogeneous situation, for +the linear scattering of waves by a weak random potential. This prob- +lem involves microscopic scales corresponding to the typical wavelengths +and periods of the waves and mesoscopic ones which are the scales +of spatial inhomogeneities in the spectral density of both the random +scatterers and the wave spectrum, and the time needed for the ran- +dom scatterers to alter the wave spectrum. The main assumption of the +kinetic regime is a large separation of these microscopic and mesoscopic +scales. For the sake of simplicity, we consider a generic model of wave +scattering by weak disorder: the Schrödinger equation with a random +potential. We derive the path large deviation principle for the local spec- +tral density and discuss its main properties. We show that the mesoscopic +process obeys a time-reversal symmetry at the level of large deviations. +This publication is part of a special issue in homage of the memory of Krzysztof +Gaw¸edzki. The subject of this work is large deviation theory applied to wave +turbulence. Large deviation theory applied to complex dynamics and turbulent +flows was one of the subjects for which Krzysztof Gaw¸edzki made a number +of important contributions during the last few years, see for instance [3–8]. He +taught many of us, including Freddy Bouchet, many aspects of large deviation +theory. We wrote a common paper on the subject of large deviation theory +and non-equilibrium quasipotentials for stochastic particles with mean field +interactions [8]. Given his scientific qualities, and his deep sense of friendship, +it is great pleasure for us to pay homage to Krzysztof Gaw¸edzki through this +modest contribution. +1 Introduction +The aim of this paper is to extend the existing kinetic theory to describe +probabilistically mesoscopic evolutions of wave fields interacting with random +potentials. We will derive for the first time a large deviation principle that +describes completely typical and rare fluctuations of the wave local spectral +density. This is also the first large deviation principle for wave kinetic theory +in an inhomogeneous setup. This work lies at the intersection of three different +active fields in theoretical and mathematical physics: the description of waves +interacting with random media and their applications to ocean and atmosphere + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +3 +dynamics, recent mathematical and theoretical advances in the kinetic theory +of wave turbulence, and large deviation theory for kinetic theories. +For the first field, we note that wave propagation in disordered media is a +ubiquitous phenomenon appearing in various areas of physics. Typical exam- +ples include light radiation through the atmosphere, acoustic or internal gravity +waves in turbulent flows, and elastic waves in solid Earth. In most cases, one +is not interested in the individual wave interference or scattering processes but +in the statistical description of the overall wave field at a mesoscopic spatial +scale much greater than the extent of disorder or wavelength. For this pur- +pose, it is customary to define the spectral density of the wave signal at each +location and investigate its statistical properties. The wave kinetic equation, +sometimes referred to as a radiative transport equation or simply as a trans- +port equation, is known as the universal model to describe the evolution of the +local spectral density. It commonly derives from elementary wave equations +and has a broad range of applications [9–14]. Recently, the evolution of wave +spectra under scattering interactions with a turbulent flow were studied in a +two-dimensional model [15]. +Wave kinetic theory is of special interest in some specific areas of ocean +and atmosphere research. Since the celebrated work by Klaus Hasselmann [16], +the kinetic description of nonlinear 4-wave interactions among water waves has +been used for estimating energy transfer rates in a wind wave spectrum and +forecasting the sea surface states. The linear counterpart of the wave kinetic +equation is relevant to surface or internal wave energy dispersion in a slowly +evolving turbulent flow [12, 13, 17–19]. In these actual problems, the scale- +separation assumption at the heart of the kinetic theory might be valid, but is +not necessarily always valid. For example, internal wave activity in the ocean +is highly heterogeneous, which is imprinted on the variability of energy dissi- +pation rates on scales of order 10 to 100 km, in the mid-depth layer [20]. For +tide or wind generated waves with 10-100 km horizontal wavelengths, devia- +tions of the spectral evolutions from that predicted by the kinetic equation +may not be negligible, and a first principle theory of fluctuation is missing. +This motivates us to revisit the theoretical basis of wave kinetic theory. Our +work can be considered as the first building block for stochastic parameteriza- +tion of the local spectral density from first principles, for the specific case of +wave interacting with random potentials. +In relation with geophysical applications, several experiments with funda- +mental scopes in wave kinetic theory have been recently devised. For instance +this led to the very first observation of the regime of inertial wave turbulence +in a rotating flow [21], the identification of regimes of weakly and strongly +nonlinear internal wave turbulence in an experiment of stratified turbulence +[22], experiments on statistical properties of water waves in a large basin [23], +the validation of the inverse cascade phenomenon [24], and extension of the +range of scales for observing pure gravity wave turbulence in the laboratory +[25] using reduced gravity experiments. + +Springer Nature 2021 LATEX template +4 +Large deviations for linear wave kinetic equation +The second field, fundamental theoretical developments in wave turbulence +theory, has seen many new advances recently. For the first time, using numeri- +cal simulations of the non-linear Schrödinger wave kinetic equation, predictions +by the wave kinetic equation were tested for several kinetic times [26, 27]. +Novel finite-size effects in wave turbulence were systematically studied in a +one-dimensional model using a combination of theory and numerics [28]. Sig- +nificant recent progress has been made to give a mathematical foundation for +wave turbulence theory: theorem about approximations of the dynamics for +times much shorter than the kinetic time [29–34], the understanding of propa- +gation of chaos [35], and a remarkable first full rigorous derivation of the wave +kinetic theory at the kinetic timescale, for the non-linear Schrödinger equation +[30] and for water waves equation [36]. From the point of view of these funda- +mental perspectives, our work gives for the first time a description of all the +cumulants of the local spectral density, through a large deviation principle, in +an inhomogeneous setting. +The third field is the development of large deviation principles in relation +with kinetic theory. Many classical equations of mathematical physics arise +from a law of large numbers, when faster and smaller scale degrees of free- +dom are averaged out. This is the case for all classical kinetic theories. It is +natural to extend all these theories to look for the statistics of fluctuations. +Generically, one expects to derive a large deviation principle that describes a +statistical field theory quantifying the probabilities of any fluctuations, either +typical or extremely rare, in a way analogous to macroscopic fluctuation the- +ory [37] for stochastic diffusive systems, or large deviation theory for stochastic +dynamics with mean field interaction [8]. Deriving such large deviation prin- +ciples from deterministic microscopic dynamics is a fundamental endeavor in +theoretical and mathematical physics. Recently, the large deviation principles +for a number of classical kinetic theories, starting from first principles, have +been uncovered: for discrete models that mimic dilute gases and with Boltz- +mann like behavior [38, 39], for dilute gases related to the Boltzmann equation +[1, 40], for the Kac model [41, 42], for plasma at length scales much smaller +than the Debye length related to the Landau equation [43], for homogeneous +systems with long range interactions related to the Balescu–Guernsey–Lenard +equation [44], for weakly interacting waves in a homogeneous setup [2] related +to the wave kinetic equation. The large deviation principles describe fluctua- +tions but also uncover gradient structure for the deterministic kinetic equation, +see [45] and a simple explanation in [1]. Several mathematical results, usually +valid for a fraction of the kinetic time in the spirit of Lanford results for the +deterministic equation, have been obtained for the large deviation principles, +for instance for the Boltzmann equation [40, 46], or for the Kac model with +unexpected corrections to the expected large deviation principle [41, 42] associ- +ated to giant concentrations and solutions of the Boltzmann equation without +energy conservation. +One aim of large deviation theory is to study rare events. In the context +of wave dynamics, large deviation theory has been used to study rare events + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +5 +for the evolution of the empirical spectrum [2] on the kinetic time scale, but +also for studying the appearance of very large amplitude waves [47], for the +non-linear Schrödinger dynamics for shorter time scales. Instantons structures +have been predicted and compared with experimental data taken from a 300 +m long wave [47]. Large deviation principles for the wave amplitude due to +short time phase mixing has also been studied [48]. +The result described in this paper is a new example of a large deviation +principle for a kinetic theory, derived from microscopic dynamics. It is the +first extension of large deviation theory for the local spectral density in an +inhomogeneous setup. It opens the way for other inhomogeneous large devia- +tion principle for wave turbulence, and for the study of new wave turbulence +phenomena where rare events play an important role. +As a generic model of waves interacting with random medium, we consider +the linear Schrödinger equation in a weak random potential +i∂ψ +∂t = −D +2 ∇2 +xψ + V ψ, +where ψ(x, t) is a wave function defined on Rd+1 and V (x) is a homogeneous +random potential. For this model, we assume a regime with a wave spectrum +which is dominated by waves of typical wavelengths λ, and with modulations +of the statistical properties of the wave spectrum on scales of order λ/µ. The +second assumption is that the typical correlation length of the potential is of +order λ and that interactions between the waves and the potential is weak, +more precise definitions are given in section 2. Then, for small value of µ, we +have a separation of scales and of the associated times, where a huge amount +of waves experience multiple scattering in domains of typical size λ/µ. It is +natural to focus on variations of the field on the mesoscopic scales of order λ/µ. +This defines a kinetic regime where such mesoscopic variations are captured by +the Wigner distribution n, that somehow measures the wave energy density in +both position and wave-vector space. After time and length rescaling t → µt +and x → µx, in the small µ limit, the wave kinetic equation is classically +derived (see for instance [9, 49]): +∂n(x, p, t) +∂t ++ p · ∇xn(x, p, t) = c +� +dησ(p, η) (n(x, η, t) − n(x, p, t)) , +where n(x, p, t) is the Wigner distribution at position x, wave vector p and +time t, and cσ(p1, p2) is the scattering cross section. +Interestingly, the evolution of the Wigner distribution predicted from the +wave kinetic equation is an irreversible relaxation process. For this problem, a +Lyapunov function, S = +� +dxdp log n(x, p), monotonically increases with time, +even though the fundamental equation of motion possesses a time-reversal +symmetry. This old irreversibility paradox has been recently revisited for the +kinetic theory of particles [1], using dynamical large deviation principles, in the +case of the Boltzmann equation. It turns out that the dynamical large deviation + +Springer Nature 2021 LATEX template +6 +Large deviations for linear wave kinetic equation +principle that quantifies the probability for the evolution of any trajectory has a +time reversal symmetry. The kinetic equation corresponds to the most probable +path of the system, while the probability of a path and its time-reversed path +is related through detailed balance, a manifestation of time reversal symmetry +for the mesoscopic stochastic process. This gives an extremely simple and +enlightening explication of the irreversibility paradox. The main result of the +present paper is a large deviation principle for the Schrödinger equation in a +weak random potential, which has also a time reversal symmetry. This gives a +new very clear explanation of the time reversal paradox. +The purpose of this paper is to formulate a path large deviation principle +for wave scattering by random disorder in spatially inhomogeneous problems. +In particular, to make the discussion as concise as possible, we restrict our +attention to the simplest Schrödinger equation model. Our fundamental results +are as follows. First, (i) for a small but finite µ we show that the probability +that a path of local spectral density {nµ(t)} evolves at a vicinity of a prescribed +specific path {n(t)} satisfies a large deviation principle: +P +� +{nµ(t) = n(t)}0⩽t⩽tf +� +≍ +µ→0 exp +� +− +1 +(2πµ)d +� tf +0 +dt sup +λ +�� +λ ˙n − H[n, λ] +�� +P0[n(0)], +where H is the large deviation Hamiltonian that generally governs the stochas- +tic fluctuations of macroscopic variables and P0[n(0)] is the probability of the +initial condition nµ(t = 0). Obtaining the explicit expression for H from the +microscopic equation is one of the main results of this paper. Next, (ii) we ver- +ify that the ordinary wave kinetic equation describes the path that minimizes +the exponent of the probability functional. Then, (iii) we establish a large +deviation principle for the microcanonical measure that defines the quasipo- +tential of the mesoscopic stochastic process of the local spectral density. We +analyze (iv) the property of the large deviation Hamiltonian, check its symme- +tries related to conservation laws and the time-reversal symmetry, and derive +an expression of the detailed balance that connects the probabilities of a path +and its time-reversed path. Finally (v) we study the diffusive limit when the +scales of variation of the random potential are much larger than the typical +wavelength of the waves. For this case we obtain a diffusive large deviation +Hamiltonian, for which we check all the desired symmetries. +The paper is organized in the following order. In section 2, we first set up +the basic problem, introduce scaling and statistical assumptions, and derive +the ordinary form of the wave kinetic equation. In section 3, we derive the +path large deviation principle for the temporal variations of the local spectral +density. The approach has some analogy with that of [2], with a slightly differ- +ent scaling assumption, and working with the Wigner distribution to describe +the wave local spectral density. We then show that this Hamiltonian satisfies +the expected properties. A remarkable point is that the quasipotential enter- +ing into the detailed balance relation is consistent with the one obtained from + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +7 +a direct computation of microcanonical ensemble, formulated in Appendix B. +Section 4 proposes several possible extensions in future studies. +2 Wave kinetic equation for a linear Schrödinger +equation +2.1 Problem setup +We consider the Schrödinger equation for a wave function ψ∗(x∗, t∗) : Rd+1 → +C, where t∗ is time and x∗ is the position vector, and with potential V ∗(x∗) : +Rd → R: +i∂ψ∗ +∂t∗ = −D +2 ∇2 +x∗ψ∗ + V ∗ψ∗. +(1) +We use an upper script ∗ to represent variables with physical dimensions. The +physical parameter D > 0 has dimension L2T −1. In absence of interaction with +the potential, the free Schrödinger equation is the dynamics of linear waves +with a dispersion relation ω(k∗) = D|k∗|2/2, where k∗ is a wave vector. A +localised wave packet propagates at group velocity ∇k∗ω(k∗) = Dk∗. +The potential V ∗(x∗) is assumed to be a spatially homogeneous random +field with zero-average, E[V ∗] = 0, with its spectral density given by +Π∗(k∗) = +1 +(2π)d +� +Rd dy∗e−ik∗·y∗E +� +V ∗ +� +x∗ + y∗ +2 +� +V ∗ +� +x∗ − y∗ +2 +�� +. +(2) +For homogeneous fields, the two-point correlation function E [V ∗(x∗ +1)V ∗(x∗ +2)] +depends only on the point separation x∗ +1 − x∗ +2. The spectral density of +the potential is the Fourier transform of the two-point correlation function +E [V ∗(x∗ +1)V ∗(x∗ +2)] with respect to x∗ +1 − x∗ +2 and thus contains the same infor- +mation. Note that a prescription of higher order cumulants would be needed +to fully characterize the potential distribution. As we will see in the following, +the higher order statistics of the potential will not affect the dynamics of the +spectral density of the waves in the kinetic regime. Hence, although we do not +specify all the cumulants, the potential needs not to be Gaussian. +In this article, we assume that the spectral density Π∗ is concentrated +around wave vectors |k∗| ∼ 2π/λ where λ is the typical wavelength. In real +space, the wavelength λ is interpreted as the typical correlation length of the +potential. For such a potential, the Schrödinger dynamics Eq. (1) may display +many different regimes, depending on the order of magnitude of λ compared +to typical wavelengths in the initial condition of ψ∗. For instance, if the initial +condition is made of waves with wavelengths much smaller than λ, the wave- +potential interaction corresponds to random but smooth refraction that leads +to diffusion in the macroscopic limit [17–19, 50]. We will see in section 3.4 +that this diffusive limit can be recovered from the wave kinetic regime. On the +other hand, if the initial waves have wavelength much greater that λ, one faces + +Springer Nature 2021 LATEX template +8 +Large deviations for linear wave kinetic equation +a homogenization problem that is not described by the wave kinetic equation +[51, 52]. In the present paper, we will focus on an intermediate regime, when +the initial condition is made of waves with typical wavelengths which are of +order λ. We also make an assumption that the potential term is very small +compared to the Laplacian term, by setting +ǫ ≡ λ2V0 +D +≪ 1, +(3) +where V0 is a constant scaling the potential, i.e., the potential spectrum is +typically Π∗ ∼ V 2 +0 λd. +Wave energy or wave action measures the local amplitude of the signal. +We denote ℓ the typical scale for spatial variation of wave action, and call it +the mesoscopic spatial scale. We introduce the second natural dimensionless +parameter +µ ≡ λ +ℓ . +(4) +The kinetic limit is the limit µ ≪ 1. In the context of wave kinetics, we are +interested in the statistical behavior of the system at the mesoscopic scale, +avoiding chasing rapid phase oscillations at scale λ. Since the group velocity +of a wave packet is D|k∗| ∝ Dλ−1, the migration time of a wave packet over +a mesoscopic distance ℓ is λ2/Dµ. We call this time the mesoscopic time. +Choosing such mesoscopic units naturally yields the following dimensionless +coordinates +x ≡ µx∗ +λ , +p ≡ λk∗, +t ≡ µDt∗ +λ2 , +(5) +where the scaled wave vector is now represented by p in a customary way of +quantum mechanics with µ corresponding to the Dirac constant. Physically, +the square of the absolute value of the wave function, |ψ∗|2, represents the +wave action density that is proportional to energy, momentum, or number of +particles contained in a unit volume. Therefore, on the dimensional ground, +the wave function should be dependent on the scaling parameters as +ψµ(x) = λd/2ψ∗(x∗) +µd/2 +. +The potential and its spectrum are scaled as +V µ(x) = V ∗(x∗) +V0 +, +Π(p) = Π∗(k∗) +V 2 +0 λd , + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +9 +such that +Π(p) = +1 +(2πµ)d +� +Rd dye−ip·y/µE +� +V µ � +x + y +2 +� +V µ � +x − y +2 +�� +. +(6) +In the end, the governing equation (1) is rewritten in the non-dimensional form +iµ∂ψµ +∂t += −µ2 +2 ∇2 +xψµ + ǫV µψµ, +(7) +which is the fundamental model of the present work. +For ǫ = 0, waves ei(p·x−ωt)/µ, with ω = |p|2/2, are exact solutions of the +equations, illustrating that the microscopic time scale and spatial scales are +tm ∼ O(µ) and xm ∼ O(µ) respectively. A wave packet a(x, t)ei(p·x−ωt)/µ with +modulation of its amplitude a on spatial scales of order one (slow modulation +compared to the microscopic scale), will actually see an evolution of a on time +scales of order one, according to the group velocity p. In the limit of small ǫ, +the effect of the inhomogeneous potential term is very small on the microscopic +time scale. Therefore, a wave packet propagates almost freely in a microscopic +time scale. Since E[V µ] = 0 has been assumed, effects of terms proportional +to ǫ will vanish on average. Accumulation of the random potential effect will +then give non zero contribution of order ǫ2. In order for this to be on the same +order as effects of free propagation on the wave action requires +ǫ = √cµ +(8) +with c > 0 a finite constant which accounts for the relative importance of the +scattering interactions with respect to propagation. The constant c is strictly +speaking not needed, and it could be absorbed in the definition of Π, but it is +useful for the physical discussion. The kinetic regime, or kinetic limit, is the +joint limit µ → 0 with ǫ = √cµ, where c is a fixed constant. Consequently, the +pertinent equation for the kinetic scaling will be +iµ∂ψµ +∂t += −µ2 +2 ∇2 +xψµ + √cµV µψµ. +(9) +In some parts of the following sections, we will perform asymptotic expansions +of the effect of the random potential by expanding (7) in power of ǫ, while +integrating out explicitly the wave propagation effects. For this reason, we +often consider (7) instead of (9) for those technical parts. +2.2 Local spectral density +In the regime of wave kinetics, one is interested in the amount of wave action +existing at each position and wave vector. This is provided by the (rescaled) +Wigner distribution of the signal that is defined, following previous work on + +Springer Nature 2021 LATEX template +10 +Large deviations for linear wave kinetic equation +inhomogeneous wave kinetics [9, 10], as +nµ(x, p, t) = +1 +(2πµ)d +� +Rd dy e−ip·y/µψµ � +x + y +2 +� +ψµ† � +x − y +2 +� +, +(10) +where we have denoted by ψµ† the complex conjugate of ψµ. The Wigner +distribution function is the local spectral density of the wave action defined +in both space of position and wave vector. Indeed, the integral of nµ over +wave-vector space coincides with the action density in position space, such +that +� +Rd dp nµ(x, p) = |ψµ(x)|2. A caution for the physical interpretation of +nµ is that it allows the existence of negative values, in contrast to wave action +or energy that should be strictly non-negative. The negativity of nµ can be +eliminated by averaging it over a scale comparable to µ [53]. +In this study, we will take the asymptotically small limit of both µ and ǫ. +Statistically, the limit of µ → 0 is regarded as a kind of thermodynamic limit; +since µ represents the typical correlation length of the wave signal, the total +number of degrees of freedom increase as µ−d. In general, the thermodynamic +limit makes sense when we specify the macroscopic or mesoscopic variables, +e.g., temperature or pressure for gas molecules. In the present model, the +potential spectrum Π(p) is a mesoscopic control parameter that should not +depend on µ, and the local spectral density nµ is a mesoscopic variable to +be determined. In our formulation, a superscript µ is put on a variable that +depends on the scaling parameter. It is worth keeping in mind that, although +the spectral density function Π(p) is fixed, the corresponding potential function +V µ(x) depends on µ because its structure becomes finer and finer for µ → 0. +2.3 Conservation properties +We consider equation (7) on a spatial domain Γ, which is either Rd or a periodic +domain V. The norm +� +Γ dx|ψµ(x, t)|2 is conserved by the dynamics. If Γ is +Rd, the norm can be either finite for localized solution, or infinite. A local +conservation law always exists for |ψµ(x, t)|2. +We now consider the other invariants, for equation (7), or the associated +local conservation laws. If not specified, integrations for positions and wave +vectors are always performed over Γ and Rd, respectively. For a polynomial +function f : R → R, we define an operator Kǫ,µ +f += f(−(µ2/2)∇2 +x + ǫV µ). Since +the Schrödinger operator is self-adjoint, a straightforward computation shows +that +⟨f⟩ ≡ +� +dxψµ†(x, t)Kǫ,µ +f ψµ(x, t) +(11) +is independent of the time t when ψµ is a solution of Eq. (7). Equation (11) is +rewritten in terms of the local spectral density as +⟨f⟩ = +� +dxdpKǫ,µ +f (x, p)nµ(x, p, t), +(12) + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +11 +where +Kǫ,µ +f (x, p) = +� +Rd dye−ip·y/µ ˆKǫ,µ +f +� +x + y +2 , x − y +2 +� +and +ˆKǫ,µ +f (x1, x2) ≡ Kǫ,µ +f δ(x1 − x2) +is the Weyl symbol of the operator Kǫ,µ +f . In general, Kǫ,µ +f +can be expanded +as a power series of ǫ and µ. The leading-order term is immediately obtained +as K0,0 +f (x, p) = f +� +|p|2/2 +� +. Therefore, for the small limit of ǫ and µ, we may +write the general invariant as +⟨f⟩ = +� +dxdp f +�|p|2 +2 +� +nµ (x, p) . +(13) +Since the choice of f is arbitrary, the present system possesses an infinite +number of invariant. This property is related to wave frequency conservation. +For a scattering of wave action in spectral space by a time-independent poten- +tial, wave frequency does not change. Therefore, the amount of wave action +with frequency less than ω remains constant. Setting f(σ) = h(ω − σ) in (13), +where h is the Heaviside function and ω ∈ R+, we define +Aω[nµ] ≡ +� +dxdp h +� +ω − |p|2 +2 +� +nµ(x, p). +(14) +Conservation of ⟨f⟩ for an arbitrary f is equivalent to the conservation of Aω +for an arbitrary ω ∈ R+, as ⟨f⟩ = +� +R+ dω dAω +dω f(ω). +2.4 Wave kinetic equation +In this subsection, we shall derive the classical form of the wave kinetic +equation. A common derivation of the wave kinetic equation starts from a +closed equation on the Wigner distribution and performs a multiple time-scale +expansion, see e.g. [9]. We rather adopt here a perturbative approach in ǫ +from the Schrödinger equation (7). This approach is also classical in the wave +turbulence literature [2, 54, 55], will appear helpful in the derivation of the +dynamical large deviation theory in section 3, and has the advantage to gen- +eralize easily to the case of the kinetic theory of non-linear waves with 3-wave +interactions. +The first step is to express the solution of the Schrödinger equation (7) +using an expansion in power of ǫ, such that ψµ = ψµ +0 + ǫψµ +1 + ǫ2ψµ +2 + O(ǫ3). +Inserting this expansion to (7), we derive the series of equations: +iµ∂ψµ +0 +∂t += −µ2 +2 ∇2 +xψµ +0 +iµ∂ψµ +1 +∂t += −µ2 +2 ∇2 +xψµ +1 + V µψµ +0 + +Springer Nature 2021 LATEX template +12 +Large deviations for linear wave kinetic equation +iµ∂ψµ +2 +∂t += −µ2 +2 ∇2 +xψµ +2 + V µψµ +1 +.... +We consider this expansion for any given initial condition ψµ(x, 0) = ψµ,0(x). +We assume that the initial condition does not depend on ǫ, i.e., ψµ +0 (x, 0) = +ψµ,0(x) and ψµ +j (x, 0) = 0 for j ≥ 1. To write down the solution, we introduce +the propagator Gµ(x, t) such that +� +iµ ∂ +∂t + µ2 +2 ∇2 +x +� +Gµ = iµδ(x)δ(t), +(15) +and Gµ = 0 for t < 0. We then obtain +ψµ +0 (x, t) = +� +Rd dξGµ(x − ξ, t)ψµ(ξ, 0) +(16a) +ψµ +1 (x, t) = 1 +iµ +� t +0 +dτ +� +Rd dξGµ(x − ξ, t − τ)V µ(ξ)ψµ +0 (ξ, τ) += 1 +iµ +� t +0 +dτ +� +R2d dξ12Gµ(x − ξ1, t − τ)V µ(ξ1)Gµ(ξ1 − ξ2, τ)ψµ(ξ2, 0) +(16b) +ψµ +2 (x, t) = 1 +iµ +� t +0 +dτ +� +Rd dξGµ(x − ξ, t − τ)V µ(ξ)ψµ +1 (ξ, τ) += +1 +−µ2 +� t +0 +dτ1 +� τ1 +0 +dτ2 +� +R3d dξ123Gµ(x − ξ1, t − τ1)V µ(ξ1) +× Gµ(ξ1 − ξ2, τ1 − τ2)V µ(ξ2)Gµ(ξ2 − ξ3, τ2)ψµ(ξ3, 0). +(16c) +The equation for the propagator (15) is analytically solved as +Gµ(x, t) = +h(t) +(2πµ)d +� +dpe−i|p|2t/2µeip·x/µ, +(17) +where h(t) is again the Heaviside function. Although the wave vector inte- +gration of this expression can be carried out, we keep this form because it is +convenient for later computations. +Importantly, the perturbation solution will be valid for not too large t. For +longer times, the higher order terms will not be small compared to ψµ +0 even +though ǫ is small. As will be clear in the following discussion, we will need +the perturbative solution to be valid up to µ ≪ t ≪ 1; an intermediate range +between the microscopic and mesoscopic time scales. +Based on the perturbation solution of ψµ derived above, we shall consider +the evolution of the local spectral density nµ. To simplify the computation, we + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +13 +introduce the Wigner transform of two functions, f(x) and g(x), as +wµ(f, g)(x, p) ≡ +1 +(2πµ)d +� +Rd dy f +� +x + y +2 +� +g† � +x − y +2 +� +e−ip·y/µ. +(18) +Its inverse is +f(x1)g†(x2) = +� +dp wµ +�x1 + x2 +2 +, p +� +eip·(x1−x2)/µ. +(19) +Basically, the local spectral density is the Wigner transform of identical wave +functions, +nµ(x, p, t) = wµ(ψµ, ψµ). +(20) +Inserting ψµ = ψµ +0 + ǫψµ +1 + ǫ2ψµ +2 + . . . to (20), and using (16), we obtain an +expansion of the local spectral density nµ in terms of ǫ as +nµ(x, p, t) = wµ(ψµ +0 , ψµ +0 ) + ǫwµ(ψµ +1 , ψµ +0 ) + ǫwµ(ψµ +0 , ψµ +1 ) ++ ǫ2wµ(ψµ +1 , ψµ +1 ) + ǫ2wµ(ψµ +2 , ψµ +0 ) + ǫ2wµ(ψµ +0 , ψµ +2 ) + O(ǫ3). +(21) +The first term on the right-hand side is easily computed as +wµ(ψµ +0 , ψµ +0 ) = nµ(x − pt, p, 0). +(22) +This expression means that, in absence of the potential, the propagation of free +waves transports the spatial distribution of wave action density at the group +velocity p. It is notable that Eq. (22) is valid without taking an ensemble +average or an asymptotic limit. +We shall consider the expectation of (21) with respect to the realization of +the random potential V µ. Since E[V µ] = 0 has been assumed, terms propor- +tional to ǫ vanish. The dominant contribution from the random potential to +the variations in local spectral density comes from terms of order ǫ2. Direct +computations, performed in Appendix C, yield the expectation values of per- +turbation terms at the leading-order, (114), (115) and (116). Consequently, we +obtain +E +� +ǫ2wµ(ψµ +1 , ψµ +1 ) + ǫ2wµ(ψµ +2 , ψµ +0 ) + ǫ2wµ(ψµ +0 , ψµ +2 ) +� +=ǫ2t +µ +� +dησ(p, η) (nµ(x, η, 0) − nµ(x, p, 0)) + h.o.t. +with +σ(p1, p2) ≡ 2πΠ(p1 − p2)δ +�|p1|2 +2 +− |p2|2 +2 +� +. +(23) + +Springer Nature 2021 LATEX template +14 +Large deviations for linear wave kinetic equation +Here, h.o.t. represents the higher-order terms in the expansion that are +negligible in the asymptotic limit. Setting ǫ = √cµ, we have +lim +µ→0 +E[nµ(x, p, t)] − nµ(x − pt, p, 0) +t += lim +µ→0 c +� +dησ(p, η) (nµ(x, η, 0) − nµ(x, p, 0)) . +Taking the limit t → 0, we obtain a differential equation for the ensemble +average of the local spectral density, limµ→0 E[nµ] = n: +∂n(x, p, 0) +∂t ++ p · ∇xn(x, p, 0) = c +� +dησ(p, η) (n(x, η, 0) − n(x, p, 0)) . +At this stage, this expression is only valid at the initial time, t = 0. One cannot +extend it to t > 0 because the wave function ψµ is a priori correlated with the +potential field V µ. However, we shall argue that the correlation between ψµ +and V µ remains always weak at any time. Indeed, in the present problem, it +is assumed that the significant modification of the wave field ψµ by scattering +on the potential occurs at a time scale of wave propagation over a mesoscopic +distance. The typical correlation length of the random potential V µ is much +shorter than this mesoscopic scale by a factor of µ. Therefore, even though +interferences of the field with the potential produce slight correlations, the +free propagation of the field makes the correlation vanishes. This situation +resembles the loss of memory for particle collision in dilute gas—the molecular +chaos hypothesis. The present weak-correlation assumption allows us to regard +the temporal evolution of nµ as a Markovian process such that the wave kinetic +equation would be valid any time as far as µ is sufficiently small. We note +that this explanation applies to d ≥ 2 cases. For one-dimensional problems, an +interesting phenomenon named localization is known to occur [56, 57]. This +localization phenomenon which invalidates the present kinetic regime [50, Sec. +5.2] is thus discarded in this paper. +Once this Markovian hypothesis is accepted, we may iterate the reasoning +expounded above for t = 0 in order to reach any time t > 0. We eventually +obtain the equation +∂n(x, p, t) +∂t ++ p · ∇xn(x, p, t) = c +� +dησ(p, η) (n(x, η, t) − n(x, p, t)) , +(24) +that is the ordinary form of the wave kinetic equation. Since the group velocity +of a free wave is now p, the second term on the left-hand side is understood +as the motion of the Wigner distribution at the group velocity due to the free +dynamics. The right-hand side represents wave scattering in wave-vector space +that occurs at microscopic scale. The function cσ(p1, p2) is the scattering cross +section determining the rate of wave action converted from wave vector p1 to +p2 per unit time. + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +15 +The wave kinetic equation inherits the conservation property of the original +Schrödinger equation, namely that dAω[n]/dt = 0. However, the wave kinetic +equation differs from the microscopic Schrödinger dynamics (7) since the for- +mer appears time-irreversible whereas the latter is time reversible. Indeed, +for a prescribed potential field V µ(x), let ψµ(x, t) be a solution of (7). Its +time-reverse counterpart is defined as ψµ +R(x, t) = ψ†(x, −t). As a consequence, +the time-reversed local spectral density reads as nµ +R(x, p, t) = nµ(x, −p, −t). +Change in sign of p is a natural outcome because the wave group veloci- +ties of the forward and the reverse paths should be opposite. If ψµ(x, t) is a +solution of the Schrödinger equation, ψµ +R(x, t) is also a solution. This is the +time-reversal symmetry. By contrast, the wave kinetic equation violates this +symmetry: if n is a solution, the time-reversed nR is not a solution of the wave +kinetic equation, unless both n and nR are an identical stationary state (see +Eq. (26) defined below). This irreversibility paradox is reminiscent to the one +raised by Boltzmann for the case of dilute gases. As explained in [1] for the +case of the Boltzmann equation, we will see in the next section that one can +recover time-reversibility for the kinetic theory at the large deviation level, as a +time-reversibility for the stochastic process of the local spectral density. Time- +irreversibility arises because the wave kinetic equation describes the evolution +of the average n = E[nµ] only, or equivalently in this case the most probable +evolution only. Other paths, including time-reversed paths, are possible: they +are just extremely unlikely. +Before moving toward the large deviation theory of wave kinetics, we +can remark that time-irreversibility can be also quantified by introducing a +Lyapunov function +S ≡ +� +dxdp log n(x, p). +(25) +Following solutions of the wave kinetic equation, S increases monotonically +with time, dS/dt ≥ 0. If the spatial domain Γ is finite, S achieves the maxi- +mum when the spectral density n(x, p) is homogeneous in x and isotropic in +p. We write this homogeneous distribution of the spectral density under the +constraint of Aω[n] = A(ω) as +nA +h (p) = +A′(|p|2/2) +� +dxdηδ(|p|2/2 − |η|2/2), +(26) +where the denominator is introduced for normalization purpose, and A′ = +dA/dω. It is obvious that n(x, p) = nA +h (p) is a stationary solution of the +wave kinetic equation. We note that the Lyapunov function S is actually the +microcanonical entropy of the macrostate specified by (nµ, Aω) = (n, A) for +the Schrödinger equation as we discuss in Appendix B, and is related to the +quasipotential that appears in the discussion of the large deviation theory, as +we discuss in section 3.2.3. + +Springer Nature 2021 LATEX template +16 +Large deviations for linear wave kinetic equation +3 Large deviation formulation for random wave +scattering +In the previous section, we derived the wave kinetic equation as an equation for +the ensemble average E [nµ] of the empirical local spectral density, with respect +to the probability measure of the random potential, in the limit µ → 0. This +kinetic limit can be understood as a law of large number: limµ→0 nµ = E [nµ], +where the limit has to be understood in a weak sense (for instance, the limit +holds when both nµ and E [nµ] are integrated over any subset U ⊂ Γ × Rd). +In statistical mechanics, it is quite common that an empirical macroscopic +quantity converges to its ensemble average for the large limit of the number +of elements. The deviation from the average is often exponentially small and +evaluated asymptotically by a large deviation principle. In this section, our +aim is to generalize the law of large number for the kinetic theory and to +compute the probability to observe any possible fluctuations for nµ, as a large +deviation principle, in the limit µ → 0. Such fluctuations are expected to be +characterized by a large deviation parameter proportional to µd, where d is +the space dimension, because the number of statistically independent degrees +of freedom is of order µ−d. +3.1 Large deviation Hamiltonian +We define the Newton ratio as the time increment for the local spectral density: +∆nµ/∆t = (nµ(∆t) − nµ(0))/∆t. We regard the temporal variations in nµ as +a stochastic process, and look for the probability to observe a value of the +Newton ratio, conditioned on the value of the local spectral density at time 0: +nµ(0) = n. Our aim is to justify that it satisfies +− lim +∆t→0 lim +µ→0 +ǫ=√cµ +(2πµ)d +∆t +log P +� ∆nµ +∆t = ˙n|nµ = n +� += L [n, ˙n] , +(27) +and to derive an explicit formula for the Lagrangian L. The limit lim µ→0 +ǫ=√cµ +corresponds to the kinetic limit µ → 0 where one has fixed ǫ = √cµ. Here, +L[n, ˙n] is the rate function of the probability of the Newton ratio P [ ˙n|n]. +Through the fast microscopic dynamics, the memory of the initial condi- +tion of the phases of ψµ are expected to be lost after some times, meaning that +two-times, or multi-times, correlation functions are expected to decay with the +time differences of two or several phase observables. Such a mixing property +is expected to be due to the conjunction of phase mixing (oscillating integrals +and the Riemann–Lebesgue lemma), spatial transport and dispersion, and the +effect of the random potential. Because the natural timescale for phase dynam- +ics is the microscopic timescale, one might expect a typical mixing time to be +much smaller than the kinetic time scale and to decay to zero in the kinetic +limit. This mixing property is required to justify a Markovian behavior and to +propagate local in time results, like the Lagrangian (27), in order to describe + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +17 +the dynamics at finite times. In none of the existing classical kinetic theories, +neither physicists nor mathematicians have been so far able to justify or prove +the requested mixing properties, in order to justify the long time validity of +kinetic theories or their probabilistic large deviation generalizations. This is +the main reason why mathematical results, for instance the celebrated Lan- +ford’s result for the Boltzmann equation [58], or its generalizations [59], are +usually valid only for a fraction of the kinetic time. As it is customary in the- +oretical physics, we will assume the validity of such a mixing property in the +following. +We now assume the natural mixing hypothesis and the related Markov +behavior of the stochastic process. As a consequence, the evolution of nµ does +not depend on its previous state. Then the path probability for the stochastic +process is Markovian, and the probability of a path of nµ for a finite time +interval, 0 ⩽ t ⩽ tf, can be derived from the local in time Lagrangian (27). +The path probability conditioned on the local spectral density at the initial +time nµ(0) = n(0) is then +Pn(0) +� +{nµ(t) = n(t)}0⩽t⩽tf +� +≍ +µ→0 exp +� +− +1 +(2πµ)d +� tf +0 +dtL[n, ˙n] +� +. +(28) +This expression is analogous to the path-integral formulation for the probabil- +ity density in quantum theory. Note that the initial conditions nµ(0) = n(0) is +fixed here, but one can easily consider a set of initial condition complemented +by an initial probability density P0. In such a case, the path probability of a +trajectory {n(t)}0⩽t⩽tf reads as +P +� +{nµ(t) = n(t)}0⩽t⩽tf +� +≍ +µ→0 exp +� +− +1 +(2πµ)d +� tf +0 +dtL[n, ˙n] +� +P0[n(0)]. +(29) +To compute the Lagrangian (27), we will use the Gärtner-Ellis theorem +that connects the rate function L to the cumulant-generating function, or the +Hamiltonian H defined by +H[n, λ] = lim +∆t→0 lim +µ→0 +ǫ=√cµ +(2πµ)d +∆t +× log E +� +exp +�� +dxdpλ(x, p)(nµ(x, p, ∆t) − n(x, p)) +(2πµ)d +�� +, +(30) +through the Legendre-Fenchel transform, +L [n, ˙n] ≡ sup +λ +�� +dxdpλ(x, p) ˙n(x, p) − H[n, λ] +� +. +(31) +Our aim here is to derive the specific form of H directly from the perturbation +solutions of the original Schrödinger equation (7). + +Springer Nature 2021 LATEX template +18 +Large deviations for linear wave kinetic equation +Following [2], we first compute the moment-generating function for the +increment of nµ, +Zµ[n, λ; ∆t] ≡ E +� +exp +�� +dxdpλ(x, p)(nµ(x, p, ∆t) − n(x, p)) +(2πµ)d +�� +. +(32) +We insert the expansion of nµ(x, p, t) in terms of ǫ into (32). Since at the lead- +ing order wµ(ψµ +0 , ψµ +0 ) is statistically independent of V µ by the weak correlation +hypothesis described before, it is possible to decompose Zµ into two parts as +Zµ = Zµ +0 Zµ +ǫ +(33a) +Zµ +0 ≡ exp +� +1 +(2πµ)d +� +dxdpλ(x, p) (wµ(ψµ +0 , ψµ +0 ) − n(x, p)) +� +(33b) +Zµ +ǫ ≡ E +� +exp +� +1 +(2πµ)d +� +dxdpλ(x, p) (nµ(x, p, ∆t) − wµ(ψµ +0 , ψµ +0 )) +�� +. (33c) +Using (22), the first part, Zµ +0 , is immediately rewritten as +Zµ +0 = exp +� +1 +(2πµ)d +� +dxdpλ(x, p) (n(x − p∆t, p, 0)) − n(x, p) +� +. +(34) +The second part is expanded in terms of ǫ to obtain +Zµ +ǫ = 1 + ǫ2Zµ +2 + o(ǫ2), +(35) +where we have used E[V µ] = 0. The Landau notation o(ǫ2) gathers all the +terms that are negligible compared to O(ǫ2) terms in the expansion. In the +following, we shall discard the higher order terms of o(ǫ2) and concentrate +on computing Zµ +2 . Because we are considering the simultaneous limit of µ, +ǫ = √cµ, neglecting o(ǫ2) terms cannot be justified a priori. This kind of +problem commonly arises in wave kinetic theory [2] but we expect o(ǫ2) to be +negligible in the kinetic limit. The direct computation of Zµ +2 yields +Zµ +2 = +1 +(2πµ)d +� +dxdpλ(x, p)E [wµ(ψµ +2 , ψµ +0 ) + wµ(ψµ +0 , ψµ +2 ) + wµ(ψµ +1 , ψµ +1 )] ++ +1 +2(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2) +× E +� +(wµ(ψµ +1 , ψµ +0 )(x1, p1) + wµ(ψµ +0 , ψµ +1 )(x1, p1)) +× (wµ(ψµ +1 , ψµ +0 )(x2, p2) + wµ(ψµ +0 , ψµ +1 )(x2, p2)) +� +. +(36) + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +19 +From the computations in Appendix C, specifically (114)-(116) and (118)- +(121), we obtain +Zµ +2 = +∆t +µ(2πµ)d +� +dxdp12 +� +(λ(x, p1) − λ(x, p2))σ(p1, p2)n(x, p2) ++ 1 +2(λ(x, p1) − λ(x, p2))2σ(p1, p2)n(x, p1)n(x, p2) +� ++ h.o.t.. +(37) +Inserting Zµ = Zµ +0 +� +1 + ǫ2Zµ +2 +� +into H = lim∆t→0 lim µ→0 +ǫ=√cµ +((2πµ)d/∆t) log Zµ, +we obtain the Hamiltonian as +H[n, λ] = HF + HS +(38a) +HF = − +� +dxdpλ(x, p)p · ∇xn(x, p) +(38b) +HS = c +� +dxdp12 +� +(λ(x, p1) − λ(x, p2))σ(p1, p2)n(x, p2) ++ 1 +2(λ(x, p1) − λ(x, p2))2σ(p1, p2)n(x, p1)n(x, p2) +� +. +(38c) +We have separated the Hamiltonian into two parts, HF and HS. The first part, +HF , represents the free wave propagation in position space and the second +one, HS, the wave scattering in wave-vector space by the random potential. +3.2 Properties of the large deviation Hamiltonian +Once the specific form of the Hamiltonian is obtained, we can discuss the +properties of the stochastic process governed by the path-integral formula (28) +and (31). Formulations in the paper [1] are simple and informative, and we +summarize several important properties of dynamical large deviation theory +in Appendix A. We now check the classical expected properties of H. +3.2.1 Weak noise Langevin dynamics and wave kinetic +equation +The first important property of the large deviation Hamiltonian H is that it is +quadratic and convex with respect to the conjugated field λ. This means that +the fluctuations of the infinitesimal current ˙ndt are, locally in time, Gaussian. +Reading the quadratic part of the Hamiltonian (38c), we see that the local +covariance of the local in time Gaussian fluctuations are given by the diffusion +kernel +Σ[n](x; p1, p2) = −cσ(p1, p2)n(x, p1)n(x, p2) ++ c +� +Rd dησ(p1, η)n(x, p1)n(x, η)δ(p1 − p2). +(39) + +Springer Nature 2021 LATEX template +20 +Large deviations for linear wave kinetic equation +As a consequence, in the kinetic limit µ ≪ 1, the dynamics of the local spectral +density in the kinetic regime corresponds to a weak noise Langevin dynamics +[60, 61] +˙n(x, p, t) = −p · ∇xn + c +� +dη σ(p, η) (n(x, η, t) − n(x, p, t)) ++ +√ +2(2πµ)d/2 +� +dη Σ1/2[n](x; p, η)ξ(η, t) +(40) +with ξ(p, t) a white noise such that E [ξ(p, t)ξ(η, s)] = δ(t − s)δ(p − η), and +where the kernel Σ1/2[n] is defined as a square root of diffusion kernel, meaning +that +� +dηΣ1/2(x, p1, η)Σ1/2(x, η, p2) = Σ(x, p1, p2). +The second expected property is that the most probable path is the solution +of the wave kinetic equation. This is easily checked by noticing that the most +probable path, for which the action +� tf +0 L[n, ˙n]dt = 0 vanishes, satisfies +∂n +∂t = δH +δλ +���� +λ=0 +(41) +(see Appendix A.1.1), and is also the linear term in λ of the Hamiltonian H and +the drift term of the Langevin dynamics (40). Then, the path large deviation +analysis confirms that the wave kinetic equation can be understood as a law +of large number at the level of trajectories. +It is enlightening to rewrite the scattering part of the Hamiltonian HS in +terms of the diffusion kernel (39) as +HS = +� +dxdp1dp2 λ(x, p1)Σ[n](x; p1, p2) +� +δS +δn(x, p2) + λ(x, p2) +� +, +(42) +where S is the entropy (25), and the Langevin equation (40) as +˙n(x, p, t) + p · ∇xn = +� +dη Σ[n](x; p, η) +δS +δn(x, η) ++ +√ +2(2πµ)d/2 +� +dη Σ1/2[n](x; p, η)ξ(η, t). +(43) +This suggestive forms immediately emphasize that S is a Lyapunov function, +and that the dynamics has a detailed balance structure, as further explained +in section 3.2.4. +3.2.2 Conservation of the wave action distribution +As we discussed in section 2.3, in the original Schrödinger equation, the quan- +tity Aω[nµ] = +� +dxdph(ω − |p|2/2)nµ(x, p) for any ω ∈ R+ is conserved. +This conservation property has to be also verified at the large deviation +level, meaning that any trajectory {n(t)}0⩽t⩽tf has to lie on the manifold + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +21 +Aω[n] = A(ω). As explained in Appendix A.2.1, this is equivalent to the +Hamiltonian symmetry +H +� +n, λ + αδAω +δn +� += H [n, λ] , +(44) +for an arbitrary α ∈ R, n and λ. For the specific form of the Hamiltonian +H = HF + HS (38a) with HS written in the symmetric form (42), one can +directly check the above Hamiltonian symmetry boils down to the following +property on the diffusive kernel (39) +� +dη Σ[n](x; p, η) +δAω +δn(x, η) = 0, +(45) +for any n, x, p, and ω. +3.2.3 Stationary quasipotential +It was shown in the previous section that the wave kinetic equation possesses +an attractor which is a homogeneous distribution, nh, with the prescribed +constraints Aω[n] = A(ω), for any ω. We now consider the fluctuations of n +from nh at the large deviation level. More precisely, we define the equilibrium +distribution of the stochastic process at a large deviation level: +Pµ +A,S[{nµ = n}] ≍ +µ→0 exp +� +− UA[n] +(2πµ)d +� +, +(46) +where Pµ +A,S is the stationary probability measure of the microcanonical ensem- +ble which is parameterized by a small constant µ as well as a function A(ω) +specifying the action conservation constraint. The rate function UA is named +the quasipotential (see Appendix A.1). +In principle, the quasipotential can be computed from the dynamics, start- +ing from the large deviation Hamiltonian, see for instance Appendix A.1.3. +Formula (69) gives an expression for the quasipotential, in the cases when +the wave kinetic equation has a single attractor. However, when one knows +explicitly the microscopic stationary distribution, for instance in the case of +equilibrium statistical mechanics, one can compute directly the quasipotential. +It is then related to the entropy. Those different expressions have to give con- +sistent results. Indeed, one may see a simple example of the relation between +microcanonical entropy and the quasipotential for the dilute gas dynamics in +[1]. The case of spatially homogeneous weakly nonlinear wave dynamics has +been discussed in [2]. For the present problem, we compute the quasipotential +from a microcanonical ensemble for the original Schrödinger equation model + +Springer Nature 2021 LATEX template +22 +Large deviations for linear wave kinetic equation +in Appendix B. One obtains +UA[n] = + + + +− +� +dxdp log +�n(x, p) +nA +h (p) +� +if +Aω[n] = A(ω) ++∞ +otherwise +. +(47) +This result is technically not obvious. As far as we know, it had never been +derived before. +As shown in Appendix A.1.4, it is a generic property of the quasipoten- +tial to play the role of a Lyapunov function for the deterministic relaxation +dynamics, in this case the wave kinetic equation. Such a property can be +derived generically from the existence of a large deviation principle, indepen- +dently on the specific form of the large deviation Hamiltonian. One can note +that the quasipotential (47) is the opposite of the entropy S (25), up to an +additive constant. The quasipotential now depends on A(ω) and satisfies the +normalization condition, minn UA[n] = 0, where the minimum is achieved when +n(x, p) = nA +h (p) (26). +From Eq. (42) and the fact that UA is equal to −S up to a constant, +it is immediately checked that UA solves the stationary Hamiltonian-Jacobi +equation, +H +� +n, δUA +δn +� += 0. +(48) +3.2.4 Time-reversal symmetry and detailed balance +Finally, we consider the time-reversal symmetry of the dynamics (see Appendix +A.2.2). For the present wave kinetic theory, we showed at the end of section +2.4 that the time-reversed local spectral density is defined as nR(x, p, t) = +n(x, −p, −t), namely that wave vector needs to change sign in addition to time- +reversal t → −t. Therefore, it is useful to introduce the involution operator I +such that I[n(x, p)] = n(x, −p). +Since the wave kinetic dynamics is an equilibrium dynamics whose sta- +tionary state is characterised by the microcanonical quasipotential (47) at the +large deviation level, we expect the fluctuating dynamics to be time-reversible. +Following Appendix A.2.2, we indeed check that the Hamiltonian satisfies the +detailed-balance condition +H +� +n, λ + δUA +δn +� += H [I[n], −I[λ]] , +(49) +thus proving the time-reversal symmetry of the dynamics. For a trajectory that +follows the wave kinetic equation, UA monotonically decreases towards 0, and +therefore the local spectral density irreversibly approaches the homogeneous +distribution nA +h . However, when µ is small but finite, there remains a possibility +that UA increases, namely that the spectrum moves away from the attractor + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +23 +nA +h . This property is quantified by the fluctuation relation that is equivalent +to the detailed balance relation (49) +Pn(0)[{nµ(t) = n(t)}0⩽t⩽tf ] +Pn(tf)[{nµ(t) = nR(t − tf)}0⩽t⩽tf ] +≍ +µ→0 exp +�UA[n(0)] − UA[n(tf)] +(2πµ)d +� +. +(50) +The fluctuation relation (50) is the fundamental answer to the irreversibil- +ity paradox that arises in any classical kinetic theory, and in particular for +the classical wave kinetic theory. Indeed, let us consider a path starting from +n(0) that follows the wave kinetic equation until it reaches a state n(tf) +at some time tf +> 0. Since the quasipotential UA is a Lyapunov func- +tion, one has UA[n(tf)] < UA[n(0)]. From the fluctuation relation Eq. (50), +the exact time-reversed path starting from the state n(tf) has a probablity +∼ exp +� +−(2πµ)−d∆U +� +(with ∆U = UA[n(0)] − UA[n(tf)] > 0) to occur in the +small µ limit. The irreversibility turns into an improbability. +3.3 Decomposition of the Hamiltonian +Since a free wave packet does not change its wave vector during a free prop- +agation, and since also wave frequency is conserved during scattering by a +time-independent potential, waves with different pulsation ω(p) = |p|2/2 (i.e. +located on distinct spherical shells) do not interfere with each other. There- +fore, the dynamics can be separated into an infinite number of subsystems in +which the degrees of freedom are mutually independent. Indeed, this expec- +tation can be verified by showing that the Hamiltonian is decomposed as an +integration over frequency, as we do now. +To do so, let us rewrite the wave vector as p = peθ and define the cor- +responding frequency, ω = p2/2. The vector eθ is a unit vector whose angle +is specified by θ. Generally, the number of degrees of freedom for the angle +θ is d − 1. For example, for a d = 3 case, elevation and azimuthal angles +would be selected as a set of representative coordinate variables. The follow- +ing consideration is not dependent on the choice of these coordinates. We +just need to assume that a pair of opposite angles, θ and −θ, are defined +such that eθ = −e−θ. We decompose the wave vector integration element as +dp = pd−1dpdθ = (2ω)(d−2)/2dωdθ with dθ a surface element on a (d − 1)- +dimensional unit sphere, Sd−1. We define new variables labeled by frequency +ω as +nω(x, θ) = pd−2n(x, peθ) +(51a) +λω(x, θ) = λ(x, peθ) +(51b) +σω(θ1, θ2) = 2πΠ(p(eθ1 − eθ2)) +(51c) +Please do not confuse nω with nµ, in the context of this section. The Hamilto- +nian is decomposed into independent subdynamics, each of which involves the + +Springer Nature 2021 LATEX template +24 +Large deviations for linear wave kinetic equation +new variables labeled by ω, +H[n, λ] = +� +Hω[nω, λω]dω = +� +(Hω +F + Hω +S)dω +(52a) +Hω +F [nω, λω] = − +� +dxdθ(2ω)1/2λω(x, θ)eθ · ∇xnω(x, θ) +(52b) +Hω +S[nω, λω] = c +� +dxdθ1dθ2 +× +� +(2ω)(d−2)/2(λω(x, θ1) − λω(x, θ2))σω(θ1, θ2)nω(x, θ1) ++ 1 +2(λω(x, θ1) − λω(x, θ2))2σω(θ1, θ2)nω(x, θ1)nω(x, θ2) +� +, (52c) +where the integration for (x, θ) is carried out over Γ × Sd−1. For this system, +the diffusion kernel and the Lyapunov function are +Σω[nω](x; θ1, θ2) = −cσω(θ1, θ2)nω(x, θ1)nω(x, θ2) ++ c +� +dθ′σω(θ1, θ′)nω(x, θ1)nω(x, θ′)δ(θ1 − θ2) +(53a) +and +Sω[nω] = +� +dxdθ(2ω)(d−2)/2 log nω(x, θ). +(53b) +The scattering part of the Hamiltonian is accordingly represented as +Hω +S = +� +dxdθ1dθ2λω(x, θ1)Σ[nω](x; θ1, θ2) +� +δSω +δnω(x, θ2) + λω(x, θ2) +� +. (54) +We may compute the probability of a path of local spectral density sepa- +rately for each frequency band. For each subsystem, the amount of wave action, +∝ +� +dxdθnω(x, θ) ≡ N[nω], is invariant. For the attractor of the wave kinetic +equation, namely the homogeneous distribution, nω is everywhere constant in +Γ × Sd−1, which we write nω +h. Finally, we find that the quasipotential, +Uω[nω] = + + + +− +� +dxdθ(2ω)(d−2)/2 log +�nω(x, θ) +nω +h +� +if +N[nω] = N[nω +h] ++∞ +otherwise +(55) +satisfies +the +detailed +balance +condition, +Hω[nω, λω + δUω/δnω] += +Hω[I[nω], −I[λω]] with the involution I defined as I[nω(x, θ)] = nω(x, −θ). +Consequently, the fluctuation relation for a path and its reverse applies to each +subsystem separately. + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +25 +3.4 Diffusive limit +In this paper, we have focused on the wave kinetic regime when the random +potential presents high oscillations at a scale comparable to a typical wave- +length of the wave field. Another relevant limit, referred to as the diffusive +approximation [17] or the Fokker-Planck limit [50], corresponds to the regime +when the spatial oscillations of the potential are at a scale larger than those +of the wave field. In this regime, a wave signal propagates along rays which +are randomly refracted by the potential, leading to an asymptotic diffusion +equation for the local spectral density [17, 50]. +Technically, the diffusion equation on the local spectral density is often +derived from a multiscale expansion from the microscopic dynamics (7) [17, 50]. +But interestingly, the diffusive limit can also be obtained from the scattering +kinetic regime that we have considered in this paper. Our goal in this sub- +section is to show how one can derive a path large deviation theory for wave +kinetics in the diffusive regime from the large deviation Hamiltonian (42). In a +recent paper, a similar weak scattering limit has been considered to derive the +path large deviation principle for plasma below the Debye length, related to +the Landau equation, from the path large deviation principle for dilute gases, +related to the Boltzmann equation [43]. +In the diffusive regime, the random potential has typical variations over +large lengthscales compared to the typical wavelength of the signal. As a con- +sequence, the spectrum of the potential (6) is localised around p = 0, which +implies from the definition of the cross section (23) that incoming wave vec- +tors are randomly refracted by an infinitesimal amount at each time step by +the potential. One thus expects to obtain the path large deviation theory of +wave transport in the diffusive regime from the scattering regime by assuming +that the potential spectrum Π is supported in the vicinity of p ≈ 0. +In order to derive the diffusive limit from the large deviation Hamiltonian, +we use Eq. (42). We show in Appendix D that the diffusion kernel Σ transforms +into a differential operator such that for any test function f and g: +� +dp1dp2f(p1)Σ(x; p1, p2)g(p2) ≈ +� +dp ∇pf(p) · +� +n(x, p)2D(p) +� +· ∇pg(p). +(56) +Here, D is the diffusion matrix computed from the potential spectrum Π as +D(p) = − c +2 +� +R +∇y ⊗ ∇yR(ps)ds +and +R(y) = +� +dη Π(η)eiη·y. +(57) +From its definition, R(y) corresponds to the rescaled two-point correlation +function of the random potential. +Using Eq. (56), the large deviation Hamiltonian in the diffusive limit reads +as +H[n, λ] = +� +dxdp λ(x, p)p · ∇xn(x, p) + +Springer Nature 2021 LATEX template +26 +Large deviations for linear wave kinetic equation ++ +� +dxdp ∇pλ(x, p) · +� +n(x, p)2D(p) +� +· ∇p +� +δS +δn(x, p) + λ(x, p) +� +. +(58) +The diffusive Hamiltonian (58) conserves the spectral density at a given +pulsation ω(p) = |p|2/2 because of the fundamental property +D(p) · p = 0, +(59) +which results from p · ∇R(ps) = ∂R(ps)/∂s. This property means that the +diffusion of the spectrum is always orthogonal to the group velocity. +Moreover, the expression of the Hamiltonian (58) makes it clear that the +quasipotential of the diffusive dynamics is the same as the quasipotential of the +scattering limit Eq. (47). Furthermore, one can easily check that the detailed +balance relation (49) is still satisfied by the Hamiltonian (58). +4 Conclusions and perspective +The linear wave kinetic equation is a statistical model that governs wave +action density spreading in position and wave-vector space through propaga- +tion and scattering in random media. Motivated by recent works on dynamical +large deviation principles for kinetic theory, this study has derived the large +deviation principle describing the probability of a finite-time evolution of the +local spectral density of wave action in an asymptotic limit of scale separa- +tion. Importantly, the large deviation principle that is derived in this paper +satisfies a time-reversal symmetry with respect to the microcanoical quasipo- +tential, that is directly (and independently) computed from the microcanonical +measure. +In this paper, we restrict our considerations to the simplest Schrödinger +model with a homogeneous random potential. The next step is to extend the +present formulation to a wider range of situations. Possible technical difficulties +encountered during such future works involve (i) generalization of the disper- +sion relation to a function of position and wave vector, ω(x, p), (ii) coping with +spatial inhomogeneity for the randomness, that leads to a space-dependent +scattering cross section, σ(x, p1, p2), (iii) consideration of a vector field that +involves polarized waves or multiple waves, e.g., elastic media holding com- +pressional and shear waves. Combining the present approach and a previous +work on wave turbulence [2], we have also derived a formula of path-large +deviations for inhomogeneous spectral density in nonlinear 4-wave interacting +systems [62]. +In the context of geophysics, the wave kinetic equation is used to discuss +energy cascades of internal waves in the oceans and atmosphere where rotation +and density stratification play key roles [13]. Because the dispersion relation +of internal waves depends not on wave vector but on its angle against the +gravity direction, even linear theory predicts interscale energy transfer. In this +process, balanced geostrophic turbulent flow acts as a random potential. If + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +27 +assuming a stationary flow state, i.e., fixing the potential field in time, the +formulation will be analogous to the present case. On the other hand, once +we allow the temporal variations in the geostrophic flow field, the situation +essentially changes. Wave frequency is no longer conserved during propagation +and scattering. Spreading of action density in frequency space associates gain +or loss of wave energy. Quantification of the energy exchange rate between +evolving turbulent eddies and waves remains an open problem. Recently, Dong +et al. [19] discussed wave frequency diffusion in geostrophic turbulence based +on a kind of kinetic equation model. As pointed out in the present study, +the ordinary kinetic equation predicts an irreversible change in the spectral +density. In the actual environmental situation, the scale separation parameter, +µ, is not necessarily small and there should be non-negligible fluctuations in +spectral density. The large deviation formulation has a possible application for +the estimation of a fluctuating energy transfer rate, in such regime where the +kinetic theory is marginally valid. +Acknowledgments. +This work was supported by JSPS Overseas Research +Fellowship as well as KAKENHI Grant Number JP20K14556 (Y.O.), and by +the Simons Foundation through the Collaboration Grant 651463 “Wave Tur- +bulence” (F.B. and J.G.) and the Targeted Grant in MPS 663054 “Revisiting +the Turbulence Problem Using Statistical Mechanics” (F.B.). We thank Gre- +gory Eyink, Laure Saint-Raymond, Jacques Vanneste, and Antoine Venaille +for fruitful discussions. +A Properties of the stochastic system specified +by a large deviation Hamiltonian +A.1 Path large deviation +This appendix presents some general properties of a stochastic process Xǫ(t) +whose probability conditioned on an initial value, Xǫ(ti) = X(ti), is specified +at the large deviation level via a formula, +P +� +{Xǫ(t) = X(t)}ti⩽t⩽tf +� +≍ +ǫ→0 exp +� +−S[X] +ǫ +� +(60a) +S[X] = +� tf +ti +dtL(X, ˙X) ≡ +� tf +ti +dt sup +P +� +P · ˙X − H(X, P) +� +. +(60b) +Here, Xǫ(t) can be a scalar, vector, or continuous function defined on some +space. Basic requirements are that an inner product is properly defined, +and the dynamical property of the system is controlled by a single non- +negative parameter ǫ. For the simplicity, we regard X as a scalar but the +following consideration can be immediately extended to general cases. The +large deviation Hamiltonian H(X, P) is a convex function of P and satis- +fies H(X, 0) = 0 for any X. From the definition, the Lagrangian L satisfies + +Springer Nature 2021 LATEX template +28 +Large deviations for linear wave kinetic equation +L(X, ˙X) ≥ P · ˙X − H(X, P) for any X, ˙X and P. Therefore, inserting P = 0, +we know L ≥ 0. +A.1.1 Relaxation path +Clearly, in the limit of ǫ → 0, the system becomes deterministic with a sin- +gle path that minimizes the action S[X] for a prescribed initial condition +X(ti)—named the relaxation path. Since L ≥ 0, if there exists a function R(X) +that satisfies L(X, R(X)) = 0, a path solving ˙X = R(X) minimizes the action +and yields minX S[X] = 0. From the facts that L(X, ˙X) = P · ˙X − H(X, P) +with P solving ˙X − ∂H/∂P = 0 and H(X, 0) = 0 for any X, we understand +that a function R(X) = ∂H/∂P|P =0 fulfills L(X, R(X)) = 0. We thus assert +that an equation +˙Xr = R(Xr) ≡ ∂H +∂P (Xr, 0) +(61) +determines the relaxation path Xr(t). +A.1.2 Optimal path +Slightly changing the situation, if we fix both the initial and final states, +X(ti) = xi and X(tf) = xf, respectively, the most probable path from xi to +xf, namely the optimal path, or the instanton, is obtained by again minimizing +S[X]. This problem is equivalent to the principle of least action in analytical +mechanics. In this context, P is called the generalized momenta and repre- +sented by P = ∂L/∂ ˙X which is no longer 0. The optimal path in phase space +is governed by a set of canonical equations, +˙X = ∂H +∂P +(62a) +˙P = − ∂H +∂X , +(62b) +and we shall write their solutions as Xo[xf, tf; xi, ti] and P o[xf, tf; xi, ti]. +For the simplicity, we fix the initial conditions, ti and xi, and rewrite the final +state as x and t. We then introduce the Hamilton’s principal function Q as an +integration of the action following the optimal path as +Q(x, t) = +� t +ti +dτL(Xo(τ), ˙Xo(τ)). +(63) +It is known in analytical mechanics that in this case the generalized momenta +at the final time is represented as P o(t) = ∂L/∂ ˙X +��� +t = ∂Q/∂x, and Q solves + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +29 +the Hamilton-Jacobi equation, +∂Q +∂t + H +� +x, ∂Q +∂x +� += 0. +(64) +In the definition of Q(x, t), a set of arguments, (x, t), is arbitrary chosen. When +we pick up an optimal path {Xo(τ), P o(τ)}, at any point on this path, the +generalized momenta P and the Hamilton’s principle function Q is related via +P o(τ) = ∂Q +∂x (Xo(τ), τ). +(65) +Therefore, combining (65) and the first line of (62), we formally obtain a single +equation determining the optimal path, +dXo +dτ += ∂H +∂P +� +Xo, ∂Q +∂x (Xo, τ) +� +. +(66) +This equation is, however, not generally useful because Q(x, t) is inaccessible +in most cases. +A.1.3 Quasipotential and fluctuation path +Going back to the original stochastic model, the meaning of Q is understood +as the rate function for the probability that Xǫ reaches xf at t = tf. Indeed, +we may derive from (60) an expression +P [Xǫ(t) = x|Xǫ(ti) = xi] ≍ +ǫ→0 exp +� +−Q(x, t) +ǫ +� +(67) +based on the contraction principle. Now, we shall consider the stationary dis- +tribution of the probability density of Xǫ. This can be done simply setting +ti = −∞ in (67). To make the discussion more specific, let us assume that the +relaxation dynamics (61) has a unique global attractor x0, where R(x0) = 0. +Then, we set xi = x0 and also t = 0, to write a large deviation formula for the +stationary distribution +Ps(x) ≍ +ǫ→0 exp +� +−U(x) +ǫ +� +(68) +with +U(x) = +inf +X(t)|X(−∞)=x0 and X(0)=x +� 0 +−∞ +dtL(X(t), ˙X(t)). +(69) +The rate function U is called the quasipotential. Since U is the special case of +Q but independent of t, it solves the stationary version of the Hamilton-Jacobi + +Springer Nature 2021 LATEX template +30 +Large deviations for linear wave kinetic equation +equation (64), +H +� +x, ∂U +∂x +� += 0. +(70) +For the present case, the optical path Xo(τ) represents the most probable route +from an attractor x0 to a specific point x. This route is called the fluctuation +path and is denoted by Xf(t). Once we obtain the quasipotential U(x), (66) +provides an equation determining the fluctuation path as +˙Xf = F(Xf) ≡ ∂H +∂P +� +Xf, ∂U +∂x (Xf) +� +. +(71) +Since the vector field F(x) does not depend on t, this equation is more useful +than the original one (66). On the fluctuation path, the generalized momenta +is computed based on (65) as P f = ∂U/∂x(Xf). Combining (70) with the fact +that H is constant along the optical path, we understand that H(Xf, P f) = 0 +always holds. +A.1.4 Quasipotential as a Lyapunov function +A relaxation path and a fluctuation path have distinct properties for the +variations in U. For a relaxation path, we have +dU +dt (Xr) = ˙Xr ∂U +∂x (Xr) = ∂H +∂P (Xr, 0)∂U +∂x (Xr) += H(Xr, 0) +� +�� +� +=0 +− H +� +Xr, ∂U +∂x (Xr)) +� +� +�� +� +=0 ++∂H +∂P (Xr, 0)∂U +∂x (Xr) ≤ 0, +where we have used the general expressions, H(X, 0) = H(X, ∂U/∂x(X)) = 0, +and the convexity of H(X, P) for P. For a fluctuation path, we have +dU +dt (Xf) = ˙Xf ∂U +∂x (Xf) = ∂H +∂P +� +Xf, ∂U +∂x (Xf) +� ∂U +∂x (Xf) += H(Xf, 0) +� +�� +� +=0 +− H +� +Xf, ∂U +∂x (Xf)) +� +� +�� +� +=0 ++∂H +∂P +� +Xf, ∂U +∂x (Xf) +� ∂U +∂x (Xf) ≥ 0, +again from the convexity of H. We have thus learned that the quasipotential is +a Lyapunov function because it monotonically decreases in a relaxation path +while increases in a fluctuation path. These results are natural consequence +from a basic property that U(x) is minimum at the attractor x0 and the +relaxation and fluctuation paths represent routes to and from the attractor. + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +31 +A.2 Properties of large deviation Hamiltonian +A.2.1 Conservation law +From now on, we shall regard X and P as vectors so that there are multiple +directions in X space. For this case, when the Hamiltonian H possesses a +kind of symmetry, it is related to the conservation law of the system. More +specifically, let us suppose that we find a function C(X) that satisfies +H +� +X, P + α ∂C +∂X +� += H(X, P) +(72) +for any X, P, and α. This condition is equivalent to +∂H +∂P (X, P) · ∂C +∂X (X) = 0 +for any X and P. Now that H(X, ·) is flat in the direction of ∂C/∂X, from the +property of the Legendre-Fenchel transform [63], the corresponding Lagrangian +has a property, +L(X, ˙X) = +∞ +if +˙X · ∂C +∂X (X) ̸= 0. +(73) +This expression indicates that the probability for a path crossing a contour of +C is strictly 0. This constraint applies not only to the optimal path but also to +any path with random fluctuations. We thus understand that (72) serves as a +condition for C being an invariant of the system. +A.2.2 Detailed balance +The detailed balance is a property of equilibrium states which asserts time- +reversibility of the process, meaning that the probabilities of any trajectory +and its reversed counterpart are equal. A basic expression of detailed balance +for a stationary stochastic process is P∆t(y; x)PS(x) = P∆t(x; y)PS(y), where +P∆t(y; x) is the transition probability from a state x to another state y during +a time interval ∆t, and PS(x) is the stationary probability distribution. +Since we are now considering a continuous Markov process, it is enough to +regard ∆t as arbitrary small. For the limit of ∆t → 0, we may write y ∼ x+ ˙x∆t +and redefine the transition probability as P∆t(x, ˙x) ∼ P∆t(x+ ˙x∆t; x). Assum- +ing the continuity of P and PS, the detailed balance condition is rewritten +as +P∆t(x, ˙x)PS(x) ∼ P∆t(x + ˙x∆t, − ˙x)PS(x + ˙x∆t). +(74) +For the present problem, the probability distribution is specified as P∆t(x, ˙x) ≍ +exp(−∆tL(x, ˙x)/ǫ) and PS(x) +≍ +exp(−U(x)/ǫ). Therefore, the detailed + +Springer Nature 2021 LATEX template +32 +Large deviations for linear wave kinetic equation +balance condition (74) is rewritten as +L(x, ˙x) − L(x, − ˙x) = ˙x · ∂U +∂x . +(75) +This condition is modified in terms of the Hamiltonian via the Legendre- +Fenchel transform as +H(x, −p) = H +� +x, p + ∂U +∂x +� +. +(76) +Because H(x, 0) = 0 in the current problem, the stationary Hamilton-Jacobi +equation (70) is a necessary (but not sufficient) condition for U being the +quasipotential. +Once the detailed balance condition (76) is verified, we understand that +the probabilities of a path and its reverse are related at the large deviation +level via the expression, +P +� +{Xǫ(t) = X(t)}ti⩽t⩽tf +� +P +� +{Xǫ(t) = X(tf + ti − t)}ti⩽t⩽tf +� ≍ +ǫ→0 exp +� +−U(xf) − U(xi) +ǫ +� +, +(77) +an equivalent form of the Crooks fluctuation theorem. Another outcome of +the detailed balance is that the fluctuation path is the time reverse of the +relaxation path. This property is derived from +R(X) = ∂H +∂P (X, 0) = −∂H +∂P +� +X, ∂U +∂X +� += −F(X). +B Microcanonical ensemble and quasi-potential +for the Schrödinger equation +In this appendix, we consider the microcanonical ensemble of the dynamics +governed by the Schrödinger equation, (7). The aim is to compute the quasipo- +tential of the local empirical spectral density, i.e., the large deviation rate +function of nµ, in the small µ limit. We will prove that +Pµ +A,m[nµ = n] ≍ +µ→0 exp +� +− UA[n] +(2πµ)d +� +, +(78) +with Pµ +A,m the probabilities with respect to the microcanonical measure with +the constraints, +Aω[nµ] ≡ +� +h +� +ω − |p|2 +2 +� +nµ(x, p)dxdp = A(ω), +(79) + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +33 +where h is the Heaviside function, and A : R+ → R+ is a prescribed function. +The expected form of the rate function, namely, the quasipotential, is +UA[n] = + + + +− +� +dxdp log +�n(x, p) +nh(p) +� +if +Aω[n] = A(ω) ++∞ +otherwise +, +(80) +where +nA +h (p) = +A′(|p|2/2) +� +dxdηδ(|p|2/2 − |η|2/2). +(81) +In the following proof, we first define the microcanonical measure and then +derive the rate function (80). +If we consider an infinite domain with a finite amount of total wave action, +� +Rd|ψµ|2dx < ∞, since the wave action density will be diluted to absolutely 0 +everywhere, an equilibrium state of the microcanonical ensemble does not make +sense. Here, we instead assume a spatial periodicity of the scaler field ψµ(x) +and concentrate our attention on a d-dimensional cubic domain, [0, 2π)d ≡ +Γ ⊂ Rd. In this setting, the empirical local spectral density (10) consists of +delta functions in wave-vector space like +nµ(x, p) = +� +k∈(1/2)Zd +nµ +k(x)δ(p − µk), +(82) +where nµ +k(x) is a discrete form of Wigner distribution adapted to periodic +domains, defined as +nµ +k(x) = +1 +(2π)d +� +Γ +dye−2ik·yψµ (x + y) ψµ† (x − y) . +(83) +We also introduce the Fourier coefficients of ψµ as +ˆψµ +k = +1 +(2π)d +� +Γ +e−ik·xψµ(x)dx +(84a) +ψµ(x) = +� +k∈Zd +eik·x ˆψµ +k. +(84b) +Perceval’s theorem allows us to write the wave action density per unit volume +in three forms, +1 +(2π)d +� +Γ×Rd nµ(x, p)dxdp = +1 +(2π)d +� +Γ +|ψµ(x)|2dx = +� +k +| ˆψµ +k|2. +(85) +We shall set µ → 0 while keeping this action density finite. When we fix a +volume element in p space, the number of k vectors which are involved there + +Springer Nature 2021 LATEX template +34 +Large deviations for linear wave kinetic equation +increases as ∼ µ−d. Therefore, the typical amplitude of the Fourier coeffi- +cients depends on µ as ˆψµ +k ∼ O(µd/2). Now we scale ˆψµ by introducing a new +coefficient, aµ +p with p ∈ µZd, as ˆψµ +k = µd/2aµ +µk, i.e, +aµ +p = +1 +(2πµ1/2)d +� +Γ +e−ip·x/µψµ(x)dx. +(86) +Note that aµ +p remains finite for µ → 0, but its norm, � +p∈µZd|aµ +p|2 = +(2πµ)−d � +Γ×Rd nµ(x, p)dxdp, diverges in the same limit. +To apply equilibrium statistical mechanics, we consider the phase space +spanned by the scaled coefficients, {aµ +p}p∈µZd. In this space, the Lebesgue +measure m is represented as +dm = +� +p +daµ +pdaµ† +p ≡ +� +p +dar +pdai +p, +(87) +where aµ +p = (ar +p + iai +p)/ +√ +2 is understood. This measure makes sense only +when an upper limit of the wave vector is set to truncate the infinite +product. As a result, the number of degrees of freedom of ψµ in physi- +cal space is also restricted. Let us define the bounded set of k as K∆ ≡ +{−1/(2∆) + 1, . . . , 1/(2∆)}d, and accordingly that of p as µK∆. The number +of elements in K∆ is N ≡ 1/∆d. Then, ψµ is specified by the values at N +points, Γ∆ ≡ {0, ∆, . . . , 2π − ∆}d, and the values in Γ \ Γ∆ are determined by +interpolation. The Lebesgue measure is now represented in either wave vector +or position space as +dm = +� +p∈µK∆ +daµ +pdaµ† +p = Jµ +∆ +� +x∈Γ∆ +dψµ(x)dψµ†(x), +(88) +where Jµ +∆ is the Jacobian of the function that maps aµ +p to ˆψµ. To compute this +Jacobian, we consider the integral, +� +dm exp +� +− +1 +(2πµ)d +� +Γ×Rd nµ(x, p)dxdp +� +(89) +that we express in two different ways: +�  + � +p∈µK∆ +daµ +pdaµ† +p + + exp + +− +� +p∈µK∆ +|aµ +p|2 + + +=Jµ +∆ +� � � +x∈Γ∆ +dψµ(x)dψµ†(x) +� +exp +� +− +� ∆ +2πµ +�d � +x∈Γ∆ +|ψµ(x)|2 +� +. +(90) + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +35 +The left-hand side turns out to be (2π)N , and the right-hand side is +Jµ +∆(2π)N (2πµ)dN N N . We therefore obtain Jµ +∆ and, accordingly, +dm = +� +x∈Γ∆ +dψµ(x)dψµ†(x) +(2πµ)dN +. +(91) +We denote by E integrals over the Lebesgue measure (91). The microcanonical +measure with constraints on A is then defined as +dmA = +Πωδ (Aω [nµ] − A(ω)) +E [Πωδ (Aω [nµ] − A(ω))]dm, +(92) +where Πωδ (Aω [nµ] − A(ω)) means that we constrain the values of all the +invariants Aω for any ω. +Our goal is to compute the probability distribution of nµ for a microcanon- +ical measure constrained by A, i.e., +Pµ +A,m[nµ = n] = E [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] +E [Πωδ (Aω [nµ] − A(ω))] +. +(93) +It will be mathematically convenient, for intermediate computations, to use in +the following a normalizable Gaussian measure dmG, +dmG = +� +p∈µK∆ +e−|aµ +p|2 daµ +pdaµ† +p +2π += exp +� +− +1 +(2πµ)d +� +Γ×Rd nµ(x, p)dxdp − N log 2π +� +dm, +(94) +which satisfies +� +dmG = 1. We note that the N-dependent term diverges in +the ∆ → 0 limit, but this divergence will be compensated in Pµ +A below. We +denote EG averages with respect to this Gaussian measure. +Then, (93) can be rewritten as +Pµ +A,m[nµ = n] += +exp +� +(2πµ)−d � +Γ×Rd n(x, p)dxdp +� +EG [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] +EG +� +exp +� +(2πµ)−d � +Γ×Rd nµ(x, p)dxdp +� +Πωδ (Aω [nµ] − A(ω)) +� +. +(95) +In the following, when we consider the Gaussian measure mG, the continuous +limit ∆ → 0 is always understood. +We look for a large deviation principle +EG [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] ≍ +µ→0 exp +� +−IG[n, A] +(2πµ)d +� +. +(96) + +Springer Nature 2021 LATEX template +36 +Large deviations for linear wave kinetic equation +Our strategy is to compute its rescaled cumulant generating function and to +apply the Gärtner-Ellis theorem. We shall then define a free energy as +fG[λ, β] ≡ − lim +µ→0 (2πµ)d log EG +� +exp +� +− +1 +(2πµ)d +� +Γ×Rd dxdp +× +�� +R+ β(ω)h +� +ω − |p|2 +2 +� +dω + λ(x, p) +� +nµ(x, p) +�� +, +(97) +where β : R+ → R is a real continuous function representing the chemical +potential and λ : Γ × Rd → R is also a real continuous function. Because nµ +is quadratic in ψµ, the expectation is a Gaussian integral. To make it explicit, +the expression in the square brackets is rewritten as +− +1 +(2πµ)d +� +ψµ†(x)L˜λψµ(x)dx, +(98) +where L˜λ is a pseudo-differential operator defined as +L˜λψµ(x) ≡ +� +L˜λ(x, x′)ψµ(x′)dx′ +(99a) +L˜λ(x, x′) = +1 +(2πµ)d +� +Rd +˜λ +�x + x′ +2 +, p +� +eip·(x−x′)/µdp +(99b) +˜λ(x, p) ≡ +� +λ(x, p) + +� +R+ β(ω)h(ω − |p|2/2)dω +(x ∈ Γ) +0 +(x /∈ Γ). +(99c) +In (98) and (99a), the integration range is unbounded. However, Riemann- +Lebesgue lemma applied to (99b) assures that the kernel function L˜λ vanishes +in the limit of µ → 0 except in the vicinity of the points of x = x′. Conse- +quently, further taking into account (99c), the range of the integration of (98) +and (99a) can be reduced from Rd to Γ. +To obtain a simpler form of fG[λ, β], we need to compute the functional +determinant of L˜λ. This is done here straightforwardly: +fG[λ, β] = − lim +µ→0 (2πµ)d log EG +� +exp +� +− +1 +(2πµ)d +� +Rd ψµ†(x)L˜λψµ(x)dx +�� += − lim +µ→0 lim +∆→0 (2πµ)d log +� +� +x∈Γ∆ +dψµ(x)dψµ†(x) +2π(2πµ)dN +× exp + +− ∆2d +(2πµ)d +� +x,x′∈Γ∆ +ψµ†(x)L˜λ(x, x′)ψµ(x′) − +∆d +(2πµ)d +� +x∈Γ∆ +|ψµ(x)|2 + + += lim +µ→0 lim +∆→0 (2πµ)d log det +� +I + L∆ +˜λ +� +. +(100) + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +37 +We have carried out the Gaussian integration. The problem thus reduces to +computing the determinant of an N × N matrix, I + L∆ +˜λ , where I is a unit +matrix and L∆ +˜λ consists of +� +∆dL˜λ(x, x′) | x, x′ ∈ Γ∆ +� +. Let us use the following +expression, +log det +� +I + L∆ +˜λ +� += tr log +� +I + L∆ +˜λ +� += − +∞ +� +j=1 +(−1)j +j +trL∆j +˜λ , +(101) +which holds for sufficiently small +˜λ. We know that L∆j +˜λ +consists of +� +∆dLj +˜λ(x, x′) | x, x′ ∈ Γ∆ +� +where Lj +˜λ(x, x′) corresponds to the kernel func- +tion of an operator, +Lj +˜λ ≡ L˜λ . . . L˜λ +� +�� +� +j +, +(102) +in the small ∆ limit. In general, a product between pseudo-differential oper- +ators corresponds to a star product, or a Moyal product, between symbols +[64, 65]. The star product is expanded in terms of µ with the leading term +equivalent to the ordinary product. Accordingly, Lj +˜λ = L˜λj +O(µ) holds, where +˜λj is the jth power of ˜λ. We thus derive +lim +∆→0 Lj +˜λ(x, x) = +1 +(2πµ)d +� +Rd +˜λj(x, p)dp + O(µ) +∴ lim +∆→0 trL∆j +˜λ += +1 +(2πµ)d +� +Γ×Rd +˜λj(x, p)dxdp + O(µ), +(103) +and hence +fG[λ, β] = +� +Γ×Rd log +� +λ(x, p) + +� +R+ β(ω)h +� +ω − |p|2 +2 +� +dω + 1 +� +dxdp. (104) +From this formula, the Gärtner-Ellis theorem yields the rate function IG (96) +as +IG[n, A] = − inf +λ,β +�� +dxdpλ(x, p)n(x, p) + +� +R+ dωβ(ω)A(ω) − f[λ, β] +� +, +(105) +which is computed as +IG[n, A] = +�� +dxdp (n − 1 − log n) +if +Aω[n] = A(ω) ++∞ +otherwise +. +(106) + +Springer Nature 2021 LATEX template +38 +Large deviations for linear wave kinetic equation +As should have been expected, the minimum value of IG[n, A] is 0, which is +realized when and only when n = 1 and A(ω) = Aω[1]. +Based on the large deviation result (96), the numerator in (95) turns out +to be +exp +� +(2πµ)−d +� +Γ×Rd n(x, p)dxdp +� +EG [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] +≍ +µ→0 exp +� SA[n] +(2πµ)d +� +(107) +with +SA[n] = +�� +dxdp (1 + log n) +if +Aω[n] = A(ω) +−∞ +otherwise +. +(108) +The finite part of this function defines the entropy for a mesoscopic state +specified by n, and it coincides with the Lyapunov function (25) for the wave +kinetic equation. Because the denominator of (95) is the integration of the +numerator over all n, the Laplace’s principle enables us to compute it as +EG +� +exp +� +(2πµ)−d +� +Γ×Rd nµ(x, p)dxdp +� +Πωδ (Aω [nµ] − A(ω)) +� +≍ +µ→0 exp +� s[A] +(2πµ)d +� +, +(109) +with s[A] = supn {SA[n]}. The supremum is achieved when Aω[n] = A(ω), and +n(x, p) = nA +h (p) = N +�|p|2 +2 +� +, +(110) +that is, n is homogeneous in space and depends only on the magnitude of its +wave vector. The function N(ω) is related to A(ω) by the condition Aω[nA +h ] = +A(ω). This gives formula (81) for nA +h . We also have +s[A] = +� +dxdp +� +1 + log nA +h +� +. +(111) +Finally, starting from (95), and using the two asymptotic relations (107) and +(109), as well as (108) and (111), we obtain +Pµ +A,m[nµ = n] ≍ +µ→0 exp +� +− UA[n] +(2πµ)d +� +(112) +where the quasipotential UA[n] is given by equation (80). We have established +the announced results. + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +39 +C Computations of scattering terms +This appendix describes somewhat intricate derivation of the scattering terms +that appear in the wave kinetic equation and the large deviation Hamiltonian. +For this purpose, we prepare some useful formulae, +� t +0 +dτ1eiωτ1/µ +� τ1 +0 +dτ2e−iωτ2/µ = πµtδ(ω) + o(µ) +(113a) +� t +0 +dτ1eiωτ1/µ +� t +0 +dτ2e−iωτ2/µ = 2πµtδ(ω) + o(µ) +(113b) +1 +(2πµ)d +� +Rd dξeip·(ξ−x)/µf(ξ) = +� +|α|≥0 +(−iµ)|α| +α! +∇α +xf(x)∇α +p δ(p). +(113c) +Equations (113a) and (113b) are often used in literature of weak turbulence +[2]. The residual terms denoted by o(µ) make negligible contributions in the +limit of µ → 0 compared to the leading-order terms when integrated with +respect to ω. In (113c), a multi-index notation is used. In the following compu- +tation, integration is always carried out over Rd, except for the basic positional +coordinates represented by x whose integration range is Γ. +C.1 Terms appearing in the classical wave kinetic +equation +We first compute the scattering terms in the wave kinetic equation, specifically +E [wµ(ψµ +2 , ψµ +0 )], E [wµ(ψµ +0 , ψµ +2 )], and E [wµ(ψµ +1 , ψµ +1 )]. These terms are common +with those linear to λ in the scattering part of the large deviation Hamiltonian, +HS. The computations are slightly involved but mostly straightforward. A +detailed procedure is presented only for the E [wµ(ψµ +2 , ψµ +0 )] case. +From (16) and (18), we have +wµ(ψµ +2 , ψµ +0 ) = +1 +−µ2(2πµ)d +� t +0 +dτ1 +� τ1 +0 +dτ2 +� +dydξ1234e−ip·y/µ +×Gµ � +x + y +2 − ξ1, t − τ1 +� +V µ(ξ1)Gµ(ξ1 − ξ2, τ1 − τ2)V µ(ξ2)Gµ(ξ2 − ξ3, τ2)ψµ(ξ3, 0) +×Gµ† � +x − y +2 − ξ4, t +� +ψµ†(ξ4, 0). +Taking ensemble average, writing the propagators Gµ as Fourier integrals (17) +with wave vectors η1, η2, η3 and η4 in this order, and setting +E [V µ(ξ1)V µ(ξ2)] = +� +dη5eiη5·(ξ1−ξ2)/µΠ(η5) +ψµ(ξ3, 0)ψµ†(ξ4, 0) = +� +dη6eiη6·(ξ3−ξ4)/µn((ξ3 + ξ4)/2, η6), + +Springer Nature 2021 LATEX template +40 +Large deviations for linear wave kinetic equation +we derive +E [wµ(ψµ +2 , ψµ +0 )] += +1 +−µ2(2πµ)5d +� +dydξ1234dη123456e−ip·y/µ +×e−i(|η1|2−|η4|2)t/2µ +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +� τ1 +0 +dτ2ei(|η2|2−|η3|2)τ2/2µ +×eiη1·(x+y/2−ξ1)/µeiη2·(ξ1−ξ2)/µeiη3·(ξ2−ξ3)/µe−iη4·(x−y/2−ξ4)/µ +×Π(η5)eiη5·(ξ1−ξ2)n((ξ3 + ξ4)/2, η6)eiη6·(ξ3−ξ4)/µ. +Integration of this expression with respect to y, ξ1 and ξ2 yields +E [wµ(ψµ +2 , ψµ +0 )] += +1 +−µ2(2πµ)2d +� +dξ34dη123456 +×e−i(|η1|2−|η4|2)t/2µ +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +� τ1 +0 +dτ2ei(|η2|2−|η3|2)τ2/2µ +×ei(η1·x−η3·ξ3−η4·x+η4·ξ4+η6·ξ3−η6·ξ4)/µ +×δ((η1 + η4)/2 − p)δ(η1 − η2 − η5)δ(η3 − η2 − η5) +×Π(η5)n((ξ3 + ξ4)/2, η6). +We change the variables as +X = ξ3 + ξ4 +2 +, +Y = ξ3 − ξ4, +and carry out the integration with respect to Y to get +E [wµ(ψµ +2 , ψµ +0 )] += +1 +−µ2(2πµ)d +� +dXdη123456 +×e−i(|η1|2−|η4|2)t/2µ +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +� τ1 +0 +dτ2ei(|η2|2−|η3|2)τ2/2µ +×ei(η1−η4)·(x−X)/µδ((η3 + η4)/2 − η6) +×δ((η1 + η4)/2 − p)δ(η1 − η2 − η5)δ(η3 − η2 − η5) +×Π(η5)n(X, η6). +We understand that η1 = η3 and η6 = p hold in the integrand. Integration +with respect to η3 and η6 yields +E [wµ(ψµ +2 , ψµ +0 )] + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +41 += +1 +−µ2(2πµ)d +� +dXdη1245 +×e−i(|η1|2−|η4|2)t/2µ +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +� τ1 +0 +dτ2ei(|η2|2−|η1|2)τ2/2µ +×ei(η1−η4)·(x−X)/µδ((η1 + η4)/2 − p)δ(η1 − η2 − η5) +×Π(η5)n(X, p). +We use (113a) to derive +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +� τ1 +0 +dτ2ei(|η2|2−|η1|2)τ2/2µ = πµtδ +�|η1|2 +2 +− |η2|2 +2 +� ++ o(µ). +We also use (113c) to derive +1 +(2πµ)d +� +dXdη4e−i(|η1|2−|η4|2)t/2µei(η1−η4)·(x−X)/µδ((η1 + η4)/2 − p)n(X, p) +=δ(η1 − p)n(X, p) + O(µ, t). +Consequently, we obtain +E [wµ(ψµ +2 , ψµ +0 )] = − t +2µ +� +dησ(p, η)n(x, p) + o +� t +µ +� +. +(114) +Because wµ(ψµ +0 , ψµ +2 ) = [wµ(ψµ +2 , ψµ +0 )]†, and σ and n are real functions, we also +have +E [wµ(ψµ +0 , ψµ +2 )] = − t +2µ +� +dησ(p, η)n(x, p) + o +� t +µ +� +. +(115) +Finally, we consider E[wµ(ψµ +1 , ψµ +1 )]. From (16) and (18), we have +wµ(ψµ +1 , ψµ +1 ) = +1 +µ2(2πµ)d +� t +0 +dτ1 +� t +0 +dτ2 +� +dydξ1234e−ip·y/µ +× Gµ � +x + y +2 − ξ1, t − τ1 +� +V µ(ξ1)Gµ(ξ1 − ξ2, τ1)ψµ(ξ2, 0) +× Gµ† � +x − y +2 − ξ3, t − τ2 +� +V µ(ξ3)Gµ†(ξ3 − ξ4, τ2)ψµ†(ξ4, 0). +Taking the ensemble average, introducing the Fourier integrals, and integrating +some variables in the same manner as the previous case, we derive +E [wµ(ψµ +1 , ψµ +1 )] += +1 +µ2(2πµ)d +� +dXdη123456 + +Springer Nature 2021 LATEX template +42 +Large deviations for linear wave kinetic equation +×e−i(|η1|2−|η3|2)t/2µ +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +� t +0 +dτ2e−i(|η3|3−|η4|2)τ2/2µ +×ei(η1−η3)·(x−X)/µδ((η2 + η4)/2 − η6) +×δ((η1 + η3)/2 − p)δ(η1 − η2 − η5)δ(η3 − η4 − η5) +×Π(η5)n(X, η6). +Applying (113c), understanding η1 = η3 and η2 = η4 in the integrand, using +(113b), and Integrating all the possible variables, we finally obtain +E [wµ(ψµ +1 , ψµ +1 )] = t +µ +� +dη2σ(p, η2)n(x, η2) + o +� t +µ +� +. +(116) +C.2 Quadratic terms in the Hamiltonian +We compute the terms in HS quadratic in λ, originating from four expressions, +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +1 , ψµ +0 )(x1, p1)wµ(ψµ +0 , ψµ +1 )(x2, p2)] +(117a) +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +0 , ψµ +1 )(x1, p1)wµ(ψµ +1 , ψµ +0 )(x2, p2)] +(117b) +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +1 , ψµ +0 )(x1, p1)wµ(ψµ +1 , ψµ +0 )(x2, p2)] +(117c) +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +0 , ψµ +1 )(x1, p1)wµ(ψµ +0 , ψµ +1 )(x2, p2)]. +(117d) +Among these, the two pairs, (117a)-(117b) and (117c)-(117d), are complex +conjugate, respectively. Therefore, we need to compute only two expressions. +Although the number of factors involved in the integration is greater than +those in the linear terms, the computation procedures are largely the same. +For the sake of conciseness, we denote time t instead of ∆t. +From (16) and (18), we write the Wigner transforms in the integrand of +(117a) as +wµ(ψµ +1 , ψµ +0 )(x1, p1) = +1 +iµ(2πµ)d +� t +0 +dτdydξ123e−ip1·y/µ +×Gµ � +x1 + y +2 − ξ1, t − τ +� +V µ(ξ1)Gµ(ξ1 − ξ2, τ)ψµ(ξ2, 0) +×Gµ† � +x1 − y +2 − ξ3, t +� +ψµ†(ξ3, 0) + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +43 +wµ(ψµ +0 , ψµ +1 )(x2, p2) = +1 +−iµ(2πµ)d +� t +0 +dτdydξ123e−ip2·y/µ +×Gµ � +x2 + y +2 − ξ1, t +� +ψµ(ξ1, 0) +×Gµ† � +x2 − y +2 − ξ2, t − τ +� +V µ(ξ2)Gµ†(ξ2 − ξ3, τ)ψµ†(ξ3, 0). +Taking the ensemble average of the product of these expressions, it would be +straightforward to have +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +1 , ψµ +0 )(x1, p1)wµ(ψµ +0 , ψµ +1 )(x2, p2)] += +1 +µ2(2πµ)4d +� +dx12dp12dX12dη123456789λ(x1, p1)λ(x2, p2) +×e−i(|η1|2−|η3|2)t/2µ +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +×e−i(|η4|2−|η5|2)t/2µ +� t +0 +dτ2e−i(|η5|2−|η6|2)τ2/2µ +×δ(η1/2 + η3/2 − p1)δ(η4/2 + η5/2 − p2)δ(η1 − η2 − η7)δ(η5 − η6 − η7) +×δ(η2/2 + η6/2 − η8)δ(η3/2 + η4/2 − η9) +×ei(η1−η3)·x1/µei(η4−η5)·x2/µe−i(η2−η6)·X1/µe−i(η4−η3)·X2/µ +×Π(η7)n(X1, η8)n(X2, η9). +Applying (113c) and (113a) and integrating all the possible variables yields +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +1 , ψµ +0 )(x1, p1)wµ(ψµ +0 , ψµ +1 )(x2, p2)] += +t +µ(2πµ)d +� +dx1dp1dη2λ(x1, p1)2σ(p1, η2)n(x1, p1)n(x1, η2) + o +� +t +µ(2πµ)d +� +. +(118) +From the the condition of complex conjugate, we also have +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +0 , ψµ +1 )(x1, p1)wµ(ψµ +1 , ψµ +0 )(x2, p2)] += +t +µ(2πµ)d +� +dx1dp1dη2λ(x1, p1)2σ(p1, η2)n(x1, p1)n(x1, η2) + o +� +t +µ(2πµ)d +� +. +(119) +For the computation of (117c), we write, +wµ(ψµ +1 , ψµ +0 )(x1, p1) = +1 +iµ(2πµ)d +� t +0 +dτdydξ123e−ip1·y/µ + +Springer Nature 2021 LATEX template +44 +Large deviations for linear wave kinetic equation +×Gµ � +x1 + y +2 − ξ1, t − τ +� +V µ(ξ1)Gµ(ξ1 − ξ2, τ)ψµ(ξ2, 0) +×Gµ† � +x1 − y +2 − ξ3, t +� +ψµ†(ξ3, 0) +Taking the ensemble average of the product of this expression, it would be +straightforward to have +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +1 , ψµ +0 )(x1, p1)wµ(ψµ +1 , ψµ +0 )(x2, p2)] += +1 +−µ2(2πµ)4d +� +dx12dp12dX12dη123456789λ(x1, p1)λ(x2, p2) +×e−i(|η1|2−|η3|2)t/2µ +� t +0 +dτ1ei(|η1|2−|η2|2)τ1/2µ +×e−i(|η4|2−|η6|2)t/2µ +� t +0 +dτ2ei(|η4|2−|η5|2)τ2/2µ +×δ(η1/2 + η3/2 − p1)δ(η4/2 + η6/2 − p2)δ(η1 − η2 − η7)δ(η4 − η5 + η7) +×δ(η2/2 + η6/2 − η8)δ(η5/2 + η3/2 − η9) +×ei(η1−η3)·x1/µei(η4−η6)·x2/µe−i(η2−η6)·X1/µe−i(η5−η3)·X2/µ +×Π(η7)n(X1, η8)n(X2, η9). +Applying (113c) and (113a) and integrating all the possible variables yields +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +1 , ψµ +0 )(x1, p1)wµ(ψµ +1 , ψµ +0 )(x2, p2)] += − +t +µ(2πµ)d +� +dx1dp12λ(x1, p1)λ(x, p2)σ(p1, p2)n(x, p1)n(x, p2) + o +� +t +µ(2πµ)d +� +. +(120) +Finally, from the the condition of complex conjugate, we have +1 +(2πµ)2d +� +dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ +0 , ψµ +1 )(x1, p1)wµ(ψµ +0 , ψµ +1 )(x2, p2)] += − +t +µ(2πµ)d +� +dx1dp12λ(x1, p1)λ(x, p2)σ(p1, p2)n(x, p1)n(x, p2) + o +� +t +µ(2πµ)d +� +. +(121) +D Derivation of the Hamiltonian in the diffusive +limit +In this appendix, we show how to obtain the large deviation Hamiltonian Eq. +(58) from Eq. (42). Since we are dealing with the scattering term that is local + +Springer Nature 2021 LATEX template +Large deviations for linear wave kinetic equation +45 +in position space, one can omit the dependence on x. As explained in the main +text, the important point is to show how the diffusion kernel Σ[n] transforms +when only infinitesimal deviation from incoming wave vector are allowed by +the cross section. For this specific purpose, we introduce a parameter ν and +write the typical correlation length of V as µ/ν. Our strategy is to expand the +Hamiltonian in terms of ν. +Because the potential spectrum Π(p) is supposed to have a finite sup- +port with the extent of O(ν), it is convenient to rewrite the cross section +as σ(p1, p2) = ν−d˜σ((p1 − p2)/ν; p1). For any test functions, f and g, the +diffusive kernel (39) satisfies +� +dp1dp2f(p1)Σ(p1, p2)g(p2) +=c +� +dpdqf(p)n(p)˜σ(q; p)n(p − νq) [g(p) − g(p − νq)] . +(122) +Taylor expanding the integrand with respect to ν, one obtains +� +dp1dp2f(p1)Σ(p1, p2)g(p2) +=c +� +dpf(p)n(p) +� +νn(p) +d +� +n=1 +�� +dqn˜σ(q; p)qn +� +∂png(p) +−ν2 +d +� +n=1 +d +� +m=1 +�� +dqndqm˜σ(q; p)qnqm +� +∂pnn(p)∂pmg(p) +−ν2 +2 n(p) +d +� +n=1 +d +� +m=1 +�� +dqndqm˜σ(q; p)qnqm +� +∂pn∂pmg(p) + O +� +ν3� +� +. +(123) +To evaluate the integrations with respect to q, we shall exponentiate the Dirac- +δ in the definition of the cross section (23) and use the inverse Fourier transform +of Π in (57). The resulting expression is +˜σ(q; p) = +1 +ν(2π)d +� +dy +� +R +dse−iq·yRν � +sp + y − ν +2 sq +� +(124) +and +Rν(y) = +� +dpΠ(p)eip·y/ν, +where Rν(y) = R(y/ν) (see Eq. (57)) is a scaled correlation function of the +random potential with the correlation length of O(1). Again Taylor expanding +Eq. (124) with respect to ν, one gets +cν +� +dq˜σ(q; p)qn = −ν +d +� +m=1 +∂pmDν +nm(p) + O(ν2) + +Springer Nature 2021 LATEX template +46 +Large deviations for linear wave kinetic equation +cν2 +� +dq˜σ(q; p)qnqm = 2νDν +nm(p) + O(ν2) +with Dν(p) = −(c/2) +� +R ∇⊗∇Rν(sp)ds. Here, +� +R ∇Rν(sp)ds vanishes because +of the point symmetry in Rν. Inserting these expressions into (123), one obtains +� +dp1dp2f(p1)Σ(p1, p2)g(p2) += − cν +� +dpfn +d +� +n=1 +d +� +m=1 +{n∂pnDν +nm∂pmg + 2Dν +nm∂pnn∂pmg + nDν +nm∂pn∂pmg} ++O(ν2). +(125) +Evidently from this result, the dominant term is of order O(ν) in the present +scaling. Therefore, in order for diffusion in wave-vector space to be comparable +to the free propagation in position space, one needs to rescale the position +coordinate. However, for the sake of simplicity here, we shall set ν = 1 while +ignoring O(ν2) terms. 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Springer series in materials science, vol. 88. +Springer, Berlin; New York (2006) +[58] Lanford, O.E.: Time evolution of large classical systems. In: Ehlers, J., +Hepp, K., Weidenmüller, H.A., Beiglböck, W., Moser, J. (eds.) Dynamical +Systems, Theory and Applications vol. 38, pp. 1–111. Springer, Berlin, +Heidelberg (1975). https://doi.org/10.1007/3-540-07171-7_1 +[59] Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltz- +mann: Hard Spheres and Short-range Potentials. European Mathematical +Society Zürich, Switzerland, ??? (2013) +[60] Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical +Systems. Grundlehren der mathematischen Wissenschaften, vol. 260. +Springer, Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642- +25847-3 +[61] Graham, R.: Macroscopic potentials, bifurcations and noise in dissipa- +tive systems. In: Moss, F., McClintock, P.V.E. (eds.) Noise in Nonlinear +Dynamical Systems, 1st edn., pp. 225–278. Cambridge University Press, +Cambridge, U.K. (1989). https://doi.org/10.1017/CBO9780511897818. +009 +[62] Guioth, J., Onuki, Y., Bouchet, F.: Path large deviations for inhomoge- +neous weak wave turbulence. To be submitted to J. Stat. Phys. +[63] Touchette, H.: The large deviation approach to statistical mechanics. +Physics Reports 478(1-3), 1–69 (2009) +[64] Polkovnikov, A.: Phase space representation of quantum dynamics. +Annals of Physics 325(8), 1790–1852 (2010) +[65] Onuki, Y.: Quasi-local method of wave decomposition in a slowly varying +medium. Journal of Fluid Mechanics 883 (2020) + diff --git a/rdE1T4oBgHgl3EQfjAQ8/content/tmp_files/load_file.txt b/rdE1T4oBgHgl3EQfjAQ8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..080eba8e8ee2a960915a65f61fb8f016308f4b0a --- /dev/null +++ b/rdE1T4oBgHgl3EQfjAQ8/content/tmp_files/load_file.txt @@ -0,0 +1,1198 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf,len=1197 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='03257v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='stat-mech] 9 Jan 2023 Springer Nature 2021 LATEX template Dynamical large deviations for an inhomogeneous wave kinetic theory: linear wave scattering by a random medium Yohei Onuki1,2*, Jules Guioth2 and Freddy Bouchet2,3 1Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 2Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, Lyon, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 3Laboratoire de Météorologie Dynamique, CNRS, Ecole Normale Supérieure, Institut Pierre-Simon Laplace, Paris Sciences Lettres Université, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' E-mail(s): onuki@riam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='kyushu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='jp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Abstract The wave kinetic equation predicts the averaged temporal evolution of a continuous spectral density of waves either randomly interacting or scattered by the fine structure of a medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In a wide range of sys- tems, the wave kinetic equation is derived from a fundamental equation of wave motion, which is symmetric through time-reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' By contrast, the corresponding wave kinetic equations is time-irreversible: its solu- tions monotonically increase an entropy-like quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A similar paradox appears whenever one make a mesoscopic description of the evolution of a very large number of microscopic degrees of freedom, the paradig- matic example being the kinetic theory of dilute gas molecules leading to the Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since Boltzmann, it has been understood that a probabilistic understanding solves the apparent paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' More recently, it has been understood that the kinetic description itself, at a mesoscopic level, should not break time reversal symmetry [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The time reversal symmetry remains a fundamental property of the meso- scopic stochastic process: without external forcing the path probabilities obey a detailed balance relation with respect to an equilibrium quasipo- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The proper theoretical or mathematical tool to derive fully this mesoscopic time reversal stochastic process is large deviation theory: a 1 Springer Nature 2021 LATEX template 2 Large deviations for linear wave kinetic equation large deviation principle uncovers a time reversible field theory, char- acterized by a large deviation Hamiltonian, for which the deterministic wave kinetic equation appears as the most probable evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Its irre- versibility appears as a consequence of an incomplete description, rather than as a consequence of the kinetic limit itself, or some related chaotic hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This paper follows [1] and a series of other works that derive the large deviation Hamiltonians of the main classical kinetic theories, for instance [2] for homogeneous wave kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We propose here a deriva- tion of the large deviation principle in an inhomogeneous situation, for the linear scattering of waves by a weak random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This prob- lem involves microscopic scales corresponding to the typical wavelengths and periods of the waves and mesoscopic ones which are the scales of spatial inhomogeneities in the spectral density of both the random scatterers and the wave spectrum, and the time needed for the ran- dom scatterers to alter the wave spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The main assumption of the kinetic regime is a large separation of these microscopic and mesoscopic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the sake of simplicity, we consider a generic model of wave scattering by weak disorder: the Schrödinger equation with a random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We derive the path large deviation principle for the local spec- tral density and discuss its main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We show that the mesoscopic process obeys a time-reversal symmetry at the level of large deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This publication is part of a special issue in homage of the memory of Krzysztof Gaw¸edzki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The subject of this work is large deviation theory applied to wave turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Large deviation theory applied to complex dynamics and turbulent flows was one of the subjects for which Krzysztof Gaw¸edzki made a number of important contributions during the last few years, see for instance [3–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' He taught many of us, including Freddy Bouchet, many aspects of large deviation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We wrote a common paper on the subject of large deviation theory and non-equilibrium quasipotentials for stochastic particles with mean field interactions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Given his scientific qualities, and his deep sense of friendship, it is great pleasure for us to pay homage to Krzysztof Gaw¸edzki through this modest contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 1 Introduction The aim of this paper is to extend the existing kinetic theory to describe probabilistically mesoscopic evolutions of wave fields interacting with random potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We will derive for the first time a large deviation principle that describes completely typical and rare fluctuations of the wave local spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This is also the first large deviation principle for wave kinetic theory in an inhomogeneous setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This work lies at the intersection of three different active fields in theoretical and mathematical physics: the description of waves interacting with random media and their applications to ocean and atmosphere Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 3 dynamics, recent mathematical and theoretical advances in the kinetic theory of wave turbulence, and large deviation theory for kinetic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the first field, we note that wave propagation in disordered media is a ubiquitous phenomenon appearing in various areas of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Typical exam- ples include light radiation through the atmosphere, acoustic or internal gravity waves in turbulent flows, and elastic waves in solid Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In most cases, one is not interested in the individual wave interference or scattering processes but in the statistical description of the overall wave field at a mesoscopic spatial scale much greater than the extent of disorder or wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this pur- pose, it is customary to define the spectral density of the wave signal at each location and investigate its statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The wave kinetic equation, sometimes referred to as a radiative transport equation or simply as a trans- port equation, is known as the universal model to describe the evolution of the local spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It commonly derives from elementary wave equations and has a broad range of applications [9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Recently, the evolution of wave spectra under scattering interactions with a turbulent flow were studied in a two-dimensional model [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Wave kinetic theory is of special interest in some specific areas of ocean and atmosphere research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since the celebrated work by Klaus Hasselmann [16], the kinetic description of nonlinear 4-wave interactions among water waves has been used for estimating energy transfer rates in a wind wave spectrum and forecasting the sea surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The linear counterpart of the wave kinetic equation is relevant to surface or internal wave energy dispersion in a slowly evolving turbulent flow [12, 13, 17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In these actual problems, the scale- separation assumption at the heart of the kinetic theory might be valid, but is not necessarily always valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For example, internal wave activity in the ocean is highly heterogeneous, which is imprinted on the variability of energy dissi- pation rates on scales of order 10 to 100 km, in the mid-depth layer [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For tide or wind generated waves with 10-100 km horizontal wavelengths, devia- tions of the spectral evolutions from that predicted by the kinetic equation may not be negligible, and a first principle theory of fluctuation is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This motivates us to revisit the theoretical basis of wave kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Our work can be considered as the first building block for stochastic parameteriza- tion of the local spectral density from first principles, for the specific case of wave interacting with random potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In relation with geophysical applications, several experiments with funda- mental scopes in wave kinetic theory have been recently devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For instance this led to the very first observation of the regime of inertial wave turbulence in a rotating flow [21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' the identification of regimes of weakly and strongly nonlinear internal wave turbulence in an experiment of stratified turbulence [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' experiments on statistical properties of water waves in a large basin [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' the validation of the inverse cascade phenomenon [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' and extension of the range of scales for observing pure gravity wave turbulence in the laboratory [25] using reduced gravity experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 Large deviations for linear wave kinetic equation The second field, fundamental theoretical developments in wave turbulence theory, has seen many new advances recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the first time, using numeri- cal simulations of the non-linear Schrödinger wave kinetic equation, predictions by the wave kinetic equation were tested for several kinetic times [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Novel finite-size effects in wave turbulence were systematically studied in a one-dimensional model using a combination of theory and numerics [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Sig- nificant recent progress has been made to give a mathematical foundation for wave turbulence theory: theorem about approximations of the dynamics for times much shorter than the kinetic time [29–34], the understanding of propa- gation of chaos [35], and a remarkable first full rigorous derivation of the wave kinetic theory at the kinetic timescale, for the non-linear Schrödinger equation [30] and for water waves equation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From the point of view of these funda- mental perspectives, our work gives for the first time a description of all the cumulants of the local spectral density, through a large deviation principle, in an inhomogeneous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The third field is the development of large deviation principles in relation with kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Many classical equations of mathematical physics arise from a law of large numbers, when faster and smaller scale degrees of free- dom are averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This is the case for all classical kinetic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is natural to extend all these theories to look for the statistics of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Generically, one expects to derive a large deviation principle that describes a statistical field theory quantifying the probabilities of any fluctuations, either typical or extremely rare, in a way analogous to macroscopic fluctuation the- ory [37] for stochastic diffusive systems, or large deviation theory for stochastic dynamics with mean field interaction [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Deriving such large deviation prin- ciples from deterministic microscopic dynamics is a fundamental endeavor in theoretical and mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' the large deviation principles for a number of classical kinetic theories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' starting from first principles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' have been uncovered: for discrete models that mimic dilute gases and with Boltz- mann like behavior [38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' for dilute gases related to the Boltzmann equation [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' for the Kac model [41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' for plasma at length scales much smaller than the Debye length related to the Landau equation [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' for homogeneous systems with long range interactions related to the Balescu–Guernsey–Lenard equation [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' for weakly interacting waves in a homogeneous setup [2] related to the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The large deviation principles describe fluctua- tions but also uncover gradient structure for the deterministic kinetic equation, see [45] and a simple explanation in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Several mathematical results, usually valid for a fraction of the kinetic time in the spirit of Lanford results for the deterministic equation, have been obtained for the large deviation principles, for instance for the Boltzmann equation [40, 46], or for the Kac model with unexpected corrections to the expected large deviation principle [41, 42] associ- ated to giant concentrations and solutions of the Boltzmann equation without energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' One aim of large deviation theory is to study rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the context of wave dynamics, large deviation theory has been used to study rare events Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 5 for the evolution of the empirical spectrum [2] on the kinetic time scale, but also for studying the appearance of very large amplitude waves [47], for the non-linear Schrödinger dynamics for shorter time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Instantons structures have been predicted and compared with experimental data taken from a 300 m long wave [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Large deviation principles for the wave amplitude due to short time phase mixing has also been studied [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The result described in this paper is a new example of a large deviation principle for a kinetic theory, derived from microscopic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is the first extension of large deviation theory for the local spectral density in an inhomogeneous setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It opens the way for other inhomogeneous large devia- tion principle for wave turbulence, and for the study of new wave turbulence phenomena where rare events play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As a generic model of waves interacting with random medium, we consider the linear Schrödinger equation in a weak random potential i∂ψ ∂t = −D 2 ∇2 xψ + V ψ, where ψ(x, t) is a wave function defined on Rd+1 and V (x) is a homogeneous random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this model, we assume a regime with a wave spectrum which is dominated by waves of typical wavelengths λ, and with modulations of the statistical properties of the wave spectrum on scales of order λ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The second assumption is that the typical correlation length of the potential is of order λ and that interactions between the waves and the potential is weak, more precise definitions are given in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Then, for small value of µ, we have a separation of scales and of the associated times, where a huge amount of waves experience multiple scattering in domains of typical size λ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is natural to focus on variations of the field on the mesoscopic scales of order λ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This defines a kinetic regime where such mesoscopic variations are captured by the Wigner distribution n, that somehow measures the wave energy density in both position and wave-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' After time and length rescaling t → µt and x → µx, in the small µ limit, the wave kinetic equation is classically derived (see for instance [9, 49]): ∂n(x, p, t) ∂t + p · ∇xn(x, p, t) = c � dησ(p, η) (n(x, η, t) − n(x, p, t)) , where n(x, p, t) is the Wigner distribution at position x, wave vector p and time t, and cσ(p1, p2) is the scattering cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Interestingly, the evolution of the Wigner distribution predicted from the wave kinetic equation is an irreversible relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this problem, a Lyapunov function, S = � dxdp log n(x, p), monotonically increases with time, even though the fundamental equation of motion possesses a time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This old irreversibility paradox has been recently revisited for the kinetic theory of particles [1], using dynamical large deviation principles, in the case of the Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It turns out that the dynamical large deviation Springer Nature 2021 LATEX template 6 Large deviations for linear wave kinetic equation principle that quantifies the probability for the evolution of any trajectory has a time reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The kinetic equation corresponds to the most probable path of the system, while the probability of a path and its time-reversed path is related through detailed balance, a manifestation of time reversal symmetry for the mesoscopic stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This gives an extremely simple and enlightening explication of the irreversibility paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The main result of the present paper is a large deviation principle for the Schrödinger equation in a weak random potential, which has also a time reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This gives a new very clear explanation of the time reversal paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The purpose of this paper is to formulate a path large deviation principle for wave scattering by random disorder in spatially inhomogeneous problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In particular, to make the discussion as concise as possible, we restrict our attention to the simplest Schrödinger equation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Our fundamental results are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (i) for a small but finite µ we show that the probability that a path of local spectral density {nµ(t)} evolves at a vicinity of a prescribed specific path {n(t)} satisfies a large deviation principle: P � {nµ(t) = n(t)}0⩽t⩽tf � ≍ µ→0 exp � − 1 (2πµ)d � tf 0 dt sup λ �� λ ˙n − H[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' λ] �� P0[n(0)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' where H is the large deviation Hamiltonian that generally governs the stochas- tic fluctuations of macroscopic variables and P0[n(0)] is the probability of the initial condition nµ(t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Obtaining the explicit expression for H from the microscopic equation is one of the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Next, (ii) we ver- ify that the ordinary wave kinetic equation describes the path that minimizes the exponent of the probability functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Then, (iii) we establish a large deviation principle for the microcanonical measure that defines the quasipo- tential of the mesoscopic stochastic process of the local spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We analyze (iv) the property of the large deviation Hamiltonian, check its symme- tries related to conservation laws and the time-reversal symmetry, and derive an expression of the detailed balance that connects the probabilities of a path and its time-reversed path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Finally (v) we study the diffusive limit when the scales of variation of the random potential are much larger than the typical wavelength of the waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this case we obtain a diffusive large deviation Hamiltonian, for which we check all the desired symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The paper is organized in the following order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In section 2, we first set up the basic problem, introduce scaling and statistical assumptions, and derive the ordinary form of the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In section 3, we derive the path large deviation principle for the temporal variations of the local spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The approach has some analogy with that of [2], with a slightly differ- ent scaling assumption, and working with the Wigner distribution to describe the wave local spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We then show that this Hamiltonian satisfies the expected properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A remarkable point is that the quasipotential enter- ing into the detailed balance relation is consistent with the one obtained from Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 7 a direct computation of microcanonical ensemble, formulated in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Section 4 proposes several possible extensions in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 2 Wave kinetic equation for a linear Schrödinger equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1 Problem setup We consider the Schrödinger equation for a wave function ψ∗(x∗, t∗) : Rd+1 → C, where t∗ is time and x∗ is the position vector, and with potential V ∗(x∗) : Rd → R: i∂ψ∗ ∂t∗ = −D 2 ∇2 x∗ψ∗ + V ∗ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (1) We use an upper script ∗ to represent variables with physical dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The physical parameter D > 0 has dimension L2T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In absence of interaction with the potential, the free Schrödinger equation is the dynamics of linear waves with a dispersion relation ω(k∗) = D|k∗|2/2, where k∗ is a wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A localised wave packet propagates at group velocity ∇k∗ω(k∗) = Dk∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The potential V ∗(x∗) is assumed to be a spatially homogeneous random field with zero-average, E[V ∗] = 0, with its spectral density given by Π∗(k∗) = 1 (2π)d � Rd dy∗e−ik∗·y∗E � V ∗ � x∗ + y∗ 2 � V ∗ � x∗ − y∗ 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (2) For homogeneous fields, the two-point correlation function E [V ∗(x∗ 1)V ∗(x∗ 2)] depends only on the point separation x∗ 1 − x∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The spectral density of the potential is the Fourier transform of the two-point correlation function E [V ∗(x∗ 1)V ∗(x∗ 2)] with respect to x∗ 1 − x∗ 2 and thus contains the same infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Note that a prescription of higher order cumulants would be needed to fully characterize the potential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As we will see in the following, the higher order statistics of the potential will not affect the dynamics of the spectral density of the waves in the kinetic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Hence, although we do not specify all the cumulants, the potential needs not to be Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this article, we assume that the spectral density Π∗ is concentrated around wave vectors |k∗| ∼ 2π/λ where λ is the typical wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In real space, the wavelength λ is interpreted as the typical correlation length of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For such a potential, the Schrödinger dynamics Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (1) may display many different regimes, depending on the order of magnitude of λ compared to typical wavelengths in the initial condition of ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For instance, if the initial condition is made of waves with wavelengths much smaller than λ, the wave- potential interaction corresponds to random but smooth refraction that leads to diffusion in the macroscopic limit [17–19, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We will see in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4 that this diffusive limit can be recovered from the wave kinetic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' On the other hand, if the initial waves have wavelength much greater that λ, one faces Springer Nature 2021 LATEX template 8 Large deviations for linear wave kinetic equation a homogenization problem that is not described by the wave kinetic equation [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the present paper, we will focus on an intermediate regime, when the initial condition is made of waves with typical wavelengths which are of order λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We also make an assumption that the potential term is very small compared to the Laplacian term, by setting ǫ ≡ λ2V0 D ≪ 1, (3) where V0 is a constant scaling the potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=', the potential spectrum is typically Π∗ ∼ V 2 0 λd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Wave energy or wave action measures the local amplitude of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We denote ℓ the typical scale for spatial variation of wave action, and call it the mesoscopic spatial scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We introduce the second natural dimensionless parameter µ ≡ λ ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (4) The kinetic limit is the limit µ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the context of wave kinetics, we are interested in the statistical behavior of the system at the mesoscopic scale, avoiding chasing rapid phase oscillations at scale λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since the group velocity of a wave packet is D|k∗| ∝ Dλ−1, the migration time of a wave packet over a mesoscopic distance ℓ is λ2/Dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We call this time the mesoscopic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Choosing such mesoscopic units naturally yields the following dimensionless coordinates x ≡ µx∗ λ , p ≡ λk∗, t ≡ µDt∗ λ2 , (5) where the scaled wave vector is now represented by p in a customary way of quantum mechanics with µ corresponding to the Dirac constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Physically, the square of the absolute value of the wave function, |ψ∗|2, represents the wave action density that is proportional to energy, momentum, or number of particles contained in a unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, on the dimensional ground, the wave function should be dependent on the scaling parameters as ψµ(x) = λd/2ψ∗(x∗) µd/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The potential and its spectrum are scaled as V µ(x) = V ∗(x∗) V0 , Π(p) = Π∗(k∗) V 2 0 λd , Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 9 such that Π(p) = 1 (2πµ)d � Rd dye−ip·y/µE � V µ � x + y 2 � V µ � x − y 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (6) In the end, the governing equation (1) is rewritten in the non-dimensional form iµ∂ψµ ∂t = −µ2 2 ∇2 xψµ + ǫV µψµ, (7) which is the fundamental model of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For ǫ = 0, waves ei(p·x−ωt)/µ, with ω = |p|2/2, are exact solutions of the equations, illustrating that the microscopic time scale and spatial scales are tm ∼ O(µ) and xm ∼ O(µ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A wave packet a(x, t)ei(p·x−ωt)/µ with modulation of its amplitude a on spatial scales of order one (slow modulation compared to the microscopic scale), will actually see an evolution of a on time scales of order one, according to the group velocity p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the limit of small ǫ, the effect of the inhomogeneous potential term is very small on the microscopic time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, a wave packet propagates almost freely in a microscopic time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since E[V µ] = 0 has been assumed, effects of terms proportional to ǫ will vanish on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Accumulation of the random potential effect will then give non zero contribution of order ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In order for this to be on the same order as effects of free propagation on the wave action requires ǫ = √cµ (8) with c > 0 a finite constant which accounts for the relative importance of the scattering interactions with respect to propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The constant c is strictly speaking not needed, and it could be absorbed in the definition of Π, but it is useful for the physical discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The kinetic regime, or kinetic limit, is the joint limit µ → 0 with ǫ = √cµ, where c is a fixed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Consequently, the pertinent equation for the kinetic scaling will be iµ∂ψµ ∂t = −µ2 2 ∇2 xψµ + √cµV µψµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (9) In some parts of the following sections, we will perform asymptotic expansions of the effect of the random potential by expanding (7) in power of ǫ, while integrating out explicitly the wave propagation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this reason, we often consider (7) instead of (9) for those technical parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2 Local spectral density In the regime of wave kinetics, one is interested in the amount of wave action existing at each position and wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This is provided by the (rescaled) Wigner distribution of the signal that is defined, following previous work on Springer Nature 2021 LATEX template 10 Large deviations for linear wave kinetic equation inhomogeneous wave kinetics [9, 10], as nµ(x, p, t) = 1 (2πµ)d � Rd dy e−ip·y/µψµ � x + y 2 � ψµ† � x − y 2 � , (10) where we have denoted by ψµ† the complex conjugate of ψµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The Wigner distribution function is the local spectral density of the wave action defined in both space of position and wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Indeed, the integral of nµ over wave-vector space coincides with the action density in position space, such that � Rd dp nµ(x, p) = |ψµ(x)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A caution for the physical interpretation of nµ is that it allows the existence of negative values, in contrast to wave action or energy that should be strictly non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The negativity of nµ can be eliminated by averaging it over a scale comparable to µ [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this study, we will take the asymptotically small limit of both µ and ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Statistically, the limit of µ → 0 is regarded as a kind of thermodynamic limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' since µ represents the typical correlation length of the wave signal, the total number of degrees of freedom increase as µ−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In general, the thermodynamic limit makes sense when we specify the macroscopic or mesoscopic variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=', temperature or pressure for gas molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the present model, the potential spectrum Π(p) is a mesoscopic control parameter that should not depend on µ, and the local spectral density nµ is a mesoscopic variable to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In our formulation, a superscript µ is put on a variable that depends on the scaling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is worth keeping in mind that, although the spectral density function Π(p) is fixed, the corresponding potential function V µ(x) depends on µ because its structure becomes finer and finer for µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='3 Conservation properties We consider equation (7) on a spatial domain Γ, which is either Rd or a periodic domain V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The norm � Γ dx|ψµ(x, t)|2 is conserved by the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' If Γ is Rd, the norm can be either finite for localized solution, or infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A local conservation law always exists for |ψµ(x, t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We now consider the other invariants, for equation (7), or the associated local conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' If not specified, integrations for positions and wave vectors are always performed over Γ and Rd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For a polynomial function f : R → R, we define an operator Kǫ,µ f = f(−(µ2/2)∇2 x + ǫV µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since the Schrödinger operator is self-adjoint, a straightforward computation shows that ⟨f⟩ ≡ � dxψµ†(x, t)Kǫ,µ f ψµ(x, t) (11) is independent of the time t when ψµ is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Equation (11) is rewritten in terms of the local spectral density as ⟨f⟩ = � dxdpKǫ,µ f (x, p)nµ(x, p, t), (12) Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 11 where Kǫ,µ f (x, p) = � Rd dye−ip·y/µ ˆKǫ,µ f � x + y 2 , x − y 2 � and ˆKǫ,µ f (x1, x2) ≡ Kǫ,µ f δ(x1 − x2) is the Weyl symbol of the operator Kǫ,µ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In general, Kǫ,µ f can be expanded as a power series of ǫ and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The leading-order term is immediately obtained as K0,0 f (x, p) = f � |p|2/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, for the small limit of ǫ and µ, we may write the general invariant as ⟨f⟩ = � dxdp f �|p|2 2 � nµ (x, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (13) Since the choice of f is arbitrary, the present system possesses an infinite number of invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This property is related to wave frequency conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For a scattering of wave action in spectral space by a time-independent poten- tial, wave frequency does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, the amount of wave action with frequency less than ω remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Setting f(σ) = h(ω − σ) in (13), where h is the Heaviside function and ω ∈ R+, we define Aω[nµ] ≡ � dxdp h � ω − |p|2 2 � nµ(x, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (14) Conservation of ⟨f⟩ for an arbitrary f is equivalent to the conservation of Aω for an arbitrary ω ∈ R+, as ⟨f⟩ = � R+ dω dAω dω f(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4 Wave kinetic equation In this subsection, we shall derive the classical form of the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A common derivation of the wave kinetic equation starts from a closed equation on the Wigner distribution and performs a multiple time-scale expansion, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We rather adopt here a perturbative approach in ǫ from the Schrödinger equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This approach is also classical in the wave turbulence literature [2, 54, 55], will appear helpful in the derivation of the dynamical large deviation theory in section 3, and has the advantage to gen- eralize easily to the case of the kinetic theory of non-linear waves with 3-wave interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The first step is to express the solution of the Schrödinger equation (7) using an expansion in power of ǫ, such that ψµ = ψµ 0 + ǫψµ 1 + ǫ2ψµ 2 + O(ǫ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Inserting this expansion to (7), we derive the series of equations: iµ∂ψµ 0 ∂t = −µ2 2 ∇2 xψµ 0 iµ∂ψµ 1 ∂t = −µ2 2 ∇2 xψµ 1 + V µψµ 0 Springer Nature 2021 LATEX template 12 Large deviations for linear wave kinetic equation iµ∂ψµ 2 ∂t = −µ2 2 ∇2 xψµ 2 + V µψµ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='. We consider this expansion for any given initial condition ψµ(x, 0) = ψµ,0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We assume that the initial condition does not depend on ǫ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=', ψµ 0 (x, 0) = ψµ,0(x) and ψµ j (x, 0) = 0 for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To write down the solution, we introduce the propagator Gµ(x, t) such that � iµ ∂ ∂t + µ2 2 ∇2 x � Gµ = iµδ(x)δ(t), (15) and Gµ = 0 for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We then obtain ψµ 0 (x, t) = � Rd dξGµ(x − ξ, t)ψµ(ξ, 0) (16a) ψµ 1 (x, t) = 1 iµ � t 0 dτ � Rd dξGµ(x − ξ, t − τ)V µ(ξ)ψµ 0 (ξ, τ) = 1 iµ � t 0 dτ � R2d dξ12Gµ(x − ξ1, t − τ)V µ(ξ1)Gµ(ξ1 − ξ2, τ)ψµ(ξ2, 0) (16b) ψµ 2 (x, t) = 1 iµ � t 0 dτ � Rd dξGµ(x − ξ, t − τ)V µ(ξ)ψµ 1 (ξ, τ) = 1 −µ2 � t 0 dτ1 � τ1 0 dτ2 � R3d dξ123Gµ(x − ξ1, t − τ1)V µ(ξ1) × Gµ(ξ1 − ξ2, τ1 − τ2)V µ(ξ2)Gµ(ξ2 − ξ3, τ2)ψµ(ξ3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (16c) The equation for the propagator (15) is analytically solved as Gµ(x, t) = h(t) (2πµ)d � dpe−i|p|2t/2µeip·x/µ, (17) where h(t) is again the Heaviside function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Although the wave vector inte- gration of this expression can be carried out, we keep this form because it is convenient for later computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Importantly, the perturbation solution will be valid for not too large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For longer times, the higher order terms will not be small compared to ψµ 0 even though ǫ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As will be clear in the following discussion, we will need the perturbative solution to be valid up to µ ≪ t ≪ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' an intermediate range between the microscopic and mesoscopic time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Based on the perturbation solution of ψµ derived above, we shall consider the evolution of the local spectral density nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To simplify the computation, we Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 13 introduce the Wigner transform of two functions, f(x) and g(x), as wµ(f, g)(x, p) ≡ 1 (2πµ)d � Rd dy f � x + y 2 � g† � x − y 2 � e−ip·y/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (18) Its inverse is f(x1)g†(x2) = � dp wµ �x1 + x2 2 , p � eip·(x1−x2)/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (19) Basically, the local spectral density is the Wigner transform of identical wave functions, nµ(x, p, t) = wµ(ψµ, ψµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (20) Inserting ψµ = ψµ 0 + ǫψµ 1 + ǫ2ψµ 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' to (20), and using (16), we obtain an expansion of the local spectral density nµ in terms of ǫ as nµ(x, p, t) = wµ(ψµ 0 , ψµ 0 ) + ǫwµ(ψµ 1 , ψµ 0 ) + ǫwµ(ψµ 0 , ψµ 1 ) + ǫ2wµ(ψµ 1 , ψµ 1 ) + ǫ2wµ(ψµ 2 , ψµ 0 ) + ǫ2wµ(ψµ 0 , ψµ 2 ) + O(ǫ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (21) The first term on the right-hand side is easily computed as wµ(ψµ 0 , ψµ 0 ) = nµ(x − pt, p, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (22) This expression means that, in absence of the potential, the propagation of free waves transports the spatial distribution of wave action density at the group velocity p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is notable that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (22) is valid without taking an ensemble average or an asymptotic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We shall consider the expectation of (21) with respect to the realization of the random potential V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since E[V µ] = 0 has been assumed, terms propor- tional to ǫ vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The dominant contribution from the random potential to the variations in local spectral density comes from terms of order ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Direct computations, performed in Appendix C, yield the expectation values of per- turbation terms at the leading-order, (114), (115) and (116).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Consequently, we obtain E � ǫ2wµ(ψµ 1 , ψµ 1 ) + ǫ2wµ(ψµ 2 , ψµ 0 ) + ǫ2wµ(ψµ 0 , ψµ 2 ) � =ǫ2t µ � dησ(p, η) (nµ(x, η, 0) − nµ(x, p, 0)) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' with σ(p1, p2) ≡ 2πΠ(p1 − p2)δ �|p1|2 2 − |p2|2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (23) Springer Nature 2021 LATEX template 14 Large deviations for linear wave kinetic equation Here, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' represents the higher-order terms in the expansion that are negligible in the asymptotic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Setting ǫ = √cµ, we have lim µ→0 E[nµ(x, p, t)] − nµ(x − pt, p, 0) t = lim µ→0 c � dησ(p, η) (nµ(x, η, 0) − nµ(x, p, 0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Taking the limit t → 0, we obtain a differential equation for the ensemble average of the local spectral density, limµ→0 E[nµ] = n: ∂n(x, p, 0) ∂t + p · ∇xn(x, p, 0) = c � dησ(p, η) (n(x, η, 0) − n(x, p, 0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' At this stage, this expression is only valid at the initial time, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' One cannot extend it to t > 0 because the wave function ψµ is a priori correlated with the potential field V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' However, we shall argue that the correlation between ψµ and V µ remains always weak at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Indeed, in the present problem, it is assumed that the significant modification of the wave field ψµ by scattering on the potential occurs at a time scale of wave propagation over a mesoscopic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The typical correlation length of the random potential V µ is much shorter than this mesoscopic scale by a factor of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, even though interferences of the field with the potential produce slight correlations, the free propagation of the field makes the correlation vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This situation resembles the loss of memory for particle collision in dilute gas—the molecular chaos hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The present weak-correlation assumption allows us to regard the temporal evolution of nµ as a Markovian process such that the wave kinetic equation would be valid any time as far as µ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We note that this explanation applies to d ≥ 2 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For one-dimensional problems, an interesting phenomenon named localization is known to occur [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This localization phenomenon which invalidates the present kinetic regime [50, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2] is thus discarded in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Once this Markovian hypothesis is accepted, we may iterate the reasoning expounded above for t = 0 in order to reach any time t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We eventually obtain the equation ∂n(x, p, t) ∂t + p · ∇xn(x, p, t) = c � dησ(p, η) (n(x, η, t) − n(x, p, t)) , (24) that is the ordinary form of the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since the group velocity of a free wave is now p, the second term on the left-hand side is understood as the motion of the Wigner distribution at the group velocity due to the free dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The right-hand side represents wave scattering in wave-vector space that occurs at microscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The function cσ(p1, p2) is the scattering cross section determining the rate of wave action converted from wave vector p1 to p2 per unit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 15 The wave kinetic equation inherits the conservation property of the original Schrödinger equation, namely that dAω[n]/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' However, the wave kinetic equation differs from the microscopic Schrödinger dynamics (7) since the for- mer appears time-irreversible whereas the latter is time reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Indeed, for a prescribed potential field V µ(x), let ψµ(x, t) be a solution of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Its time-reverse counterpart is defined as ψµ R(x, t) = ψ†(x, −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As a consequence, the time-reversed local spectral density reads as nµ R(x, p, t) = nµ(x, −p, −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Change in sign of p is a natural outcome because the wave group veloci- ties of the forward and the reverse paths should be opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' If ψµ(x, t) is a solution of the Schrödinger equation, ψµ R(x, t) is also a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This is the time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' By contrast, the wave kinetic equation violates this symmetry: if n is a solution, the time-reversed nR is not a solution of the wave kinetic equation, unless both n and nR are an identical stationary state (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (26) defined below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This irreversibility paradox is reminiscent to the one raised by Boltzmann for the case of dilute gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As explained in [1] for the case of the Boltzmann equation, we will see in the next section that one can recover time-reversibility for the kinetic theory at the large deviation level, as a time-reversibility for the stochastic process of the local spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Time- irreversibility arises because the wave kinetic equation describes the evolution of the average n = E[nµ] only, or equivalently in this case the most probable evolution only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Other paths, including time-reversed paths, are possible: they are just extremely unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Before moving toward the large deviation theory of wave kinetics, we can remark that time-irreversibility can be also quantified by introducing a Lyapunov function S ≡ � dxdp log n(x, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (25) Following solutions of the wave kinetic equation, S increases monotonically with time, dS/dt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' If the spatial domain Γ is finite, S achieves the maxi- mum when the spectral density n(x, p) is homogeneous in x and isotropic in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We write this homogeneous distribution of the spectral density under the constraint of Aω[n] = A(ω) as nA h (p) = A′(|p|2/2) � dxdηδ(|p|2/2 − |η|2/2), (26) where the denominator is introduced for normalization purpose, and A′ = dA/dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is obvious that n(x, p) = nA h (p) is a stationary solution of the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We note that the Lyapunov function S is actually the microcanonical entropy of the macrostate specified by (nµ, Aω) = (n, A) for the Schrödinger equation as we discuss in Appendix B, and is related to the quasipotential that appears in the discussion of the large deviation theory, as we discuss in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 16 Large deviations for linear wave kinetic equation 3 Large deviation formulation for random wave scattering In the previous section, we derived the wave kinetic equation as an equation for the ensemble average E [nµ] of the empirical local spectral density, with respect to the probability measure of the random potential, in the limit µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This kinetic limit can be understood as a law of large number: limµ→0 nµ = E [nµ], where the limit has to be understood in a weak sense (for instance, the limit holds when both nµ and E [nµ] are integrated over any subset U ⊂ Γ × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In statistical mechanics, it is quite common that an empirical macroscopic quantity converges to its ensemble average for the large limit of the number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The deviation from the average is often exponentially small and evaluated asymptotically by a large deviation principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this section, our aim is to generalize the law of large number for the kinetic theory and to compute the probability to observe any possible fluctuations for nµ, as a large deviation principle, in the limit µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Such fluctuations are expected to be characterized by a large deviation parameter proportional to µd, where d is the space dimension, because the number of statistically independent degrees of freedom is of order µ−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1 Large deviation Hamiltonian We define the Newton ratio as the time increment for the local spectral density: ∆nµ/∆t = (nµ(∆t) − nµ(0))/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We regard the temporal variations in nµ as a stochastic process, and look for the probability to observe a value of the Newton ratio, conditioned on the value of the local spectral density at time 0: nµ(0) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Our aim is to justify that it satisfies − lim ∆t→0 lim µ→0 ǫ=√cµ (2πµ)d ∆t log P � ∆nµ ∆t = ˙n|nµ = n � = L [n, ˙n] , (27) and to derive an explicit formula for the Lagrangian L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The limit lim µ→0 ǫ=√cµ corresponds to the kinetic limit µ → 0 where one has fixed ǫ = √cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Here, L[n, ˙n] is the rate function of the probability of the Newton ratio P [ ˙n|n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Through the fast microscopic dynamics, the memory of the initial condi- tion of the phases of ψµ are expected to be lost after some times, meaning that two-times, or multi-times, correlation functions are expected to decay with the time differences of two or several phase observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Such a mixing property is expected to be due to the conjunction of phase mixing (oscillating integrals and the Riemann–Lebesgue lemma), spatial transport and dispersion, and the effect of the random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Because the natural timescale for phase dynam- ics is the microscopic timescale, one might expect a typical mixing time to be much smaller than the kinetic time scale and to decay to zero in the kinetic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This mixing property is required to justify a Markovian behavior and to propagate local in time results, like the Lagrangian (27), in order to describe Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 17 the dynamics at finite times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In none of the existing classical kinetic theories, neither physicists nor mathematicians have been so far able to justify or prove the requested mixing properties, in order to justify the long time validity of kinetic theories or their probabilistic large deviation generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This is the main reason why mathematical results, for instance the celebrated Lan- ford’s result for the Boltzmann equation [58], or its generalizations [59], are usually valid only for a fraction of the kinetic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As it is customary in the- oretical physics, we will assume the validity of such a mixing property in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We now assume the natural mixing hypothesis and the related Markov behavior of the stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As a consequence, the evolution of nµ does not depend on its previous state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Then the path probability for the stochastic process is Markovian, and the probability of a path of nµ for a finite time interval, 0 ⩽ t ⩽ tf, can be derived from the local in time Lagrangian (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The path probability conditioned on the local spectral density at the initial time nµ(0) = n(0) is then Pn(0) � {nµ(t) = n(t)}0⩽t⩽tf � ≍ µ→0 exp � − 1 (2πµ)d � tf 0 dtL[n, ˙n] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (28) This expression is analogous to the path-integral formulation for the probabil- ity density in quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Note that the initial conditions nµ(0) = n(0) is fixed here, but one can easily consider a set of initial condition complemented by an initial probability density P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In such a case, the path probability of a trajectory {n(t)}0⩽t⩽tf reads as P � {nµ(t) = n(t)}0⩽t⩽tf � ≍ µ→0 exp � − 1 (2πµ)d � tf 0 dtL[n, ˙n] � P0[n(0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (29) To compute the Lagrangian (27), we will use the Gärtner-Ellis theorem that connects the rate function L to the cumulant-generating function, or the Hamiltonian H defined by H[n, λ] = lim ∆t→0 lim µ→0 ǫ=√cµ (2πµ)d ∆t × log E � exp �� dxdpλ(x, p)(nµ(x, p, ∆t) − n(x, p)) (2πµ)d �� , (30) through the Legendre-Fenchel transform, L [n, ˙n] ≡ sup λ �� dxdpλ(x, p) ˙n(x, p) − H[n, λ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (31) Our aim here is to derive the specific form of H directly from the perturbation solutions of the original Schrödinger equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 18 Large deviations for linear wave kinetic equation Following [2], we first compute the moment-generating function for the increment of nµ, Zµ[n, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ∆t] ≡ E � exp �� dxdpλ(x, p)(nµ(x, p, ∆t) − n(x, p)) (2πµ)d �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (32) We insert the expansion of nµ(x, p, t) in terms of ǫ into (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since at the lead- ing order wµ(ψµ 0 , ψµ 0 ) is statistically independent of V µ by the weak correlation hypothesis described before, it is possible to decompose Zµ into two parts as Zµ = Zµ 0 Zµ ǫ (33a) Zµ 0 ≡ exp � 1 (2πµ)d � dxdpλ(x, p) (wµ(ψµ 0 , ψµ 0 ) − n(x, p)) � (33b) Zµ ǫ ≡ E � exp � 1 (2πµ)d � dxdpλ(x, p) (nµ(x, p, ∆t) − wµ(ψµ 0 , ψµ 0 )) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (33c) Using (22), the first part, Zµ 0 , is immediately rewritten as Zµ 0 = exp � 1 (2πµ)d � dxdpλ(x, p) (n(x − p∆t, p, 0)) − n(x, p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (34) The second part is expanded in terms of ǫ to obtain Zµ ǫ = 1 + ǫ2Zµ 2 + o(ǫ2), (35) where we have used E[V µ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The Landau notation o(ǫ2) gathers all the terms that are negligible compared to O(ǫ2) terms in the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the following, we shall discard the higher order terms of o(ǫ2) and concentrate on computing Zµ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Because we are considering the simultaneous limit of µ, ǫ = √cµ, neglecting o(ǫ2) terms cannot be justified a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This kind of problem commonly arises in wave kinetic theory [2] but we expect o(ǫ2) to be negligible in the kinetic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The direct computation of Zµ 2 yields Zµ 2 = 1 (2πµ)d � dxdpλ(x, p)E [wµ(ψµ 2 , ψµ 0 ) + wµ(ψµ 0 , ψµ 2 ) + wµ(ψµ 1 , ψµ 1 )] + 1 2(2πµ)2d � dx12dp12λ(x1, p1)λ(x2, p2) × E � (wµ(ψµ 1 , ψµ 0 )(x1, p1) + wµ(ψµ 0 , ψµ 1 )(x1, p1)) × (wµ(ψµ 1 , ψµ 0 )(x2, p2) + wµ(ψµ 0 , ψµ 1 )(x2, p2)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (36) Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 19 From the computations in Appendix C, specifically (114)-(116) and (118)- (121), we obtain Zµ 2 = ∆t µ(2πµ)d � dxdp12 � (λ(x, p1) − λ(x, p2))σ(p1, p2)n(x, p2) + 1 2(λ(x, p1) − λ(x, p2))2σ(p1, p2)n(x, p1)n(x, p2) � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='. (37) Inserting Zµ = Zµ 0 � 1 + ǫ2Zµ 2 � into H = lim∆t→0 lim µ→0 ǫ=√cµ ((2πµ)d/∆t) log Zµ, we obtain the Hamiltonian as H[n, λ] = HF + HS (38a) HF = − � dxdpλ(x, p)p · ∇xn(x, p) (38b) HS = c � dxdp12 � (λ(x, p1) − λ(x, p2))σ(p1, p2)n(x, p2) + 1 2(λ(x, p1) − λ(x, p2))2σ(p1, p2)n(x, p1)n(x, p2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (38c) We have separated the Hamiltonian into two parts, HF and HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The first part, HF , represents the free wave propagation in position space and the second one, HS, the wave scattering in wave-vector space by the random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2 Properties of the large deviation Hamiltonian Once the specific form of the Hamiltonian is obtained, we can discuss the properties of the stochastic process governed by the path-integral formula (28) and (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Formulations in the paper [1] are simple and informative, and we summarize several important properties of dynamical large deviation theory in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We now check the classical expected properties of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1 Weak noise Langevin dynamics and wave kinetic equation The first important property of the large deviation Hamiltonian H is that it is quadratic and convex with respect to the conjugated field λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This means that the fluctuations of the infinitesimal current ˙ndt are, locally in time, Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Reading the quadratic part of the Hamiltonian (38c), we see that the local covariance of the local in time Gaussian fluctuations are given by the diffusion kernel Σ[n](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1, p2) = −cσ(p1, p2)n(x, p1)n(x, p2) + c � Rd dησ(p1, η)n(x, p1)n(x, η)δ(p1 − p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (39) Springer Nature 2021 LATEX template 20 Large deviations for linear wave kinetic equation As a consequence, in the kinetic limit µ ≪ 1, the dynamics of the local spectral density in the kinetic regime corresponds to a weak noise Langevin dynamics [60, 61] ˙n(x, p, t) = −p · ∇xn + c � dη σ(p, η) (n(x, η, t) − n(x, p, t)) + √ 2(2πµ)d/2 � dη Σ1/2[n](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p, η)ξ(η, t) (40) with ξ(p, t) a white noise such that E [ξ(p, t)ξ(η, s)] = δ(t − s)δ(p − η), and where the kernel Σ1/2[n] is defined as a square root of diffusion kernel, meaning that � dηΣ1/2(x, p1, η)Σ1/2(x, η, p2) = Σ(x, p1, p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The second expected property is that the most probable path is the solution of the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This is easily checked by noticing that the most probable path, for which the action � tf 0 L[n, ˙n]dt = 0 vanishes, satisfies ∂n ∂t = δH δλ ���� λ=0 (41) (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1), and is also the linear term in λ of the Hamiltonian H and the drift term of the Langevin dynamics (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Then, the path large deviation analysis confirms that the wave kinetic equation can be understood as a law of large number at the level of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is enlightening to rewrite the scattering part of the Hamiltonian HS in terms of the diffusion kernel (39) as HS = � dxdp1dp2 λ(x, p1)Σ[n](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1, p2) � δS δn(x, p2) + λ(x, p2) � , (42) where S is the entropy (25), and the Langevin equation (40) as ˙n(x, p, t) + p · ∇xn = � dη Σ[n](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p, η) δS δn(x, η) + √ 2(2πµ)d/2 � dη Σ1/2[n](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p, η)ξ(η, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (43) This suggestive forms immediately emphasize that S is a Lyapunov function, and that the dynamics has a detailed balance structure, as further explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2 Conservation of the wave action distribution As we discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='3, in the original Schrödinger equation, the quan- tity Aω[nµ] = � dxdph(ω − |p|2/2)nµ(x, p) for any ω ∈ R+ is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This conservation property has to be also verified at the large deviation level, meaning that any trajectory {n(t)}0⩽t⩽tf has to lie on the manifold Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 21 Aω[n] = A(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1, this is equivalent to the Hamiltonian symmetry H � n, λ + αδAω δn � = H [n, λ] , (44) for an arbitrary α ∈ R, n and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the specific form of the Hamiltonian H = HF + HS (38a) with HS written in the symmetric form (42), one can directly check the above Hamiltonian symmetry boils down to the following property on the diffusive kernel (39) � dη Σ[n](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p, η) δAω δn(x, η) = 0, (45) for any n, x, p, and ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='3 Stationary quasipotential It was shown in the previous section that the wave kinetic equation possesses an attractor which is a homogeneous distribution, nh, with the prescribed constraints Aω[n] = A(ω), for any ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We now consider the fluctuations of n from nh at the large deviation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' More precisely, we define the equilibrium distribution of the stochastic process at a large deviation level: Pµ A,S[{nµ = n}] ≍ µ→0 exp � − UA[n] (2πµ)d � , (46) where Pµ A,S is the stationary probability measure of the microcanonical ensem- ble which is parameterized by a small constant µ as well as a function A(ω) specifying the action conservation constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The rate function UA is named the quasipotential (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In principle, the quasipotential can be computed from the dynamics, start- ing from the large deviation Hamiltonian, see for instance Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Formula (69) gives an expression for the quasipotential, in the cases when the wave kinetic equation has a single attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' However, when one knows explicitly the microscopic stationary distribution, for instance in the case of equilibrium statistical mechanics, one can compute directly the quasipotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' It is then related to the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Those different expressions have to give con- sistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Indeed, one may see a simple example of the relation between microcanonical entropy and the quasipotential for the dilute gas dynamics in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The case of spatially homogeneous weakly nonlinear wave dynamics has been discussed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the present problem, we compute the quasipotential from a microcanonical ensemble for the original Schrödinger equation model Springer Nature 2021 LATEX template 22 Large deviations for linear wave kinetic equation in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' One obtains UA[n] = \uf8f1 \uf8f2 \uf8f3 − � dxdp log �n(x, p) nA h (p) � if Aω[n] = A(ω) +∞ otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (47) This result is technically not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As far as we know, it had never been derived before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4, it is a generic property of the quasipoten- tial to play the role of a Lyapunov function for the deterministic relaxation dynamics, in this case the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Such a property can be derived generically from the existence of a large deviation principle, indepen- dently on the specific form of the large deviation Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' One can note that the quasipotential (47) is the opposite of the entropy S (25), up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The quasipotential now depends on A(ω) and satisfies the normalization condition, minn UA[n] = 0, where the minimum is achieved when n(x, p) = nA h (p) (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (42) and the fact that UA is equal to −S up to a constant, it is immediately checked that UA solves the stationary Hamiltonian-Jacobi equation, H � n, δUA δn � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (48) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4 Time-reversal symmetry and detailed balance Finally, we consider the time-reversal symmetry of the dynamics (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the present wave kinetic theory, we showed at the end of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4 that the time-reversed local spectral density is defined as nR(x, p, t) = n(x, −p, −t), namely that wave vector needs to change sign in addition to time- reversal t → −t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, it is useful to introduce the involution operator I such that I[n(x, p)] = n(x, −p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since the wave kinetic dynamics is an equilibrium dynamics whose sta- tionary state is characterised by the microcanonical quasipotential (47) at the large deviation level, we expect the fluctuating dynamics to be time-reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Following Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2, we indeed check that the Hamiltonian satisfies the detailed-balance condition H � n, λ + δUA δn � = H [I[n], −I[λ]] , (49) thus proving the time-reversal symmetry of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For a trajectory that follows the wave kinetic equation, UA monotonically decreases towards 0, and therefore the local spectral density irreversibly approaches the homogeneous distribution nA h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' However, when µ is small but finite, there remains a possibility that UA increases, namely that the spectrum moves away from the attractor Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 23 nA h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This property is quantified by the fluctuation relation that is equivalent to the detailed balance relation (49) Pn(0)[{nµ(t) = n(t)}0⩽t⩽tf ] Pn(tf)[{nµ(t) = nR(t − tf)}0⩽t⩽tf ] ≍ µ→0 exp �UA[n(0)] − UA[n(tf)] (2πµ)d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (50) The fluctuation relation (50) is the fundamental answer to the irreversibil- ity paradox that arises in any classical kinetic theory, and in particular for the classical wave kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Indeed, let us consider a path starting from n(0) that follows the wave kinetic equation until it reaches a state n(tf) at some time tf > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since the quasipotential UA is a Lyapunov func- tion, one has UA[n(tf)] < UA[n(0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From the fluctuation relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (50), the exact time-reversed path starting from the state n(tf) has a probablity ∼ exp � −(2πµ)−d∆U � (with ∆U = UA[n(0)] − UA[n(tf)] > 0) to occur in the small µ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The irreversibility turns into an improbability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='3 Decomposition of the Hamiltonian Since a free wave packet does not change its wave vector during a free prop- agation, and since also wave frequency is conserved during scattering by a time-independent potential, waves with different pulsation ω(p) = |p|2/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' located on distinct spherical shells) do not interfere with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' There- fore, the dynamics can be separated into an infinite number of subsystems in which the degrees of freedom are mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Indeed, this expec- tation can be verified by showing that the Hamiltonian is decomposed as an integration over frequency, as we do now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To do so, let us rewrite the wave vector as p = peθ and define the cor- responding frequency, ω = p2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The vector eθ is a unit vector whose angle is specified by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Generally, the number of degrees of freedom for the angle θ is d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For example, for a d = 3 case, elevation and azimuthal angles would be selected as a set of representative coordinate variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The follow- ing consideration is not dependent on the choice of these coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We just need to assume that a pair of opposite angles, θ and −θ, are defined such that eθ = −e−θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We decompose the wave vector integration element as dp = pd−1dpdθ = (2ω)(d−2)/2dωdθ with dθ a surface element on a (d − 1)- dimensional unit sphere, Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We define new variables labeled by frequency ω as nω(x, θ) = pd−2n(x, peθ) (51a) λω(x, θ) = λ(x, peθ) (51b) σω(θ1, θ2) = 2πΠ(p(eθ1 − eθ2)) (51c) Please do not confuse nω with nµ, in the context of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The Hamilto- nian is decomposed into independent subdynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' each of which involves the Springer Nature 2021 LATEX template 24 Large deviations for linear wave kinetic equation new variables labeled by ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' H[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' λ] = � Hω[nω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' λω]dω = � (Hω F + Hω S)dω (52a) Hω F [nω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' λω] = − � dxdθ(2ω)1/2λω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ)eθ · ∇xnω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ) (52b) Hω S[nω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' λω] = c � dxdθ1dθ2 × � (2ω)(d−2)/2(λω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ1) − λω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ2))σω(θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ2)nω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ1) + 1 2(λω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ1) − λω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ2))2σω(θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ2)nω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ1)nω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (52c) where the integration for (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ) is carried out over Γ × Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this system, the diffusion kernel and the Lyapunov function are Σω[nω](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ1, θ2) = −cσω(θ1, θ2)nω(x, θ1)nω(x, θ2) + c � dθ′σω(θ1, θ′)nω(x, θ1)nω(x, θ′)δ(θ1 − θ2) (53a) and Sω[nω] = � dxdθ(2ω)(d−2)/2 log nω(x, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (53b) The scattering part of the Hamiltonian is accordingly represented as Hω S = � dxdθ1dθ2λω(x, θ1)Σ[nω](x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' θ1, θ2) � δSω δnω(x, θ2) + λω(x, θ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (54) We may compute the probability of a path of local spectral density sepa- rately for each frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For each subsystem, the amount of wave action, ∝ � dxdθnω(x, θ) ≡ N[nω], is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the attractor of the wave kinetic equation, namely the homogeneous distribution, nω is everywhere constant in Γ × Sd−1, which we write nω h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Finally, we find that the quasipotential, Uω[nω] = \uf8f1 \uf8f2 \uf8f3 − � dxdθ(2ω)(d−2)/2 log �nω(x, θ) nω h � if N[nω] = N[nω h] +∞ otherwise (55) satisfies the detailed balance condition, Hω[nω, λω + δUω/δnω] = Hω[I[nω], −I[λω]] with the involution I defined as I[nω(x, θ)] = nω(x, −θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Consequently, the fluctuation relation for a path and its reverse applies to each subsystem separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4 Diffusive limit In this paper, we have focused on the wave kinetic regime when the random potential presents high oscillations at a scale comparable to a typical wave- length of the wave field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Another relevant limit, referred to as the diffusive approximation [17] or the Fokker-Planck limit [50], corresponds to the regime when the spatial oscillations of the potential are at a scale larger than those of the wave field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this regime, a wave signal propagates along rays which are randomly refracted by the potential, leading to an asymptotic diffusion equation for the local spectral density [17, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Technically, the diffusion equation on the local spectral density is often derived from a multiscale expansion from the microscopic dynamics (7) [17, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' But interestingly, the diffusive limit can also be obtained from the scattering kinetic regime that we have considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Our goal in this sub- section is to show how one can derive a path large deviation theory for wave kinetics in the diffusive regime from the large deviation Hamiltonian (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In a recent paper, a similar weak scattering limit has been considered to derive the path large deviation principle for plasma below the Debye length, related to the Landau equation, from the path large deviation principle for dilute gases, related to the Boltzmann equation [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the diffusive regime, the random potential has typical variations over large lengthscales compared to the typical wavelength of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As a con- sequence, the spectrum of the potential (6) is localised around p = 0, which implies from the definition of the cross section (23) that incoming wave vec- tors are randomly refracted by an infinitesimal amount at each time step by the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' One thus expects to obtain the path large deviation theory of wave transport in the diffusive regime from the scattering regime by assuming that the potential spectrum Π is supported in the vicinity of p ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In order to derive the diffusive limit from the large deviation Hamiltonian, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We show in Appendix D that the diffusion kernel Σ transforms into a differential operator such that for any test function f and g: � dp1dp2f(p1)Σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1, p2)g(p2) ≈ � dp ∇pf(p) · � n(x, p)2D(p) � ∇pg(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (56) Here, D is the diffusion matrix computed from the potential spectrum Π as D(p) = − c 2 � R ∇y ⊗ ∇yR(ps)ds and R(y) = � dη Π(η)eiη·y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (57) From its definition, R(y) corresponds to the rescaled two-point correlation function of the random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (56), the large deviation Hamiltonian in the diffusive limit reads as H[n, λ] = � dxdp λ(x, p)p · ∇xn(x, p) Springer Nature 2021 LATEX template 26 Large deviations for linear wave kinetic equation + � dxdp ∇pλ(x, p) · � n(x, p)2D(p) � ∇p � δS δn(x, p) + λ(x, p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (58) The diffusive Hamiltonian (58) conserves the spectral density at a given pulsation ω(p) = |p|2/2 because of the fundamental property D(p) · p = 0, (59) which results from p · ∇R(ps) = ∂R(ps)/∂s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This property means that the diffusion of the spectrum is always orthogonal to the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Moreover, the expression of the Hamiltonian (58) makes it clear that the quasipotential of the diffusive dynamics is the same as the quasipotential of the scattering limit Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Furthermore, one can easily check that the detailed balance relation (49) is still satisfied by the Hamiltonian (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 4 Conclusions and perspective The linear wave kinetic equation is a statistical model that governs wave action density spreading in position and wave-vector space through propaga- tion and scattering in random media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Motivated by recent works on dynamical large deviation principles for kinetic theory, this study has derived the large deviation principle describing the probability of a finite-time evolution of the local spectral density of wave action in an asymptotic limit of scale separa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Importantly, the large deviation principle that is derived in this paper satisfies a time-reversal symmetry with respect to the microcanoical quasipo- tential, that is directly (and independently) computed from the microcanonical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this paper, we restrict our considerations to the simplest Schrödinger model with a homogeneous random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The next step is to extend the present formulation to a wider range of situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Possible technical difficulties encountered during such future works involve (i) generalization of the disper- sion relation to a function of position and wave vector, ω(x, p), (ii) coping with spatial inhomogeneity for the randomness, that leads to a space-dependent scattering cross section, σ(x, p1, p2), (iii) consideration of a vector field that involves polarized waves or multiple waves, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=', elastic media holding com- pressional and shear waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Combining the present approach and a previous work on wave turbulence [2], we have also derived a formula of path-large deviations for inhomogeneous spectral density in nonlinear 4-wave interacting systems [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the context of geophysics, the wave kinetic equation is used to discuss energy cascades of internal waves in the oceans and atmosphere where rotation and density stratification play key roles [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Because the dispersion relation of internal waves depends not on wave vector but on its angle against the gravity direction, even linear theory predicts interscale energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this process, balanced geostrophic turbulent flow acts as a random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' If Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 27 assuming a stationary flow state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=', fixing the potential field in time, the formulation will be analogous to the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' On the other hand, once we allow the temporal variations in the geostrophic flow field, the situation essentially changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Wave frequency is no longer conserved during propagation and scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Spreading of action density in frequency space associates gain or loss of wave energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Quantification of the energy exchange rate between evolving turbulent eddies and waves remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Recently, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' [19] discussed wave frequency diffusion in geostrophic turbulence based on a kind of kinetic equation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As pointed out in the present study, the ordinary kinetic equation predicts an irreversible change in the spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the actual environmental situation, the scale separation parameter, µ, is not necessarily small and there should be non-negligible fluctuations in spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The large deviation formulation has a possible application for the estimation of a fluctuating energy transfer rate, in such regime where the kinetic theory is marginally valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This work was supported by JSPS Overseas Research Fellowship as well as KAKENHI Grant Number JP20K14556 (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ), and by the Simons Foundation through the Collaboration Grant 651463 “Wave Tur- bulence” (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=') and the Targeted Grant in MPS 663054 “Revisiting the Turbulence Problem Using Statistical Mechanics” (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We thank Gre- gory Eyink, Laure Saint-Raymond, Jacques Vanneste, and Antoine Venaille for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A Properties of the stochastic system specified by a large deviation Hamiltonian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1 Path large deviation This appendix presents some general properties of a stochastic process Xǫ(t) whose probability conditioned on an initial value, Xǫ(ti) = X(ti), is specified at the large deviation level via a formula, P � {Xǫ(t) = X(t)}ti⩽t⩽tf � ≍ ǫ→0 exp � −S[X] ǫ � (60a) S[X] = � tf ti dtL(X, ˙X) ≡ � tf ti dt sup P � P · ˙X − H(X, P) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (60b) Here, Xǫ(t) can be a scalar, vector, or continuous function defined on some space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Basic requirements are that an inner product is properly defined, and the dynamical property of the system is controlled by a single non- negative parameter ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the simplicity, we regard X as a scalar but the following consideration can be immediately extended to general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The large deviation Hamiltonian H(X, P) is a convex function of P and satis- fies H(X, 0) = 0 for any X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From the definition, the Lagrangian L satisfies Springer Nature 2021 LATEX template 28 Large deviations for linear wave kinetic equation L(X, ˙X) ≥ P · ˙X − H(X, P) for any X, ˙X and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, inserting P = 0, we know L ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1 Relaxation path Clearly, in the limit of ǫ → 0, the system becomes deterministic with a sin- gle path that minimizes the action S[X] for a prescribed initial condition X(ti)—named the relaxation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since L ≥ 0, if there exists a function R(X) that satisfies L(X, R(X)) = 0, a path solving ˙X = R(X) minimizes the action and yields minX S[X] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From the facts that L(X, ˙X) = P · ˙X − H(X, P) with P solving ˙X − ∂H/∂P = 0 and H(X, 0) = 0 for any X, we understand that a function R(X) = ∂H/∂P|P =0 fulfills L(X, R(X)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We thus assert that an equation ˙Xr = R(Xr) ≡ ∂H ∂P (Xr, 0) (61) determines the relaxation path Xr(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2 Optimal path Slightly changing the situation, if we fix both the initial and final states, X(ti) = xi and X(tf) = xf, respectively, the most probable path from xi to xf, namely the optimal path, or the instanton, is obtained by again minimizing S[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This problem is equivalent to the principle of least action in analytical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this context, P is called the generalized momenta and repre- sented by P = ∂L/∂ ˙X which is no longer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The optimal path in phase space is governed by a set of canonical equations, ˙X = ∂H ∂P (62a) ˙P = − ∂H ∂X , (62b) and we shall write their solutions as Xo[xf, tf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' xi, ti] and P o[xf, tf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' xi, ti].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the simplicity, we fix the initial conditions, ti and xi, and rewrite the final state as x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We then introduce the Hamilton’s principal function Q as an integration of the action following the optimal path as Q(x, t) = � t ti dτL(Xo(τ), ˙Xo(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (63) It is known in analytical mechanics that in this case the generalized momenta at the final time is represented as P o(t) = ∂L/∂ ˙X ��� t = ∂Q/∂x, and Q solves Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 29 the Hamilton-Jacobi equation, ∂Q ∂t + H � x, ∂Q ∂x � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (64) In the definition of Q(x, t), a set of arguments, (x, t), is arbitrary chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' When we pick up an optimal path {Xo(τ), P o(τ)}, at any point on this path, the generalized momenta P and the Hamilton’s principle function Q is related via P o(τ) = ∂Q ∂x (Xo(τ), τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (65) Therefore, combining (65) and the first line of (62), we formally obtain a single equation determining the optimal path, dXo dτ = ∂H ∂P � Xo, ∂Q ∂x (Xo, τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (66) This equation is, however, not generally useful because Q(x, t) is inaccessible in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='3 Quasipotential and fluctuation path Going back to the original stochastic model, the meaning of Q is understood as the rate function for the probability that Xǫ reaches xf at t = tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Indeed, we may derive from (60) an expression P [Xǫ(t) = x|Xǫ(ti) = xi] ≍ ǫ→0 exp � −Q(x, t) ǫ � (67) based on the contraction principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Now, we shall consider the stationary dis- tribution of the probability density of Xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This can be done simply setting ti = −∞ in (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To make the discussion more specific, let us assume that the relaxation dynamics (61) has a unique global attractor x0, where R(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Then, we set xi = x0 and also t = 0, to write a large deviation formula for the stationary distribution Ps(x) ≍ ǫ→0 exp � −U(x) ǫ � (68) with U(x) = inf X(t)|X(−∞)=x0 and X(0)=x � 0 −∞ dtL(X(t), ˙X(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (69) The rate function U is called the quasipotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since U is the special case of Q but independent of t, it solves the stationary version of the Hamilton-Jacobi Springer Nature 2021 LATEX template 30 Large deviations for linear wave kinetic equation equation (64), H � x, ∂U ∂x � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (70) For the present case, the optical path Xo(τ) represents the most probable route from an attractor x0 to a specific point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This route is called the fluctuation path and is denoted by Xf(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Once we obtain the quasipotential U(x), (66) provides an equation determining the fluctuation path as ˙Xf = F(Xf) ≡ ∂H ∂P � Xf, ∂U ∂x (Xf) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (71) Since the vector field F(x) does not depend on t, this equation is more useful than the original one (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' On the fluctuation path, the generalized momenta is computed based on (65) as P f = ∂U/∂x(Xf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Combining (70) with the fact that H is constant along the optical path, we understand that H(Xf, P f) = 0 always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='4 Quasipotential as a Lyapunov function A relaxation path and a fluctuation path have distinct properties for the variations in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For a relaxation path, we have dU dt (Xr) = ˙Xr ∂U ∂x (Xr) = ∂H ∂P (Xr, 0)∂U ∂x (Xr) = H(Xr, 0) � �� � =0 − H � Xr, ∂U ∂x (Xr)) � � �� � =0 +∂H ∂P (Xr, 0)∂U ∂x (Xr) ≤ 0, where we have used the general expressions, H(X, 0) = H(X, ∂U/∂x(X)) = 0, and the convexity of H(X, P) for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For a fluctuation path, we have dU dt (Xf) = ˙Xf ∂U ∂x (Xf) = ∂H ∂P � Xf, ∂U ∂x (Xf) � ∂U ∂x (Xf) = H(Xf, 0) � �� � =0 − H � Xf, ∂U ∂x (Xf)) � � �� � =0 +∂H ∂P � Xf, ∂U ∂x (Xf) � ∂U ∂x (Xf) ≥ 0, again from the convexity of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We have thus learned that the quasipotential is a Lyapunov function because it monotonically decreases in a relaxation path while increases in a fluctuation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' These results are natural consequence from a basic property that U(x) is minimum at the attractor x0 and the relaxation and fluctuation paths represent routes to and from the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 31 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2 Properties of large deviation Hamiltonian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1 Conservation law From now on, we shall regard X and P as vectors so that there are multiple directions in X space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this case, when the Hamiltonian H possesses a kind of symmetry, it is related to the conservation law of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' More specifically, let us suppose that we find a function C(X) that satisfies H � X, P + α ∂C ∂X � = H(X, P) (72) for any X, P, and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This condition is equivalent to ∂H ∂P (X, P) · ∂C ∂X (X) = 0 for any X and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Now that H(X, ·) is flat in the direction of ∂C/∂X, from the property of the Legendre-Fenchel transform [63], the corresponding Lagrangian has a property, L(X, ˙X) = +∞ if ˙X · ∂C ∂X (X) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (73) This expression indicates that the probability for a path crossing a contour of C is strictly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This constraint applies not only to the optimal path but also to any path with random fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We thus understand that (72) serves as a condition for C being an invariant of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2 Detailed balance The detailed balance is a property of equilibrium states which asserts time- reversibility of the process, meaning that the probabilities of any trajectory and its reversed counterpart are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A basic expression of detailed balance for a stationary stochastic process is P∆t(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' x)PS(x) = P∆t(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' y)PS(y), where P∆t(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' x) is the transition probability from a state x to another state y during a time interval ∆t, and PS(x) is the stationary probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since we are now considering a continuous Markov process, it is enough to regard ∆t as arbitrary small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the limit of ∆t → 0, we may write y ∼ x+ ˙x∆t and redefine the transition probability as P∆t(x, ˙x) ∼ P∆t(x+ ˙x∆t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Assum- ing the continuity of P and PS, the detailed balance condition is rewritten as P∆t(x, ˙x)PS(x) ∼ P∆t(x + ˙x∆t, − ˙x)PS(x + ˙x∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (74) For the present problem, the probability distribution is specified as P∆t(x, ˙x) ≍ exp(−∆tL(x, ˙x)/ǫ) and PS(x) ≍ exp(−U(x)/ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, the detailed Springer Nature 2021 LATEX template 32 Large deviations for linear wave kinetic equation balance condition (74) is rewritten as L(x, ˙x) − L(x, − ˙x) = ˙x · ∂U ∂x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (75) This condition is modified in terms of the Hamiltonian via the Legendre- Fenchel transform as H(x, −p) = H � x, p + ∂U ∂x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (76) Because H(x, 0) = 0 in the current problem, the stationary Hamilton-Jacobi equation (70) is a necessary (but not sufficient) condition for U being the quasipotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Once the detailed balance condition (76) is verified, we understand that the probabilities of a path and its reverse are related at the large deviation level via the expression, P � {Xǫ(t) = X(t)}ti⩽t⩽tf � P � {Xǫ(t) = X(tf + ti − t)}ti⩽t⩽tf � ≍ ǫ→0 exp � −U(xf) − U(xi) ǫ � , (77) an equivalent form of the Crooks fluctuation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Another outcome of the detailed balance is that the fluctuation path is the time reverse of the relaxation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This property is derived from R(X) = ∂H ∂P (X, 0) = −∂H ∂P � X, ∂U ∂X � = −F(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' B Microcanonical ensemble and quasi-potential for the Schrödinger equation In this appendix, we consider the microcanonical ensemble of the dynamics governed by the Schrödinger equation, (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The aim is to compute the quasipo- tential of the local empirical spectral density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=', the large deviation rate function of nµ, in the small µ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We will prove that Pµ A,m[nµ = n] ≍ µ→0 exp � − UA[n] (2πµ)d � , (78) with Pµ A,m the probabilities with respect to the microcanonical measure with the constraints, Aω[nµ] ≡ � h � ω − |p|2 2 � nµ(x, p)dxdp = A(ω), (79) Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 33 where h is the Heaviside function, and A : R+ → R+ is a prescribed function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The expected form of the rate function, namely, the quasipotential, is UA[n] = \uf8f1 \uf8f2 \uf8f3 − � dxdp log �n(x, p) nh(p) � if Aω[n] = A(ω) +∞ otherwise , (80) where nA h (p) = A′(|p|2/2) � dxdηδ(|p|2/2 − |η|2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (81) In the following proof, we first define the microcanonical measure and then derive the rate function (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' If we consider an infinite domain with a finite amount of total wave action, � Rd|ψµ|2dx < ∞, since the wave action density will be diluted to absolutely 0 everywhere, an equilibrium state of the microcanonical ensemble does not make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Here, we instead assume a spatial periodicity of the scaler field ψµ(x) and concentrate our attention on a d-dimensional cubic domain, [0, 2π)d ≡ Γ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this setting, the empirical local spectral density (10) consists of delta functions in wave-vector space like nµ(x, p) = � k∈(1/2)Zd nµ k(x)δ(p − µk), (82) where nµ k(x) is a discrete form of Wigner distribution adapted to periodic domains, defined as nµ k(x) = 1 (2π)d � Γ dye−2ik·yψµ (x + y) ψµ† (x − y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (83) We also introduce the Fourier coefficients of ψµ as ˆψµ k = 1 (2π)d � Γ e−ik·xψµ(x)dx (84a) ψµ(x) = � k∈Zd eik·x ˆψµ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (84b) Perceval’s theorem allows us to write the wave action density per unit volume in three forms, 1 (2π)d � Γ×Rd nµ(x, p)dxdp = 1 (2π)d � Γ |ψµ(x)|2dx = � k | ˆψµ k|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (85) We shall set µ → 0 while keeping this action density finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' When we fix a volume element in p space, the number of k vectors which are involved there Springer Nature 2021 LATEX template 34 Large deviations for linear wave kinetic equation increases as ∼ µ−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, the typical amplitude of the Fourier coeffi- cients depends on µ as ˆψµ k ∼ O(µd/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Now we scale ˆψµ by introducing a new coefficient, aµ p with p ∈ µZd, as ˆψµ k = µd/2aµ µk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='e, aµ p = 1 (2πµ1/2)d � Γ e−ip·x/µψµ(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (86) Note that aµ p remains finite for µ → 0, but its norm, � p∈µZd|aµ p|2 = (2πµ)−d � Γ×Rd nµ(x, p)dxdp, diverges in the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To apply equilibrium statistical mechanics, we consider the phase space spanned by the scaled coefficients, {aµ p}p∈µZd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In this space, the Lebesgue measure m is represented as dm = � p daµ pdaµ† p ≡ � p dar pdai p, (87) where aµ p = (ar p + iai p)/ √ 2 is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This measure makes sense only when an upper limit of the wave vector is set to truncate the infinite product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As a result, the number of degrees of freedom of ψµ in physi- cal space is also restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Let us define the bounded set of k as K∆ ≡ {−1/(2∆) + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' , 1/(2∆)}d, and accordingly that of p as µK∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The number of elements in K∆ is N ≡ 1/∆d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Then, ψµ is specified by the values at N points, Γ∆ ≡ {0, ∆, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' , 2π − ∆}d, and the values in Γ \\ Γ∆ are determined by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The Lebesgue measure is now represented in either wave vector or position space as dm = � p∈µK∆ daµ pdaµ† p = Jµ ∆ � x∈Γ∆ dψµ(x)dψµ†(x), (88) where Jµ ∆ is the Jacobian of the function that maps aµ p to ˆψµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To compute this Jacobian, we consider the integral, � dm exp � − 1 (2πµ)d � Γ×Rd nµ(x, p)dxdp � (89) that we express in two different ways: � \uf8eb \uf8ed � p∈µK∆ daµ pdaµ† p \uf8f6 \uf8f8 exp \uf8ee \uf8f0− � p∈µK∆ |aµ p|2 \uf8f9 \uf8fb =Jµ ∆ � � � x∈Γ∆ dψµ(x)dψµ†(x) � exp � − � ∆ 2πµ �d � x∈Γ∆ |ψµ(x)|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (90) Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 35 The left-hand side turns out to be (2π)N , and the right-hand side is Jµ ∆(2π)N (2πµ)dN N N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We therefore obtain Jµ ∆ and, accordingly, dm = � x∈Γ∆ dψµ(x)dψµ†(x) (2πµ)dN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (91) We denote by E integrals over the Lebesgue measure (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The microcanonical measure with constraints on A is then defined as dmA = Πωδ (Aω [nµ] − A(ω)) E [Πωδ (Aω [nµ] − A(ω))]dm, (92) where Πωδ (Aω [nµ] − A(ω)) means that we constrain the values of all the invariants Aω for any ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Our goal is to compute the probability distribution of nµ for a microcanon- ical measure constrained by A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=', Pµ A,m[nµ = n] = E [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] E [Πωδ (Aω [nµ] − A(ω))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (93) It will be mathematically convenient, for intermediate computations, to use in the following a normalizable Gaussian measure dmG, dmG = � p∈µK∆ e−|aµ p|2 daµ pdaµ† p 2π = exp � − 1 (2πµ)d � Γ×Rd nµ(x, p)dxdp − N log 2π � dm, (94) which satisfies � dmG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We note that the N-dependent term diverges in the ∆ → 0 limit, but this divergence will be compensated in Pµ A below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We denote EG averages with respect to this Gaussian measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Then, (93) can be rewritten as Pµ A,m[nµ = n] = exp � (2πµ)−d � Γ×Rd n(x, p)dxdp � EG [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] EG � exp � (2πµ)−d � Γ×Rd nµ(x, p)dxdp � Πωδ (Aω [nµ] − A(ω)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (95) In the following, when we consider the Gaussian measure mG, the continuous limit ∆ → 0 is always understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We look for a large deviation principle EG [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] ≍ µ→0 exp � −IG[n, A] (2πµ)d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (96) Springer Nature 2021 LATEX template 36 Large deviations for linear wave kinetic equation Our strategy is to compute its rescaled cumulant generating function and to apply the Gärtner-Ellis theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We shall then define a free energy as fG[λ, β] ≡ − lim µ→0 (2πµ)d log EG � exp � − 1 (2πµ)d � Γ×Rd dxdp × �� R+ β(ω)h � ω − |p|2 2 � dω + λ(x, p) � nµ(x, p) �� , (97) where β : R+ → R is a real continuous function representing the chemical potential and λ : Γ × Rd → R is also a real continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Because nµ is quadratic in ψµ, the expectation is a Gaussian integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To make it explicit, the expression in the square brackets is rewritten as − 1 (2πµ)d � ψµ†(x)L˜λψµ(x)dx, (98) where L˜λ is a pseudo-differential operator defined as L˜λψµ(x) ≡ � L˜λ(x, x′)ψµ(x′)dx′ (99a) L˜λ(x, x′) = 1 (2πµ)d � Rd ˜λ �x + x′ 2 , p � eip·(x−x′)/µdp (99b) ˜λ(x, p) ≡ � λ(x, p) + � R+ β(ω)h(ω − |p|2/2)dω (x ∈ Γ) 0 (x /∈ Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (99c) In (98) and (99a), the integration range is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' However, Riemann- Lebesgue lemma applied to (99b) assures that the kernel function L˜λ vanishes in the limit of µ → 0 except in the vicinity of the points of x = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Conse- quently, further taking into account (99c), the range of the integration of (98) and (99a) can be reduced from Rd to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' To obtain a simpler form of fG[λ, β], we need to compute the functional determinant of L˜λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This is done here straightforwardly: fG[λ, β] = − lim µ→0 (2πµ)d log EG � exp � − 1 (2πµ)d � Rd ψµ†(x)L˜λψµ(x)dx �� = − lim µ→0 lim ∆→0 (2πµ)d log � � x∈Γ∆ dψµ(x)dψµ†(x) 2π(2πµ)dN × exp \uf8ee \uf8f0− ∆2d (2πµ)d � x,x′∈Γ∆ ψµ†(x)L˜λ(x, x′)ψµ(x′) − ∆d (2πµ)d � x∈Γ∆ |ψµ(x)|2 \uf8f9 \uf8fb = lim µ→0 lim ∆→0 (2πµ)d log det � I + L∆ ˜λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (100) Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 37 We have carried out the Gaussian integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The problem thus reduces to computing the determinant of an N × N matrix, I + L∆ ˜λ , where I is a unit matrix and L∆ ˜λ consists of � ∆dL˜λ(x, x′) | x, x′ ∈ Γ∆ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Let us use the following expression, log det � I + L∆ ˜λ � = tr log � I + L∆ ˜λ � = − ∞ � j=1 (−1)j j trL∆j ˜λ , (101) which holds for sufficiently small ˜λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We know that L∆j ˜λ consists of � ∆dLj ˜λ(x, x′) | x, x′ ∈ Γ∆ � where Lj ˜λ(x, x′) corresponds to the kernel func- tion of an operator, Lj ˜λ ≡ L˜λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' L˜λ � �� � j , (102) in the small ∆ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In general, a product between pseudo-differential oper- ators corresponds to a star product, or a Moyal product, between symbols [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The star product is expanded in terms of µ with the leading term equivalent to the ordinary product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Accordingly, Lj ˜λ = L˜λj +O(µ) holds, where ˜λj is the jth power of ˜λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We thus derive lim ∆→0 Lj ˜λ(x, x) = 1 (2πµ)d � Rd ˜λj(x, p)dp + O(µ) ∴ lim ∆→0 trL∆j ˜λ = 1 (2πµ)d � Γ×Rd ˜λj(x, p)dxdp + O(µ), (103) and hence fG[λ, β] = � Γ×Rd log � λ(x, p) + � R+ β(ω)h � ω − |p|2 2 � dω + 1 � dxdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (104) From this formula, the Gärtner-Ellis theorem yields the rate function IG (96) as IG[n, A] = − inf λ,β �� dxdpλ(x, p)n(x, p) + � R+ dωβ(ω)A(ω) − f[λ, β] � , (105) which is computed as IG[n, A] = �� dxdp (n − 1 − log n) if Aω[n] = A(ω) +∞ otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (106) Springer Nature 2021 LATEX template 38 Large deviations for linear wave kinetic equation As should have been expected, the minimum value of IG[n, A] is 0, which is realized when and only when n = 1 and A(ω) = Aω[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Based on the large deviation result (96), the numerator in (95) turns out to be exp � (2πµ)−d � Γ×Rd n(x, p)dxdp � EG [δ (nµ − n) Πωδ (Aω [nµ] − A(ω))] ≍ µ→0 exp � SA[n] (2πµ)d � (107) with SA[n] = �� dxdp (1 + log n) if Aω[n] = A(ω) −∞ otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (108) The finite part of this function defines the entropy for a mesoscopic state specified by n, and it coincides with the Lyapunov function (25) for the wave kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Because the denominator of (95) is the integration of the numerator over all n, the Laplace’s principle enables us to compute it as EG � exp � (2πµ)−d � Γ×Rd nµ(x, p)dxdp � Πωδ (Aω [nµ] − A(ω)) � ≍ µ→0 exp � s[A] (2πµ)d � , (109) with s[A] = supn {SA[n]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The supremum is achieved when Aω[n] = A(ω), and n(x, p) = nA h (p) = N �|p|2 2 � , (110) that is, n is homogeneous in space and depends only on the magnitude of its wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The function N(ω) is related to A(ω) by the condition Aω[nA h ] = A(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' This gives formula (81) for nA h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We also have s[A] = � dxdp � 1 + log nA h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (111) Finally, starting from (95), and using the two asymptotic relations (107) and (109), as well as (108) and (111), we obtain Pµ A,m[nµ = n] ≍ µ→0 exp � − UA[n] (2πµ)d � (112) where the quasipotential UA[n] is given by equation (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We have established the announced results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 39 C Computations of scattering terms This appendix describes somewhat intricate derivation of the scattering terms that appear in the wave kinetic equation and the large deviation Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this purpose, we prepare some useful formulae, � t 0 dτ1eiωτ1/µ � τ1 0 dτ2e−iωτ2/µ = πµtδ(ω) + o(µ) (113a) � t 0 dτ1eiωτ1/µ � t 0 dτ2e−iωτ2/µ = 2πµtδ(ω) + o(µ) (113b) 1 (2πµ)d � Rd dξeip·(ξ−x)/µf(ξ) = � |α|≥0 (−iµ)|α| α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ∇α xf(x)∇α p δ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (113c) Equations (113a) and (113b) are often used in literature of weak turbulence [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The residual terms denoted by o(µ) make negligible contributions in the limit of µ → 0 compared to the leading-order terms when integrated with respect to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In (113c), a multi-index notation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' In the following compu- tation, integration is always carried out over Rd, except for the basic positional coordinates represented by x whose integration range is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='1 Terms appearing in the classical wave kinetic equation We first compute the scattering terms in the wave kinetic equation, specifically E [wµ(ψµ 2 , ψµ 0 )], E [wµ(ψµ 0 , ψµ 2 )], and E [wµ(ψµ 1 , ψµ 1 )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' These terms are common with those linear to λ in the scattering part of the large deviation Hamiltonian, HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The computations are slightly involved but mostly straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' A detailed procedure is presented only for the E [wµ(ψµ 2 , ψµ 0 )] case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From (16) and (18), we have wµ(ψµ 2 , ψµ 0 ) = 1 −µ2(2πµ)d � t 0 dτ1 � τ1 0 dτ2 � dydξ1234e−ip·y/µ ×Gµ � x + y 2 − ξ1, t − τ1 � V µ(ξ1)Gµ(ξ1 − ξ2, τ1 − τ2)V µ(ξ2)Gµ(ξ2 − ξ3, τ2)ψµ(ξ3, 0) ×Gµ† � x − y 2 − ξ4, t � ψµ†(ξ4, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Taking ensemble average,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' writing the propagators Gµ as Fourier integrals (17) with wave vectors η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η3 and η4 in this order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' and setting E [V µ(ξ1)V µ(ξ2)] = � dη5eiη5·(ξ1−ξ2)/µΠ(η5) ψµ(ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0)ψµ†(ξ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0) = � dη6eiη6·(ξ3−ξ4)/µn((ξ3 + ξ4)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 40 Large deviations for linear wave kinetic equation we derive E [wµ(ψµ 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )] = 1 −µ2(2πµ)5d � dydξ1234dη123456e−ip·y/µ ×e−i(|η1|2−|η4|2)t/2µ � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ � τ1 0 dτ2ei(|η2|2−|η3|2)τ2/2µ ×eiη1·(x+y/2−ξ1)/µeiη2·(ξ1−ξ2)/µeiη3·(ξ2−ξ3)/µe−iη4·(x−y/2−ξ4)/µ ×Π(η5)eiη5·(ξ1−ξ2)n((ξ3 + ξ4)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η6)eiη6·(ξ3−ξ4)/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Integration of this expression with respect to y, ξ1 and ξ2 yields E [wµ(ψµ 2 , ψµ 0 )] = 1 −µ2(2πµ)2d � dξ34dη123456 ×e−i(|η1|2−|η4|2)t/2µ � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ � τ1 0 dτ2ei(|η2|2−|η3|2)τ2/2µ ×ei(η1·x−η3·ξ3−η4·x+η4·ξ4+η6·ξ3−η6·ξ4)/µ ×δ((η1 + η4)/2 − p)δ(η1 − η2 − η5)δ(η3 − η2 − η5) ×Π(η5)n((ξ3 + ξ4)/2, η6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We change the variables as X = ξ3 + ξ4 2 , Y = ξ3 − ξ4, and carry out the integration with respect to Y to get E [wµ(ψµ 2 , ψµ 0 )] = 1 −µ2(2πµ)d � dXdη123456 ×e−i(|η1|2−|η4|2)t/2µ � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ � τ1 0 dτ2ei(|η2|2−|η3|2)τ2/2µ ×ei(η1−η4)·(x−X)/µδ((η3 + η4)/2 − η6) ×δ((η1 + η4)/2 − p)δ(η1 − η2 − η5)δ(η3 − η2 − η5) ×Π(η5)n(X, η6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We understand that η1 = η3 and η6 = p hold in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Integration with respect to η3 and η6 yields E [wµ(ψµ 2 , ψµ 0 )] Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 41 = 1 −µ2(2πµ)d � dXdη1245 ×e−i(|η1|2−|η4|2)t/2µ � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ � τ1 0 dτ2ei(|η2|2−|η1|2)τ2/2µ ×ei(η1−η4)·(x−X)/µδ((η1 + η4)/2 − p)δ(η1 − η2 − η5) ×Π(η5)n(X, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We use (113a) to derive � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ � τ1 0 dτ2ei(|η2|2−|η1|2)τ2/2µ = πµtδ �|η1|2 2 − |η2|2 2 � + o(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' We also use (113c) to derive 1 (2πµ)d � dXdη4e−i(|η1|2−|η4|2)t/2µei(η1−η4)·(x−X)/µδ((η1 + η4)/2 − p)n(X, p) =δ(η1 − p)n(X, p) + O(µ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Consequently, we obtain E [wµ(ψµ 2 , ψµ 0 )] = − t 2µ � dησ(p, η)n(x, p) + o � t µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (114) Because wµ(ψµ 0 , ψµ 2 ) = [wµ(ψµ 2 , ψµ 0 )]†, and σ and n are real functions, we also have E [wµ(ψµ 0 , ψµ 2 )] = − t 2µ � dησ(p, η)n(x, p) + o � t µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (115) Finally, we consider E[wµ(ψµ 1 , ψµ 1 )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From (16) and (18), we have wµ(ψµ 1 , ψµ 1 ) = 1 µ2(2πµ)d � t 0 dτ1 � t 0 dτ2 � dydξ1234e−ip·y/µ × Gµ � x + y 2 − ξ1, t − τ1 � V µ(ξ1)Gµ(ξ1 − ξ2, τ1)ψµ(ξ2, 0) × Gµ† � x − y 2 − ξ3, t − τ2 � V µ(ξ3)Gµ†(ξ3 − ξ4, τ2)ψµ†(ξ4, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Taking the ensemble average, introducing the Fourier integrals, and integrating some variables in the same manner as the previous case, we derive E [wµ(ψµ 1 , ψµ 1 )] = 1 µ2(2πµ)d � dXdη123456 Springer Nature 2021 LATEX template 42 Large deviations for linear wave kinetic equation ×e−i(|η1|2−|η3|2)t/2µ � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ � t 0 dτ2e−i(|η3|3−|η4|2)τ2/2µ ×ei(η1−η3)·(x−X)/µδ((η2 + η4)/2 − η6) ×δ((η1 + η3)/2 − p)δ(η1 − η2 − η5)δ(η3 − η4 − η5) ×Π(η5)n(X, η6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Applying (113c), understanding η1 = η3 and η2 = η4 in the integrand, using (113b), and Integrating all the possible variables, we finally obtain E [wµ(ψµ 1 , ψµ 1 )] = t µ � dη2σ(p, η2)n(x, η2) + o � t µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (116) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content='2 Quadratic terms in the Hamiltonian We compute the terms in HS quadratic in λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' originating from four expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 1 (2πµ)2d � dx12dp12λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)E[wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)wµ(ψµ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 1 )(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)] (117a) 1 (2πµ)2d � dx12dp12λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)E[wµ(ψµ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 1 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)] (117b) 1 (2πµ)2d � dx12dp12λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)E[wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)] (117c) 1 (2πµ)2d � dx12dp12λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)E[wµ(ψµ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 1 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)wµ(ψµ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 1 )(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (117d) Among these, the two pairs, (117a)-(117b) and (117c)-(117d), are complex conjugate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, we need to compute only two expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Although the number of factors involved in the integration is greater than those in the linear terms, the computation procedures are largely the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For the sake of conciseness, we denote time t instead of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' From (16) and (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' we write the Wigner transforms in the integrand of (117a) as wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1) = 1 iµ(2πµ)d � t 0 dτdydξ123e−ip1·y/µ ×Gµ � x1 + y 2 − ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' t − τ � V µ(ξ1)Gµ(ξ1 − ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' τ)ψµ(ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0) ×Gµ† � x1 − y 2 − ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' t � ψµ†(ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0) Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 43 wµ(ψµ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 1 )(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2) = 1 −iµ(2πµ)d � t 0 dτdydξ123e−ip2·y/µ ×Gµ � x2 + y 2 − ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' t � ψµ(ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0) ×Gµ† � x2 − y 2 − ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' t − τ � V µ(ξ2)Gµ†(ξ2 − ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' τ)ψµ†(ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Taking the ensemble average of the product of these expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' it would be straightforward to have 1 (2πµ)2d � dx12dp12λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)E[wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)wµ(ψµ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 1 )(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)] = 1 µ2(2πµ)4d � dx12dp12dX12dη123456789λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2) ×e−i(|η1|2−|η3|2)t/2µ � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ ×e−i(|η4|2−|η5|2)t/2µ � t 0 dτ2e−i(|η5|2−|η6|2)τ2/2µ ×δ(η1/2 + η3/2 − p1)δ(η4/2 + η5/2 − p2)δ(η1 − η2 − η7)δ(η5 − η6 − η7) ×δ(η2/2 + η6/2 − η8)δ(η3/2 + η4/2 − η9) ×ei(η1−η3)·x1/µei(η4−η5)·x2/µe−i(η2−η6)·X1/µe−i(η4−η3)·X2/µ ×Π(η7)n(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η8)n(X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Applying (113c) and (113a) and integrating all the possible variables yields 1 (2πµ)2d � dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ 1 , ψµ 0 )(x1, p1)wµ(ψµ 0 , ψµ 1 )(x2, p2)] = t µ(2πµ)d � dx1dp1dη2λ(x1, p1)2σ(p1, η2)n(x1, p1)n(x1, η2) + o � t µ(2πµ)d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (118) From the the condition of complex conjugate, we also have 1 (2πµ)2d � dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ 0 , ψµ 1 )(x1, p1)wµ(ψµ 1 , ψµ 0 )(x2, p2)] = t µ(2πµ)d � dx1dp1dη2λ(x1, p1)2σ(p1, η2)n(x1, p1)n(x1, η2) + o � t µ(2πµ)d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (119) For the computation of (117c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' we write,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1) = 1 iµ(2πµ)d � t 0 dτdydξ123e−ip1·y/µ Springer Nature 2021 LATEX template 44 Large deviations for linear wave kinetic equation ×Gµ � x1 + y 2 − ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' t − τ � V µ(ξ1)Gµ(ξ1 − ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' τ)ψµ(ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0) ×Gµ† � x1 − y 2 − ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' t � ψµ†(ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' 0) Taking the ensemble average of the product of this expression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' it would be straightforward to have 1 (2πµ)2d � dx12dp12λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)E[wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)wµ(ψµ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' ψµ 0 )(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2)] = 1 −µ2(2πµ)4d � dx12dp12dX12dη123456789λ(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1)λ(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p2) ×e−i(|η1|2−|η3|2)t/2µ � t 0 dτ1ei(|η1|2−|η2|2)τ1/2µ ×e−i(|η4|2−|η6|2)t/2µ � t 0 dτ2ei(|η4|2−|η5|2)τ2/2µ ×δ(η1/2 + η3/2 − p1)δ(η4/2 + η6/2 − p2)δ(η1 − η2 − η7)δ(η4 − η5 + η7) ×δ(η2/2 + η6/2 − η8)δ(η5/2 + η3/2 − η9) ×ei(η1−η3)·x1/µei(η4−η6)·x2/µe−i(η2−η6)·X1/µe−i(η5−η3)·X2/µ ×Π(η7)n(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η8)n(X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' η9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Applying (113c) and (113a) and integrating all the possible variables yields 1 (2πµ)2d � dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ 1 , ψµ 0 )(x1, p1)wµ(ψµ 1 , ψµ 0 )(x2, p2)] = − t µ(2πµ)d � dx1dp12λ(x1, p1)λ(x, p2)σ(p1, p2)n(x, p1)n(x, p2) + o � t µ(2πµ)d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (120) Finally, from the the condition of complex conjugate, we have 1 (2πµ)2d � dx12dp12λ(x1, p1)λ(x2, p2)E[wµ(ψµ 0 , ψµ 1 )(x1, p1)wµ(ψµ 0 , ψµ 1 )(x2, p2)] = − t µ(2πµ)d � dx1dp12λ(x1, p1)λ(x, p2)σ(p1, p2)n(x, p1)n(x, p2) + o � t µ(2πµ)d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (121) D Derivation of the Hamiltonian in the diffusive limit In this appendix, we show how to obtain the large deviation Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (58) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Since we are dealing with the scattering term that is local Springer Nature 2021 LATEX template Large deviations for linear wave kinetic equation 45 in position space, one can omit the dependence on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' As explained in the main text, the important point is to show how the diffusion kernel Σ[n] transforms when only infinitesimal deviation from incoming wave vector are allowed by the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For this specific purpose, we introduce a parameter ν and write the typical correlation length of V as µ/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Our strategy is to expand the Hamiltonian in terms of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Because the potential spectrum Π(p) is supposed to have a finite sup- port with the extent of O(ν), it is convenient to rewrite the cross section as σ(p1, p2) = ν−d˜σ((p1 − p2)/ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' For any test functions, f and g, the diffusive kernel (39) satisfies � dp1dp2f(p1)Σ(p1, p2)g(p2) =c � dpdqf(p)n(p)˜σ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p)n(p − νq) [g(p) − g(p − νq)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (122) Taylor expanding the integrand with respect to ν, one obtains � dp1dp2f(p1)Σ(p1, p2)g(p2) =c � dpf(p)n(p) � νn(p) d � n=1 �� dqn˜σ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p)qn � ∂png(p) −ν2 d � n=1 d � m=1 �� dqndqm˜σ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p)qnqm � ∂pnn(p)∂pmg(p) −ν2 2 n(p) d � n=1 d � m=1 �� dqndqm˜σ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p)qnqm � ∂pn∂pmg(p) + O � ν3� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (123) To evaluate the integrations with respect to q, we shall exponentiate the Dirac- δ in the definition of the cross section (23) and use the inverse Fourier transform of Π in (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' The resulting expression is ˜σ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p) = 1 ν(2π)d � dy � R dse−iq·yRν � sp + y − ν 2 sq � (124) and Rν(y) = � dpΠ(p)eip·y/ν, where Rν(y) = R(y/ν) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (57)) is a scaled correlation function of the random potential with the correlation length of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Again Taylor expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (124) with respect to ν, one gets cν � dq˜σ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p)qn = −ν d � m=1 ∂pmDν nm(p) + O(ν2) Springer Nature 2021 LATEX template 46 Large deviations for linear wave kinetic equation cν2 � dq˜σ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' p)qnqm = 2νDν nm(p) + O(ν2) with Dν(p) = −(c/2) � R ∇⊗∇Rν(sp)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Here, � R ∇Rν(sp)ds vanishes because of the point symmetry in Rν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Inserting these expressions into (123), one obtains � dp1dp2f(p1)Σ(p1, p2)g(p2) = − cν � dpfn d � n=1 d � m=1 {n∂pnDν nm∂pmg + 2Dν nm∂pnn∂pmg + nDν nm∂pn∂pmg} +O(ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (125) Evidently from this result, the dominant term is of order O(ν) in the present scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Therefore, in order for diffusion in wave-vector space to be comparable to the free propagation in position space, one needs to rescale the position coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' However, for the sake of simplicity here, we shall set ν = 1 while ignoring O(ν2) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' Consequently, after reorganizing products of derivatives and performing integration by parts, we confirm that (125) transforms into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE1T4oBgHgl3EQfjAQ8/content/2301.03257v1.pdf'} +page_content=' (56).' 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https://git-lfs.github.com/spec/v1 +oid sha256:adb0b5d6379331ccc3d506bcf812367499b47e5803fc97085b5605ce1d22e889 +size 2118366 diff --git a/ytE1T4oBgHgl3EQf4AXn/content/tmp_files/2301.03497v1.pdf.txt b/ytE1T4oBgHgl3EQf4AXn/content/tmp_files/2301.03497v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d61d15d3faffdb202051d3b6ead0e2e786c9943 --- /dev/null +++ b/ytE1T4oBgHgl3EQf4AXn/content/tmp_files/2301.03497v1.pdf.txt @@ -0,0 +1,1042 @@ +Phase transitions in inorganic halide perovskites from machine learning potentials +Erik Fransson, Julia Wiktor, and Paul Erhart +Department of Physics, Chalmers University of Technology, SE-41296, Gothenburg, Sweden ∗ +The atomic scale dynamics of halide perovskites have a direct impact not only on their thermal +stability but their optoelectronic properties. Progress in machine learned potentials has only recently +enabled modeling the finite temperature behavior of these material using fully atomistic methods +with near first-principles accuracy. +Here, we systematically analyze the impact of heating and +cooling rate, simulation size, model uncertainty, and the role of the underlying exchange-correlation +functional on the phase behavior of CsPbX3 with X=Cl, Br, and I, including both the perovskite +and the δ-phases. We show that rates below approximately 30 K/ns and system sizes of at least a +few ten thousand atoms are indicated to achieve convergence with regard to these parameters. By +controlling these factors and constructing models that are specific for different exchange-correlation +functionals we then show that the semi-local functionals considered in this work (SCAN, vdW- +DF-cx, PBEsol, and PBE) systematically underestimate the transition temperatures separating the +perovskite phases while overestimating the lattice parameters. Among the considered functionals the +vdW-DF-cx functional yields the closest agreement with experiment, followed by SCAN, PBEsol, +and PBE. Our work provides guidelines for the systematic analysis of dynamics and phase transitions +in inorganic halide perovskites and similar systems. It also serves as a benchmark for the further +development of machine-learned potentials as well as exchange-correlation functionals. +INTRODUCTION +Halide perovskites are among the most intensively +studied materials of the last decade due to their attrac- +tive properties for applications in, for example, solar en- +ergy harvesting and lighting [1–4]. Similar to their ox- +ide counterparts many of these materials exhibit sev- +eral different phases that are connected through soft +modes and continuous or weak first-order phase transi- +tions [5, 6]. This complex dynamic behavior turns out +to be intimately connected to their remarkable optoelec- +tronic properties. +In this context, electronic structure calculations play a +crucial role as they can provide detailed insight into the +atomic scale dynamics and microscopic coupling mech- +anisms. Static calculations can, however, only provide +limited information due the strong anharmonicity asso- +ciated with the soft modes [7–10]. This has motivated a +number of dynamic studies based on ab-initio molecular +dynamics (MD) simulations [11–18] and, more recently, +machine learning potentials (MLPs) [19–28]. From such +simulations one can then obtain, for example, transition +temperatures [19] or structural information at finite tem- +peratures [11, 18, 29]. +The accuracy of such simulations is, however, limited +by several factors, most notably (1) sampling time, (2) +system size, (3) the quality of the exchange-correlation +(XC) functional, and, in the case of MLPs, (4) the model +uncertainty. Sampling time and system size are in partic- +ular a problem for ab-initio MD simulations, which are +typically limited to time scales of a few ten picoseconds +and system sizes on the order of a 1000 atoms. Previ- +ous MLP based studies were able to extend these ranges +∗ erhart@chalmers.se +to total run times of a few nanoseconds using about a +thousand atoms [19, 20, 26]. +Here, we carry out a systematic analysis of the four fac- +tors described above. We consider CsPbX3 with X=Cl, +Br, and I and the strongly constrained and appropriately +normed (SCAN) [30], van-der-Waals-density functional +with consistent exchange (vdW-DF-cx) [31], PBEsol [32], +and PBE [33] XC functionals. The potential energy sur- +face (PES) is mapped using third-generation neuroevo- +lution potentials (NEPs) models and sampled using the +gpumd package. The latter provides an efficient NEP +implementation that enables us to routinely sample sys- +tems comprising on the order of 60 000 atoms for 100 or +more nanoseconds. +We show that well converged results can be achieved +using systems containing several ten thousand atoms and +heating/cooling rates on the order of 30 K/ns or lower. +Using bootstrapping and ensembles of models we are able +to readily generate accurate NEP models with an un- +certainty that is comparable or lower than the training +errors. +By controlling rate and size effects as well as model +errors, we are able to isolate the impact of the under- +lying XC functionals and thus to assess quantitatively +the quality of different XC functionals for the description +of phase transitions and finite temperature properties of +halide perovskites. We find the vdW-DF-cx functional to +perform the best among the XC functionals considered +here when comparing transition temperatures and lattice +constants to experimental data. +In the following section we analyze in order rate and +size effects (Sect. Rate and size effects), mode uncertainty +(Sect. Model uncertainty), the impact of the XC func- +tional (Sect. Impact of XC functional and extension to +other halides), and finally the transition temperature be- +tween the δ and perovskite phases (Sect. Transition to +δ-phase). We then summarize and discuss the outcome +arXiv:2301.03497v1 [cond-mat.mtrl-sci] 9 Jan 2023 + +2 +of this analysis (Sect. Discussion). +RESULTS +Rate and size effects +The different perovskite phases are structurally closely +related and connected through phase transitions with +mixed continuous-first order character [34–37]. For the +CsPbX3 (with X=Cl, Br, or I) materials considered in +this study the perovskite lattice transforms with increas- +ing temperature from an orthorhombic phase (Pnma) via +a tetragonal phase (P4/mbm) to a cubic phase (Pm¯3m). +Since these transitions do not involve a switch in the sign +of the Glazer angles between the orthorhombic (a−a−c+) +and tetragonal phases (a0a0c+), unlike, e.g., MAPbI3 +[19], it is possible to observe these transitions in MD +simulations. Due to the remaining first-order character +and the extreme heating/cooling rates that can be real- +ized in MD simulations, one can, however, nonetheless +anticipate some degree of hysteresis. +A further aggravating factor is the finite system size. +For small supercells the fluctuations are naturally larger, +which renders it more challenging to achieve converged +results. +In this section, we discriminate the effects of +heating/cooling rate and system size by considering in +detail MD simulations for CsPbI3 using the full model +based on the vdW-DF-cx functional (Sect. Sect. ). +Rate effects +To separate rate from size effects, we first consider the +former in the large size limit, using a supercell compris- +ing 61 440 atoms, equivalent to 16x16x12 primitive or- +thorhombic (20-atom) unit cells. +On heating all simulations yield the correct (exper- +imentally observed) sequence of phases irrespective of +heating rate. On cooling, this sequence is reversed again +regardless of rate. At the cubic-to-tetragonal transition, +for a small number of simulations, one can, however, ob- +serve the simultaneous formation of multiple tetragonal +domains with incompatible orientations, which can lead +to the formation of domain boundaries. Since the mov- +ing of these boundaries involves a nucleation-and-growth +mechanism, they remain in the simulated structure upon +cooling. +The temperatures for the transitions between the per- +ovskite phases can be readily obtained from the lattice +parameters (Fig. 1a,b), revealing a strong dependence +on the heating/cooling rate. For the rates below approx- +imately 30 K/ns, the hysteresis between heating and cool- +ing runs is 15 K are less and no longer vary systematically +with the rate (Fig. 1c). By contrast, for the largest rate +of 600 K/ns considered here, which is only slightly larger +than values used in some earlier MLP studies [19, 26], +one observes a hysteresis of 100 K or more for both the +300 +400 +500 +Temperature (K) +6.1 +6.2 +6.3 +6.4 +Lattice parameters ( ˚A) +a) heating +300 +400 +500 +Temperature (K) +b) cooling +600 K/ns +60 K/ns +6 K/ns +6 +60 +600 +Rate (K/ns) +250 +350 +450 +550 +Transition temperature (K) +c) +cubic (Pm3m, a0a0a0) +tetragonal +(P4/mbm, a0a0c+) +orthorhombic +(Pnma, a−a−c+) +heating +cooling +FIG. 1. +Convergence with heating/cooling rate. Lat- +tice parameters for CsPbI3 as a function of temperature from +MD simulations with varying (a) heating and (b) cooling +rates for supercells comprising 61 440 atoms. The transition +temperatures extracted from these data are indicated by dia- +monds. The gray lines show the raw data whereas the colored +lines show the data after application of a Hamming window of +0.6 K (see Fig. S3). (c) Transition temperatures as a function +of heating/cooling rate. All results were obtained using the +full model (Sect. Model construction) based on the vdW-DF- +cx exchange-correlation functional. +lower and higher temperature transitions. In addition, +the transition itself is smeared out in temperature, which +is particular apparent for the orthorhombic-to-tetragonal +transition (Fig. 1a). +We observed similar trends also for the other materi- +als and models studied in this work. We therefore con- +clude that rates below approximately 30 K/ns are recom- +mended in order to achieve convergence of the transition +temperatures for this class of materials. +Size effects +Next we examine the impact of supercell size on the +temperature dependence of the lattice parameters and +the transition temperatures. +First, simulations were +carried out at a heating/cooling rate of 6 K/ns using +structures comprising 1280 atoms (4x4x4 primitive or- +thorhombic unit cells), 7680 atoms (8x8x6), 23 040 atoms +(12x12x8) or 61 440 atoms (16x16x12). + +3 +300 +400 +500 +Temperature (K) +6.1 +6.2 +6.3 +6.4 +Lattice parameters ( ˚A) +a) heating +300 +400 +500 +Temperature (K) +b) cooling +1280 atoms +7680 atoms +23040 atoms +61440 atoms +1280 7680 +23040 +61440 +Number of atoms +250 +350 +450 +550 +Transition temperature (K) +c) +cubic (Pm3m, a0a0a0) +tetragonal +(P4/mbm, a0a0c+) +orthorhombic +(Pnma, a−a−c+) +FIG. 2. +Convergence with supercell size. Lattice pa- +rameters for CsPbI3 as a function of temperature from MD +simulations during (a) heating and (b) cooling at a rate of +R = 6 K/ns for different supercells. The transition tempera- +tures extracted from these data are indicated by diamonds. A +Hamming window of 0.6 K was applied. (c) Transition tem- +peratures as a function of supercell size. All results were ob- +tained using the full model (Sect. Model construction) based +on the vdW-DF-cx exchange-correlation functional. +For the smallest supercell (7680 atoms) one notices a +marked deviation from the (reference) lattice parameter +parameter data from the largest supercell (61 440 atoms) +around the cubic-tetragonal phase transition (Fig. 2a,b). +This deviation is, however, absent for the next larger +structure of 7680 atoms and at this size the transition +temperatures are already converged to within 10 K of the +results for the largest supercell (Fig. 2c). +A key characteristic of a phase transition that is at +least partly continuous is a peak or kink in the heat ca- +pacity. The heat capacity can be readily extracted from +the fluctuations of the energy in MD simulations. For +the present purpose this is, however, impractical as the +transition is very sharp in temperature and the temper- +ature range of interest is wide. Here, we therefore com- +pute the heat capacity instead by numerically differen- +tiating the potential energy from heating/cooling runs, +i.e., Cp = dH/dT ≈ ∆H/∆T. This requires averaging +of the data in order to obtain numerically well behaved +results. To this end, we first apply a Hamming window of +0.6 K to the energy-vs-temperature data. The resulting +data is numerically differentiated, after which the data +300 +400 +500 +Temperature (K) +1.0 +1.5 +2.0 +2.5 +3.0 +Heat capacity (kB) +a) heating +7680 atoms +23040 atoms +61440 atoms +300 +400 +500 +Temperature (K) +b) cooling +FIG. 3. +Convergence of heat capacity with super- +cell size. +Isobaric heat capacity of CsPbI3 as a function +of temperature for different supercell sizes obtained through +numerical differentiation as detailed in the text. The phase +transitions are visible as peaks in the heat capacity. All results +were obtained using the full model (Sect. Model construction) +based on the vdW-DF-cx exchange-correlation functional. +is smoothened again using a Hamming window of 6 K. +The Hamming window sizes are chosen to be sufficiently +large to remove noise and small enough to avoid removing +features (Fig. S4). +For the largest system size (61 440 atoms) the phase +transitions are clearly visible as peaks in the temperature +dependence of the heat capacity (Fig. 3). These features +become, however, less distinct with decreasing system +size as fluctuations increase. +The analysis presented in this section suggests that +supercells with at least about 10 000 atoms can be ex- +pected to yield accurate lattice parameters and transi- +tion temperatures within about 10 K of the converged +results. Extracting the temperature dependence of the +heat capacity requires larger systems still. Even for the +largest systems considered here (61 440 atoms) the noise +level is rather high, but the data still allows one to ac- +curately extract phase transition temperatures from the +heat capacity data. +Model uncertainty +Having established heating/cooling rates and system +sizes that yield converged results for lattice parameters +and transition temperatures, we can now address the +model uncertainty. +To this end, we resort to ensem- +ble models. +The latter comprise five separate models +constructed using five distinct 90-10 splits of the train- +ing data (Sect. Model construction). All simulations in +this section were carried out using supercells with 61 440 +atoms and a heating/cooling rate of 6 K/ns. Once again +we use CsPbI3 and models trained using reference data +generated by the vdW-DF-cx functional as a representa- +tive example. + +4 +6.0 +6.2 +6.4 +Lattice parameters ( ˚A) +a) heating +b) cooling +200 +400 +Temperature (K) +1.0 +1.5 +2.0 +2.5 +3.0 +Heat capacity (kB) +c) +ensemble +average +full +200 +400 +Temperature (K) +d) +FIG. 4. +Assessing model uncertainty through en- +semble of models. (a,b) Lattice parameters and (c,d) heat +capacity as a function of temperature during (a,c) heating +and (b,d) cooling from full (blue lines) and ensemble models +(orange lines). All results are for CsPbI3 using models based +on the vdW-DF-cx exchange-correlation functional. +The uncertainty of the model predictions can be es- +timated by the considering the standard deviation over +the model ensemble (Fig. 4a,b). At 100 K this approach +yields an uncertainty of up to 0.02 ˚A depending on di- +rection. +This value diminishes, however, quickly with +temperature to a level of 0.003 ˚A in the tetragonal and +even less than 0.002 ˚A in the cubic phases (Fig. S5). +Away from the phase transitions the heat capacity +curves from the different models agree well with each +other (Fig. 4). The actual transition temperatures, cor- +responding to the position of the peaks in the heat capac- +ity curves, obtained from the full model (and the model +ensemble) are 340 K (341(9) K) and 487 K (496(7) K) on +heating and 331 K (323(17) K) and 470 K (478(5) K) on +cooling. The agreement between the results obtained us- +ing the full model and the model ensemble support the +good convergence of the models with respect to training +data. The remaining hysteresis can be attributed to the +mixed first/continuous-order character, which is evident +from the very small but non-zero latent heat associated +with these transitions [34–37]. +We can thus conservatively estimate the error in the +transition temperatures due to model uncertainty to be +on the order of 20 K. +In combination with the model +performance measures (Sect. Sect. ) and the very good +agreement with the density functional theory (DFT) ref- +erence data, this provides strong evidence that the NEP +models constructed are accurate representations of the +DFT potential energy landscape in the regions of config- +uration space included here. They can thus be used to +analyze the performance of different XC functionals with +CX +SCAN PBEsol PBE +200 +300 +400 +500 +600 +Transition temperature (K) +a) CsPbI3 +tetra-cubic +ortho-tetra +CX +SCAN +b) CsPbBr3 +CX +SCAN +c) CsPbCl3 +FIG. 5. +Transition temperatures from MD simula- +tions in comparison with experiment. +Data were ob- +tained using heating (upward triangles) and cooling rates +(downward triangles) of 6 ns for supercells comprising 61 440 +atoms. Colored and gray symbols indicate results obtained +using full models and model ensembles, respectively. In the +latter case the uncertainty calculated as the standard devia- +tion over the ensemble is indicated by vertical bars. Exper- +imental transition temperatures taken from Refs. 9, 38, and +39 are shown by horizontal dashed lines. +respect to the finite temperature behavior of CsPbI3 and +the other materials considered here (Sect. Impact of XC +functional and extension to other halides). +Impact of XC functional and extension to other +halides +We can now apply the framework established above +to predict transition temperatures for CsPbI3 using dif- +ferent XC functionals for generating reference data, and +compare the results to experimental data. To this end, +we use the same computational settings in terms of sys- +tem size and heating/cooling rate as in the analysis of +the model uncertainty. +The results show that all XC functionals consid- +ered here systematically underestimate the transition +temperatures for both the orthorhombic-tetragonal and +tetragonal-cubic transitions by as much as 200 K (Fig. 5). +The agreement improves in the sequence PBE, PBEsol, +SCAN, and vdW-DF-cx, which mirrors the trends ob- +served previously for transition metals [40]. +Conversely for the lattice parameters one observes a +systematic overestimation relative to experiment (Fig. 6) +with a similar trend. The closest agreement is obtained +using the vdW-DF-cx functional, for which the lattice +parameters of the cubic phase are still overestimated by +0.02 to 0.04 ˚A in the high-temperature limit. +Finally, we extend our investigation to CsPbBr3 and +CsPbCl3 including the SCAN and vdW-DF-cx function- +als. As for CsPbI3 the transition temperatures (Fig. 5b,c) +and lattice parameters (Fig. 6b,c) are systematically un- +derestimated and overestimated, respectively. The ther- + +5 +100 +200 +300 +400 +500 +5.9 +6.0 +6.1 +6.2 +6.3 +6.4 +6.5 +Lattice parameters ( ˚A) +a) CsPbI3 +a) CsPbI3 +a) CsPbI3 +a) CsPbI3 +experiment +PBE +PBEsol +SCAN +vdW-DF-cx +100 +200 +300 +400 +500 +5.7 +5.8 +5.9 +6.0 +Lattice parameters ( ˚A) +b) CsPbBr3 +b) CsPbBr3 +100 +200 +300 +400 +500 +Temperature (K) +5.5 +5.6 +5.7 +c) CsPbCl3 +c) CsPbCl3 +FIG. 6. +Lattice parameters as a function of tempera- +ture from simulation in comparison with experiment. +Data were obtained using “full” NEP models trained using +different XC functionals in comparison. Experimental data +from Refs. 9, 38, and 39. +mal expansion is, however, captured well by all function- +als. +We conclude that while the XC functionals considered +here yield the correct sequence of phases, none of them +are quantitatively in satisfactory agreement with experi- +ment, systematically underestimating the transition tem- +peratures (Fig. 5) and overestimating the lattice param- +eters (Fig. 6). +Transition to δ-phase +It is experimentally well established that the perovskite +phases of CsPbI3 are only metastable at low tempera- +tures as the actual ground state structure of the material +is the so-called δ-phase [41]. We therefore also computed +the transition temperature between the cubic perovskite +and the δ-phase by free energy integration (see Fig. S6 +for the free energy curves). +The transition temperature obtained by the vdW-DF- +cx model of 635 K with an estimated uncertainty of 20 +to 40 K is in rather good agreement with experimental +values of around 600 K (Table I). The SCAN and PBEsol +models yield lower values that are considerably smaller +than the experimental data. +TABLE I. +Temperatures for the transition between the δ- +phase and the cubic perovskite phase. +Experimental data +from Refs. 42–44. +vdW-DF-cx SCAN PBEsol PBE Experiment +CsPbI3 +635 +432 +380 +310 +∼600 +CsPbBr3 +310 +151 +According to DFT (and NEP) calculations the δ-phase +is the most stable structure for CsPbBr3. The energy +difference (Table S3) is, however, much smaller than for +CsPbI3 (Table S4), leading to lower transition tempera- +tures that are predicted to be 310 K and 151 K according +to the vdW-DF-cx and SCAN models, respectively. We +are unaware of experimental measurements of the tran- +sition temperature, which given the present predictions +might actually be close to or below room temperature +and thus difficult to observe. +For CsPbCl3 the DFT calculations yield energy dif- +ferences between perovskite and δ-phases close to zero +(Table S2), suggesting that the δ-phase is actually not +the most stable phase under any conditions or only at +extremely low temperatures. +DISCUSSION +We have systematically analyzed four key sources of +error in atomic scale simulations of phase transitions of +inorganic halide perovskites, related respectively to (1) +sampling time, (2) system size, (3) model uncertainty, +and (4) the underlying XC functional. Based on these +results, it is recommended to use heating/cooling rates +of at most approximately 30 K/ns but preferably even +lower and system sizes comprising at least a couple ten +thousand atoms, corresponding to a few thousand prim- +itive unit cells. We expect that these guidelines are not +limited to inorganic halide perovskites but should also be +heeded when modeling the dynamics of other perovskites +and related systems. +The model uncertainty was assessed using ensembles +of models, from which uncertainty estimates for, e.g., +transition temperatures and lattice parameters can be +estimated. The results show that by careful model con- +struction the model uncertainty can be reduced to a level +that allows one to quantitatively discriminate the perfor- +mance of different XC functionals. +Using this approach, we found that the semi-local XC +functionals considered here systematically underestimate +the transition temperatures and overestimate the lat- +tice parameters at finite temperature compared to ex- +periment. +The best overall agreement is obtained for +the vdW-DF-cx functional, which also outperforms the +SCAN functional. In pioneering work on the use of MLPs +for probing the dynamics of halide perovskite, the latter +functional had been suggested as achieving a good match +with experiment [19, 26, 45, 46]. Our analysis suggests + +6 +that the good agreement was likely fortuitous and likely +the result of using high rates and small system sizes (see +Fig. S6 for a more detailed comparison). +This raises the question how, for example, hybrid func- +tionals or the random phase approximation would per- +form with regard to the finite temperature properties +and dynamics of these materials [13, 27, 47]. +In the +present work we included a relatively large set of struc- +tures and supercells that would pose a considerable com- +putational challenge for either one of these methods. As +noted above (Sect. Model construction), one can, how- +ever, expect that the number of training structures can +be considerably reduced without a notable loss in model +accuracy by using active learning. This strategy could be +combined with principal component analysis to identify +regions with very dense sampling [48] or entropy max- +imization [49] to reduce the training set size even fur- +ther, eventually allowing one to build NEP or other MLP +models that can represent such more accurate electronic +structure methods. +Finally, we note that the accuracy of the models pre- +sented here in combination with the very high computa- +tional efficiency provide by the implementation on graph- +ical processing units (GPUs), now enables one to sample +the dynamics of these and related materials with unprece- +dented time resolution and accuracy. +METHODS +Machine learned potentials +Neuroevolution potentials +Here, we use the third generation of the NEP scheme +(NEP3) [48] to build MLPs for CsPbX3 with X=Cl, Br, +and I. The NEP format employs a simple multi-layer per- +ceptron neural network architecture with a single hidden +layer [50]. In NEP3 the radial part of the atomic environ- +ment descriptor is constructed from linear combinations +of Chebyshev basis functions while the three-body an- +gular part is similarly build from Legendre polynomials. +Four and five-body terms of the atomic cluster expan- +sion form [51] can be included as well but here we limit +ourselves to two and three-body terms. +For the present purpose it is crucial that the NEP +scheme is not only accurate but has been implemented +on GPUs in the gpumd package [48]. For the models de- +scribed in the following this allows us to achieve a speed +of 2 × 107 atom step/s on an NVidia A100 card, i.e., we +can simulate a system of 60 000 atoms for about 150 ns +per day using a time step of 5 fs. +Computational parameters +In this study we used the same hyperparameters for +all models, which were chosen based on experience and +pre-trials [48]. The cutoffs for two and three-body inter- +actions are 8 ˚A and 4 ˚A, respectively. There are 8 radial +and 6 angular descriptor components, 8 basis functions +for building both the radial and angular descriptor func- +tions, and the angular components are expanded up to +fourth order. The hidden layer contains 50 neurons. +The weights for energies, forces, and virials in the loss +function were set to 1, 5, and 0.2 in gpumd units, respec- +tively, while the weights for the ℓ1 and ℓ2 regularization +terms were set to 0.1. The neuroevolution strategy [52] +used for optimizing the parameters used a population size +of 50 and was run for 200 000 generations. +Model construction +To construct NEP models we employed a boot- +strapping strategy. First we identified potentially rele- +vant phases. This included the cubic perovskite structure +(Pm¯3m, Glazer notation a0a0a0), two tetragonal struc- +tures (I4/mcm → a0a0c−, P4/mbm → a0a0c+), repre- +senting out-of-phase and in-phase tilts relative to the c- +axis, respectively, one orthorhombic structure (Pnma → +a−a−c+) as well as the so-called delta-phase (Pnma), +which is experimentally known to be the most stable +structure at least for CsPbI3 and CsPbBr3. +We then calculated energy-volume curves for these five +prototype structures using DFT calculations (Sect. Ref- +erence calculations) allowing both the ionic coordinates +and the cell shape to relax under the constraint of con- +stant volume until the maximum force on any atom fell +below 30 meV/˚A. +Subsequently we generated supercells for each proto- +type with random atomic displacements using the Monte +Carlo rattle procedure from the hiphive package [53] +with a standard deviation of 0.04 ˚A. The supercell size +was chosen to be between 120 and 160 atoms and the vol- +ume was varied between 85% and 110% of the respective +equilibrium volume with five structures per volume and +prototype. +Using these data we generated a first iteration of NEP +models using the gpumd package for the optimization +[48] and the calorine package for data preparation +and analysis [54]. One model was generated using the +full data set (“full model”) and five additional models +(“model ensemble”) were generated by using five differ- +ent 90-10 splits of the available data. +Using the full +model we generated new structures for each prototype by +running short MD simulations at pressures between −1 +and 10 GPa using a temperature ramp from 20 to 620 K +over 3 ns. From each trajectory we selected 12 configura- +tions. For each of these configurations we then computed +the standard deviation of the energy and forces using +the model ensemble. +The standard deviation over the +ensemble predictions provided a measure for the uncer- +tainty of the current model generation for the respective +conditions (temperature, pressure, structure). We then +computed energy and forces for the new structures using + +7 +TABLE II. +RMSE scores for the final NEP models ob- +tained by training against the full data set. Additional per- +formance measures including RMSE and Pearson correlation +coefficients for model ensembles can be found in Table S1. +Energy +Force +Virial +meV/atom meV/˚A meV/atom +CsPbCl3 +vdW-DF-cx +1.2 +46.4 +13.0 +SCAN +1.1 +47.8 +14.0 +CsPbBr3 +vdW-DF-cx +1.1 +45.1 +11.1 +SCAN +1.2 +47.6 +15.1 +CsPbI3 +vdW-DF-cx +1.8 +47.5 +14.5 +SCAN +2.1 +51.4 +16.0 +PBEsol +1.9 +50.6 +15.2 +PBE +1.0 +43.1 +12.6 +DFT calculations, added these to the training set and +repeated the procedure. Typically after four generations +we found that the uncertainty in the energy and forces +was comparable are smaller than the respective training +error indicating convergence of the model construction. +We note that in principle one could have adapted an +active learning strategy based on the model ensemble and +only included configurations with high uncertainty as ad- +ditional reference structures. Here, we decided to include +rather more data in the training set but we expect that +the number of structures can be reduced considerably +without a notable decrease in model performance. +The final models yield RMSE scores for training and +validation sets of about 2 meV/atom, 50 meV/˚A, and +15 meV/atom or better for energies, forces, and viri- +als, respectively (Table II and Table S1). Importantly +the models closely reproduce the energy differences and +energy-volume curves of all the structures of interest in +the present study (Fig. S1; Fig. S2; Table S2 to Table S5). +The final models were subsequently used in large scale +MD simulations to predict, for example, transition tem- +peratures or lattice parameters (Sect. MD simulations). +MD simulations +All MD simulations were carried out using the gpumd +code. Temperature and pressure were controlled using +stochastic velocity [55] and cell rescaling [56] and the +time step was 5 ps, where all simulations were run at +zero pressure. +For studying the convergence with size (Sect. Size +effects), we considered system sizes between 1280 and +61 440 atoms, equivalent to 4x4x4 to 16x16x12 primitive +orthorhombic perovskite (20-atom) unit cells. To analyze +the impact of heating and cooling rates (Sect. Rate ef- +fects) the temperature was linearly varied between 20 K +and 620 K over 1 ns to 100 ns. +The production runs used to quantify model uncer- +tainty (Sect. Model uncertainty) and the impact of the +XC functional (Sect. Impact of XC functional and exten- +sion to other halides) were carried out using supercells +comprising 16x16x12 primitive orthorhombic unit cells +(61 440 atoms). +The total simulation time was set to +100 ns and the temperature was varied over a range of +400 to 600 K corresponding to a heating/cooling rate of +4 to 6 K/ns. +Free energy calculations +For CsPbI3 and possibly CsPbBr3 the perovskite +phases are only metastable at lower temperatures. Pro- +vided sufficient kinetic activation, below a certain tem- +perature the perovskite structure transforms into the so- +called δ-phase via a first order transition. +To deter- +mine the transition temperature from the NEP models +we calculated the free energies of the δ and cubic per- +ovskite phases through thermodynamic integration using +the classical method by Frenkel and Ladd [57–59], as im- +plemented in ase [60]. In these calculations, we used an +Einstein solid as reference system, for which the free en- +ergy can be computed analytically, and used supercells +containing about 1500 atoms for each phase. For each +temperature the integration was carried out over 50 ps +and the results were averaged over ten independent runs. +Reference calculations +DFT calculations were performed using the projector +augmented-wave method [61] as implemented in the Vi- +enna ab-initio simulation package [62, 63]. The exchange- +correlation contribution was represented using the vdW- +DF-cx method [31], the SCAN density functional [30], +the PBEsol functional [32], and the PBE functional [33]. +The Brillouin zone was sampled with a Γ-centered grid +with a k-point density of 0.18/˚A and Gaussian smearing +with a width of 0.1 eV. For the calculation of the forces +a finer support grid was employed to improve their nu- +merical accuracy. +ACKNOWLEDGMENTS +This work was funded by the Swedish Research Coun- +cil (grant numbers 2018-06482, 2019-03993, 2020-04935, +2021-05072) and the Chalmers Initiative for Advance- +ment of Neutron and Synchrotron Techniques. 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Furthm¨uller, Computational Materials +Science 6, 15 (1996). + diff --git a/ytE1T4oBgHgl3EQf4AXn/content/tmp_files/load_file.txt b/ytE1T4oBgHgl3EQf4AXn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28af27be54a2e0e3b3434777fb8064318e109968 --- /dev/null +++ b/ytE1T4oBgHgl3EQf4AXn/content/tmp_files/load_file.txt @@ -0,0 +1,858 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf,len=857 +page_content='Phase transitions in inorganic halide perovskites from machine learning potentials Erik Fransson, Julia Wiktor, and Paul Erhart Department of Physics, Chalmers University of Technology, SE-41296, Gothenburg, Sweden ∗ The atomic scale dynamics of halide perovskites have a direct impact not only on their thermal stability but their optoelectronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Progress in machine learned potentials has only recently enabled modeling the finite temperature behavior of these material using fully atomistic methods with near first-principles accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Here, we systematically analyze the impact of heating and cooling rate, simulation size, model uncertainty, and the role of the underlying exchange-correlation functional on the phase behavior of CsPbX3 with X=Cl, Br, and I, including both the perovskite and the δ-phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We show that rates below approximately 30 K/ns and system sizes of at least a few ten thousand atoms are indicated to achieve convergence with regard to these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' By controlling these factors and constructing models that are specific for different exchange-correlation functionals we then show that the semi-local functionals considered in this work (SCAN, vdW- DF-cx, PBEsol, and PBE) systematically underestimate the transition temperatures separating the perovskite phases while overestimating the lattice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Among the considered functionals the vdW-DF-cx functional yields the closest agreement with experiment, followed by SCAN, PBEsol, and PBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Our work provides guidelines for the systematic analysis of dynamics and phase transitions in inorganic halide perovskites and similar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' It also serves as a benchmark for the further development of machine-learned potentials as well as exchange-correlation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' INTRODUCTION Halide perovskites are among the most intensively studied materials of the last decade due to their attrac- tive properties for applications in, for example, solar en- ergy harvesting and lighting [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Similar to their ox- ide counterparts many of these materials exhibit sev- eral different phases that are connected through soft modes and continuous or weak first-order phase transi- tions [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This complex dynamic behavior turns out to be intimately connected to their remarkable optoelec- tronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In this context, electronic structure calculations play a crucial role as they can provide detailed insight into the atomic scale dynamics and microscopic coupling mech- anisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Static calculations can, however, only provide limited information due the strong anharmonicity asso- ciated with the soft modes [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This has motivated a number of dynamic studies based on ab-initio molecular dynamics (MD) simulations [11–18] and, more recently, machine learning potentials (MLPs) [19–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' From such simulations one can then obtain, for example, transition temperatures [19] or structural information at finite tem- peratures [11, 18, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The accuracy of such simulations is, however, limited by several factors, most notably (1) sampling time, (2) system size, (3) the quality of the exchange-correlation (XC) functional, and, in the case of MLPs, (4) the model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Sampling time and system size are in partic- ular a problem for ab-initio MD simulations, which are typically limited to time scales of a few ten picoseconds and system sizes on the order of a 1000 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Previ- ous MLP based studies were able to extend these ranges ∗ erhart@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='se to total run times of a few nanoseconds using about a thousand atoms [19, 20, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Here, we carry out a systematic analysis of the four fac- tors described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We consider CsPbX3 with X=Cl, Br, and I and the strongly constrained and appropriately normed (SCAN) [30], van-der-Waals-density functional with consistent exchange (vdW-DF-cx) [31], PBEsol [32], and PBE [33] XC functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The potential energy sur- face (PES) is mapped using third-generation neuroevo- lution potentials (NEPs) models and sampled using the gpumd package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The latter provides an efficient NEP implementation that enables us to routinely sample sys- tems comprising on the order of 60 000 atoms for 100 or more nanoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We show that well converged results can be achieved using systems containing several ten thousand atoms and heating/cooling rates on the order of 30 K/ns or lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Using bootstrapping and ensembles of models we are able to readily generate accurate NEP models with an un- certainty that is comparable or lower than the training errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' By controlling rate and size effects as well as model errors, we are able to isolate the impact of the under- lying XC functionals and thus to assess quantitatively the quality of different XC functionals for the description of phase transitions and finite temperature properties of halide perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We find the vdW-DF-cx functional to perform the best among the XC functionals considered here when comparing transition temperatures and lattice constants to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In the following section we analyze in order rate and size effects (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Rate and size effects), mode uncertainty (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model uncertainty), the impact of the XC func- tional (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Impact of XC functional and extension to other halides), and finally the transition temperature be- tween the δ and perovskite phases (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Transition to δ-phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We then summarize and discuss the outcome arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='03497v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='mtrl-sci] 9 Jan 2023 2 of this analysis (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' RESULTS Rate and size effects The different perovskite phases are structurally closely related and connected through phase transitions with mixed continuous-first order character [34–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the CsPbX3 (with X=Cl, Br, or I) materials considered in this study the perovskite lattice transforms with increas- ing temperature from an orthorhombic phase (Pnma) via a tetragonal phase (P4/mbm) to a cubic phase (Pm¯3m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Since these transitions do not involve a switch in the sign of the Glazer angles between the orthorhombic (a−a−c+) and tetragonal phases (a0a0c+), unlike, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=', MAPbI3 [19], it is possible to observe these transitions in MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Due to the remaining first-order character and the extreme heating/cooling rates that can be real- ized in MD simulations, one can, however, nonetheless anticipate some degree of hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' A further aggravating factor is the finite system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For small supercells the fluctuations are naturally larger, which renders it more challenging to achieve converged results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In this section, we discriminate the effects of heating/cooling rate and system size by considering in detail MD simulations for CsPbI3 using the full model based on the vdW-DF-cx functional (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Rate effects To separate rate from size effects, we first consider the former in the large size limit, using a supercell compris- ing 61 440 atoms, equivalent to 16x16x12 primitive or- thorhombic (20-atom) unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' On heating all simulations yield the correct (exper- imentally observed) sequence of phases irrespective of heating rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' On cooling, this sequence is reversed again regardless of rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' At the cubic-to-tetragonal transition, for a small number of simulations, one can, however, ob- serve the simultaneous formation of multiple tetragonal domains with incompatible orientations, which can lead to the formation of domain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Since the mov- ing of these boundaries involves a nucleation-and-growth mechanism, they remain in the simulated structure upon cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The temperatures for the transitions between the per- ovskite phases can be readily obtained from the lattice parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 1a,b), revealing a strong dependence on the heating/cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the rates below approx- imately 30 K/ns, the hysteresis between heating and cool- ing runs is 15 K are less and no longer vary systematically with the rate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' By contrast, for the largest rate of 600 K/ns considered here, which is only slightly larger than values used in some earlier MLP studies [19, 26], one observes a hysteresis of 100 K or more for both the 300 400 500 Temperature (K) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='4 Lattice parameters ( ˚A) a) heating 300 400 500 Temperature (K) b) cooling 600 K/ns 60 K/ns 6 K/ns 6 60 600 Rate (K/ns) 250 350 450 550 Transition temperature (K) c) cubic (Pm3m, a0a0a0) tetragonal (P4/mbm, a0a0c+) orthorhombic (Pnma, a−a−c+) heating cooling FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Convergence with heating/cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Lat- tice parameters for CsPbI3 as a function of temperature from MD simulations with varying (a) heating and (b) cooling rates for supercells comprising 61 440 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The transition temperatures extracted from these data are indicated by dia- monds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The gray lines show the raw data whereas the colored lines show the data after application of a Hamming window of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='6 K (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' (c) Transition temperatures as a function of heating/cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' All results were obtained using the full model (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model construction) based on the vdW-DF- cx exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' lower and higher temperature transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In addition, the transition itself is smeared out in temperature, which is particular apparent for the orthorhombic-to-tetragonal transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We observed similar trends also for the other materi- als and models studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We therefore con- clude that rates below approximately 30 K/ns are recom- mended in order to achieve convergence of the transition temperatures for this class of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Size effects Next we examine the impact of supercell size on the temperature dependence of the lattice parameters and the transition temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' First, simulations were carried out at a heating/cooling rate of 6 K/ns using structures comprising 1280 atoms (4x4x4 primitive or- thorhombic unit cells), 7680 atoms (8x8x6), 23 040 atoms (12x12x8) or 61 440 atoms (16x16x12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 3 300 400 500 Temperature (K) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='4 Lattice parameters ( ˚A) a) heating 300 400 500 Temperature (K) b) cooling 1280 atoms 7680 atoms 23040 atoms 61440 atoms 1280 7680 23040 61440 Number of atoms 250 350 450 550 Transition temperature (K) c) cubic (Pm3m, a0a0a0) tetragonal (P4/mbm, a0a0c+) orthorhombic (Pnma, a−a−c+) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Convergence with supercell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Lattice pa- rameters for CsPbI3 as a function of temperature from MD simulations during (a) heating and (b) cooling at a rate of R = 6 K/ns for different supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The transition tempera- tures extracted from these data are indicated by diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' A Hamming window of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='6 K was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' (c) Transition tem- peratures as a function of supercell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' All results were ob- tained using the full model (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model construction) based on the vdW-DF-cx exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the smallest supercell (7680 atoms) one notices a marked deviation from the (reference) lattice parameter parameter data from the largest supercell (61 440 atoms) around the cubic-tetragonal phase transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 2a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This deviation is, however, absent for the next larger structure of 7680 atoms and at this size the transition temperatures are already converged to within 10 K of the results for the largest supercell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' A key characteristic of a phase transition that is at least partly continuous is a peak or kink in the heat ca- pacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The heat capacity can be readily extracted from the fluctuations of the energy in MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the present purpose this is, however, impractical as the transition is very sharp in temperature and the temper- ature range of interest is wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Here, we therefore com- pute the heat capacity instead by numerically differen- tiating the potential energy from heating/cooling runs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=', Cp = dH/dT ≈ ∆H/∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This requires averaging of the data in order to obtain numerically well behaved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' To this end, we first apply a Hamming window of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='6 K to the energy-vs-temperature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The resulting data is numerically differentiated, after which the data 300 400 500 Temperature (K) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 Heat capacity (kB) a) heating 7680 atoms 23040 atoms 61440 atoms 300 400 500 Temperature (K) b) cooling FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Convergence of heat capacity with super- cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Isobaric heat capacity of CsPbI3 as a function of temperature for different supercell sizes obtained through numerical differentiation as detailed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The phase transitions are visible as peaks in the heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' All results were obtained using the full model (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model construction) based on the vdW-DF-cx exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' is smoothened again using a Hamming window of 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The Hamming window sizes are chosen to be sufficiently large to remove noise and small enough to avoid removing features (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the largest system size (61 440 atoms) the phase transitions are clearly visible as peaks in the temperature dependence of the heat capacity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' These features become, however, less distinct with decreasing system size as fluctuations increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The analysis presented in this section suggests that supercells with at least about 10 000 atoms can be ex- pected to yield accurate lattice parameters and transi- tion temperatures within about 10 K of the converged results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Extracting the temperature dependence of the heat capacity requires larger systems still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Even for the largest systems considered here (61 440 atoms) the noise level is rather high, but the data still allows one to ac- curately extract phase transition temperatures from the heat capacity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model uncertainty Having established heating/cooling rates and system sizes that yield converged results for lattice parameters and transition temperatures, we can now address the model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' To this end, we resort to ensem- ble models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The latter comprise five separate models constructed using five distinct 90-10 splits of the train- ing data (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' All simulations in this section were carried out using supercells with 61 440 atoms and a heating/cooling rate of 6 K/ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Once again we use CsPbI3 and models trained using reference data generated by the vdW-DF-cx functional as a representa- tive example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='4 Lattice parameters ( ˚A) a) heating b) cooling 200 400 Temperature (K) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 Heat capacity (kB) c) ensemble average full 200 400 Temperature (K) d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Assessing model uncertainty through en- semble of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' (a,b) Lattice parameters and (c,d) heat capacity as a function of temperature during (a,c) heating and (b,d) cooling from full (blue lines) and ensemble models (orange lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' All results are for CsPbI3 using models based on the vdW-DF-cx exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The uncertainty of the model predictions can be es- timated by the considering the standard deviation over the model ensemble (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 4a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' At 100 K this approach yields an uncertainty of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='02 ˚A depending on di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This value diminishes, however, quickly with temperature to a level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='003 ˚A in the tetragonal and even less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='002 ˚A in the cubic phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Away from the phase transitions the heat capacity curves from the different models agree well with each other (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The actual transition temperatures, cor- responding to the position of the peaks in the heat capac- ity curves, obtained from the full model (and the model ensemble) are 340 K (341(9) K) and 487 K (496(7) K) on heating and 331 K (323(17) K) and 470 K (478(5) K) on cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The agreement between the results obtained us- ing the full model and the model ensemble support the good convergence of the models with respect to training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The remaining hysteresis can be attributed to the mixed first/continuous-order character, which is evident from the very small but non-zero latent heat associated with these transitions [34–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We can thus conservatively estimate the error in the transition temperatures due to model uncertainty to be on the order of 20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In combination with the model performance measures (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' ) and the very good agreement with the density functional theory (DFT) ref- erence data, this provides strong evidence that the NEP models constructed are accurate representations of the DFT potential energy landscape in the regions of config- uration space included here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' They can thus be used to analyze the performance of different XC functionals with CX SCAN PBEsol PBE 200 300 400 500 600 Transition temperature (K) a) CsPbI3 tetra-cubic ortho-tetra CX SCAN b) CsPbBr3 CX SCAN c) CsPbCl3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Transition temperatures from MD simula- tions in comparison with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Data were ob- tained using heating (upward triangles) and cooling rates (downward triangles) of 6 ns for supercells comprising 61 440 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Colored and gray symbols indicate results obtained using full models and model ensembles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In the latter case the uncertainty calculated as the standard devia- tion over the ensemble is indicated by vertical bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Exper- imental transition temperatures taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 9, 38, and 39 are shown by horizontal dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' respect to the finite temperature behavior of CsPbI3 and the other materials considered here (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Impact of XC functional and extension to other halides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Impact of XC functional and extension to other halides We can now apply the framework established above to predict transition temperatures for CsPbI3 using dif- ferent XC functionals for generating reference data, and compare the results to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' To this end, we use the same computational settings in terms of sys- tem size and heating/cooling rate as in the analysis of the model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The results show that all XC functionals consid- ered here systematically underestimate the transition temperatures for both the orthorhombic-tetragonal and tetragonal-cubic transitions by as much as 200 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The agreement improves in the sequence PBE, PBEsol, SCAN, and vdW-DF-cx, which mirrors the trends ob- served previously for transition metals [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Conversely for the lattice parameters one observes a systematic overestimation relative to experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 6) with a similar trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The closest agreement is obtained using the vdW-DF-cx functional, for which the lattice parameters of the cubic phase are still overestimated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='04 ˚A in the high-temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Finally, we extend our investigation to CsPbBr3 and CsPbCl3 including the SCAN and vdW-DF-cx function- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' As for CsPbI3 the transition temperatures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 5b,c) and lattice parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 6b,c) are systematically un- derestimated and overestimated, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The ther- 5 100 200 300 400 500 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 Lattice parameters ( ˚A) a) CsPbI3 a) CsPbI3 a) CsPbI3 a) CsPbI3 experiment PBE PBEsol SCAN vdW-DF-cx 100 200 300 400 500 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 Lattice parameters ( ˚A) b) CsPbBr3 b) CsPbBr3 100 200 300 400 500 Temperature (K) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='7 c) CsPbCl3 c) CsPbCl3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Lattice parameters as a function of tempera- ture from simulation in comparison with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Data were obtained using “full” NEP models trained using different XC functionals in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 9, 38, and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' mal expansion is, however, captured well by all function- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We conclude that while the XC functionals considered here yield the correct sequence of phases, none of them are quantitatively in satisfactory agreement with experi- ment, systematically underestimating the transition tem- peratures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 5) and overestimating the lattice param- eters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Transition to δ-phase It is experimentally well established that the perovskite phases of CsPbI3 are only metastable at low tempera- tures as the actual ground state structure of the material is the so-called δ-phase [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We therefore also computed the transition temperature between the cubic perovskite and the δ-phase by free energy integration (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' S6 for the free energy curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The transition temperature obtained by the vdW-DF- cx model of 635 K with an estimated uncertainty of 20 to 40 K is in rather good agreement with experimental values of around 600 K (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The SCAN and PBEsol models yield lower values that are considerably smaller than the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Temperatures for the transition between the δ- phase and the cubic perovskite phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 42–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' vdW-DF-cx SCAN PBEsol PBE Experiment CsPbI3 635 432 380 310 ∼600 CsPbBr3 310 151 According to DFT (and NEP) calculations the δ-phase is the most stable structure for CsPbBr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The energy difference (Table S3) is, however, much smaller than for CsPbI3 (Table S4), leading to lower transition tempera- tures that are predicted to be 310 K and 151 K according to the vdW-DF-cx and SCAN models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We are unaware of experimental measurements of the tran- sition temperature, which given the present predictions might actually be close to or below room temperature and thus difficult to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For CsPbCl3 the DFT calculations yield energy dif- ferences between perovskite and δ-phases close to zero (Table S2), suggesting that the δ-phase is actually not the most stable phase under any conditions or only at extremely low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' DISCUSSION We have systematically analyzed four key sources of error in atomic scale simulations of phase transitions of inorganic halide perovskites, related respectively to (1) sampling time, (2) system size, (3) model uncertainty, and (4) the underlying XC functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Based on these results, it is recommended to use heating/cooling rates of at most approximately 30 K/ns but preferably even lower and system sizes comprising at least a couple ten thousand atoms, corresponding to a few thousand prim- itive unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We expect that these guidelines are not limited to inorganic halide perovskites but should also be heeded when modeling the dynamics of other perovskites and related systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The model uncertainty was assessed using ensembles of models, from which uncertainty estimates for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=', transition temperatures and lattice parameters can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The results show that by careful model con- struction the model uncertainty can be reduced to a level that allows one to quantitatively discriminate the perfor- mance of different XC functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Using this approach, we found that the semi-local XC functionals considered here systematically underestimate the transition temperatures and overestimate the lat- tice parameters at finite temperature compared to ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The best overall agreement is obtained for the vdW-DF-cx functional, which also outperforms the SCAN functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In pioneering work on the use of MLPs for probing the dynamics of halide perovskite, the latter functional had been suggested as achieving a good match with experiment [19, 26, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Our analysis suggests 6 that the good agreement was likely fortuitous and likely the result of using high rates and small system sizes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' S6 for a more detailed comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This raises the question how, for example, hybrid func- tionals or the random phase approximation would per- form with regard to the finite temperature properties and dynamics of these materials [13, 27, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In the present work we included a relatively large set of struc- tures and supercells that would pose a considerable com- putational challenge for either one of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' As noted above (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model construction), one can, how- ever, expect that the number of training structures can be considerably reduced without a notable loss in model accuracy by using active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This strategy could be combined with principal component analysis to identify regions with very dense sampling [48] or entropy max- imization [49] to reduce the training set size even fur- ther, eventually allowing one to build NEP or other MLP models that can represent such more accurate electronic structure methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Finally, we note that the accuracy of the models pre- sented here in combination with the very high computa- tional efficiency provide by the implementation on graph- ical processing units (GPUs), now enables one to sample the dynamics of these and related materials with unprece- dented time resolution and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' METHODS Machine learned potentials Neuroevolution potentials Here, we use the third generation of the NEP scheme (NEP3) [48] to build MLPs for CsPbX3 with X=Cl, Br, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The NEP format employs a simple multi-layer per- ceptron neural network architecture with a single hidden layer [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In NEP3 the radial part of the atomic environ- ment descriptor is constructed from linear combinations of Chebyshev basis functions while the three-body an- gular part is similarly build from Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Four and five-body terms of the atomic cluster expan- sion form [51] can be included as well but here we limit ourselves to two and three-body terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the present purpose it is crucial that the NEP scheme is not only accurate but has been implemented on GPUs in the gpumd package [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the models de- scribed in the following this allows us to achieve a speed of 2 × 107 atom step/s on an NVidia A100 card, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=', we can simulate a system of 60 000 atoms for about 150 ns per day using a time step of 5 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Computational parameters In this study we used the same hyperparameters for all models, which were chosen based on experience and pre-trials [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The cutoffs for two and three-body inter- actions are 8 ˚A and 4 ˚A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' There are 8 radial and 6 angular descriptor components, 8 basis functions for building both the radial and angular descriptor func- tions, and the angular components are expanded up to fourth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The hidden layer contains 50 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The weights for energies, forces, and virials in the loss function were set to 1, 5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 in gpumd units, respec- tively, while the weights for the ℓ1 and ℓ2 regularization terms were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The neuroevolution strategy [52] used for optimizing the parameters used a population size of 50 and was run for 200 000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model construction To construct NEP models we employed a boot- strapping strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' First we identified potentially rele- vant phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' This included the cubic perovskite structure (Pm¯3m, Glazer notation a0a0a0), two tetragonal struc- tures (I4/mcm → a0a0c−, P4/mbm → a0a0c+), repre- senting out-of-phase and in-phase tilts relative to the c- axis, respectively, one orthorhombic structure (Pnma → a−a−c+) as well as the so-called delta-phase (Pnma), which is experimentally known to be the most stable structure at least for CsPbI3 and CsPbBr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We then calculated energy-volume curves for these five prototype structures using DFT calculations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Ref- erence calculations) allowing both the ionic coordinates and the cell shape to relax under the constraint of con- stant volume until the maximum force on any atom fell below 30 meV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Subsequently we generated supercells for each proto- type with random atomic displacements using the Monte Carlo rattle procedure from the hiphive package [53] with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='04 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The supercell size was chosen to be between 120 and 160 atoms and the vol- ume was varied between 85% and 110% of the respective equilibrium volume with five structures per volume and prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Using these data we generated a first iteration of NEP models using the gpumd package for the optimization [48] and the calorine package for data preparation and analysis [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' One model was generated using the full data set (“full model”) and five additional models (“model ensemble”) were generated by using five differ- ent 90-10 splits of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Using the full model we generated new structures for each prototype by running short MD simulations at pressures between −1 and 10 GPa using a temperature ramp from 20 to 620 K over 3 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' From each trajectory we selected 12 configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For each of these configurations we then computed the standard deviation of the energy and forces using the model ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The standard deviation over the ensemble predictions provided a measure for the uncer- tainty of the current model generation for the respective conditions (temperature, pressure, structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We then computed energy and forces for the new structures using 7 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' RMSE scores for the final NEP models ob- tained by training against the full data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Additional per- formance measures including RMSE and Pearson correlation coefficients for model ensembles can be found in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Energy Force Virial meV/atom meV/˚A meV/atom CsPbCl3 vdW-DF-cx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 SCAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 CsPbBr3 vdW-DF-cx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 SCAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 CsPbI3 vdW-DF-cx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='8 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5 SCAN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 PBEsol 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='2 PBE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='6 DFT calculations, added these to the training set and repeated the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Typically after four generations we found that the uncertainty in the energy and forces was comparable are smaller than the respective training error indicating convergence of the model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' We note that in principle one could have adapted an active learning strategy based on the model ensemble and only included configurations with high uncertainty as ad- ditional reference structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Here, we decided to include rather more data in the training set but we expect that the number of structures can be reduced considerably without a notable decrease in model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The final models yield RMSE scores for training and validation sets of about 2 meV/atom, 50 meV/˚A, and 15 meV/atom or better for energies, forces, and viri- als, respectively (Table II and Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Importantly the models closely reproduce the energy differences and energy-volume curves of all the structures of interest in the present study (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Table S2 to Table S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The final models were subsequently used in large scale MD simulations to predict, for example, transition tem- peratures or lattice parameters (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' MD simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' MD simulations All MD simulations were carried out using the gpumd code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Temperature and pressure were controlled using stochastic velocity [55] and cell rescaling [56] and the time step was 5 ps, where all simulations were run at zero pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For studying the convergence with size (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Size effects), we considered system sizes between 1280 and 61 440 atoms, equivalent to 4x4x4 to 16x16x12 primitive orthorhombic perovskite (20-atom) unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' To analyze the impact of heating and cooling rates (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Rate ef- fects) the temperature was linearly varied between 20 K and 620 K over 1 ns to 100 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The production runs used to quantify model uncer- tainty (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Model uncertainty) and the impact of the XC functional (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Impact of XC functional and exten- sion to other halides) were carried out using supercells comprising 16x16x12 primitive orthorhombic unit cells (61 440 atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The total simulation time was set to 100 ns and the temperature was varied over a range of 400 to 600 K corresponding to a heating/cooling rate of 4 to 6 K/ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Free energy calculations For CsPbI3 and possibly CsPbBr3 the perovskite phases are only metastable at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Pro- vided sufficient kinetic activation, below a certain tem- perature the perovskite structure transforms into the so- called δ-phase via a first order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' To deter- mine the transition temperature from the NEP models we calculated the free energies of the δ and cubic per- ovskite phases through thermodynamic integration using the classical method by Frenkel and Ladd [57–59], as im- plemented in ase [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' In these calculations, we used an Einstein solid as reference system, for which the free en- ergy can be computed analytically, and used supercells containing about 1500 atoms for each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For each temperature the integration was carried out over 50 ps and the results were averaged over ten independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Reference calculations DFT calculations were performed using the projector augmented-wave method [61] as implemented in the Vi- enna ab-initio simulation package [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The exchange- correlation contribution was represented using the vdW- DF-cx method [31], the SCAN density functional [30], the PBEsol functional [32], and the PBE functional [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The Brillouin zone was sampled with a Γ-centered grid with a k-point density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='18/˚A and Gaussian smearing with a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' For the calculation of the forces a finer support grid was employed to improve their nu- merical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was funded by the Swedish Research Coun- cil (grant numbers 2018-06482, 2019-03993, 2020-04935, 2021-05072) and the Chalmers Initiative for Advance- ment of Neutron and Synchrotron Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' The com- putations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC, C3SE, and PDC partially funded by the Swedish Research Council (grant number 2018-05973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' 8 DATA AVAILABILITY The DFT data and NEP models generated in this study are openly available via Zenodo at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content='7454224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' COMPETING INTERESTS The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS All authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf'} +page_content=' Kojima, K.' metadata={'source': 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b/ytE1T4oBgHgl3EQfQwPe/content/tmp_files/2301.03045v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b05a65e203363fc8cd2a34f1fa414478de1214d --- /dev/null +++ b/ytE1T4oBgHgl3EQfQwPe/content/tmp_files/2301.03045v1.pdf.txt @@ -0,0 +1,19344 @@ +Seamless Multimodal Biometrics for +Continuous Personalised Wellbeing +Monitoring +João Tiago Ribeiro Pinto +Doctoral Programme in Electrical and Computer Engineering +Supervisor: Professor Jaime dos Santos Cardoso +Co-Supervisor: Professor Miguel Velhote Correia +December 21, 2022 +arXiv:2301.03045v1 [cs.CV] 8 Jan 2023 + +© João Tiago Ribeiro Pinto, 2022 + +Seamless Multimodal Biometrics for Continuous +Personalised Wellbeing Monitoring +João Tiago Ribeiro Pinto +Doctoral Programme in Electrical and Computer Engineering +Approved in public examination by the Jury: +President: +Professor Luís Miguel Pinho de Almeida +Referee: +Professor Hugo Filipe Silveira Gamboa +Referee: +Professor Armando José Formoso de Pinho +Referee: +Professor Jaime dos Santos Cardoso +Referee: +Professor Ana Maria Rodrigues de Sousa Faria de Mendonça +Referee: +Professor Luís Filipe Pinto de Almeida Teixeira +December 21, 2022 + + +Resumo +A perceção através da inteligência artificial está cada vez mais presente nas nossas vidas. Os +veículos não são exceção, uma vez que sistemas avançados de assistência ao condutor auxiliam +no cumprimento de limites de velocidade, na manutenção dentro das faixas e na prevenção de +acidentes. Num futuro próximo, o reconhecimento de padrões terá um paper ainda mais prepon- +derante nos veículos, uma vez que o automóvel autónomo necessitará de meios automáticos para +compreender o que acontece ao seu redor (e no seu interior) para agir de forma adequada. +Em reconhecimento de padrões, a biometria oferece aplicações promissoras para veículos, do +acesso keyless à personalização automática de opções de condução com base no condutor recon- +hecido. De igual modo, as tecnologias de reconhecimento de bem-estar têm atraído atenção pela +possibilidade de reconhecer atividade, emoções, sonolência ou stress em condutores e passageiros. +No entanto, estes dois tópicos são diametralmente opostos, uma vez que o reconhecimento de bem- +estar usa a variabilidade intra-sujeito, enquanto a biometria se baseia na variabilidade inter-sujeito. +Apesar das diferenças, a biometria e o reconhecimento de bem-estar poderiam (e deveriam) +co-existir. O reconhecimento contínuo de identidade em dados adquiridos de forma impercetível +poderiam ser usados para personalizar modelos de reconhecimento de bem-estar e obter melhor +desempenho. Estes modelos personalizados poderiam ser a chave para meios mais robustos de +monitorizar sonolência e atenção em condutores e evitar acidentes. Num sentido mais amplo, +estes poderiam ser aplicados a todos os ocupantes, abrindo o caminho em direção ao reconheci- +mento eficaz de atividade, emoções, conforto e até episódios de violência em veículos autónomos +partilhados. +Este doutoramento focou-se em avançar o tópico de perceção em veículos através do es- +tudo de novas metodologias de visão computacional e reconhecimento de padrões para biome- +tria e reconhecimento de bem-estar. O foco principal foi na biometria com electrocardiograma +(ECG), um traço reconhecido pelo seu potencial em monitorização impercetível de condutores. +Esforços foram dedicados à obtenção de desempenho melhorado em identificação e verificação +de identidade em cenários off-the-person, reconhecidos pelo elevado teor de ruído e variabilidade. +Aqui, foram propostas soluções deep learning end-to-end e analisados tópicos importantes como +o desempenho cross-database e a longo prazo, a importância relativa das ondas através da inter- +pretabilidade, e a conversão entre canais. +A biometria com face, um complemento natural ao ECG em cenários impercetíveis, foi tam- +bém estudada nesta tese. Os desafios em reconhecimento de faces com máscaras e na interpretabil- +idade em biometria foram abordados com o intuito de avançar para algoritmos mais transparentes, +confiáveis e robustos a oclusões significativas. Dentro do tópico de reconhecimento de bem-estar, +foram propostas soluções melhoradas para o reconhecimento multimodal de emoções em grupos +de pessoas e de atividade/violência dentro de veículos partilhados. Por fim, foram propostos ainda +uma forma inovadora de aprender segurança de templates em modelos end-to-end, evitando pro- +cessos adicionais de encriptação, e um método auto-supervisionado adaptado a dados sequenciais, +para garantir segurança de dados e desempenho otimizado. +i + +ii +Segundo os resultados deste trabalho, é possível concluir que o ideal de reconhecimento per- +sonalizado de bem-estar está ainda por atingir. No entanto, este trabalho construiu uma base sólida +para suportar trabalho futuro em direção à integração da biometria com o reconhecimento de bem- +estar de forma multimodal, impercetível, contínua e realista. Em geral, este doutoramento levou +a múltiplas contribuições para os tópicos de biometria e reconhecimento de bem-estar, resultando +diretamente em vinte e quatro publicações científicas em fóruns de renome em biometria e recon- +hecimento de padrões. A sua qualidade e impacto foram reconhecidas pela comunidade científica +com mais de trezentas citações e múltiplos prémios, incluindo o prémio EAB Max Snijder 2022. +Palavras-chave: Aprendizagem Computacional; Atividade; Áudio; Biometria; Electrocardio- +grama; Emoção; Face; Monitorização de Bem-Estar; Reconhecimento de Padrões; Processamento +de Sinal; Veículos Autónomos; Vídeo; Visão Computacional. + +Abstract +Artificially intelligent perception is increasingly present in the lives of every one of us. Vehicles +are no exception, as advanced driver assistance systems (ADAS) help us comply with speed limits, +keep within the lanes, and avoid accidents. In the near future, pattern recognition will have an even +stronger role in vehicles, as self-driving cars will require automated ways to understand what is +happening around (and within) them and act accordingly. +Within pattern recognition, biometrics offer promising applications in vehicles, from keyless +access control to the automatic personalisation of driving and environmental conditions based on +the recognised driver. Similarly, wellbeing monitoring technologies have long attracted attention +to the possibility of recognising activity, emotions, sleepiness, or stress from drivers and pas- +sengers. However, these two topics are starkly opposed, since wellbeing recognition relies on +intrasubject variability while biometrics thrives on intersubject variability. +Despite their differences, biometric recognition and wellbeing monitoring could (and should) +coexist. Continuous identity recognition from seamlessly acquired data could be used to person- +alise wellbeing monitoring models and attain improved performance. These personalised models +could be the key to more robust ways of monitoring drivers’ drowsiness and attention and avoid- +ing accidents. In a broader sense, they could be applied to all vehicle occupants, paving the way +towards the accurate recognition of activity, emotions, comfort, and even violence episodes in +shared autonomous vehicles. +This doctoral work focused on advancing in-vehicle sensing through the research of novel +computer vision and pattern recognition methodologies for both biometrics and wellbeing moni- +toring. The main focus has been on electrocardiogram (ECG) biometrics, a trait well-known for +its potential for seamless driver monitoring. Major efforts were devoted to achieving improved +performance in identification and identity verification in off-the-person scenarios, well-known for +increased noise and variability. Here, end-to-end deep learning ECG biometric solutions were +proposed and important topics were addressed such as cross-database and long-term performance, +waveform relevance through explainability, and interlead conversion. +Face biometrics, a natural complement to the ECG in seamless unconstrained scenarios, was +also studied in this work. The open challenges of masked face recognition and interpretability in +biometrics were tackled in an effort to evolve towards algorithms that are more transparent, trust- +worthy, and robust to significant occlusions. Within the topic of wellbeing monitoring, improved +solutions to multimodal emotion recognition in groups of people and activity/violence recognition +in in-vehicle scenarios were proposed. At last, we also proposed a novel way to learn template +security within end-to-end models, dismissing additional separate encryption processes, and a +self-supervised learning approach tailored to sequential data, in order to ensure data security and +optimal performance. +Following the results of this work, one can conclude that truly personalised wellbeing is yet +to be achieved. However, this work has built a strong framework to support future work towards +the goal of integrating biometric recognition and wellbeing monitoring in a multimodal, seamless, +iii + +iv +continuous, and realistic way. Overall, this doctoral work led to numerous contributions to biomet- +rics and wellbeing monitoring in general, resulting directly in twenty-four scientific publications +in major biometrics and pattern recognition venues. Its quality and impact have been recognised +by the scientific community with over three hundred citations and multiple awards, including the +EAB Max Snijder Award 2022. +Keywords: Activity; Audio; Autonomous Vehicles; Biometrics; Computer Vision; Electrocardio- +gram; Emotion; Face; Machine Learning; Pattern Recognition; Signal Processing; Video; Wellbe- +ing Monitoring. + +Acknowledgements +Doutor. Finalmente. Volvidos cinco anos desde o início do meu percurso em investigação, a es- +crita deste documento dá-me a oportunidade de reviver os sucessos, fracassos, ideias, desafios, +pessoas e momentos que marcaram este meu doutoramento. Foi uma excelente aventura, mas +não foi fácil. As vozes do impostor syndrome fizeram-me duvidar se realmente teria a inteligên- +cia e a capacidade suficientes para completar um doutoramento. Foi inevitável, por vezes, tentar +comparar com os doutoramentos de outros, o que frequentemente resultou em desilusão. Mas +fui aprendendo a evitar os obstáculos no caminho e a compreender os avisos de quem já passara +por eles. Diziam que um doutoramento não é um sprint, mas sim uma maratona. Que não re- +quer capacidades extraordinárias, mas sim resiliência, motivação e determinação. Duvidei, mas +depois deste longo percurso reconheço a veracidade dessa afirmação. Podemos duvidar da nossa +inteligência e capacidades, mas o importante é continuar, e insistir, e procurar até encontrar a meta. +Com tempo percebi que cada doutoramento é único, e cada um tem o seu caminho a traçar, com +desafios e dificuldades específicos. A comparação com os doutoramentos dos outros, apesar de +inevitável, será sempre incompleta e injusta. Aprendi ainda que um doutoramento completo deverá +ir bem para além da investigação científica. Não devemos ser apenas “máquinas de fazer artigos”, +mas sim procurar explorar todas as outras vertentes de um investigador completo, como o ensino, +a mentoria, as colaborações, e a organização de eventos científicos. É natural que rapidamente +esgotemos as horas do dia (e a nossa energia) quando nos dividimos entre tantas atividades. Foram +frequentes as longas noites de trabalho e os fins-de-semana que não o foram. Mas considero que +encontrei o verdadeiro caminho para um bom doutoramento, e fico feliz por ter decidido segui-lo. +Foi perfeito? Não. Em retrospetiva, reconheço escolhas menos certas, certamente frutos da +minha inexperiência, e imensos caminhos alternativos que teriam sido mais proveitosos. Relembro +com pesar as ideias que faziam sentido mas não funcionaram. E imagino o potencial de tantas +outras que não foram além de um rabisco esquecido numa folha qualquer. Mas não estou só. +Ainda não conheci um único aluno de doutoramento que tenha chegado ao fim do seu trajeto +plenamente contente e verdadeiramente confiante no resultado. Resta a ténue calma na ideia de +que esta tese de doutoramento não é a minha “obra-prima” nem aquilo que irá definir toda a +minha carreira. Mas sim, apenas, um livro de esboços, uma coleção de rascunhos de alguém que +tropeçou, escorregou e errou inúmeras vezes durante quase cinco anos na tentativa de mapear um +caminho inexplorado e, com sorte, tornar-se um investigador completo e autónomo. +Mudaria alguma coisa? Também não. Suponho que teria conseguido publicar mais artigos se +não tivesse dividido a minha atenção entre tantas atividades paralelas. Talvez tivesse um trabalho +com mais impacto se não tivesse orientado tantas teses de mestrado e estágios. Deveria talvez ter +dado prioridade à importante experiência de dar aulas. Mas tudo isto é incerto. Certos são alguns +momentos marcantes que deram um brilho especial a estes cinco anos de esforço e dedicação. +Relembro o convívio e a partilha de conhecimento no BTAS, no IbPRIA, no BIOSIG, nos vários +RECPADs e em tantas outras conferências. O prazer de poder contribuir para a organização do +IWBF 2020 e dos workshops xAI4Biometrics. O orgulho em ver o meu esforço reconhecido pela +v + +vi +comunidade europeia de biometria através do prémio Max Snijder. O entusiasmo ao ver (e poder +demostrar a tantas pessoas) o meu trabalho a funcionar, ao vivo, no fim do projeto Easy Ride. A +felicidade ao assistir a cada uma das provas públicas dos meus alunos de mestrado, concluindo com +sucesso as suas teses, depois do privilégio de os acompanhar ao longo de um ano de aprendizagem, +dedicação, esforço e evolução. A honra enorme de ser o escolhido da Inês, da Sofia e do Duarte +para os cartolar na Imposição das Insígnias, um momento tão solene e significante a marcar o fim +dos seus caminhos pela academia. A satisfação dos últimos dias de cada VISUM, onde depois de +tanto trabalho nos damos conta do imenso valor da nossa escola de verão (e ainda a minha estranha +proficiência na apresentação de quizzes). E ainda, a amizade e entreajuda que encontrei todos os +dias no grupo VCMI. Ao relembrar as pessoas encontradas e estes momentos vividos, chego à +conclusão que não mudaria absolutamente nada, no receio de que perdesse sequer um deles. E +afinal, talvez o meu doutoramento não esteja assim tão longe da perfeição. +Por tudo isto tenho a agradecer, em primeiro lugar, ao Professor Jaime Cardoso. O melhor +professor que tive a sorte de conhecer durante o meu percurso académico. Ao aproximar-se o fim +do meu mestrado, muitas foram as dúvidas relativamente à possibilidade de um doutoramento. Há +muito que era algo que almejava fazer, mas a magnitude da tarefa impunha respeito. Recebi muitas +e variadas opiniões sobre o doutoramento e como fazer um com qualidade. No meio de expectável +discórdia, um ponto de consenso: “o orientador é, de longe, o mais importante”. Mais vale o +orientador certo numa universidade qualquer que o orientador errado na Ivy League, diziam em +uníssono. Estavam certos, todos eles, e sem dúvida estava certo eu também quando escolhi fazer +o doutoramento consigo, Professor. Um verdadeiro exemplo de integridade, profunda dedicação, +impressionante inteligência e contagiante amor pelo que faz. Apesar das diferenças em experiência +e currículo, tão comumente enfatizadas na academia, sempre me deixou à vontade para expressar +livremente todas as ideias, dúvidas e problemas como se fosse um simples colega. E tudo isto +vale o mundo para um aluno de doutoramento a percorrer a sua maratona. Recordo-me de certos +momentos, como quando chegou a Professor Catedrático, em que me senti tão alegre como se +tivesse sido eu próprio a conseguir essa conquista. E sei que não fui o único. Deixar tal marca +nos alunos é prova definitiva da sua capacidade, dedicação e entrega como Professor e orientador. +Desejo apenas que todos os seus objetivos se continuem a cumprir, com sucesso e felicidade, e +que eu possa continuar a colaborar consigo e a tê-lo como meu mentor. +Ao Professor Miguel Velhote Correia, por ter aceitado o desafio de fazer parte deste projeto. +Apesar dos nossos planos ambiciosos para este doutoramento, acabei por não ter muitas oportu- +nidades para colaborar consigo. No entanto, fico feliz por ter conseguido que deixasse a sua marca +naquele que considero ser o meu melhor trabalho durante este doutoramento. Continuo a acreditar +no valor da investigação em instrumentação para o futuro da biometria com electrocardiograma, e +espero um dia voltar a poder contar com o seu conhecimento e a sua experiência neste e em outros +tópicos. +Ao grupo VCMI e aos que dele fizeram parte e contribuíram para a sua história, plena de +inúmeros sucessos. Tal como ninguém vive numa bolha, também ninguém faz um doutoramento +sozinho (apesar de ser uma aventura bastante solitária). Recordo a afirmação atribuída a Isaac +Newton - “if I have seen further, it is by standing on the shoulders of giants”. Os que por aqui +passaram antes de mim abriram-me e mostraram-me o caminho com os seus sucessos, e aqueles +com os quais tive o prazer de percorrer esta jornada ajudaram-me a encontrar novas oportunidades +e a aprender com diversos desafios em biometria e tantos outros tópicos. Não teria certamente +conseguido metade do que consegui neste doutoramento se não tivesse como alicerce este grupo +de verdadeiros gigantes, repleto de genialidade, capacidade e dedicação. Entre eles, um agradeci- +mento especial ao Professor Jaime pela criação e constante dedicação ao grupo e à Filipa Sequeira +pelos excelentes esforços recentes para aumentar a colaboração do subgrupo de biometria com + +vii +a comunidade internacional. Obrigado por fazerem do grupo VCMI um símbolo de qualidade e +excelência em biometria além-fronteiras, e por encherem o meu doutoramento de desafios, opor- +tunidades, sucessos e ainda muitos momentos de felicidade. Sem dúvida, não podia ter escolhido +um melhor grupo para o meu doutoramento. +À VISUM, indubitavelmente a melhor escola de verão por esse mundo fora. Obrigado pela +oportunidade de organizar estes momentos de confluência entre culturas e aprendizagem, com tal +magnitude e visibilidade internacional. Agradeço a todos os que deram um pouco de si para tornar +a VISUM possível (e excelente), em especial à Ana, à Sara, e ao Wilson, que sacrificaram bem +mais que todos os restantes e puxaram a nossa escola de verão para uma 10ª edição que ficará para +a história. Cada edição foi única, mas foram todas fantásticas, e fico feliz de ter tido a oportunidade +de fazer parte de quatro delas. +A todos os que confiaram em mim para co-orientar as suas tese de mestrado. Ao Gabriel, à +Carolina, ao Leonardo, ao Arthur, ao João, à Inês, à Sofia, à Telma, à Mariana, ao Duarte, ao +Guilherme, ao Vítor e ao Erfan. Cada um aceitou um tema único, com objetivos e dificuldades +diferentes, e cada um o abordou com perspetivas e ideias diversas. Mas todos estiveram em sin- +tonia na vontade de aprender, na dedicação à procura do caminho certo, e no ânimo firme mesmo +em momentos de maior dificuldade. Sei que fui excecionalmente sortudo em ter tantos e tão ex- +celentes alunos durante estes anos de doutoramento, e adorei trabalhar com cada um de vocês. +Espero que tenham também gostado de trabalhar comigo (ou que pelo menos não se arrependam +da vossa escolha) e que eu tenha estado à altura das vossas expectativas. Espero ainda que tenham +aprendido algo comigo, tal como eu aprendi com cada um de vocês. Obrigado pelas vossas exce- +lentes contribuições para este projeto, que na verdade também é um pouco vosso, e lembrem-se +que continuarei aqui para vos ajudar, sempre que precisarem. +A todos aqueles que colaboraram comigo durante este doutoramento, tanto em temas rela- +cionados com biometria como naqueles que me permitiram alargar os meus horizontes e adquirir +conhecimentos em tópicos diferentes, incluindo os projetos AUTOMOTIVE, Easy Ride e Au- +rora. A todos na CardioID Technologies, em especial ao André, ao Carlos, ao Roberto, ao Pedro +e ao Lourenço, pela ajuda em biometria com ECG e no projeto AUTOMOTIVE. À Bosch Car +Multimedia, em especial ao Joaquim, ao Filipe, à Carolina, ao Ricardo, à Margarida, ao Niklas +e ao Jochen pela excelente colaboração em in-vehicle monitoring no projeto Easy Ride. Aos da +Fraunhofer IGD, em especial ao Fadi, ao Naser, ao Florian, ao Marco, ao Juan e à Meiling pela +colaboração em masked face recognition e ainda pela calorosa receção em Darmstadt. Entre to- +dos, um agradecimento especial ao André Lourenço pela colaboração, apoio e amizade já desde a +minha tese de mestrado. Espero um dia poder voltar a colaborar com todos vós. +E por fim, mas certamente não em último lugar, à minha família e aos meus amigos, por tudo +o resto que fez de mim o que sou hoje e que me ajudou a chegar aqui. +Obrigado a todos, +João Tiago + +viii +Funding +This work was financed by the Portuguese science and technology foundation, Fundação para a +Ciência e a Tecnologia – FCT – and co-financed by the European Social Fund through the North +Regional Operational Programme (NORTE 2020), under the grant “SFRH/BD/137720/2018”. +Data +The author wishes to thank the creators, contributors, and administrators of all databases and data +collections used in the research work presented in this thesis. +ECG signal databases and collections: the author acknowledges the creators of the PTB [49] and +PTB-XL [443; 444] databases at the Physikalisch-Technische Bundesanstalt, Germany, the cre- +ators of the University of Toronto ECG Database (UofTDB) [445] at the University of Toronto, +Canada, the creators of the Check Your Biosignals Here initiative [394] at Instituto de Teleco- +municações, Portugal, the creators of the INCART database at the St. Petersburg Institute of +Cardiological Technics, Russia, as well as the creators and administrators of the Physionet online +data repository [146]. E-HOL-03-0202-003 (E-HOL) Data used for this research was provided by +the Telemetric and Holter ECG Warehouse of the University of Rochester (THEW), NY. +Face biometric databases and collections: the author wishes to acknowledge the creators of the +YouTube Faces [460] database at Tel Aviv University, Israel, the creators of the ROSE Youtu [249] +database at the Nanyang Technological University, Singapore, the creators of the MFRC-21 [50] +database at Fraunhofer IGD, Germany, the creators of the VGGFace2 [58] at the University of +Oxford, UK, the creators of the MS1MV2 [98] dataset at Imperial College London, UK, and the +creators of the Labelled Faces in the Wild (LFW) [178] dataset at the University of Massachusetts, +USA. +Other databases: the author wishes to acknowledge the EmotiW 2020 Grand Challenge [101] +organisers, and the authors of the Multi-Moments in Time (MMIT) [300] dataset and pretrained +models at the Massachusetts Institute of Technology, USA. +Credits +All previously published copyrighted content reused, reprinted, or adapted in this thesis is appro- +priately referenced and acknowledged and has been licensed as detailed below: +Content in Chapter 2: Adapted from João Ribeiro Pinto, “Continuous Biometric Identification +on the Steering Wheel”, M.Sc. Thesis, University of Porto, Portugal, 2017. Figure 2.12 was +reprinted from Cognition, vol. 121, Rob Jenkins, David White, Xandra Van Montfort, A. Mike +Burton, “Variability in photos of the same face”, pp. 313–323, Copyright 2011, with permission +from Elsevier. +Content in Chapter 3 and Appendix A: © 2018 IEEE. Reprinted, with permission, from João +Ribeiro Pinto, Jaime S. Cardoso, André Lourenço, “Evolution, Current Challenges, and Future +Possibilities in ECG Biometrics”, IEEE Access, June 2018. + +Fundagao +FCT +REPUBLICA +ORTUGAI +UNIAO EUROPEIA +NORTE2O2O +**** +CENSIN TEUNRIORIA +para a Ciencia +PORTUGUESA +e a Tecnologia +***** +ROGRAMA OPERACIONAL REGIONAL DO NORTix +Content in Chapter 4 and Figure 3.2: Reproduced with permission of Taylor and Francis Group +LLC (Books) US through PLSclear. +Content in Chapter 5: © 2019 IEEE. Reprinted, with permission, from João Ribeiro Pinto, Jaime S. +Cardoso, “An End-to-End Convolutional Neural Network for ECG-Based Biometric Authentica- +tion”, 2019 IEEE 10th International Conference on Biometrics Theory, Applications and Systems +(BTAS), September 2019. +Content in Chapter 6: Reprinted/adapted by permission from Springer Nature: Springer Na- +ture “Don’t You Forget About Me: A Study on Long-Term Performance in ECG Biometrics” +by Gabriel Lopes, João Ribeiro Pinto, Jaime S. Cardoso © 2019. +Content in Chapter 9: Figures 9.3 and 9.4 were reprinted from Neurocomputing, vol. 429, Mei +Wang, Weihong Deng, “Deep face recognition: A survey”, pp. 30, Copyright 2021, with permis- +sion from Elsevier. +Content in Chapter 10: © 2021 IEEE. Reprinted, with permission, from Pedro C. Neto, Fadi +Boutros, João Ribeiro Pinto, Mohsen Saffari, Naser Damer, Ana F. Sequeira, Jaime S. Cardoso, +“My Eyes Are Up Here: Promoting Focus on Uncovered Regions in Masked Face Recognition”, +2021 International Conference of the Biometrics Special Interest Group (BIOSIG), September +2021. +Content in Chapter 12: © 2020 IEEE. Reprinted, with permission, from João Ribeiro Pinto, Tiago +Gonçalves, Carolina Pinto, Luís Sanhudo, Joaquim Fonseca, Filipe Gonçalves, Pedro Carvalho, +Jaime S. Cardoso, “Audiovisual Classification of Group Emotion Valence Using Activity Recogni- +tion Networks”, 2020 IEEE 4th International Conference on Image Processing, Applications and +Systems (IPAS), December 2020. +Content in Chapter 14: © 2021 IEEE. Reprinted, with permission, from João Ribeiro Pinto, +Miguel V. Correia, Jaime S. Cardoso, “Secure Triplet Loss: Achieving Cancelability and Non- +Linkability in End-to-End Deep Biometrics”, IEEE Transactions on Biometrics, Behavior, and +Identity Science, April 2021. +Content in Chapter 15: © 2020 IEEE. Reprinted, with permission, from João Ribeiro Pinto, Jaime +S. Cardoso, “Self-Learning with Stochastic Triplet Loss”, 2020 International Joint Conference on +Neural Networks (IJCNN), July 2020. + + +“In theory, there is no difference between theory and practice, +while in practice, there is.” +Benjamin Brewster +xi + + +Contents +I +Prologue +1 +1 +Introduction +3 +1.1 +Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.2 +Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.3 +Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +1.4 +Dissemination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +1.5 +Collaborations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +1.5.1 +Research projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +1.5.2 +Synergies within the research group . . . . . . . . . . . . . . . . . . . . +14 +1.5.3 +Organisation of scientific events . . . . . . . . . . . . . . . . . . . . . . +14 +1.5.4 +Supervision of dissertations and internships . . . . . . . . . . . . . . . . +15 +1.6 +Awards and Distinctions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +1.7 +Document Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +2 +Fundamental Concepts +19 +2.1 +Biometric Systems +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.1.1 +General structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.1.2 +Operation modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +2.1.3 +Security and privacy concerns . . . . . . . . . . . . . . . . . . . . . . . +22 +2.2 +Biometric Traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +2.2.1 +General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +2.2.2 +Electrocardiogram +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +2.2.3 +Face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +2.3 +Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +2.3.1 +System design considerations +. . . . . . . . . . . . . . . . . . . . . . . +36 +2.3.2 +Recognition accuracy measurement . . . . . . . . . . . . . . . . . . . . +37 +2.3.3 +Time-based performance measurement +. . . . . . . . . . . . . . . . . . +40 +2.3.4 +Wellbeing monitoring performance measurement . . . . . . . . . . . . . +42 +II +Electrocardiogram Biometrics +45 +3 +Prior Art in Electrocardiogram Biometrics +47 +3.1 +Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +3.1.1 +Building a complete ECG data collection +. . . . . . . . . . . . . . . . . +47 +3.1.2 +Publicly available data . . . . . . . . . . . . . . . . . . . . . . . . . . . +48 +3.2 +Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +3.2.1 +Signal denoising +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +3.2.2 +Signal preparation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +xiii + +xiv +CONTENTS +3.2.3 +Feature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +3.2.4 +Decision +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +3.2.5 +Deep learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +3.3 +Open Challenges and Opportunities +. . . . . . . . . . . . . . . . . . . . . . . . +66 +4 +End-to-End Models and Augmentation Strategies for Identification +69 +4.1 +Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +4.2 +Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +70 +4.2.1 +Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +70 +4.2.2 +Data augmentation strategies . . . . . . . . . . . . . . . . . . . . . . . . +71 +4.3 +Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +4.4 +Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +4.5 +Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +77 +5 +Triplet Loss and Transfer Learning for Identity Verification +79 +5.1 +Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +79 +5.2 +Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +80 +5.2.1 +Model architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +80 +5.2.2 +Model training +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +81 +5.3 +Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +82 +5.3.1 +Data and reference methods +. . . . . . . . . . . . . . . . . . . . . . . . +82 +5.3.2 +Evaluation scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . +83 +5.4 +Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +83 +5.4.1 +Single-database scenario . . . . . . . . . . . . . . . . . . . . . . . . . . +83 +5.4.2 +Varying identity set size scenario . . . . . . . . . . . . . . . . . . . . . . +85 +5.4.3 +Cross-database scenario +. . . . . . . . . . . . . . . . . . . . . . . . . . +88 +5.4.4 +Fine-tuning scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . +88 +5.5 +Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +88 +6 +Long-Term Performance and Template Update +91 +6.1 +Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +91 +6.2 +Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +92 +6.3 +Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +94 +6.3.1 +Biometric identification methods . . . . . . . . . . . . . . . . . . . . . . +94 +6.3.2 +Template update methods . . . . . . . . . . . . . . . . . . . . . . . . . . +95 +6.4 +Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +96 +6.4.1 +Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +96 +6.4.2 +Experiments +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +96 +6.5 +Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +97 +6.5.1 +Handcrafted methodologies +. . . . . . . . . . . . . . . . . . . . . . . . +97 +6.5.2 +Deep convolutional network . . . . . . . . . . . . . . . . . . . . . . . . +100 +6.6 +Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +101 +7 +Leveraging Explainability to Understand ECG Biometrics +103 +7.1 +Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +103 +7.2 +The Electrocardiogram as a Biometric Trait . . . . . . . . . . . . . . . . . . . . +104 +7.3 +Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +105 +7.3.1 +Biometric identification model . . . . . . . . . . . . . . . . . . . . . . . +105 +7.3.2 +Interpretability tools +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +105 + +CONTENTS +xv +7.3.3 +Visualisation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +107 +7.4 +Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +107 +7.5 +Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +107 +7.6 +Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +111 +8 +Interlead Conversion of Electrocardiographic Signals +113 +8.1 +Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +113 +8.2 +Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +114 +8.3 +Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +115 +8.3.1 +General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +115 +8.3.2 +Model architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +116 +8.3.3 +Shared vs. individual encoders . . . . . . . . . . . . . . . . . . . . . . . +117 +8.4 +Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +118 +8.4.1 +Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +118 +8.4.2 +Model training and evaluation . . . . . . . . . . . . . . . . . . . . . . . +119 +8.5 +Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +120 +8.5.1 +Architecture comparison . . . . . . . . . . . . . . . . . . . . . . . . . . +120 +8.5.2 +One-to-all leads conversion . . . . . . . . . . . . . . . . . . . . . . . . . +121 +8.5.3 +Comparison with the state-of-the-art . . . . . . . . . . . . . . . . . . . . +124 +8.5.4 +Cross-database evaluation +. . . . . . . . . . . . . . . . . . . . . . . . . +125 +8.5.5 +Influence of medical conditions +. . . . . . . . . . . . . . . . . . . . . . +127 +8.6 +Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +127 +III +Face Biometrics +133 +9 +Prior Art in Face Biometrics +135 +9.1 +Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +135 +9.2 +Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +138 +9.2.1 +Face detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +138 +9.2.2 +Feature extraction and recognition . . . . . . . . . . . . . . . . . . . . . +140 +9.2.3 +Presentation attack detection . . . . . . . . . . . . . . . . . . . . . . . . +142 +9.2.4 +Robustness and trustworthiness +. . . . . . . . . . . . . . . . . . . . . . +143 +9.3 +Open Challenges and Opportunities +. . . . . . . . . . . . . . . . . . . . . . . . +146 +10 Masked Face Recognition +149 +10.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +149 +10.2 Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +150 +10.2.1 Adapted triplet loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +150 +10.2.2 Multi-task contrastive learning . . . . . . . . . . . . . . . . . . . . . . . +151 +10.3 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +153 +10.3.1 Adapted triplet loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +153 +10.3.2 Multi-task contrastive learning . . . . . . . . . . . . . . . . . . . . . . . +155 +10.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +159 +11 Interpretability for Face Biometrics +161 +11.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +161 +11.2 Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +162 +11.2.1 Implemented face PAD network . . . . . . . . . . . . . . . . . . . . . . +162 + +xvi +CONTENTS +11.2.2 Interpretability method . . . . . . . . . . . . . . . . . . . . . . . . . . . +163 +11.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +163 +11.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +163 +11.3.2 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . +164 +11.3.3 Experimental scenarios and evaluation . . . . . . . . . . . . . . . . . . . +164 +11.4 Conducted Studies on PAD Interpretability . . . . . . . . . . . . . . . . . . . . . +165 +11.4.1 Representation of a model’s explanations +. . . . . . . . . . . . . . . . . +165 +11.4.2 Semantic representation of explanations . . . . . . . . . . . . . . . . . . +165 +11.4.3 Comparison of explanations across different scenarios +. . . . . . . . . . +166 +11.4.4 Interclass comparison in the unseen-attack scenario . . . . . . . . . . . . +168 +11.4.5 Intraclass comparison across different samples +. . . . . . . . . . . . . . +169 +11.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +170 +11.5.1 Performance of the face PAD algorithm . . . . . . . . . . . . . . . . . . +170 +11.5.2 Comparison of explanations across different scenarios +. . . . . . . . . . +170 +11.5.3 Interclass comparison in the unseen-attack scenario . . . . . . . . . . . . +175 +11.5.4 Intraclass comparison across different samples +. . . . . . . . . . . . . . +176 +11.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +177 +IV +Wellbeing Monitoring +179 +12 Emotion Valence Classification in the Wild +181 +12.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +181 +12.2 Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +182 +12.2.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +182 +12.2.2 Video-based emotion recognition +. . . . . . . . . . . . . . . . . . . . . +182 +12.2.3 Audio-based emotion recognition +. . . . . . . . . . . . . . . . . . . . . +183 +12.2.4 Score-level ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . +184 +12.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +185 +12.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +185 +12.3.2 Baseline algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +186 +12.3.3 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +186 +12.3.4 Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +186 +12.3.5 Experiments +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +187 +12.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +187 +12.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +190 +13 Activity and Violence Recognition in Shared Vehicles +193 +13.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +193 +13.2 Methodology +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +194 +13.2.1 Multimodal pipeline +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +194 +13.2.2 Cascade strategy +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +197 +13.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +198 +13.3.1 Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +198 +13.3.2 Data preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +199 +13.3.3 Model training +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +200 +13.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +200 +13.4.1 Generic scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +201 +13.4.2 In-vehicle scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +202 + +CONTENTS +xvii +13.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +203 +V +Broader Topics on Biometrics and Pattern Recognition +205 +14 Learning Template Security on End-to-End Biometric Models +207 +14.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +207 +14.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +208 +14.3 The Secure Triplet Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +209 +14.3.1 Original triplet loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +209 +14.3.2 Learning cancelability +. . . . . . . . . . . . . . . . . . . . . . . . . . . +210 +14.3.3 Promoting unlinkability +. . . . . . . . . . . . . . . . . . . . . . . . . . +211 +14.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +212 +14.4.1 ECG identity verification . . . . . . . . . . . . . . . . . . . . . . . . . . +212 +14.4.2 Face identity verification . . . . . . . . . . . . . . . . . . . . . . . . . . +214 +14.4.3 Evaluation frameworks and metrics +. . . . . . . . . . . . . . . . . . . . +214 +14.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +216 +14.5.1 Verification performance . . . . . . . . . . . . . . . . . . . . . . . . . . +217 +14.5.2 Cancelability evaluation +. . . . . . . . . . . . . . . . . . . . . . . . . . +221 +14.5.3 Unlinkability evaluation +. . . . . . . . . . . . . . . . . . . . . . . . . . +221 +14.5.4 Non-invertibility and secrecy leakage +. . . . . . . . . . . . . . . . . . . +222 +14.5.5 Comparison with state-of-the-art approaches +. . . . . . . . . . . . . . . +223 +14.5.6 Effects of varying γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +223 +14.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +223 +15 Self-Supervised Learning with Sequential Data +225 +15.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +225 +15.2 The Stochastic Triplet Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +226 +15.3 Application Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +229 +15.3.1 ECG identity verification . . . . . . . . . . . . . . . . . . . . . . . . . . +230 +15.3.2 Face identity verification . . . . . . . . . . . . . . . . . . . . . . . . . . +231 +15.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +232 +15.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +232 +15.4.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +232 +15.4.3 Evaluation metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +233 +15.4.4 Experiments’ description . . . . . . . . . . . . . . . . . . . . . . . . . . +233 +15.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +234 +15.5.1 ECG identity verification . . . . . . . . . . . . . . . . . . . . . . . . . . +234 +15.5.2 Face identity verification . . . . . . . . . . . . . . . . . . . . . . . . . . +235 +15.5.3 Stress experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +236 +15.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +238 +VI +Epilogue +239 +16 Summary and Conclusions +241 +17 Future Work Considerations +245 + +xviii +CONTENTS +VII +Appendices +249 +A ECG Biometrics Literature Methods +251 +References +269 + +List of Figures +1.1 +Schema of the topics covered during this doctoral project, their interconnec- +tions, and their link to the central topic of personalised wellbeing monitoring. . +7 +2.1 +General structure of a biometric recognition system. +. . . . . . . . . . . . . . +20 +2.2 +Schematics of the operation of a biometric recognition system in identification +and identity verification modes, and in the enrollment phase. . . . . . . . . . . +21 +2.3 +Attack points on a biometric system. . . . . . . . . . . . . . . . . . . . . . . . +22 +2.4 +The sequence of depolarisation and depolarisation events in the heart, and their +relationship with the different heartbeat waveforms in an ECG signal. . . . . . +27 +2.5 +Medical acquisition settings: electrode placement and leads on the standard 12- +lead configuration and Frank leads. +. . . . . . . . . . . . . . . . . . . . . . . +28 +2.6 +Acquisition settings with movement: example of a five-electrode Holter system +for ambulatory recordings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +28 +2.7 +Examples of off-the-person ECG acquisition configurations, using thumb elec- +trodes, index finger electrodes, metallic rods grabbed by the subjects, or elec- +trodes mounted on a table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +2.8 +Wearable and seamless acquisition: examples of surveyed configurations. . . . +30 +2.9 +Variability in off-the-person ECG heartbeats from the same subjects. . . . . . . +31 +2.10 +Comparison of face images acquired on the visible light, short-wave infrared, +mid-wave infrared, and long-wave infrared spectra. . . . . . . . . . . . . . . . +33 +2.11 +Examples of tridimensional face models. +. . . . . . . . . . . . . . . . . . . . +34 +2.12 +Variability in unconstrained face images of a subject. . . . . . . . . . . . . . . +35 +2.13 +Example of a Receiver Operating Characteristic (ROC) curve for an identity +verification system and the evolution of False Acceptance and False Rejection +rates with the threshold value. . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +2.14 +Examples of a Cumulative Match Characteristic (CMC) curve, and a Receiver +Operating Characteristic (ROC) curve for an identification system. . . . . . . . +41 +2.15 +Example of a Usability-Security characteristic curve. . . . . . . . . . . . . . . +42 +2.16 +Illustration of the bidimensional valence-arousal space with example emotion +categories. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +3.1 +Currently available ECG collections and the number of surveyed publications +that have used them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +3.2 +General structure of an ECG-based recognition system. . . . . . . . . . . . . . +53 +3.3 +A panorama of ECG biometrics across time: the past, present, and future trends +in ECG acquisition for biometrics and the corresponding research challenges. . +67 +4.1 +Architecture of the proposed CNN model for ECG-based identification. . . . . +70 +4.2 +Illustration of the effects of the different data augmentation techniques on an +example five-second ECG segment. . . . . . . . . . . . . . . . . . . . . . . . +72 +xix + +xx +LIST OF FIGURES +4.3 +Illustration of the progressive phases of integration of the traditional pipeline +stages into the CNN architecture. +. . . . . . . . . . . . . . . . . . . . . . . . +73 +4.4 +Results of the proposed and baseline algorithms, when using DCT features as +input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 +4.5 +Results of the proposed and baseline algorithms, when using ensemble heart- +beats as input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 +4.6 +Results of the proposed and baseline algorithms, when using five-second ECG +segments as input, raw or denoised. . . . . . . . . . . . . . . . . . . . . . . . +75 +4.7 +Results of the proposed algorithm receiving raw five-second segments, with +each technique of data augmentation, on the datasets of 25 and 100 subjects. +. +75 +4.8 +Results of the proposed algorithm, receiving raw five-second segments as input, +with combinations of data augmentation techniques. +. . . . . . . . . . . . . . +76 +4.9 +Direct benchmarking between the proposed architecture with the best baseline +algorithm and the two implemented state-of-the-art algorithms. +. . . . . . . . +76 +5.1 +Schemes of the proposed identity verification model, including the weight trans- +fer between networks for both proposed training methodologies. . . . . . . . . +80 +5.2 +Network outputs for all training samples of five example subjects from the +UofTDB collection. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +84 +5.3 +Varying identity set size scenario: EER evolution with number of subjects re- +served for training, for diverse enrollment durations, for the proposed methods +IT-CNN and TL-CNN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +86 +5.4 +Cross-database scenario: EER for the proposed methods IT-CNN and TL-CNN +when trained with UofTDB data and directly applied to CYBHi or PTB, and +comparison with state-of-the-art methods. . . . . . . . . . . . . . . . . . . . . +86 +5.5 +Fine-tuning scenario: EER results for the proposed methods IT-CNN and TL- +CNN when directly trained with CYBHi or PTB data from 20 subjects, and +comparison with state-of-the-art methods. . . . . . . . . . . . . . . . . . . . . +87 +5.6 +Fine-tuning scenario: EER results for the proposed methods when (DT) trained, +from scratch, with data from CYBHi or PTB, or when (FT) trained with +UofTDB data and fine-tuned to CYBHi/PTB. . . . . . . . . . . . . . . . . . . +87 +6.1 +Dendrogram representing the taxonomy of template update techniques. . . . . +93 +6.2 +Illustration of the search for the ideal threshold. . . . . . . . . . . . . . . . . . +96 +6.3 +Schema illustrating the use of each E-HOL record for training and testing. . . . +97 +6.4 +Identification +performance +over +time +corresponding +to +(a) +the +La- +bati et al. method, and (b) the implemented state-of-the-art methods. . . . . . . +98 +6.5 +Comparison of the FIFO method applied with different thresholds to different +identification methodologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . +99 +6.6 +Results using FIFO update with different thresholds. . . . . . . . . . . . . . . +99 +6.7 +Results using Fixation update. . . . . . . . . . . . . . . . . . . . . . . . . . . +100 +6.8 +Performance results over time for the CNN model with fine-tuning-based model +update. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +101 +6.9 +Performance results over time for the CNN model adapted with kNN decision +and FIFO template update, for several threshold criteria. . . . . . . . . . . . . +101 +7.1 +Illustration of the ECG waveforms on a sample PTB signal segment. . . . . . . +104 +7.2 +Architecture of the biometric identification model. . . . . . . . . . . . . . . . +106 +7.3 +Explanations over an example five-second ECG segment from PTB. . . . . . . +109 + +LIST OF FIGURES +xxi +7.4 +Explanations over an example five-second ECG segment from UofTDB. . . . . +109 +7.5 +Average explanations over heartbeat waveforms of subjects #1 and #2 on the +subsets of the PTB database. . . . . . . . . . . . . . . . . . . . . . . . . . . . +110 +7.6 +Average explanations over heartbeat waveforms of subjects #1 and #2 on the +subsets of the UofTDB database. . . . . . . . . . . . . . . . . . . . . . . . . . +110 +8.1 +Schema of the U-Net architecture. . . . . . . . . . . . . . . . . . . . . . . . . +117 +8.2 +Schema of the convolutional autoencoder (AE) architecture. . . . . . . . . . . +117 +8.3 +Schema of the architecture based on label refinement networks (LRN). +. . . . +118 +8.4 +Example result of lead II to all conversion on the PTB test dataset. . . . . . . . +122 +8.5 +Example result of lead I to all conversion on the PTB test dataset. . . . . . . . +123 +8.6 +Example cross-database result of lead II to all conversion on INCART. +. . . . +128 +8.7 +Example cross-database result of lead I to all conversion on INCART. . . . . . +129 +8.8 +Example cross-database result of lead II to all conversion on PTB-XL. . . . . . +130 +8.9 +Example cross-database result of lead I to all conversion on PTB-XL. . . . . . +131 +9.1 +Stages of a biometric recognition algorithm based on face images. . . . . . . . +138 +9.2 +Examples of face detection in unconstrained settings. . . . . . . . . . . . . . . +139 +9.3 +Evolution of face recognition approaches, from holistic to deep learning. +. . . +141 +9.4 +Recent history of face recognition, from deep learning to tailored objective func- +tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +141 +9.5 +Example of how a mask can significantly occlude a face and limit the informa- +tion that can be used by a face recognition algorithm. . . . . . . . . . . . . . . +144 +9.6 +Illustration of how interpretability/explainability can be used to understand and +improve a biometric model. +. . . . . . . . . . . . . . . . . . . . . . . . . . . +145 +10.1 +Expected effect of the original triplet loss on the output embedding space. . . . +150 +10.2 +Expected effect of the proposed adapted triplet loss on the output embedding +space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +151 +10.3 +Schema of the proposed multi-task contrastive learning approach. . . . . . . . +152 +10.4 +Explanations obtained for each trained model with the Smooth Grad-CAM++ +explainability tool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +155 +10.5 +Example of masked face images generated to validate the multi-task contrastive +learning approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +156 +10.6 +Receiver operating characteristic (ROC) curve for mask detection on the LFW +dataset with simulated masks. . . . . . . . . . . . . . . . . . . . . . . . . . . +159 +11.1 +Architecture of the PAD end-to-end deep model used in this work. . . . . . . . +163 +11.2 +Example of the approach used to quantify the difference between two explana- +tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +166 +11.3 +Comparison of explanations for a bona fide sample Ik, on the evaluation scenario +x, and fixing Attack #i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +167 +11.4 +Comparison of explanations for a presentation attack sample Ik, on the evalua- +tion scenario x, and fixing Attack #i. . . . . . . . . . . . . . . . . . . . . . . . +167 +11.5 +Pairwise comparison of explanations produced by the models in unseen-attack +scenarios, for a bona fide sample Ik. . . . . . . . . . . . . . . . . . . . . . . . +169 +11.6 +Pairwise comparison of explanations produced by the models in unseen-attack +scenarios, for the presentation attack sample Ik of type #i. +. . . . . . . . . . . +169 + +xxii +LIST OF FIGURES +11.7 +Image Average mean and standard deviation (StD) results for bona fide (BF) +and presentation attack (PA) samples in the comparison across the one-attack +(OA) and unseen-attack (UA) scenarios. . . . . . . . . . . . . . . . . . . . . . +172 +11.8 +Comparison of explanations in intraclass one-attack for an example PA sample +of type 5 presenting high Iµ value. . . . . . . . . . . . . . . . . . . . . . . . . +173 +11.9 +Comparison of explanations in intraclass unseen-attack for an example PA sam- +ple of type 5 presenting low Iµ value. . . . . . . . . . . . . . . . . . . . . . . +173 +11.10 +Mean Aµ results for bona fide and presentation attack samples in the one-attack +(OA) and unseen-attack (UA, i = 1,...,7) scenarios and respective mean value. +174 +11.11 +Explanations for an example bona fide samples with pairwise distance close to +the obtained average ( ¯dBF = 0.54). . . . . . . . . . . . . . . . . . . . . . . . . +175 +11.12 +Explanations for an example presentation attack sample of type #2 with pairwise +distance close to the obtained average ( ¯dPA = 0.52). . . . . . . . . . . . . . . . +175 +11.13 +Explanations for an example bona fide sample with pairwise distance above the +obtained average ( ¯dBF = 0.54). . . . . . . . . . . . . . . . . . . . . . . . . . . +176 +11.14 +Explanations for an example presentation attack sample of type #7 with pairwise +distance above the obtained average ( ¯dPA = 0.52). . . . . . . . . . . . . . . . . +177 +11.15 +Bona fide and presentation attack intraclass comparison mean and standard de- +viation results for the one-attack and unseen-attack (i = 1,...,7) scenarios and +respective overall results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +178 +12.1 +Illustration of the structure of the proposed method for audiovisual group emo- +tion recognition. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +183 +12.2 +Structure of the video-based group emotion recognition module, based on an +inflated ResNet-50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +184 +12.3 +Structure of the audio-based group emotion recognition module, based on a Bi- +LSTM network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +185 +12.4 +Some examples of validation set videos where the model offered unsuccessful +predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +190 +13.1 +Diagram of the full multimodal pipeline for activity recognition. . . . . . . . . +195 +13.2 +Diagram of the visual submodule. . . . . . . . . . . . . . . . . . . . . . . . . +196 +13.3 +Diagram of the audio submodule. . . . . . . . . . . . . . . . . . . . . . . . . +197 +13.4 +Example frames from the in-vehicle dataset, depicting normal activities (top +row) and violence between passengers (bottom row). . . . . . . . . . . . . . . +199 +13.5 +Rank accuracy results for the 21 selected classes from the MMIT database. . . +201 +13.6 +Cascade results for the 21 selected classes from the MMIT database. . . . . . . +201 +13.7 +Rank accuracy results for the in-vehicle scenario with 42 classes. +. . . . . . . +203 +13.8 +Cascade results in the in-vehicle scenario with 42 classes. +. . . . . . . . . . . +203 +13.9 +Cascade results in the in-vehicle scenario with 3 classes. . . . . . . . . . . . . +204 +14.1 +Comparison between the model training schemes of the original triplet loss and +the proposed Secure Triplet Loss method. . . . . . . . . . . . . . . . . . . . . +210 +14.2 +Illustration of the expected results when training with the proposed Secure +Triplet Loss. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +211 +14.3 +Architecture of the models used for ECG and face identity verification. +. . . . +213 +14.4 +Detection Error Tradeoff (DET) curves for the ECG identity verification model +when trained with the original triplet loss vs. the proposed formulations of the +Secure Triplet Loss. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +218 + +LIST OF FIGURES +xxiii +14.5 +Detection Error Tradeoff (DET) curves for the face identity verification model +when trained with the original triplet loss vs. the proposed formulations of the +Secure Triplet Loss. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +218 +14.6 +False match rate (FMR) and false non-match rate (FNMR) curves w.r.t. the +distance comparison threshold t, for ECG identity verification with triplet loss +and the proposed Secure Triplet Loss formulations. . . . . . . . . . . . . . . . +219 +14.7 +False match rate (FMR) and false non-match rate (FNMR) curves w.r.t. the +distance comparison threshold t, for face identity verification with triplet loss +and the proposed Secure Triplet Loss formulations. . . . . . . . . . . . . . . . +220 +14.8 +Template linkability analysis for the ECG and face identity verification models. +222 +14.9 +Results with the proposed loss when varying the γ parameter. +. . . . . . . . . +224 +15.1 +Architecture of the ECG identity verification model that was trained with the +proposed methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +231 +15.2 +Architecture of the face identity verification model that was trained with the +proposed methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +231 +15.3 +Comparison of the Receiver-Operating Characteristic curves on ECG identity +verification for supervised training, unsupervised training, and recording-based +supervision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +235 +15.4 +Comparison of the Receiver-Operating Characteristic curves on face verification +for supervised training, unsupervised training, and recording-based supervision. 235 +15.5 +Receiver-Operating Characteristic curve for negative selection error based on +the number of database subjects. . . . . . . . . . . . . . . . . . . . . . . . . . +236 +15.6 +Receiver-Operating Characteristic curves for varying positive sample selection +error probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +237 +15.7 +Receiver-Operating Characteristic curves for varying negative sample selection +error probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +237 +15.8 +Equal Error Rate (EER) results when using more enrollment data from each +subject. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +238 + + +List of Tables +2.1 +Main benefits and drawbacks of different biometrics traits. . . . . . . . . . . . +25 +2.2 +Definition of the commonly used metrics for performance evaluation in identi- +fication and identity verification tasks. . . . . . . . . . . . . . . . . . . . . . . +38 +3.1 +Summary of the technical specificities of the most relevant publicly available +ECG collections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +3.2 +Results of surveyed approaches evaluated with PTB. . . . . . . . . . . . . . . +61 +3.3 +Results of surveyed approaches evaluated with ECG-ID. . . . . . . . . . . . . +62 +3.4 +Results of surveyed approaches evaluated with MIT-BIH NSR. . . . . . . . . . +63 +3.5 +Results of surveyed approaches evaluated with MIT-BIH Arrhythmia. . . . . . +64 +3.6 +Results of surveyed approaches evaluated with UofTDB. . . . . . . . . . . . . +65 +3.7 +Results of surveyed approaches evaluated with CYBHi. +. . . . . . . . . . . . +65 +4.1 +Comparison of the proposed and baseline algorithms with recent state-of-the-art +methods. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +76 +5.1 +Single-database scenario: EER results (%) when trained with data from 100 +UofTDB subjects and tested with 919 UofTDB subjects. . . . . . . . . . . . . +84 +5.2 +Single-database scenario: Mean and standard deviation of the EER results (%) +obtained on 100 random data divisions. . . . . . . . . . . . . . . . . . . . . . +85 +6.1 +Graph-based template update methods and their respective loss and regulariser +functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +94 +7.1 +True positive identification rate results (%) on the test data. . . . . . . . . . . . +108 +8.1 +Comparison of encoder-decoder architectures on one-to-one lead conversion. . +120 +8.2 +Average correlation between lead II signals and the remaining leads on the PTB, +INCART, and PTB-XL databases. . . . . . . . . . . . . . . . . . . . . . . . . +120 +8.3 +Test results of the U-Net used for multi-lead conversion from lead II, with +shared or individual encoders. . . . . . . . . . . . . . . . . . . . . . . . . . . +120 +8.4 +Average correlation between lead I signals and the remaining leads on the PTB, +INCART, and PTB-XL databases. . . . . . . . . . . . . . . . . . . . . . . . . +121 +8.5 +Test results of the U-Net used for multi-lead conversion from lead I, with shared +or individual encoders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +121 +8.6 +Cross-database test results for INCART conversion from lead II. . . . . . . . . +125 +8.7 +Cross-database test results for INCART conversion from lead I. +. . . . . . . . +125 +8.8 +Cross-database test results for PTB-XL conversion from lead II. . . . . . . . . +126 +8.9 +Cross-database test results for PTB-XL conversion from lead I. +. . . . . . . . +126 +8.10 +Average correlation results for PTB-XL conversion from lead II according to +medical condition class. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +132 +xxv + +xxvi +LIST OF TABLES +8.11 +Average correlation results for PTB-XL conversion from lead I according to +medical condition class. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +132 +9.1 +Details on the main face recognition databases that are currently available. +. . +136 +10.1 +Results with the adapted triplet loss on synthetic masked face data (SMFD). . . +154 +10.2 +Results with the adapted triplet loss on real masked face data (RMFD). +. . . . +154 +10.3 +Ablation results with the multi-task contrastive learning approach on the test +dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +157 +10.4 +Comparison of the modules of the proposed approach by the total number of +parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +157 +10.5 +Comparison of the FMR100 results of the methods presented in the MFR com- +petition, the official baseline, the adapted triplet loss, and the multi-task con- +trastive learning approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +158 +11.1 +PAI species in the ROSE Youtu DB and the respective number of extracted +samples. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +164 +11.2 +Overview of the strategy to compare explanations for a sample across different +evaluation scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +168 +11.3 +Performance of the PAD models in the one-attack and unseen-attack evaluation +scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +171 +12.1 +Accuracy (%) of the proposed method on the validation set. +. . . . . . . . . . +188 +12.2 +Accuracy (%) of the proposed method on the test set. . . . . . . . . . . . . . . +188 +12.3 +Confusion matrix of audio-based recognition on the validation set. . . . . . . . +188 +12.4 +Confusion matrix of video-based recognition on the validation set. . . . . . . . +188 +12.5 +Confusion matrix of multimodal recognition on the validation set. . . . . . . . +188 +12.6 +Accuracy (%) on the validation set for videos of small groups vs. large groups. +189 +13.1 +Summary of the accuracy (%) results obtained in the various experimental sce- +narios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +200 +13.2 +Summary of the size, total number of parameters, and average run times per +instance of the three pipeline submodules for the in-lab scenario. . . . . . . . . +202 +14.1 +Summary of the test results for ECG identity verification. . . . . . . . . . . . . +216 +14.2 +Summary of the test results for face identity verification. . . . . . . . . . . . . +217 +A.1 +Summary of the surveyed state-of-the-art unimodal methods proposed for ECG +biometrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +252 + +Abbreviations +1D Unidimensional +2D Bidimensional +2.5D Bidimensional with Depth Information +3D Tridimensional +AC Autocorrelation +ADAS Advanced Driver Assistance Systems +AE Autoencoder +AHA American Heart Association (dataset) +AMAE Average Mean Absolute Error +ANN Artificial Neural Network +APCER Attack Presentation Classification +Error Rate +AR Autoregressive (coefficients) +AUC Area Under the Curve +BF Bona Fide +BPCER Bona fide Presentation Classification +Error Rate +BPF Bandpass Filter +BIOSIG Biometrics Special Interest Group +Bi-LSTM Bidirectional Long Short-Term +Memory (network) +CAM Class Activation Mapping +CASIA Chinese Academy of Sciences +CC (Pearson’s) Correlation Coefficient +CC0 Creative Commons “No Rights +Reserved” License +CCC Concordance Correlation Coefficient +CE Cross-Entropy +CGFSI Cross-GAN Filter Similarity Index +CIBB International Conference on +Computational Intelligence Methods for +Bioinformatics and Biostatistics +CinC Computers in Cardiology (conference) +CMC Cumulative Match Characteristic +CNN Convolutional Neural Network +CSIST Chung-Shan Institute of Science and +Technology (datasets) +CTM Centre for Telecommunications and +Multimedia +CYBHi Check Your Biosignals Here initiative +CWT Continuous Wavelet Transform +DBNN Decision-based Neural Network +DCT Discrete Cosine Transform +DET Detection Error Trade-off (curve) +DNA Deoxyribonucleic Acid +DT Direct Training (experimental scenario) +DTW Dynamic Time Warping +DWT Discrete Wavelet Transform +EAB European Association for Biometrics +ECCV European Conference on Computer +Vision +xxvii + +xxviii +ABBREVIATIONS +ECG Electrocardiogram +EEG Electroencephalogram +EEMD Ensemble Empirical Mode +Decomposition +EER Equal Error Rate +EMD Empirical Mode Decomposition +EMG Electromyogram +ERDF European Regional Development Fund +ETs Extremely Randomised Trees +FAR False Acceptance Rate +FC Fully-Connected (network layer) +FCT Fundação para a Ciência e a Tecnologia +FDDB Face Detection Database & Benchmark +FEC Forward Error Control +FERET Facial Recognition Technology +FG4COVID19 Face and Gesture Analysis for +COVID-19 (workshop) +FLDA Fisher Linear Discriminant Analysis +FLRP Focused Layer-wise Relevance +Propagation +FMR False Match Rate +FNMR False Non-Match Rate +FOGD First-Order Gaussian Derivative (filter) +FPIR False Positive Identification Rate +FRR False Rejection Rate +FRVT Face Recognition Vendor Test +FT Fine-Tuning +FTDNN Focused Time-Delay Neural Network +GAN Generative Adversarial Network +GBFS Greedy Best-First Search +GMM Gaussian Mixture Model +GMean Genuine Mean (scores) +GNMF Graph-regularised Non-negative +Matrix Factorisation +GPU Graphics Processing Unit +GRU Gated Recurrent Unit +Grad-CAM Gradient-weighted Class +Activation Mapping +HASLab High-Assurance Software +Laboratory +HE Homomorphic Encryption +HLDA Heteroscedastic Linear Discriminant +Analysis +HPF Highpass Filter +HRV Heart Rate Variability +I3D Two-Stream Inflated Tridimensional +Convolutional Network +IARPA Intelligence Advanced Research +Projects Activity +ICA Independent Component Analysis +IDR Identification Rate +IJB-C IARPA Janus Benchmark-C +IJCB International Joint Conference on +Biometrics +IJCNN International Joint Conference on +Neural Networks +IMDb Internet Movie Database +IMean Impostor Mean (scores) +IMF Intrinsic Mode Function +IOMBA Interval Optimised Mapping Bit +Allocation +IoU Intersection over Union +IPAS International Conference on Image +Processing, Applications and Systems +ISEL Instituto Superior de Engenharia de +Lisboa + +ABBREVIATIONS +xxix +ISM In-Vehicle Sensing Monitorisation +(workshop) +IT-CNN Identification Training Convolutional +Neural Network +IWBF International Workshop on Biometrics +and Forensics +KLD Kullback-Leibler Divergence +kNN k-Nearest Neighbours +KPCA Kernel Principal Component Analysis +KSS Karolinska Sleepiness Scale +LBP Local Binary Patterns +LDA Linear Discriminant Analysis +LDP Local Difference Patterns +LFW Labeled Faces in the Wild (dataset) +LLR Log-Likelihood Ratio +LPF Lowpass Filter +LRN Label Refinement Network +LSTM Long Short-Term Memory (network) +LWIR Long-Wave Infrared +MB Megabyte +MFCC Mel-Frequency Cepstral Coefficients +MFR Masked Face Recognition +MFRC-21 2021 Masked Face Recognition +Competition Dataset +MIDR Misidentification Rate +MIT-BIH Massachusetts Institute of +Technology - Beth Israel Hospital +MLP Multi-Layer Perceptron +MMAE Maximum Mean Absolute Error +MMIT Multi-Moments in Time (dataset) +MRLBP Multi-Resolution Local Binary +Patterns +MSDF-1DMRLBP Multi-Scale Differential +Features fusion of Unidimensional +Multi-Resolution Local Binary Patterns +MSE Mean Squared Error +MSFS Multisession Feature Selection +MTCNN Multi-Task Convolutional Neural +Network +MWIR Mid-Wave Infrared +NCC Normalised Cross-Correlation +NCCC Normalised Cross-Correlation +Clustering +NF Notch Filter +NIR Near-Infrared +NIS Networked Intelligent Systems (cluster) +NIST National Institute of Standards and +Technology +NN Neural Network +NRC Normalised Relative Compression +NSR Normal Sinus Rhythm +OA One-Attack (evaluation scenario) +OCFR Advanced Occluded Face Recognition +(competition) +PA Presentation Attack +PAD Presentation Attack Detection +PAI Presentation Attack Instrument +PAISp Presentation Attack Instrument Species +PaSC Point and Shoot Face Recognition +Challenge +PCA Principal Component Analysis +PLI Powerline Interference +PNN Probabilistic Neural Network +PPG Photoplethysmogram +PRD Percent Root-Mean-Square Difference +PSD Power Spectral Density + +xxx +ABBREVIATIONS +PTB Physicalisch-Technische Bundesanstalt +(dataset) +PTCD Probability of Time to Correct +Decision (score) +RBF Radial Basis Function +RCNN Region-based Convolutional Neural +Network +ReLU Rectified Linear Unit +RF Random Forest +RGB Red-Green-Blue (colour model) +RMFD Real Masked Face Data (evaluation +scenario) +RMSE Root Mean Squared Error +ROC Receiver Operating Characteristic +(curve) +RR Reject Rate +RSA Recurrent Scale Approximation +S3FD Single Shot Scale-Invariant Face +Detector +SAGR Sign Agreement Metric +SAX Symbolic Aggregate Approximation +SCG Seismocardiogram +SecureTL Secure Triplet Loss +SFA Simplified Fuzzy ARTMAP +SGD Stochastic Gradient Descent +SHAP Shapley Addictive Explanation +SIMCA Soft Independent Modelling by Class +Analogy +SL Self-Learning +SMFD Synthetic Masked Face Data +(evaluation scenario) +SSIM Structural Similarity Index Measure +SVM Support Vector Machine +StD Standard Deviation +STFT Short-Time Fourier Transform +SWIR Short-Wave Infrared +TCD Time to Correct Decision (score) +TL-CNN Transfer-Learning Convolutional +Neural Network +TPIR True Positive Identification Rate +TVCG Tridimensional Vectorcardiogram +t-SNE t-Distributed Stochastic Neighbour +Embedding +UA Unseen-Attack (evaluation scenario) +UBM Universal Background Model +ULHT Universidade Lusófona de +Humanidades e Tecnologias +UMAP Face and Gesture Analysis for +COVID-19 +UMD University of Maryland (dataset) +USC Usability-Security Curve +UofTDB University of Toronto ECG Database +Var Variance +VCMI Visual Computing and Machine +Intelligence (research group) +VGAF Video-level Group Affect (dataset) +VGG Visual Geometry Group (dataset) +VISAPP International Conference on +Computer Vision Theory and +Applications +VGA Video Graphics Array +WACV Winter Conference on Applications of +Computer Vision +WDIST Wavelet Distance +WWPRD Wavelet-Weighted Percent +Root-Mean-Square Difference +xaFCM Extended-alphabet Finite-Context +Model +xAI4Biometrics Explainable Artificial +Intelligence for Biometrics (workshop) + +Part I +Prologue +1 + + +Chapter 1 +Introduction +1.1 +Context and Motivation +The interactions between humans and machines are increasingly mediated by intelligent sensing +technologies. Nowadays, the default way to unlock a smartphone is using face or fingerprint bio- +metrics. Highly-populated countries like India and China are building unprecedentedly massive +national identity networks relying entirely on biometric characteristics (and dealing with the so- +cietal intricacies of such colossal endeavours) [198; 257; 404]. Sophisticated algorithms monitor +our attention levels while we drive [119; 176]. Meanwhile, the gaming and entertainment indus- +tries are investing heavily in using affective computing to continuously personalise and enhance +user experience [24; 97; 259]. +Among all of these applications, few have been so revolutionised (and so quickly) as vehicles, +and there are plenty of great reasons for that. Whether in personal cars or public transport, people +can typically spend up to multiple hours of their day inside vehicles, especially those with long +daily commutes or those living/working in large urban centres. Moreover, a considerable fraction +of preventable deaths and serious injuries occur in accidents involving vehicles, caused mainly by +driver fatigue or the influence of psychotropic substances [370; 399]. +While we wait for fully autonomous vehicles, advanced driver assistance systems (ADAS) +based on artificial intelligence help drivers comply with speed limits, stay in their lanes, beware +of their blind spots, and avoid accidents [81; 82; 410]. One of the most interesting applications of +this is drowsiness detection, where biometric data such as face video or physiological signals can +be used to detect episodes of sleepiness and infer the fatigue levels of the driver [116; 320; 399]. +When done continuously (or at least frequently) during vehicle usage, it can be used to warn the +driver or even trigger the automatic safe interruption of the vehicle’s operation. +Drowsiness, just like most other wellbeing parameters, reveals itself on biometric data as in- +trasubject variability: the way a person’s data varies over time and across diverse conditions [116]. +This stands in stark opposition to biometric recognition applications, which rely on intersubject +variability (the way a person’s data differs from another’s) to distinguish identities [213; 343]. In +3 + +4 +Introduction +biometric recognition, intrasubject variability is typically regarded as a nuisance, as it blurs the +boundaries between individuals’ data and hampers accurate and robust identity recognition [199]. +In spite of their differences, biometric recognition and wellbeing monitoring can coexist in +a vehicle scenario. Applications of biometric recognition for vehicles are typically focused on +access control or the automatic personalisation of driving and environmental conditions (such as +seat position, mirror adjustments, or infotainment settings) based on the recognised driver and +occupants [91; 342]. However, in this thesis, we defend that this coexistence could (and should) +be intensified, and eventually evolve to a level of symbiotic integration. +Despite the efforts devoted to wellbeing monitoring technologies, one challenge remains un- +vanquished: the fact that wellbeing patterns are deeply personal. For example, no two individuals +experience drowsiness in the exact same way, and similar levels of fatigue can have dramatically +different effects on each person. This is also true for the way wellbeing parameters reflect on bio- +metric data. A drowsiness monitoring system can present acceptable accuracy levels for a given +set of subjects and, simultaneously, be inadequate at recognising the specific drowsiness patterns +of other people [116; 399]. +Biometric recognition could be the key to unlocking the next generation of wellbeing mon- +itoring technologies. Instead of using generalist models, identity predictions obtained through +biometric recognition would enable the selection of specific models for each of the users. These +models could also benefit from the influx of new data, continuously learning the subject-specific +patterns of wellbeing. Thus, combining biometrics with wellbeing could enable the creation of +more accurate and robust solutions for wellbeing monitoring [116]. +In the future, fully autonomous driving may put an end to the need for drivers, but not to +the need for automatic intelligent sensing technologies inside vehicles [23; 302; 348]. The same +biometric algorithms and wellbeing monitoring systems that once were focused on the driver may +easily be adapted to target the occupants. In fact, self-driving vehicles unveil a new scenario +for automatic passenger monitoring: since there is no driver, shared autonomous vehicles (such +as autonomous taxis) lack an authority figure, responsible for the integrity of the vehicle and +the comfort and security of the passengers. The driver could be replaced by pattern recognition +solutions to monitor the shared vehicle interior and its passengers. +Overall, the advantages of introducing intelligent sensing technologies in vehicles are plenty. +From a narrower subject-centric perspective, integrating the use of inter and intrasubject variability +would enable the development of continuously personalisable models for more robust monitoring +of wellbeing parameters. From a broader perspective, taking advantage of seamless multimodal +data acquisitions for automatic monitoring of the interior and passengers is of utmost importance +to ensure security and comfort inside autonomous shared vehicles. +1.2 +Objectives +This doctoral work focused on advancing in-vehicle sensing technology through the conceptual- +isation and development of novel computer vision and pattern recognition methodologies. The + +1.2 Objectives +5 +goal has been to create automatic solutions for in-vehicle monitoring, robustly and efficiently. To +achieve this, we targeted two specific scenarios, as follows: +• Personalised wellbeing monitoring systems using biometrics: Here, the objective was to +advance biometric recognition technologies to be integrated with wellbeing monitoring +methodologies, especially for driver assistance systems. We build upon the ECG biometrics +research conducted in [337], with additional work on face recognition and other impor- +tant topics such as biometric security and learning from data streams. With this work, we +aimed to pave the way towards a robust multimodal system to recognise the driver using +ECG and face information. The automatic identity predictions enable the use of the drivers’ +data to continuously learn their personal patterns of wellbeing for more accurate monitor- +ing. Within wellbeing, this research work focused on driver drowsiness and emotions, as +part of the AUTOMOTIVE project, but it stands to reason that biometrics could be used for +personalised monitoring of any wellbeing parameter; +• Occupant monitoring for autonomous shared vehicles: Forecasting the advent of fully auto- +nomous vehicles, this work aimed towards the use of data streams for monitoring occupants. +Integrated within the Easy Ride project, we targeted the monitoring of emotions among pas- +senger groups, as well as the recognition of activities and violence inside shared autonomous +vehicles. Just like the first scenario, this aims towards contributions to more intelligent and +robust wellbeing monitoring systems using multimodal data sources, albeit in a less personal +subject-centric way, focused instead on passenger groups as a whole. Although activity re- +cognition is a relatively mature research topic, the in-vehicle environment offers very spe- +cific challenges, mainly regarding perspective, lighting, and occlusions, which enabled the +study of innovative solutions for robust passenger monitoring. Ultimately, the developed +occupant wellbeing monitoring solutions could also be personalised, using the biometric +recognition of individual passengers as additional information for improved accuracy. +Despite these two scenarios, used to contextualise and motivate the work conducted during this +work, we aimed to build solutions that are applicable outside the target applications and beyond +the fields of biometrics and wellbeing monitoring. This work also aimed to result in significant +advances to the target fields, which are ready for real-life applications and capable to withstand the +test of time. As such, throughout the entirety of this doctoral project, there was a constant concern +to ensure the proposed methodologies were: +• Multimodal: The diversification of data sources is the key to truly robust models. On the +topic of biometric recognition, considering the strengths and shortcomings of the ECG sig- +nal as a biometric trait, the face is the best complementary trait for better performance and +robustness. Beyond biometric recognition, the fusion of ECG and face results in a larger +availability of anatomical and physiological measurements, which enable the more compre- +hensive monitoring of wellbeing parameters; + +6 +Introduction +• Seamless: Regardless of the possible consequences to accuracy, user comfort should be +of utmost importance. Hence, subjects should be as unaware of the acquisition process +as possible, to avoid attention or behaviour changes that could impact their comfort or the +realism of the collected data. As such, this work focused, as extensively as possible, on +seamlessly acquired data, in nearly unconstrained settings. When drivers are the target, and +thus the subject is in physical contact with the system nearly continuously (e. g., driving +the car), ECG and other physiological signals can be acquired unobtrusively at the steering +wheel. Face video can also be easily and inexpensively acquired with cameras. On the other +hand, for shared vehicle occupant monitoring, physiological signals have been avoided in +favour of less contact-intensive alternatives, namely video and audio; +• Continuous: Continuous biometric methodologies offer unique advantages in usability and +effectiveness, both for recognition and wellbeing monitoring. On the other hand, having +a continuous stream of biometric data opens up new possibilities for improved accuracy, +immersive systems, and error management. As such, beyond novelty and performance, +the algorithms developed during this work aimed towards real-time operation, reflecting a +constant preference for efficiency and simplicity whenever possible; +• Realistic: Overall, performance results in ECG-based biometrics literature are unrealistic, +mainly due to inadequate train-test splits and overly clean signal databases. The same hap- +pens in wellbeing monitoring when the problem of subject-independence is highlighted. +This reveals the deeply flawed nature of typical evaluation frameworks, which should more +realistically resemble actual application conditions and ensure reproducible results. +To +achieve this, this doctoral work included the reformulation of testing procedures, especially +for ECG-based biometrics, through the definition of adequate test protocols, the benchmark- +ing of literature methods, and the development of more robust recognition and monitoring +algorithms. +1.3 +Contributions +On the quest for personalised wellbeing monitoring, focused on the objectives detailed in the pre- +vious section, this doctoral project comprised several research topics related to both biometric +recognition and wellbeing monitoring. Fig. 1.1 presents them and illustrates their interconnec- +tions. Most of these were the focus of research work during this doctoral project and resulted in +innovative contributions towards the target of in-vehicle personalised wellbeing monitoring. +This thesis presents the innovative contributions of this work organised throughout four parts. +The first two of these are focused on biometric recognition, aiming towards the development of +personalised wellbeing monitoring solutions. The third part targets the scenario of passenger +monitoring inside shared autonomous vehicles, specifically for emotion, activity, and violence re- +cognition. The fourth presents broader contributions applicable to multiple scenarios in biometrics +and pattern recognition. These contributions are concisely enumerated below. + +1.3 Contributions +7 +PERSONALISED +WELLBEING +MONITORING +BIOMETRIC +RECOGNITION +WELLBEING +MONITORING +FACE +BIOMETRICS +ECG +BIOMETRICS +MASKED FACE +RECOGNITION +PRESENTATION +ATTACK DETECTION +OCCLUSIONS +INTERPRETABILITY +INTERPRETABILITY +END-TO-END +DEEP MODELS +LONG-TERM +PERFORMANCE +LEARNED 2D +TRANSFORMS +TEMPLATE UPDATE +INTERLEAD +CONVERSION +TEMPLATE +SECURITY +SELF-SUPERVISED +LEARNING +ORDINAL +CLASSIFICATION +EMOTION +RECOGNITION +ACTIVITY +RECOGNITION +DROWSINESS +MONITORING +FACE-BASED +ECG-BASED +MULTIMODAL +AUDIO + VIDEO +GROUP EMOTION +FACE-BASED +ECG-BASED +MULTIMODAL +MULTIMODAL +AUDIO + VIDEO +OPTIMIZATION +& EFFICIENCY +IN-VEHICLE +SCENARIOS +VIOLENCE +DETECTION +DRIVERS’ +MONITORING +MULTIMODAL +FACE + ECG +DATA AUGMENTATION +TRANSFER +LEARNING +OPEN WORLD +FACE RECOGNITION +Figure 1.1: Schema of the topics covered during this doctoral project, their interconnections, and +their link to the central topic of personalised wellbeing monitoring (black topics in bold correspond +to the strongest direct contributions, which are presented in this thesis). +In Part II, Electrocardiogram Biometrics: +• a comprehensive survey of one hundred and twenty-five state-of-the-art methodologies for +ECG-based biometric recognition, describing the evolution of the topic from 1999 to 2022 +and open challenges and opportunities for future research (Chapter 3); +• the first end-to-end methodology for ECG biometrics, alongside tailored data augmentation +strategies for ECG signals and a study on the advantages of integrating typically separate +processes inside a single deep architecture (Chapter 4); +• an application of triplet loss and transfer learning for ECG-based identity verification, +aiming towards higher robustness under realistic evaluation setups using off-the-person +databases (Chapter 5); +• a study on long-term performance with multiple state-of-the-art ECG biometric methodolo- +gies, including an assessment of the effect of diverse template and model update strategies +(Chapter 6); +• a study on the relative importance of ECG waveforms for identification under diverse scenar- +ios, less to more challenging, using explainability tools on our deep learning identification +methodology (Chapter 7); +• a methodology for recovering missing ECG leads based on single-lead blindly-segmented +input signals, paving the way towards more complete applications (even in clinical scenar- +ios) with less obtrusive signal collection setups (Chapter 8). +In Part III, Face Biometrics: + +8 +Introduction +• a custom training methodology tailored for face recognition with masks, aiming to promote +the use of unmasked parts of the face and close the performance gap relative to typical face +recognition (Chapter 10); +• a study using interpretability tools to understand the use of face image information in pre- +sentation attack detection (PAD), alongside a discussion on the need for explainability and +transparency in biometric recognition (Chapter 11). +In Part IV, Wellbeing Monitoring: +• an approach for classifying emotion valence in groups of people, based on the late fusion of +parallel visual and audio data streams using deep neural networks (Chapter 12); +• a cascade strategy for improved efficiency in continuous audiovisual activity recognition +and violence detection inside vehicles (Chapter 13). +In Part V, Broader Topics on Biometrics and Pattern Recognition: +• the Secure Triplet Loss, a training methodology that promotes template cancelability and +unlinkability, alongside identity discrimination, on end-to-end biometric algorithms without +the need for separate encryption or hashing processes (Chapter 14); +• a methodology for self-supervised learning based on the triplet loss, taking advantage of +the nature of balanced multiclass datasets, especially those composed of sequential data, for +more adequate learning of the target tasks (Chapter 15). +These research works may be categorised according to the specific contributions by the author +of this thesis. The presented work in ECG-based biometric identification (Chapter 4), authenti- +cation (Chapter 5), and explainability (Chapter 7), as well as emotion recognition (Chapter 12), +activity recognition (Chapter 13), biometric template security (Chapter 14), and self-supervised +learning (Chapter 15) comprise the main contributions, resulting completely or mostly from the +work performed by the author. The research on long-term performance in ECG biometrics (Chap- +ter 6), ECG interlead conversion (Chapter 8), masked face recognition (Chapter 10), and inter- +pretability for face PAD (Chapter 11) are secondary contributions that resulted from collaborations +within the scope of this thesis and benefitted partially from the work of the author. Further details +on the specific contributions can be found at the start of each chapter. Other topics have been +addressed which are not presented in this thesis, due to the relevance of the respective topics or +the relative weight of the author’s contributions. They are, however, also mapped in Fig. 1.1 and +listed in the sections below. +1.4 +Dissemination +The contributions of the doctoral research to biometrics, wellbeing monitoring, and broader topics +described in this thesis have been disseminated as part of twenty-four scientific publications. These +are (clustered by type and in reverse chronological order): + +1.4 Dissemination +9 +• Articles in journals: +5. S. Beco, J. R. Pinto, and J. S. Cardoso, “Electrocardiogram Lead Conversion from +Single-Lead Blindly-Segmented Signals,” BMC Medical Informatics and Decision +Making, 22: 314, 2022. [31] +4. T. Esteves, J. R. Pinto, P. M. Ferreira, P. Costa, L. A. Rodrigues, I. Antunes, G. Lopes, +P. Gamito, A. Abrantes, P. M. Jorge, A. Lourenço, A. F. Sequeira, J. S. Cardoso, and +A. Rebelo, “AUTOMOTIVE: A case study on AUTOmatic multiMOdal drowsiness +detecTIon for smart VEhicles,” IEEE Access, 9: 153678–153700, 2021. [116] +3. A. F. Sequeira, T. Gonçalves, W. Silva, J. R. Pinto, and J. S. Cardoso, “An Exploratory +Study of Interpretability for Face Presentation Attack Detection,” IET Biometrics, 10 +(4): 441–455, 2021. [386] +2. J. R. Pinto, M. V. Correia, and J. S. Cardoso, “Secure Triplet Loss: Achieving Cance- +lability and Non-Linkability in End-to-End Deep Biometrics,” IEEE Transactions on +Biometrics, Behavior, and Identity Science, 3 (2): 180–189, 2021. [347] +1. J. R. Pinto, J. S. Cardoso, and A. Lourenço, “Evolution, Current Challenges, and Fu- +ture Possibilities in ECG Biometrics,” IEEE Access, 6: 34746–34776, 2018. [343] +• Articles in international conference proceedings: +11. J. R. Pinto, P. Carvalho, C. Pinto, A. Sousa, L. G. Capozzi, and J. S. Cardoso, “Stream- +lining Action Recognition in Autonomous Shared Vehicles with an Audiovisual Cas- +cade Strategy,” in 17th International Conference on Computer Vision Theory and Ap- +plications (VISAPP), Feb. 2022. [348] +10. P. C. Neto, F. Boutros, J. R. Pinto, N. Damer, A. F. Sequeira, J. S. Cardoso, “Focus- +Face: Multi-task Contrastive Learning for Masked Face Recognition,” in Workshop on +Face and Gesture Analysis for COVID-19 (FG4COVID19), Dec. 2021. [308] +9. P. C. Neto, F. Boutros, J. R. Pinto, M. Saffari, N. Damer, A. F. Sequeira, and J. S. Car- +doso, “My Eyes Are Up Here: Promoting Focus on Uncovered Regions in Masked +Face Recognition,” in International Conference of the Biometrics Special Interest +Group (BIOSIG 2021), Sep. 2021. [309] +8. F. Boutros, N. Damer, J. Kolf, K. Raja, F. Kirchbuchner, R. Ramachandra, A. Kui- +jper, P. Fang, C. Zhang, F. Wang, D. M. Martin, N. Aginako, B. Sierra, M. Nieto, +M. E. Erakin, U. Demir, H. Ekenel, A. Kataoka, K. Ichikawa, S. Kubo, J. Zhang, M. +He, D. Han, S. Shan, K. Grm, V. Struc, S. Seneviratne, N. Kasthuriarachchi, S. Ras- +nayaka, P. C. Neto, A. F. Sequeira, J. R. Pinto, M. Saffari, and J. S. Cardoso, “MFR +2021: Masked Face Recognition Competition,” in International Joint Conference on +Biometrics (IJCB 2021), Aug. 2021. [50] + +10 +Introduction +7. J. R. Pinto, T. Gonçalves, C. Pinto, L. Sanhudo, J. Fonseca, F. Gonçalves, P. Carvalho, +and J. S. Cardoso, “Audiovisual Classification of Group Emotion Valence Using Ac- +tivity Recognition Networks,” in Fourth IEEE International Conference on Image Pro- +cessing, Applications and Systems (IPAS 2020), Dec. 2020. [346] +6. J. R. Pinto and J. S. Cardoso, “Explaining ECG Biometrics: Is It All In The QRS?,” +in International Conference of the Biometrics Special Interest Group (BIOSIG 2020), +Sep. 2020. [339] +5. J. R. Pinto and J. S. Cardoso, “Self-Learning with Stochastic Triplet Loss,” in Interna- +tional Joint Conference on Neural Networks (IJCNN 2020), Jul. 2020. [340] +4. J. R. Pinto, J. S. Cardoso, and M. V. Correia, “Secure Triplet Loss for End-to-End +Deep Biometrics,” in 8th International Workshop on Biometrics and Forensics (IWBF +2020), Apr. 2020. [345] +3. A. F. Sequeira, W. Silva, J. R. Pinto, T. Gonçalves, and J. S. Cardoso, “Inter- +pretable Biometrics: Should We Rethink How Presentation Attack Detection is Eval- +uated?,” in 8th International Workshop on Biometrics and Forensics (IWBF 2020), +Apr. 2020. [385] +2. J. R. Pinto and J. S. Cardoso, “An End-to-End Convolutional Neural Network for +ECG-Based Biometric Authentication,” in 10th IEEE International Conference on +Biometrics: Theory, Applications and Systems (BTAS 2019), Sep. 2019. [338] +1. G. Lopes, J. R. Pinto, and J. S. Cardoso, “Don’t You Forget About Me: A Study on +Long-Term Performance in ECG Biometrics,” in IbPRIA 2019: 9th Iberian Confer- +ence on Pattern Recognition and Image Analysis, Jul. 2019. [269] +• Short papers presented in international conferences: +1. S. Beco, J. R. Pinto, and J. S. Cardoso, “Interlead Conversion of Single-Lead Blindly- +Segmented Electrocardiogram Signals,” in 17th International Conference on Com- +putational Intelligence Methods for Bioinformatics and Biostatistics (CIBB 2021), +Nov. 2021. [30] +• Chapters in books: +1. J. R. Pinto, J. S. Cardoso, and A. Lourenço, “Deep Neural Networks for Biomet- +ric Identification Based on Non-Intrusive ECG Acquisitions,” in K. V. Arya and +R. S. Bhadoria, Eds., The Biometric Computing: Recognition and Registration, CRC +Press, 2019. [344] +• Encyclopaedia entries: +1. J. R. Pinto and J. S. Cardoso, “ECG Biometrics,” in S. Jajodia, P. Samarati, and +M. Yung, Eds., Encyclopedia of Cryptography, Security and Privacy, Springer, +2021. [341] + +1.4 Dissemination +11 +• Abstracts in national conference proceedings: +5. J. R. Pinto and J. S. Cardoso, “xECG: Using Interpretability to Understand Deep ECG +Biometrics,” in 27th Portuguese Conference on Pattern Recognition (RECPAD 2021), +Nov. 2021. +4. J. R. Pinto, M. V. Correia, and J. S. Cardoso, “Achieving Cancellability in End-to-End +Deep Biometrics with the Secure Triplet Loss,” in 26th Portuguese Conference on +Pattern Recognition (RECPAD 2020), Oct. 2020. +3. W. Silva, J. R. Pinto, T. Gonçalves, A. F. Sequeira, and Jaime S. Cardoso, “Explain- +able Artificial Intelligence for Face Presentation Attack Detection,” in 26th Portuguese +Conference on Pattern Recognition (RECPAD 2020), Oct. 2020. +2. G. Lopes, J. R. Pinto, J. S. Cardoso, and A. Rebelo, “Long-Term Performance of a +Convolutional Neural Network for ECG-Based Biometrics,” in 25th Portuguese Con- +ference on Pattern Recognition (RECPAD 2019), Oct. 2019. +1. J. R. Pinto, J. S. Cardoso, and A. Lourenço, “Improving ECG-Based Biometric Iden- +tification Using End-to-End Convolutional Networks,” in 24th Portuguese Conference +on Pattern Recognition (RECPAD 2018), Oct. 2018. +Beyond the aforementioned publications, the author has contributed to fourteen other scientific +publications related to diverse pattern recognition and computer vision research topics not covered +in this thesis. These are listed below: +• Articles in journals: +4. P. C. Neto, T. Gonçalves, J. R. Pinto, W. Silva, A. F. Sequeira, A. Ross, and J. S. +Cardoso, “Explainable Biometrics in the Age of Deep Learning,” ACM Computing +Surveys, 2022. [311] (submitted) +3. L. G. Capozzi, V. Barbosa, C. Pinto, J. R. Pinto, A. Pereira, P. M. Carvalho, and J. S. +Cardoso, “Toward Vehicle Occupant-Invariant Models for Activity Characterization,” +IEEE Access, 10: 104215–104225, 2022. [64] +2. P. C. Neto, J. R. Pinto, F. Boutros, N. Damer, A. F. Sequeira, and J. S. Cardoso, “Be- +yond Masks: On the Generalization of Masked Face Recognition Models to Occluded +Face Recognition,” IEEE Access, 10: 86222–86233, 2022. [312] +1. S. P. Oliveira, J. R. Pinto, T. Gonçalves, R. C. Marques, M. J. Cardoso, H. P. Oliveira, +and J. S. Cardoso, “Weakly-Supervised Classification of HER2 Expression in Breast +Cancer Haematoxylin and Eosin Stained Slides,” Applied Sciences, 10 (14): 4728, +2020. [321] +• Articles in international conference proceedings: + +12 +Introduction +6. P. C. Neto, F. Boutros, J. R. Pinto, N. Damer, A. F. Sequeira, J. S. Cardoso, M. +Bengherabi, A. Bousnat, S. Boucheta, N. Hebbadj, B. Yahya-Zoubir, M. E. Erakın, +U. Demir, H. K. Ekenel, P. B. Q. Vidal, and D. Menotti, “IJCB OCFR 2022: Compe- +tition on Occluded Face Recognition From Synthetically Generated Structure-Aware +Occlusions,” in International Joint Conference on Biometrics (IJCB 2022), Oct. 2022. +(in press) [310] +5. L. G. Capozzi, P. Carvalho, A. Sousa, C. Pinto, J. R. Pinto, and J. S. Cardoso, “Impact +of visual noise in activity recognition using deep neural networks - an experimen- +tal approach,” in 2nd International Conference on Pattern Recognition and Machine +Learning (PRML 2021), Jul. 2021. [63] +4. L. G. Capozzi, J. R. Pinto, J. S. Cardoso, and A. Rebelo, “End-to-End Deep Sketch- +to-Photo Matching Enforcing Realistic Photo Generation,” in 25th Iberoamerican +Congress on Pattern Recognition (CIARP’21), May 2021. [60] +3. L. G. Capozzi, J. R. Pinto, J. S. Cardoso, and A. Rebelo, “Optimizing Person Re- +Identification using Generated Attention Masks,” in 25th Iberoamerican Congress on +Pattern Recognition (CIARP’21), May 2021. [61] +2. A. Matta, J. R. Pinto, and J. S. Cardoso, “Mixture-Based Open World Face Recog- +nition,” in 9th World Conference on Information Systems and Technologies (World- +CIST’21), Apr. 2021. [290] +1. W. Silva, J. R. Pinto, and J. S. Cardoso, “A Uniform Performance Index for Ordinal +Classification with Imbalanced Classes,” in International Joint Conference on Neural +Networks (IJCNN 2018), Jul. 2018. [396] +• Abstracts in national conference proceedings: +4. L. Capozzi, J. R. Pinto, J. S. Cardoso, and A. Rebelo, “Sketch-to-Photo Matching En- +forcing Realistic Rendering Generation,” in 27th Portuguese Conference on Pattern +Recognition (RECPAD 2021), Nov. 2021. +3. S. P. Oliveira, J. R. Pinto, T. Gonçalves, H. P. Oliveira, and Jaime S. Cardoso, “IHC +Classification in Breast Cancer H&E Slides with a Weakly-Supervised Approach,” in +26th Portuguese Conference on Pattern Recognition (RECPAD 2020), Oct. 2020. +2. P. Costa, P. Silva, J. R. Pinto, A. F. Sequeira, and A. Rebelo, “Face Anti Spoofing: +Handcrafted and Learned Features for Face Liveness Detection,” in 25th Portuguese +Conference on Pattern Recognition (RECPAD 2019), Oct. 2019. +1. J. R. Pinto and J. S. Cardoso, “Fine Segmentation of Head and Torso Using Label Re- +finement Networks,” in 25th Portuguese Conference on Pattern Recognition (RECPAD +2019), Oct. 2019. +The research conducted during these doctoral studies has also been partially presented to the +scientific community at the Doctoral Consortium of the 2019 IEEE International Conference on + +1.5 Collaborations +13 +Biometrics: Theory, Applications and Systems (Tampa, FL, USA) and at the 2020 International +Summer School for Advanced Studies on Biometrics for Secure Authentication (Alghero, Italy). +1.5 +Collaborations +The doctoral work presented in this thesis included close collaborations with researchers from +several institutions within the AUTOMOTIVE, Easy Ride, and Aurora projects. The author also +collaborated frequently with VCMI group colleagues within their research work and master disser- +tations related to diverse biometrics, pattern recognition, and computer vision topics, as described +below. +1.5.1 +Research projects +The AUTOMOTIVE project1 was focused on ushering in the next generation of driver drowsiness +monitoring technologies. Led by the VCMI research group at INESC TEC, this project featured +the participation of CardioID Technologies, Instituto Superior de Engenharia de Lisboa (ISEL), +and Universidade Lusófona de Humanidades e Tecnologias (ULHT). The author of this thesis has +participated in the AUTOMOTIVE project from June 2019 to its conclusion in November 2021. +He mainly contributed to the development of novel algorithms for ECG-based biometric recogni- +tion to enable personalised drowsiness models. The results of this project are nicely summed up +in [116]. +The Easy Ride project2, under the motto “Experience is Everything”, was a large effort led +by Bosch Car Multimedia and Universidade do Minho aiming to improve passenger comfort and +safety in autonomous shared vehicles. It featured INESC TEC’s participation in the SP5 sub- +project, in close cooperation with Bosch Car Multimedia, which focused on occupant emotional +monitoring. The author of this thesis has participated in writing this project’s proposal, and then +in the research work from its start in February 2020 to its conclusion in December 2021. He has +mainly contributed to the design, implementation, and evaluation of efficient multimodal deep +learning algorithms using audio and RGB video for in-vehicle emotion, activity, and violence +recognition. The work encompassed all stages of research and development from scratch to in- +vehicle deployment and is presented in [346; 348]. +The Aurora project inherited Easy Ride’s goals of bringing forth the future of shared autono- +mous vehicles. Specifically, it brought together Bosch Car Multimedia and INESC TEC’s Cen- +tre for Telecommunications and Multimedia (CTM) and the High-Assurance Software Labora- +tory (HASLab) to continue SP5’s mission of contributing towards accurate and efficient occupant +emotional and activity monitoring. The author of this thesis participated in this project since its +1AUTOMOTIVE (“POCI-01-0145-FEDER-030707”) was financed by the European Regional Development Fund +(ERDF) through the Operational Programme for Competitiveness and Internationalisation (COMPETE 2020 Pro- +gramme), and by national funds through the Portuguese funding agency, Fundação para a Ciência e a Tecnologia +(FCT). +2Easy Ride (“POCI-01-0247-FEDER-039334”) was supported by European Structural and Investment Funds in the +FEDER component, through the Operational Competitiveness and Internationalisation Programme (COMPETE 2020). + +14 +Introduction +beginning in December 2021, having collaborated on the definition of the project’s objectives and +task planning. +1.5.2 +Synergies within the research group +Within the VCMI research group, the author collaborated, in 2018, with Wilson Silva to develop a +metric for ordinal classification which takes into account both accuracy and label ranking while re- +taining robustness to class imbalance [396]. In 2020, the author collaborated with Sara P. Oliveira, +Tiago Gonçalves, and Hélder Oliveira, alongside Rita C. Marques and Maria João Cardoso at +Champalimaud Foundation (Lisboa, Portugal), on a weakly-supervised methodology based on +multiple instance learning to classify HER2 expression in breast cancer histology slides [321]. +Since 2020, the author has also been collaborating with Ana F. Sequeira, Wilson Silva, +Tiago Gonçalves, and Pedro C. Neto, alongside Arun Ross from Michigan State University +(East Lansing, MI, USA) to study biometrics from the perspective of interpretability, rethink- +ing how model accuracy should be measured, and calling for more transparent biometric algo- +rithms [311; 385; 386]. +In 2021, the author collaborated with Leonardo G. Capozzi and Ana Rebelo in their work +related to person re-identification and scene geolocation for automatic missing person search- +ing [60; 61]. Also in 2021 and 2022, the author has collaborated with Pedro C. Neto, Mohsen +Saffari, and Ana F. Sequeira, alongside Fadi Boutros and Naser Damer at Fraunhofer IGD (Darm- +stadt, Germany), on novel strategies for masked face recognition to uphold state-of-the-art bio- +metric performance amidst a global pandemic [50; 308; 309]. +1.5.3 +Organisation of scientific events +This doctoral project and the aforementioned collaborations and synergies motivated the contribu- +tion to the organisation of multiple scientific events. +In regards to biometrics, the main example is that of the 2020 edition of the International +Workshop on Biometrics and Forensics (IWBF), organised by INESC TEC and NTNU, where the +author of this thesis collaborated as Demo Chair. He has also helped organise the Workshop on +Explainable & Interpretable Artificial Intelligence for Biometrics (xAI4Biometrics), hosted yearly +at the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), as Publicity +Chair in 2021 and 2022 and Programme Committee member in 2023. +The collaboration with Pedro C. Neto and Ana F. Sequeira from INESC TEC and Fadi Boutros +and Naser Damer from Fraunhofer IGD was enhanced by the co-organisation of the Advanced +Occluded Face Recognition (OCFR) competition at the 2022 International Joint Conference on +Biometrics (IJCB) [310]. +Aligned with the topic of wellbeing monitoring, and as an extension to the Easy Ride and Au- +rora projects, the author has also co-organised the In-Vehicle Sensing and Monitorization Work- +shop (ISM). This first edition of the workshop was hosted at the European Conference on Com- +puter Vision (ECCV) in October 2022. + +1.5 Collaborations +15 +In a broader scope, the author of this thesis has also co-organised the special session on Ma- +chine Learning in Healthcare Informatics and Medical Biology at the International Conference +on Computational Intelligence Methods for Bioinformatics and Biostatistics (CIBB), in 2019 and +2021. He was also a member of the Technical and Programme committee of this conference for +its 2021 edition. +Since 2019, the author has also helped organise the VISUM Summer School on computer vision +and machine intelligence. In the 2019 and 2020 editions, he was a member of the project team, +while in 2021 and 2022 he was part of the main organisation team. +1.5.4 +Supervision of dissertations and internships +The author of this thesis has collaborated, as co-supervisor or external supervisor, on the following +master dissertations related to his doctoral studies (in reverse chronological order): +13. Mariana Silva Xavier (2022), “Inside Out: Fusing ECG and Face Information to Recognise +Emotions”, Master in Bioengineering, Universidade do Porto - as co-supervisor, alongside +Jaime S. Cardoso (supervisor) [464]; +12. Guilherme Augusto Tiritan Romano Barbosa (2022), “Going 2D: Exploring Learnable Bidi- +mensional Approaches for ECG Biometrics”, Master in Bioengineering, Universidade do +Porto - as co-supervisor, alongside Jaime S. Cardoso (supervisor) [26]; +11. Pedro Duarte da Cunha Nunes Lopes (2022), “Deep Neural Networks for Face-based +Emotion Recognition”, Master in Bioengineering, Universidade do Porto - as external +institution supervisor, alongside Jaime S. Cardoso (supervisor) and Ana F. Sequeira (co- +supervisor) [271]; +10. Erfan Omidvar (2022), “Single-Wrist Electrocardiogram Acquisition Application in +Biometrics”, Master in Biomedical Engineering, Universidade do Porto - as second +co-supervisor, alongside Miguel V. Correia (supervisor) and Duarte Dias (first co- +supervisor) [323]; +9. Vítor Hugo Pereira Barbosa (2022), “Robust occupant action classification in shared au- +tonomous vehicles”, Master in Informatics and Computing Engineering, Universidade do +Porto - as external institution supervisor, alongside Jaime S. Cardoso (supervisor) and Pe- +dro Carvalho (co-supervisor) [27]; +8. Telma Sofia Caldeira Esteves (2021), “Sleepy Drivers: Drowsiness Monitoring Using ECG +and Face Video”, Master in Biomedical Engineering, Universidade Nova de Lisboa - as +external institution supervisor, alongside Ricardo Vigário (supervisor), André Lourenço (co- +supervisor), and Ana Rebelo (external institution supervisor) [115]; +7. Sofia Cardoso Beco (2021), “Make My Heartbeat: Generation and Interlead Conversion of +ECG Signals”, Master in Bioengineering, Universidade do Porto - as co-supervisor, along- +side Jaime S. Cardoso (supervisor) [29]; + +16 +Introduction +6. Inês Alexandra Teixeira Antunes de Magalhães (2021), “Feel My Heart: Emotion Recog- +nition Using the Electrocardiogram”, Master in Bioengineering, Universidade do Porto - as +co-supervisor, alongside Jaime S. Cardoso (supervisor) [285]; +5. Arthur Johas Matta (2020), “Open-World Face Recognition”, Master in Informatics and +Computing Engineering, Universidade do Porto - as co-supervisor, alongside Jaime S. Car- +doso (supervisor) [291]; +4. Leonardo Gomes Capozzi (2020), “Face Recognition For Forensic Applications: Methods +for Matching Facial Sketches to Mugshot Pictures”, Master in Informatics and Comput- +ing Engineering, Universidade do Porto - as co-supervisor, alongside Ana Rebelo (supervi- +sor) [62]; +3. João Manuel Guedes Ferreira (2020), “Head Pose Estimation for Facial Biometric Re- +cognition Systems”, Master in Informatics and Computing Engineering, Universidade do +Porto - as co-supervisor, alongside Ana F. Sequeira (supervisor) and Jaime S. Cardoso (co- +supervisor) [130]; +2. Carolina Martins Barbosa Rodrigues Afonso (2020), “Changing Perspectives: Interlead +Conversion in Electrocardiographic Signals”, Master in Network and Information Systems +Engineering, Universidade do Porto - as co-supervisor, alongside Miguel Coimbra (super- +visor) [5]; +1. Gabriel Carneiro Lopes (2019), “Don’t You Forget About Me: Enhancing Long Term Per- +formance in Electrocardiogram Biometrics”, Master in Bioengineering, Universidade do +Porto - as co-supervisor, alongside Jaime S. Cardoso (supervisor) [270]. +Beyond these aforementioned dissertations, the author of this thesis also collaborated in the +supervision of more than twenty students on curricular, extra-curricular, and summer internships +related to biometrics, pattern recognition, and computer vision topics. +1.6 +Awards and Distinctions +Beyond the aforelisted peer-reviewed publications and presentations to the scientific community, +the doctoral research work presented in this dissertation has also been the recipient of multiple +awards and distinctions. These are listed below: +• The author of this thesis was granted the EAB Max Snijder Award at the European Biomet- +rics Awards 2022 organised by the European Association for Biometrics (EAB). This award +recognised the wider perspective and applicability of his work on ECG biometrics, which +contributed towards a complete deep learning solution encompassing end-to-end models, +learnable template security, and explainability; + +1.7 Document Structure +17 +• The initial work on the Secure Triplet Loss [345], focused on biometric template cancela- +bility for end-to-end deep models, received the Computers Journal Best Paper Award at the +2020 International Workshop on Biometrics and Forensics (IWBF); +• The work on audiovisual group emotion valence recognition [346] received the Best Session +Paper Award at the 2020 IEEE International Conference on Image Processing, Applications +and Systems (IPAS); +• The extended work on the Secure Triplet Loss [347], presented at the 2021 NIS Workshop +organised by INESC TEC’s Networked Intelligent Systems cluster, received the Best Pre- +sentation Award by the official jury. +1.7 +Document Structure +This thesis is composed of six parts. Part I is the prologue, which includes this introduction, Chap- +ter 1, and offers an overview of the fundamental concepts related to biometric systems, biometric +traits, and wellbeing monitoring in Chapter 2. +Part II focuses on electrocardiogram biometrics, presenting the contributions to this topic pro- +duced during the doctoral work. It begins with an overview of the existing data, related literature +methodologies, and a discussion of open challenges and opportunities, in Chapter 3. Chapter 4 +presents our study on end-to-end deep learning for ECG-based identification, including tailored +unidimensional data augmentation strategies. Chapter 5 showcases the work on triplet loss and +transfer learning for ECG-based identity verification. A study on long-term performance and +template update for identification is presented in Chapter 6. Chapter 7 delves into the topic of +explainability for ECG biometrics, aiming to better understand which parts of the signal are best +for identification. Finally, Chapter 8 proposes a methodology for recovering missing ECG leads +based on blindly-segmented single-lead segments. +Part III deals with face biometrics. Chapter 9 offers an overview of the data, existing method- +ologies, and open challenges in this topic. Two methodologies to close the performance gap +in masked face recognition are presented in Chapter 10. Chapter 11 describes a study on inter- +pretability to understand the decisions of deep learning models in face presentation attack detection +and motivate a more widespread usage of interpretability for more transparent biometrics. +Part IV is centred on wellbeing monitoring and covers the topics of emotion recognition, activ- +ity recognition, and violence detection. Chapter 12 describes an audiovisual approach developed +to classify emotion valence in groups of people. Chapter 13 presents an adaptation of the afore- +mentioned approach for activity recognition and violence detection, alongside a cascade strategy +for increased efficiency in in-vehicle scenarios. +Part V covers broader topics related to biometrics and pattern recognition which have been +addressed during the doctoral work. Specifically, Chapter 14 introduces the Secure Triplet Loss, a +novel approach to ensure biometric template security on end-to-end deep learning models. Lastly, + +18 +Introduction +a methodology for self-supervised learning formulated for minimal performance gaps when using +sequential data is presented in Chapter 15. +Part VI is the epilogue, which concludes this thesis. It includes an overview of the conducted +work and the conclusions drawn from it, in Chapter 16. Chapter 17 offers a discussion on future +work opportunities related to the results of this doctoral thesis. + +Chapter 2 +Fundamental Concepts +Biometric systems are, in several ways, different from other pattern recognition applications. The +need for storage of personal data from users and the different modes on which the systems can +operate are only some of the special characteristics of biometric systems. When developing one, +one should be aware of these specificities to ensure the best performance and robustness. +Hence, this chapter presents the fundamental concepts needed to build a biometric system, +either for identity recognition or the monitoring of wellbeing parameters. It includes an overview +of the general structure and operation of biometric systems, their security vulnerabilities, the dif- +ferent biometric traits (with a special focus on the electrocardiogram and face), and the metrics for +thorough performance evaluations. +2.1 +Biometric Systems +2.1.1 +General structure +2.1.1.1 +Biometric recognition systems +Biometric recognition systems are tools that use hardware and pattern recognition algorithms to +compare the identity of a user with that of registered individuals based on their attributes (des- +ignated as biometric characteristics or traits). Like traditional identification systems based on +keys, cards, or codes, biometric systems are mostly used for access control to restricted places, +confidential information, or personal data and belongings [96]. +A biometric system is typically composed of an acquisition module, a storage module, and +a biometric algorithm. The algorithm can, in turn, be divided into three modules: quality as- +sessment, feature extraction, and decision (see Fig. 2.1) [46; 201]. These modules are described +below: +• Acquisition: The acquisition module is the interface between the system and the subject and +is responsible for the measurement of the biometric characteristic. The sensors used in this +module should be carefully designed to fit the expected application settings and avoid, as +much as possible, the noise and artefacts from environmental interference; +19 + +20 +Fundamental Concepts +ACQUISITION +QUALITY +ASSESSMENT +FEATURE +EXTRACTION +Enrollment +Recognition +Stored +Data +STORAGE +DECISION +Extraction of +meaningful +attributes +Quality check +and enhancement +of the collected +trait +Acquisition and +quantifcation +of biometric +traits +Identifcation or +acceptance/rejection +of identity claim +Figure 2.1: General structure of a biometric recognition system (from [343], based on [46; 201; +351]). +• Quality Assessment: This module aims to evaluate the quality of the trait measurement and +either accept it in its current form, enhance it to reduce noise and variability effects, or +discard it if the quality is unacceptably low; +• Feature Extraction: The feature extraction module is focused on the processing of the ac- +quired measurements, using pattern recognition tools, to extract the most meaningful at- +tributes of the biometric trait and thus enable a robust decision. The feature extraction pro- +cess should be designed to provide attributes that present high intersubject discrimination +power and low intrasubject variability; +• Decision: This module uses the output from the feature extraction module and the stored +information from registered users to identify the user, or validate or reject their identity +claim. To achieve this, it compares the processed traits of the current user and the registered +individuals; +• Storage: The storage module is typically composed of a database that stores biometric tem- +plates (processed biometric trait measurements) from all individuals registered on the sys- +tem. For security purposes, it can include template protection measures, such as hashing, to +prevent leaks of sensitive personal information. +2.1.1.2 +Wellbeing monitoring systems +Wellbeing monitoring systems are very diverse due to the variety of parameters these can monitor, +which include emotions, stress, fatigue, and health conditions. However, most of these systems +follow a general structure that is very similar to that of most biometric recognition systems. +Just like biometric recognition systems, an acquisition module performs the recording of data, +which are checked and processed by the quality assessment module. After this, the feature extrac- +tion module extracts meaningful attributes from the trait measurements to enable accurate labelling +by the decision module. +Most wellbeing monitoring systems do not require a storage module, as biometric templates +from the specific set of enrolled subjects will not be required, unlike in recognition systems. This +is an advantage in terms of data security and privacy, which will be discussed later. + +2.1 Biometric Systems +21 +Enrollment +True +identity +Biometric +Trait(s) +Database +Identity Verifcation +Claimed +identity +Biometric +Trait(s) +Database +Feature +Extraction +Decision +Claimed identity’s +template +Acceptance/Rejection +Biometric +Trait(s) +Database +All stored +templates +Identity/Rejection +Quality +Check +Feature +Extraction +Quality +Check +Feature +Extraction +Acquisition +Quality +Check +Acquisition +Decision +Acquisition +Figure 2.2: Schematics of the operation of a biometric recognition system in identification and +identity verification modes, and in the enrollment phase (adapted from [337], based on [133; 201; +351; 413]). +2.1.2 +Operation modes +2.1.2.1 +Biometric recognition systems +Depending on the application context and its requirements, a biometric system can either operate +in identification mode or identity verification mode [2; 7; 46; 201; 351]. +In identity verification mode, also commonly called authentication, the biometric system will +receive an identity claim along with the biometric measurement (the current user will claim to +be a specific enrolled individual). Hence, the decision module will only compare the current +measurement with the stored data from the claimed identity, performing a one-vs-one comparison, +and either accept or reject the claim. +In the identification mode, the biometric system will only receive the biometric measurement. +Thus, the decision module performs a one-vs-all comparison between the current biometric mea- +surement and the data stored for each enrolled individual. Ultimately, the system will either assign +one of the enrolled identities (corresponding to the strongest comparison) to the current user or +reject to identify (if no comparison was strong enough). +Additionally, biometric systems include the enrollment phase, which comprises the acquisi- +tion, processing, and storage of a biometric template of a subject for its registration on the system. + +22 +Fundamental Concepts +Acquisition +Processing +Decision +1 +2 +3 +4 +5 +7 +Database +7 +6 +8 +Figure 2.3: Attack points on a biometric system (based on [138; 361]). +After this, the system will be able to correctly perform identification or identity verification when +used by the subject [7; 351]. +2.1.2.2 +Wellbeing monitoring systems +As aforementioned, wellbeing monitoring systems like emotion recognition devices rarely require +the storage of personal information from the users. Hence, they dismiss enrollment phases and +only operate in one mode: inference. In this mode, a trait acquisition is performed by the system, +which will output a corresponding label. The output is composed of discrete categories (e. g., +sad, happy, or angry, in emotion recognition) or continuous scores (e. g., from very awake to very +drowsy, in drowsiness recognition). +2.1.3 +Security and privacy concerns +As key protectors of sensitive data, prized possessions, or restricted locations, biometric re- +cognition systems are a prime target for attackers. +The literature defines eight attack points +(see Fig. 2.3) that sum up the different ways to unlawfully gain access to a biometric sys- +tem [202; 297; 317]. Considering a general structure (that groups the quality assessment and +feature extraction into a single processing module) are described below: +• Type 1 – At the acquisition module: Such attacks are commonly called presentation attacks, +as they consist of physically forcing the biometric system to grant access to the attacker, +either through the use of fake biometric traits (such as fake fingers, voice recordings, or +prerecorded face videos) or even through the physical destruction of the system; +• Type 2 – Between the acquisition and the processing modules: In these attacks, called re- +play attacks, the attackers will target the communication link between the sensor and the +processing module, to steal the trait acquisition. They can then bypass the sensor module +by injecting the stolen trait measurements directly into the processing module; + +2.1 Biometric Systems +23 +• Type 3 – At the processing module: The processing module can be attacked and overridden +by another program, controlled by the attacker, that sends the desired features to the decision +module upon request; +• Type 4 – Between the processing and decision modules: These attacks are similar to replay +attacks. The attacker will target the link between the processing module and the decision +module and steal the features sent between them, to be later injected, bypassing the acquisi- +tion and processing modules; +• Type 5 – At the decision module: Here, the attackers can replace the decision algorithms so +they can generate high matching scores as requested, thus granting them access whenever +desired, or to always output negative decisions, amounting to a denial-of-service attack; +• Type 6 – At the storage module: This consists in exploiting database security flaws to add, +modify, or delete templates, to ultimately grant access to unauthorised individuals or deny +access to enrolled users; +• Type 7 – Between the storage and decision modules: Here, the attackers intercept the com- +munications between the database and the decision module, to steal biometric templates and +replay them later; +• Type 8 – Between the decision module and the application: These attacks consist in the +manipulation of the data transmitted between the decision module and the application, e. g., +to override a rejection decision. +In ECG-based biometrics, security vulnerabilities are still to be adequately addressed. Despite +the pioneer studies of Eberz et al. [110] and Karimian et al. [211], no efforts have yet been devoted +to better protect such systems. In general, biometric systems offer undeniable advantages when +compared with traditional authentication systems, but this would be meaningless if the system +introduced new vulnerabilities that paved the way to successful attacks. Hence, it is very important +to remember the attack points presented above and their specificities throughout all stages of the +development and deployment of a biometric system. +Of all modules, storage is one of the most sensitive, as it stores personal data that could be +used to unlawfully access private information and belongings. These intimate data are not specific +to a single application and can be used by their legitimate owner as a single credential on several +biometric systems (e. g., a user could use fingerprint-based access on two different computers). +This is an obvious vulnerability as, just like using the same password for several online services, +a single security failure can risk the privacy of the user on several applications [200]. +Regardless of how sophisticated the database is, it can still be accessed or hacked by intruders +who exert enough effort. Besides working towards more secure databases, it is paramount to +prepare for possible successful attacks and ensure biometric templates cannot be retrieved in those +cases [186; 200]. Hence, regarding biometric template security, it is important to take into account +the following factors: + +24 +Fundamental Concepts +• Non-Invertibility: The processing module should be designed in a way that eases the cre- +ation of templates from biometric trait acquisitions. However, the retrieval of a close ap- +proximation of the original trait measurement or feature set, from a stored template, should +be difficult and sufficiently time-consuming to render the process unfeasible or unattractive +for attackers; +• Revokability/Cancelability: Keys can be changed when moving to a new house, and users +can easily change e-mail passwords after they have become compromised. However, bio- +metric systems rely on intrinsic personal characteristics, such as facial features or fingerprint +minutiæ, which are very difficult (or even impossible) to be changed. Hence, biometric sys- +tems should include measures that allow the easy invalidation of templates when these have +become compromised. Thus, intruders will be denied access when using those credentials, +but legitimate users will still be allowed to authenticate using their unchanged biometric +trait; +• Unlinkability: For improved performance, a single biometric system can store, separately, +more than one template for each user. The data protection scheme should ensure that the +comparison between stored templates does not enable an attacker to cluster them by iden- +tity. This should also be difficult with templates from the same user in different biometric +systems so that attackers are not able to attack multiple systems with a single stolen tem- +plate [186; 304]. +Besides these factors, the biometric system should also be able to deal with the characteristic +variability of biometric traits and their measurements. Methods like hashing are commonly used +for passwords, but such traditional credentials do not present variability. With biometric systems, +small variations of the input should be considered normal and acceptable (e. g., with the face, +different haircuts or beard styles), and should not influence the final decision. When designing +a secure biometric system, it is necessary (although hard) to find an equilibrium between non- +invertibility, unlinkability, and the controlled acceptance of natural intrasubject variability. +2.2 +Biometric Traits +2.2.1 +General overview +Biometric traits are human attributes that include enough personal information to reliably serve as +the basis for the recognition and discrimination of individuals [9; 213]. According to the identity +information they carry and the performance they can offer, biometric traits can either be considered +hard traits, strong enough to be standalone traits in a reliable biometric system, or soft traits, which +need further traits or information to offer acceptable recognition performance (see Table 2.1). +Traits can be categorised according to their nature, as anatomical, physiological, or behaviour- +al traits. Anatomical traits result from measurements of parts of the human body and include the +fingerprints, the face, and the iris. Physiological traits are those that originate from physiological + +2.2 Biometric Traits +25 +Table 2.1: Main benefits and drawbacks of different biometric traits (from [343], based on [2; +213]). +Trait +Benefits +Drawbacks +Electrocardiogram +(ECG) +Universality +Hidden nature +Simple acquisition +Requires contact +Variability over time +Electroencephalogram +(EEG) +Universality +Hidden nature +Expensive equipment +Vulnerability to noise +Variability over time +Face +Easily measurable +Affordable equipment +Easy circumvention +Depends on face visibility +and lighting +Fingerprint +High performance +Permanent over time +Requires contact +Gait +Easy to measure +Affordable equipment +Low performance +Variability over time +Iris +High performance +Expensive equipment +Palmprint +High measurability +Permanent over time +Requires contact +Photoplethysmogram +(PPG) +Easy to acquire +Hidden nature +Affordable equipment +Low performance +Variability over time +Voice +Affordable equipment +Low performance +events in the body and include the heart rate, facial or hand thermography, the electrocardiogram +(ECG), and the electroencephalogram (EEG). Behavioural traits originate from a person’s actions +or behaviours, such as their gait (walking cadence), their signature, or their voice [2; 9]. +The quality of a biometric trait can be defined, as proposed by Jain et al. [199], through seven +different aspects: +1. Universality: the trait should be present in all subjects using the system; +2. Uniqueness: the trait should include enough personal information to present differences +between all subjects, and thus allow their identification; +3. Permanence: despite the intersubject variability desired (uniqueness), the trait should be suf- +ficiently stable over time (reduced intrasubject variability) to allow the identification through +the comparison of measurements in different instances; +4. Measurability: the trait should be easily and comfortably acquired and digitised, and its +representation should allow easy processing and measurement; + +26 +Fundamental Concepts +5. Performance: a system based on such a trait should meet or exceed the recognition accuracy +requirements, set by the context in which it will be applied; +6. Acceptability: there should be no foreseeable reservations that could make the subjects un- +willing to allow the trait acquisition; +7. Circumvention: the trait should be as hard as possible to mimic or counterfeit, in any way, +to prevent spoofing of the biometric system. +Abo-Zahhad et al. [2] compared sixteen biometric traits according to their compliance with +each of the seven defined qualities. In that comparison, it is possible to verify that the traits +with the lowest overall quality are the behavioural ones (gait, keystroke, signature, and voice) +and phonocardiogram (heart sounds), with low performance, permanence, and distinctiveness. +Low circumvention and universality are also downsides of behavioural traits. The traits with +reported highest overall quality are the DNA, facial thermogram, fingerprint, iris, palm print, and +ECG. Overall, the electrocardiogram excels in most factors, with just two ‘average’ scores (for +collectability and acceptability). +Traits like fingerprint, face, signature, iris, and voice have been the most studied. However, +these have long seen a quick growth and evolution of spoofing methods (methods of counterfeit- +ing a certain user’s trait to unlawfully gain access through the biometric system), which urges +researchers to find more robust alternatives [13; 213]. +Throughout this doctoral work, the focus will be on the electrocardiogram (ECG), an emerging +biometric trait that offers unique advantages regarding inherent liveness, anti-spoofing abilities, +and wellbeing insight. Its performance and robustness drawbacks shall be mitigated through its +fusion with face, a well-established and robust biometric trait. Hence, the next subsections consist +of a presentation of both biometric traits, the ways to measure them, and the variability factors that +provide them with identity and wellbeing information. +2.2.2 +Electrocardiogram +The electrocardiogram (ECG) is a physiological signal generated from the contraction and the +recovery of the heart, that has been gaining traction as a biometric trait [343]. The heart has three +main functions: generate blood pressure to keep blood circulating, route venous and arterial blood +to the respective parts of the body, and regulate blood supply according to the metabolic demands +[428]. To do this, the heart needs to contract and relax its muscle, the myocardium, through the +controlled generation and flow of depolarisation and repolarisation currents [377; 428]. +The measurement of such currents using electrodes placed on the body is designated as elec- +trocardiography and results in the electrocardiographic (ECG) signal. In normal conditions, the +ECG is a cyclic repetition of five easily recognisable deflections: the P, Q, R, S, and T waves (see +Fig. 2.4). A group of these deflections comprises a single heartbeat and each deflection can be +traced back to the phase that originated it [286; 377; 428]. + +2.2 Biometric Traits +27 +DEPOLARISATION +DEPOLARISED +REPOLARISATION +Figure 2.4: The sequence of depolarisation and depolarisation events in the heart, and their rela- +tionship with the different heartbeat waveforms in an ECG signal (from [343], based on [286]). +2.2.2.1 +Acquisition +The configurations used for the acquisition of ECG signals for biometric purposes have +greatly evolved. From the first ECG-based biometric research works, considerable efforts have +been devoted to more usable and comfortable acquisition technologies. This aims to place the +ECG as a more attractive alternative to established biometric traits, mitigating the main disadvan- +tage of the ECG, the obtrusive measurement techniques [343]. +In early ECG-based biometrics research, recordings from the standard 12-lead or Frank leads +were commonly used for the development and evaluation of algorithms [144; 349; 463]. These +are two defined and established configurations of electrodes for standardised and comparable ECG +measurement (see Fig. 2.5), widely used for the diagnosis of cardiac disorders. Authors frequently +selected certain leads for their biometric algorithms, especially Lead I [298; 327; 490] (because of +its higher acceptability due to the electrode placement on the wrists), but also Lead II [233; 234; +303; 332], or chest leads [122; 235; 472]. +Some researchers opted for acquisitions without movement restrictions and with fewer elec- +trodes. Prominent choices in the literature include Holter systems (see Fig. 2.6), which are pre- +pared to acquire ECG signals for several hours while the subjects move and perform their daily +activities. These were first used for ECG biometrics by Shen et al. [389], using ambulatory record- +ings from the MIT-BIH Normal Sinus Rhythm database, acquired for thirty minutes using Holter +equipment. Labati et al. [236, 237] used 24-hour-long Holter acquisitions, from the E-HOL 24h +signal collection, and seized the opportunity to study the effect of ECG variability over time on +identification performance. Similarly, Zhou et al. [493] used a mini-Holter system to continuously +record ECG signals. +Medical and Holter systems are designated as on-the-person acquisition settings. +These +present considerable drawbacks for biometric purposes, mainly concerning user comfort during + +28 +Fundamental Concepts +MEDICAL ACQUISITION SETTINGS +z +x +y +Lead I +Lead II +Lead III +1 +2 +3 +45 +6 +RA +LA +LL +RL +Standard 12-Lead +Confguration +Orthogonal/Frank +Leads +I +E +C A +aVF +aVR +aVL +F +H +Figure 2.5: Medical acquisition settings: electrode placement and leads on the standard 12-lead +configuration and Frank leads (from [343], anterior electrodes depicted in blue, posterior elec- +trodes depicted in lighter blue). +HOLTER ACQUISITION EQUIPMENT +Figure 2.6: Acquisition settings with movement: example of a five-electrode Holter system for +ambulatory recordings (from [343], electrodes depicted in blue). + +2.2 Biometric Traits +29 +OFF-THE-PERSON CONFIGURATIONS +Thumb Button +Electrodes +Index Finger +Electrodes +Metallic Rod +electrodes +Electrodes mounted +on a table +Figure 2.7: Examples of off-the-person ECG acquisition configurations, using thumb electrodes +[69], index finger electrodes [289], metallic rods grabbed by the subjects [32; 33; 261; 390], or +electrodes mounted on a table [394] (from [343], electrodes depicted in blue). +acquisition due to the high number of electrodes and their placement on the chest and legs of +the users. Although allowing for longer acquisitions with movement and activity, Holter acquisi- +tions still require the placement of electrodes on the torso. This significantly reduces acquisition +acceptability and comfort and damages the ECG strength as a biometric trait. +To improve acceptability and acquisition comfort, and get closer to biometric systems deploy- +able in real settings, wet electrodes are being replaced by dry metallic electrodes, their number has +been reduced to two or three, and their placement has been confined to the upper limbs, especially +the on wrists, hands, or fingers (see Fig. 2.7). These acquisition configurations were designated +as off-the-person settings. The first research works in ECG biometrics to use such signals were, to +the best of our knowledge, Molina et al. [298], who used commercial metallic electrodes strapped +to the wrists of the subjects, and Chan et al. [69], who acquired ECG signals using dry button +electrodes held by the subjects in contact with their thumbs. +Since then, ECG signals have been recorded using metallic rod electrodes [32; 33; 261; 390], +and dry metallic electrodes mounted on plaques [275] or attached to the users’ fingers [289], which +offer increased comfort over on-the-person techniques. Nevertheless, off-the-person systems still +require the user to hold the electrodes or deliberately place the fingers or palms over them. This +prevents us from designating them as unconstrained systems, which puts the ECG at a disadvan- +tage over other biometric traits. Besides this, the use of dry electrodes in farther placements makes +the acquisition more vulnerable to interference, thus affecting the quality of the signal [32; 394]. +Recently, some researchers have tried to improve off-the-person configurations and approach +unconstrained settings in ECG biometrics. They aimed to close the gap to real, commercial appli- +cations by developing wearable technologies for ECG acquisition or embedding the sensors into +common objects (see Fig. 2.8). In research, the first example of this highly acceptable acquisition + +30 +Fundamental Concepts +WEARABLES AND SEAMLESS +ACQUISITION +Nymi Band +CardioWheel +miBEAT +Figure 2.8: Wearable and seamless acquisition: examples of surveyed configurations (from [343], +electrodes depicted in blue). +was a sensor pad to be used alongside a computer keyboard, to acquire ECG signals continuously +during computer use [86; 87; 393]. More recently, Zhang et al. [484] have shown it is possible to +acquire ECG signals from a single arm and successfully used them for biometric recognition. +As for commercial applications, the Nymi Band [11], a wearable wristband, acquires the ECG +using two metallic electrodes on its inner and outer surfaces. Identity verification is performed +when the band is put on and the session remains open until the band is taken off. While a ses- +sion is open, the Nymi Band broadcasts an identity signal to authenticate the user in other nearby +systems. The CardioWheel [279] is a steering wheel cover that uses conductive leather for seam- +less and continuous biometric recognition and health monitoring of drivers, focused on automatic +personalisation of driving settings and remote fleet supervision. +The miBEAT [471] is a versatile platform for the simultaneous acquisition of ECG and photo- +plethysmography (PPG) signals, which can be used for seamlessly integrated signal acquisition in +smartphones or tablets. AliveCor provides a set of commercial solutions for easy ECG acquisition +in the Kardia 1 lineup, including the KardiaMobile, to be used with typical smartphones, and the +KardiaMobile Card, a credit card-sized slim single-lead acquisition device with integrated metallic +electrodes. +These recent efforts have brought ECG biometrics closer to viable and unconstrained appli- +cability. However, these newer technologies still require the users to wear certain products or +perform specific actions and need contact with both limbs during acquisition. Besides this, the +quality of the acquired signals is typically very low, because of the loose contact with the subject’s +skin, suffering from wide impedance variations, sensor saturation, and contact loss artefacts. +Some researchers have already started to address these issues. The single-arm acquisition +settings studied by Zhang et al. [484] and the contactless electrodes developed by Chi et al. [75] +1Kardia by AliveCor. Available on: https://www.kardia.com/. + +2.2 Biometric Traits +31 +Figure 2.9: Variability in off-the-person ECG heartbeats from the same subjects (from [337], +individual heartbeats superposed after denoising, amplitude normalisation, and outlier removal). +raise new and inspiring possibilities for wearable ECG devices. For applications based solely on +heart rate, techniques have been proposed to measure it at a distance, using microwave Doppler +sensors [48; 316; 376]. +These efforts pave the way for better ECG acquisition technologies. Such systems could con- +sist of seamlessly integrated biometric systems that can acquire ECG signals at short distances +from one hand of the user, without requiring contact and thus suffering from signal loss. For +wearables, the future could reside in products that can continuously monitor the users’ ECG while +only contacting with one of their wrists, or when inside their pockets separated from the body by +clothes. Despite all efforts devoted so far to ECG biometrics, much work is still needed to reach +true applicability in the form of real, comfortable, and easy-to-use ECG-based biometric systems. +2.2.2.2 +Variability +Although the ECG signals present, in normal conditions, the same deflections for all subjects at all +times, these are characterised by a high degree of variability (see Fig. 2.9). Variability in the ECG +can be designated as intrasubject, the variations between cycles (heartbeats) in the electrocardio- +gram of a single subject, or intersubject, the variations between heartbeats of different subjects. +Intrasubject variations on the ECG signal are mainly explored for health monitoring and medical + +32 +Fundamental Concepts +diagnosis [8; 164; 370], while intersubject variations are especially useful to discriminate between +subjects in biometric recognition. Both intrasubject and intersubject variability can originate from +several factors, such as: +• Heart Geometry: Heart size, cardiac muscle thickness, and the overall shape of the heart +influence the trajectories of electrical currents throughout the heart, the number of muscle +cells that will conduct those currents, and the time to do it across the whole heart. Athletes, +because of intensive physical training, commonly have thicker myocardia, which affects the +ECG with higher voltages in the QRS complex, and lower basal heart rates [171; 172; 441]; +• Individual Attributes: Age, weight, and pregnancy are some individual attributes that can +cause shifts in the heart position and orientation. These shifts will change the orientation of +the electrical current conduction vectors across the heart, meaning the electrodes will detect +the signal from a different perspective, thus altering the ECG waveforms [379]; +• Physical Exercise or Meditation: The duration of, and intervals between the different de- +flections of the heartbeats in an ECG signal, vary with the heart rate. These changes are +especially visible on the interval between the QRS complex and the T wave in situations of +tachycardia (higher heart rates) or bradycardia (lower heart rates). Changes in the heart rate +caused by physical exercise or meditation are reflected on the electrocardiogram [10]; +• Cardiac Conditions: Medical conditions of the heart can also interfere with the dynamics of +electrical pulse conduction and generate variability. In biometrics, one of the most studied +conditions is Arrhythmia, which causes wide variations in the heart rate across time and, +as reported by several researchers, can consistently shrink the performance of ECG-based +biometric systems [9; 369; 472]. +• Posture: Postures such as standing or laying down change the position and shape of internal +organs. The heart is also affected by this, changing its position in the thorax, and thus its +position in reference to the electrode placement, which will cause variations in the collected +ECG signal [379]; +• Emotions and Fatigue: The sympathetic and parasympathetic systems of the autonomous +nervous system work to, respectively, increase or reduce the heart rate. These systems are +under the direct influence of psychological states and thus, under stress, fear and other strong +emotions, fatigue or drowsiness, the heart rate and the ECG signal can be affected [10; 370]; +• Electrode characteristics and placement: The type, the size, and the number of electrodes, +whether they are wet or dry, and the positioning on the chest or limbs, can influence the +dominance of noise on the signal. The mispositioning of electrodes and reversal of leads are +also sources of variability, as they change the perspective of detection of the electrocardio- +graphic signal [171; 379]. + +2.2 Biometric Traits +33 +Figure 2.10: Comparison of face images acquired on the (a) visible light, (b) short-wave infrared, +(c) mid-wave infrared, and (d) long-wave infrared spectra (from [41]). +All the previously presented factors change the morphology of the electrocardiographic sig- +nals acquired from an individual. The first two factors contribute more to intersubject variability +and the biometric potential of the ECG signals. The remaining factors are the main origins of +intrasubject variability, which may undermine the process of biometric recognition, but offer the +ECG information on several health and wellbeing parameters. When considering the acquisition +of ECG for a specific application, whether for medical or biometric recognition, it is paramount +to consider these factors, the way they can ease or difficult the task at hand, and how to mitigate +their negative effects. +2.2.3 +Face +The face may be considered the most intuitive of all biometric traits, since humans use facial +features as the main clues for the identification of other people. Despite the challenges in its +use for biometric recognition, the face offers the unique advantage of being the only trait that +can be acquired using sensors at a considerable distance from the user, possibly without their +knowledge [55; 166]. Below, the ways to measure the face trait and the factors that may affect the +measurement are described and discussed. +2.2.3.1 +Acquisition +Among widespread and inexpensive cameras and sophisticated thermal sensors, the face trait can +be acquired in the following settings [21; 123]: +• Visible Spectrum: This is the most common setting in face biometrics. Visible light, with +a wavelength in [400,750] nm, generally from natural and unconstrained sources in the ap- +plication environment, is reflected by the face of the user and captured by the sensor. Using +the visible spectrum of light has some disadvantages, discussed in subsubsection 2.2.3.2, +but it is commonly less expensive than the alternatives and, thus, the most fitting option for +widespread applications. + +(a) +(b) +(c) +(d)34 +Fundamental Concepts +Figure 2.11: Examples of tridimensional face models (from the Bosphorus 3D Face Data- +base [375]). +• Infrared Spectrum: To overcome some limitations of the use of visible light, some sensors +acquire the face trait using the infrared spectrum. These use thermal cameras that can ob- +tain reliable face images in a much wider range of lighting conditions, capturing the heat +from blood vessels and tissues on the user’s face. Based on the wavelength range, these +can be designated as Near-Infrared (NIR, [750,1400] nm wavelength), Short-Wave Infrared +(SWIR, [1400,3000] nm), Mid-Wave Infrared (MWIR, [3000,8000] nm), or Long-Wave In- +frared (LWIR, [8000,15000] nm). These are compared with visible light images in Fig. 2.10. +Depending on the frequency spectra used, the face trait can be categorised as either anatomical +(visible light), or physiological (infrared spectrum). This is because the former mostly captures +anatomical features of the face, while the latter is much more dependent on physiological factors +that affect blood flow and face heat patterns. +The face trait measurements can be acquired simultaneously with depth information, resulting +in an upgrade of 2D images to 2.5D or 3D. While 2D data only includes colour or greyscale pixel +information, 2.5D data combines that with depth information about the face of the subject, which +is useful to increase recognition accuracy or avoid attack attempts. 3D data is a further step, where +both sources of information are fused to build tridimensional models of the subject’s face (see +Fig. 2.11) [55; 77]. +2.2.3.2 +Variability +The measurement of the face trait is affected by several intrinsic, environmental, or operational +factors (see Fig. 2.12). These can enhance intersubject or intrasubject variability, and thus make + +2.2 Biometric Traits +35 +Figure 2.12: Variability in unconstrained face images of a subject (from [205]). +the recognition task easier or more difficult [4; 21; 55; 102; 103]. These factors are: +• Illumination and Heat Sources: This factor is most relevant for visible spectrum acquisition +settings, as the illumination with visible light will directly affect the light received by the +sensor and change the face image. In infrared images, the influence of illumination can also +be felt, due to the heat generated by light and heat sources, although generally in a lesser +degree than in the visual spectrum; +• Pose Variations: Variations in pitch, roll, and yaw of the subjects head, relative to the sensor, +will result in the capture of facial features in different perspectives, which may encumber +pattern recognition tasks. Moreover, the relative position of those facial features will also +suffer from pose variations. Some authors argue infrared imaging is less susceptible to pose +variations than visible spectrum sensors [135]; +• Facial Expressions: Like pose variations, facial expressions change the relative positions of +the facial features, which may encumber the task at hand. Furthermore, it may create new +features (e. g., wrinkles) or hide facial features that would be needed for the task. As with +the previous factor, facial expressions were found to have less impact in infrared imaging +[135]; +• Stress, Emotions, and Illnesses: Stress, emotions like happiness and fear, or illnesses like +headaches or tooth infections can influence facial expressions. But more than that, they +change the heat patterns on a subject’s head, which will influence the face images captured +by infrared sensors; +• Age: Growing old is generally accompanied by expression wrinkles, grey hair, among other +changes that may affect someone’s appearance and make it more difficult for them to be +recognised; + +36 +Fundamental Concepts +• Cosmetics, Tattoos, Disguises, and Accessories: Makeup, face tattoos, disguises, and acces- +sories like rings or piercings can change, hide, or create new facial features. Recognition +algorithms may not be prepared to deal with this new information and be induced into errors; +• Occlusions and Surrounding Objects: Like disguises and accessories, objects surrounding +the person can sometimes occlude part of the face of the subject, and thus hide certain facial +features that could be key for the recognition task at hand; +• Sensor Quality and Stability: The sensor is another important variability source. Using +different cameras, with different characteristics, will result in different face images. Fur- +thermore, the movement of the cameras (just like the movement of the subject) can cause +blurred images that will disable the retrieval of finer facial features. +Like with the electrocardiogram, it is important to appropriately weigh all these factors when +designing a biometric system. Some will contribute more to intersubject variability, and will be +desirable for biometric recognition, while others will enhance intrasubject variability, and em- +power wellbeing monitoring. Considering the task at hand, it is key to design the acquisition and +processing stages to focus on the factors that fit the application needs. +2.3 +Performance Evaluation +2.3.1 +System design considerations +When designing, developing, and deploying a biometric system, it is important to be aware of +how it will behave in realistic conditions. Hence, designers and developers should consider some +central aspects, as described by Bolle et al. [46]: +• System accuracy: This measures the frequency with which the biometric system makes +correct or wrong decisions; +• Computation time: This corresponds to the time required by the system to output a deci- +sion, starting from the moment the user initiates contact with the acquisition module. This +time depends on the processes of trait acquisition, quality assurance, feature extraction, and +decision, and should be as low as possible, to improve usability. This aspect is especially +important for continuous recognition systems, where decisions should be quickly output and +frequently renewed; +• Exception handling: Because of universality issues and sensor flaws or unavailability, some +users may find the system unable to acquire their biometric traits. Also, software errors +may occur and render the system incapable of adequately performing its function. These +possibilities should be addressed during system design and development, to ensure the ap- +plication works even without the biometric system; + +2.3 Performance Evaluation +37 +• System cost: The system cost includes all expenses related to acquisition and processing +equipment needed, algorithm development and implementation, routine maintenance, and +operational costs. It should be as low as possible, to make the biometric system more af- +fordable; +• Security: Biometric systems decisions can serve as proof of the actions of a certain indi- +vidual, and this, in some settings, can escalate to serious legal implications. Thus, it is +important to minimise the possibility of decision flaws that allow impostors to act under the +identity of an authorised person; +• Privacy: Biometric systems require the storage of templates, consisting in discriminating +information about each of the enrolled individuals. That information, in the interest of +anonymity and security, should be kept as safe as possible, resorting to encryption tech- +niques that allow matching but minimise the possibility of reconstructing the original ac- +quired trait data. +Some conflicts may exist between these aspects. Using more sophisticated approaches (such +as deep learning) often leads to higher system accuracy, but at the expense of higher computational +complexity. Using passwords or credentials as a fallback in case of an exception weakens system +security. Tighter security measures generally lead to more frequent rejection of legitimate users, +which translates into reduced accuracy. +Ultimately, trying to fully cover all aspects will increase costs. Hence, all these aspects should +be carefully analysed and weighted, considering the expected application to get an affordable, +efficient, accurate, secure, and usable system. In the next subsection, the most relevant metrics are +presented, regarding the specific aspect of system accuracy. +2.3.2 +Recognition accuracy measurement +2.3.2.1 +Identity verification +As described before, identity verification consists in accepting or rejecting an identity claim made +by a user based on their biometric trait measurements. As such, there are four outcomes: +1. The claim is true and the system correctly accepts it; +2. The claim is false and the system correctly rejects it; +3. The claim is true, but the system incorrectly rejects it; +4. The claim is false, but the system incorrectly accepts it. +In identity verification, the main goal is to minimise the frequency of outcomes three and +four. Outcome three corresponds to a False Reject or False Non-Match error, and outcome four +corresponds to a False Accept or False Match error. These errors are the foundation for most +identity verification accuracy metrics (see Table 2.2), among which the two most commonly used +are [7; 46; 155]: + +38 +Fundamental Concepts +Table 2.2: Definition of the commonly used metrics for performance evaluation in identification +and identity verification tasks. +Metric +Definition +Identity verification +False Acceptance Rate +FAR(T) = Number of impostor trials with prediction score above T +Total number of impostor trials +False Rejection Rate +FRR(T) = Number of legitimate trials with prediction score below T +Total number of legitimate trials +Equal Error Rate +EER = FAR(T), for T that gives FAR(T) = FRR(T) +Area Under the Curve +AUC = +� 1 +0 1−FRR(FAR(T)) dT +Identification +True Positive Identification Rate +TPIR(R) = No. of trials where one of the strongest R predictions is correct +Total number of trials +Identification Rate +IDR = TPIR(1) = No. of trials where the strongest prediction is correct +Total number of trials +Misidentification Rate +MIDR = 1−IDR = No. of trials where the strongest prediction is incorrect +Total number of trials +• False Acceptance Rate (FAR or FMR, False Match Rate): measures the fraction of trials +where the system accepted the identity claims, even though they were false. It represents +how frequently the system erroneously grants access to impostors. The complement of FAR +is called Convenience; +• False Rejection Rate (FRR or FNMR, False Non-Match Rate): measures the fraction of +trials where the system rejected true identity claims. As such, it measures how frequently +the system denies access to legitimate users. The complement of FRR is designated as +Security. +For a given system, these two metrics commonly depend on the chosen operation point. For +example, in similarity or dissimilarity-based matching algorithms, the criterion for accepting or +rejecting a claim is generally a threshold. Different threshold values will correspond to different +FAR and FRR results. Hence, for a more complete performance evaluation, it is common to use +performance characteristic curves such as the Receiver Operating Characteristic (ROC), which +plots 1−FRR versus FAR for varying threshold values (see Fig. 2.13). +From the ROC curve, it is also common to extract two metrics that combine all results into a +single performance value, easing performance comparison between algorithms. The Equal Error +Rate (EER) is the error that corresponds to the operation point where FAR = FRR and represents +an equilibrium between convenience and security (see Fig. 2.13). The Area Under the Curve +(AUC) measures the area under the ROC curve and serves as a measure of overall quality of a +biometric system. +2.3.2.2 +Identification +In identification, the system does not receive an identity claim. Hence, it will compare the current +biometric measurements with the stored templates to assign one of the enrolled identities to the + +2.3 Performance Evaluation +39 +AUC +FAR + +R +R +F- +1 +EER +0 +1 +1 +Threshold +0 +EER +1 +1 +FAR +FRR +Figure 2.13: Example of a Receiver Operating Characteristic (ROC) curve for an identity veri- +fication system (left) and the evolution of False Acceptance and False Rejection rates with the +threshold value (right) (from [343], adapted from [419], example for a similarity-based matching +method including the Equal Error Rate point and the Area Under the Curve). +subject. Additionally, it can reject to identify when the strongest match is, still, not strong enough, +which is generally asserted based on a threshold (as in identity verification mode). +Thus, in the identification mode, there are five possible outcomes: +1. The subject is enrolled and the system correctly identifies them, granting them access; +2. The subject is enrolled, but the system mistakes their identity for another enrolled subject, +granting them access under a wrong identity; +3. The subject is enrolled, but the system fails to identify them with any enrolled subject, +rejecting access; +4. The subject is not enrolled and the system correctly rejects access; +5. The subject is not enrolled, but the system erroneously identifies them as one of the enrolled +subjects and grants them access. +A good biometric system would maximise the frequency of outcomes 1 and 4 and minimise +the frequency of the remaining outcomes. Most metrics for identification mode (see Table 2.2) +are based on these frequencies [7; 46; 155]. Regarding the situations where the subject is enrolled +(designated as legitimate trials), the most common metrics are: +• True-Positive Identification Rate (TPIR or Hit Rate): For a total number of legitimate trials, +TPIR corresponds to the fraction of those where one of the system’s R strongest predictions +corresponds to the true subject identity. As most other metrics, TPIR depends on the de- +fined threshold, the selected number of top ranks R, and the list of enrolled candidates (in +identification, each enrolled subject is considered a candidate, and TPIR, as most metrics, +varies not only with the size of L but also with the variety and individual characteristics of +each subject in L); + +40 +Fundamental Concepts +• Identification Rate (IDR or Accuracy): IDR corresponds to TPIR when only the single high- +est ranking prediction is considered (R = 1). It corresponds to the fraction of the legitimate +trials where the true identity was the method’s strongest prediction above the threshold. In +the literature, this is one of the most used metrics; +• Reliability: Reliability corresponds to TPIR with R = N (where N is the number of enrolled +subjects), we get the reliability metric. This measures how frequently the true identity sat- +isfies the minimum threshold constraint, regardless of its ranking; +• False-Negative Identification Rate (FNIR or Miss Rate): Represents the fraction of trials +where the true identity does not correspond to one of the R strongest predictions above the +threshold T; +• Reject Rate (RR): This metric pertains to the very specific situations where all identity +predictions stand below the defined threshold T, and the system has no choice but to reject +to identify (RR = 1−TPIR−FNIR); +• Misidentification Rate (MIDR or, commonly, Misclassification Error): MIDR is the comple- +ment of IDR (MIDR = 1−IDR), equivalent to FNIR with R = 1, and measures the fraction +of legitimate trials where the true identity is not the system’s top ranking prediction above +T. +Regarding situations where the subject is not enrolled in the biometric system (designated as +impostor trials), the most common metrics are: +• False Positive Identification Rate (FPIR): It is the fraction of impostor trials where at least +one of the system’s predictions meets the threshold criterion, and the system thus grants +access to the unenrolled subject; +• Selectivity: Similar to FPIR, selectivity counts the average number of predictions above the +threshold T across all impostor trials. +As in identity verification mode, some characteristic curves can be drawn based on these met- +rics and the defined thresholds, to help evaluate the algorithms more robustly. The first is the Cu- +mulative Match Characteristic (CMC), which plots TPIR with threshold T = 0 against R, varying +R ∈ [1,N]. The second is the Receiver Operating Characteristic (ROC) which, in the case of iden- +tification, plots Reliability against FPIR, for various threshold values (see Fig. 2.14). From these +plots, we can also extract the AUC and EER metrics. +2.3.3 +Time-based performance measurement +There are biometric systems that perform recognition upon request, for example in computers or +smartphones, keeping the session open until the user ends it or it reaches an idle time limit. This +creates security issues, specifically when the user leaves the device unattended and forgets to close +the session. + +2.3 Performance Evaluation +41 +AUC +Cumulative Match Characteristic +Receiver Operating Characteristic +R +FPIR +y +tili +b +aile +R +1 +1 +0 +0 +1 +N +Figure 2.14: Examples of a Cumulative Match Characteristic (CMC) curve, and a Receiver Operat- +ing Characteristic (ROC) curve for an identification system (from [337], including a representation +of the Area Under the Curve, AUC). +To solve this issue, there are continuous (or online) biometric systems, that aim to perform +biometric recognition in real-time, acquire traits continuously, and renew decisions as frequently +as possible based on the most recent acquisition. With such systems, if the user leaves and is +replaced by an attacker, the system would be able to detect this and close the session before any +harm could have been done. +Besides accuracy, an important facet of performance evaluation for continuous biometric sys- +tems is timeliness. Sim et al. [400] have addressed this issue and proposed some time-based +metrics for the evaluation of continuous identity verification systems. These can be adapted for +both identification and identity verification, as described below: +• Time to Correct Decision (TCD): TCD measures the time the system takes to detect an +impostor, and take an appropriate decision, relative to the moment the impostor replaces a +legitimate user. For an ideal system, this should be zero, but that is virtually impossible to +achieve. Hence, Sim et al. [400] state that this window should at least be always lower than +W, called Window of Vulnerability (the minimum access time required for the impostor or +wrong individual to cause any kind of damage); +• Probability of Time to Correct Decision (PTCD): PTCD measures the probability, for a +given system, of TCD being lower than W. The higher this value, the lesser the probability +of an impostor having time to cause damage before the system acts; +• Usability: In a normal continuous biometric system, we should expect some decisions to be +incorrect. Over t seconds of usage, Usability measures the fraction of t where the legitimate +user is deprived of access due to wrong decisions of the system. For any biometric system, +this should be as close to zero as possible. + +42 +Fundamental Concepts +AUC +Usability-Security Curve +Security (PTCD) +y +tili +b +a +s +U +1 +1 +0 +Figure 2.15: Example of a Usability-Security characteristic curve (from [337], adapted from Sim +et al. [400]). +From these metrics, the authors also define the Usability-Security curve (USC), a new char- +acteristic curve that plots Usability vs. PTCD for a varying threshold T (see Fig. 2.15). USC is +similar to a ROC curve and, thus, AUC can also be computed, being considered a good metric to +evaluate the timeliness of a biometric system. +2.3.4 +Wellbeing monitoring performance measurement +For wellbeing monitoring systems, several metrics have been used to measure their perfor- +mance [299]. The choices depend on the nature of the ground-truths and system outputs. +For example, emotion recognition or affective computing algorithms generally focus either +on a limited set of discrete emotion categories or on continuous ranges of emotion qualifiers. In +the former case, they typically cluster all emotions onto six categories designated as the six basic +emotions by Ekman [112]: happiness, sadness, anger, fear, surprise, and disgust. +For systems based on categorical labels such as these, the most common metrics are: +• Accuracy: The accuracy is the fraction of test samples that have been correctly classified by +the algorithm. It is widely used in wellbeing monitoring applications; +• F1-score: The F 1-score is the harmonic mean of precision p (the fraction of positive predic- +tions that are truly positive) and recall r (the fraction of positive samples that are classified +as such), through the expression F1 = 2(pr)/(p + r). It is commonly used to evaluate per- +formance in binary tasks; +• AUC: The Area Under the ROC Curve, which plots sensitivity (the fraction of positive sam- +ples correctly classified) versus the complement of specificity (1−specificity, the fraction of +negative samples incorrectly classified as positive) for several thresholds, is also commonly +used for binary classification tasks. + +2.3 Performance Evaluation +43 +Figure 2.16: Illustration of the bidimensional valence-arousal space with example emotion cate- +gories (adapted from [392], with valence on the horizontal axis and arousal on the vertical axis). +In the case of continuous labels, emotion recognition systems typically consider a bidimen- +sional space with two main emotion qualifier variables: valence and arousal [240]. +Valence +measures the pleasure of the emotion being felt, ranging from negative (unpleasant) to positive +(pleasant). Arousal measures the level of activation or intensity of the emotion, ranging from low +(passive) to high (active). Fig. 2.16 illustrates these concepts by offering examples of categorical +emotions in the bidimensional valence-arousal space. +For systems focused on regression tasks such as these, outputting continuous scores, the most +common metrics are: +• Root Mean Squared Error (RMSE): RMSE is the root square of the mean of all squared +differences between corresponding predictions and ground-truths; +• Pearson’s Correlation Coefficient (CC): To correct the limitations of RMSE, the Pearson’s +correlation coefficient uses the covariance between predictions ˆθ and ground-truths θ. It +follows the expression CC = COV{ ˆθ,θ}/(σ ˆθσθ); +• Concordance Correlation Coefficient (CCC): This metric is used for time-series predictions, +based on the Pearson’s correlation coefficient and the mean value of each time series, fol- +lowing the expression ρc = {2CCσ ˆθσθ}/{σ2 +ˆθ + σ2 +θ(µ ˆθ − µθ)2}. This is a very common +metric for performance evaluation of emotion recognition over time; +• Sign Agreement Metric (SAGR): The sign agreement metric combines the magnitude of +the prediction error with a penalisation for sign errors. It can be computed with SAGR = +1 +n ∑n +i=1 δ(sign( ˆθi),sign(θi)), where δ is the Kronecker delta function. This metric has been +mainly used for valence and arousal prediction, where the concordance of signs between the +predictions and the ground-truths can be more important than the magnitude of the scoring +errors. + +ACTIVE +ALARMED +EXCITED +AFRAID +AMUSED +ANGRY +GLAD +PLEASED +NEGATIVE +SAD +SATISFIED POSITIVE +GLOOMY +RELAXED +BORED +TIRED +SLEEPY +PASSIVE44 +Fundamental Concepts +Among these alternatives, it is important to consider the task at hand when selecting metrics +for an adequate performance evaluation that also enables a simple and thorough comparison with +the state-of-the-art. + +Part II +Electrocardiogram Biometrics +45 + + +Chapter 3 +Prior Art in Electrocardiogram +Biometrics +3.1 +Data +Numerous researchers, when working with ECG signals, for biometric recognition purposes or +automatic diagnosis of medical cardiac conditions, opt for private acquisitions of data. However, +as the needs grow for more complete datasets, with more subjects, including medical conditions, +on more sessions, spread across wider time frames, and under different posture and activity con- +ditions, researchers became more aware of the importance of public signal collections [394]. +Moreover, public ECG databases are needed to enable the comparison and benchmarking of al- +gorithms in challenging conditions, across different publications, without requiring authors to im- +plement algorithms and evaluate them again. Below, we delve into the important aspects behind a +well-structured ECG signal collection, we present the most relevant publicly available collections, +and we discuss the current needs and future possibilities regarding data in ECG biometrics. +3.1.1 +Building a complete ECG data collection +A well-structured ECG signal collection is key to appropriately guiding the development towards +the exploitation of the best possibilities for the system, and accurately predicting its performance +upon real-life application. To achieve such a complete collection, a few aspects have to be consid- +ered: +• Number of electrodes: Fewer electrodes and leads have been shown to provide more chal- +lenging settings for biometrics [122; 350]; +• Electrode placement: As shown by [490], the use of chest leads is less challenging than +limb leads, and the distance of the electrodes to the heart has a significant negative impact +on the system’s performance; +47 + +48 +Prior Art in Electrocardiogram Biometrics +• Sampling frequency: Sampling causes the loss of fine details that influence the recognition +process [350]. The lower the sampling frequency, the larger the amount of detail that can +be lost, and the higher the risk of aliasing of high-frequency noise (such as electromyogram +interference); +• Subject posture, activity, and fatigue: Several studies have shown that fatigue, exercise, or +different postures have a negative effect on recognition performance if the systems have not +been trained accordingly [332; 350; 445]; +• Subject health: Some health issues, mainly arrhythmia, can generate intrasubject signal +variability that encumbers the recognition process [95; 96; 369]. Thus, systems should be +made robust against this, by including subjects with heart conditions in the datasets used +during the development and validation of the methods; +• Number of subjects: The diversity of individuals and their own characteristics may ease or +difficult the job of the biometric systems [104; 468], and successful state-of-the-art algo- +rithms have been shown to be significantly worse when evaluated on larger datasets [319]. +The use of a collection with a large number of subjects ensures the presence of subject di- +versity, increasing the thoroughness of the performance assessment. As discussed in [343], +the vast majority of literature in ECG biometrics reports the use of data from less than 100 +subjects; +• Acquisition sessions: The ECG signal varies enough to cause recognition errors in most +biometric systems, even over a short 24-hour period [236; 237]. Systems should be prepared +with data from several sessions, weeks or months apart, to ensure their robustness [350; +393]. +All these factors can have an impact on the performance of an ECG-based biometric system. +In order to correctly assess the capabilities of such systems, it is of the highest relevance to not +only build a database that fits the system’s expected application context, but also one that reflects +all possibilities mentioned above, in order to study the use of the same biometric system in a wider +set of contexts. +3.1.2 +Publicly available data +Currently, there are several collections, publicly available for ECG biometrics research1, which try +to cover the aforementioned factors to create a challenging environment for the development of +robust biometric systems. Many are stored by Physionet2, while others are ceded by their owners +upon request. Below, we present and characterise the most relevant of the currently available +1Some of these databases may require prospective users to contact the respective administrators to request access to +the data and/or sign agreements beforehand. Nevertheless, all presented databases are made available by the creators +for research purposes. +2Physionet ECG databases. Available on: https://www.physionet.org/physiobank/database/#ecg. + +3.1 Data +49 +Table 3.1: Summary of the technical specificities of the most relevant publicly available ECG collections (from [343], OP – off-the-person; NS – number +of subjects; Fs – sampling frequency (Hz); L / E – number of leads/electrodes). +Collection +OP +NS +Fs +Electrode +Placement +L / E +Health +Conditions +Activity/Posture +Sessions +AHA +No +154 +250 +Chest +2 / - +Various +- +3 h +CEBSDB [139] +No +20 +5000 +Chest +2 / - +None +At rest, listening to music +60 min. +CYBHi [394] +Yes +128 +1000 +Palms + Fingers +2 / 4 +None +Reactions triggered by +sound and video +Up to two 5 min. sessions, 3 +months apart +DriveDB [168] +No +9 +456 +Chest +1 / - +- +Rest, highway, and city +driving +50 min. to 1.5 h +ECG-ID [281; 307] +No +90 +500 +Wrists +1 / - +- +Sitting, unrestrained +movement +Various 20 s rec. per subject +over 6 months +E-HOL 24h +No +203 +200 +Chest +3 / 4 +None +Ambulatory recordings +24 h +European ST-T [421] +No +79 +250 +Chest +2 / - +Various +Ambulatory recordings +2 h sessions +FANTASIA [195] +No +40 +250 +- +1 / - +None +Supine, at rest, watching a +movie +120 min. +LTST [196] +No +80 +250 +Chest +2-3 / - +Arrhythmia and +ischaemia +Ambulatory recordings +21-24 h +MIT-BIH +Arrhythmia +[287; 301] +No +47 +360 +Chest +2 / - +None +Ambulatory recordings +30 min. +MIT-BIH NSR +[287; 301] +No +18 +360 +Chest +2 / - +None +Ambulatory recordings +30 min. +Physionet 2017 CinC +Challenge +Yes +8528 +300 +Fingers +1 / 2 +Various +At rest +10-60s +PTB [49] +No +290 +1000 +Chest + Limbs +15 / - +Various +At rest only +1-5 per subject, 38.4-104.2 s +PTB-XL [443; 444] +No +18 885 +500 +Chest + Limbs +12 / 10 +Various +At rest only +1-2 per subject, 10 s +QT [239] +No +105 +250 +Chest +- / - +Various +Rest and exercise +15 min. +UofTDB [445] +Yes +1019 +200 +Fingers +1 / 2 +None +Sit, stand, supine, exercise, +and tripod +Up to six 2-5 min. recordings +over 6 months + +50 +Prior Art in Electrocardiogram Biometrics +ECG collections (see Fig. 3.1 for the number of publications that have used them), and Table 3.1 +summarises the characteristics of each. +• AHA: The AHA ECG database3 was created by the American Heart Association to guide +the training of health professionals on the diagnosis of arrhythmias. It includes 154 ECG +recordings from real patients, donated by various institutions, each three hours long and +composed of two lead signals. The last 30 minutes of each recording are annotated for +seven types of arrhythmia; +• CEBSDB: The Combined measurement of ECG, Breathing and Seismocardiograms (CEB- +SDB) database [139] is a multimodal database available on Physionet. It includes two chan- +nels of ECG (standard leads I and II), thoracic respiratory signals, and seismocardiograms +(SCG) from twenty healthy subjects at rest and in the supine position. Recordings include +50 minutes of classical music listening preceded and followed by five minutes at rest. ECG +signals are sampled at 5 kHz and were acquired with foam tape and gel electrodes; +• CYBHi: The Check Your Biosignals Here initiative4 [394] is a collection of off-the-person +ECG signals acquired with two dry electrodes at the palms, and two electrolycras at the +middle and index fingers. It consists of a short-term dataset, with single-session recordings +of 65 volunteers; and a long-term dataset, where 63 subjects were recorded in two sessions, +three months apart. In each session, for 5 minutes, the subjects were exposed to videos +designed to cause emotional reactions; +• DriveDB: Resulting from the Stress Recognition in Automobile Drivers initiative, this data- +base was created with the purpose of monitoring stress in drivers [168]. Various physiolog- +ical parameters (electrocardiogram, electromyogram, and skin conductivity) were recorded +from 9 subjects over a total of 18 driving sessions, including periods of rest (lower stress +levels), highway driving, and city driving (higher stress levels); +• ECG-ID: The ECG-ID is a database entirely focused on biometrics [281; 307]. 20-second +ECG recordings were collected from 90 subjects, and are currently available on Physionet. +For each subject, the database has between 2 and 20 recordings (a total of 310) collected +over six months. The signals were acquired from Lead I using limb-clamp electrodes at the +wrists; +• E-HOL 24h Holter: This is an ECG database, focused on biometrics, from the University +of Rochester5. A total of 203 healthy subjects were recorded using a Holter monitor for 24 +hours, with four electrodes placed on the chest, from 3 leads following a pseudo-orthogonal +configuration; +3American Heart Association ECG database. Available on: https://www.ecri.org/components/Pages/AHA_ECG_ +USB.aspx. +4CYBHi dataset for off-the-person ECG biometrics. +Available on: +https://zenodo.org/record/2381823# +.YwDLmHbMKMp. +5University of Rochester Medical Center, Telemetric and Holter ECG Warehouse. Database E-HOL-03-0202-003. +Available on: http://thew-project.org/Database/E-HOL-03-0202-003.html. + +3.1 Data +51 +• European ST-T: The European ST-T database [421] was originally intended for the analysis +of ST and T-wave changes. The database is composed of 90 two-hour excerpts of record- +ings from 79 subjects, from 2 leads, and includes abnormalities with origin in myocardial +ischaemia, hypertension, ventricular dyskinesia, and effects of medication; +• FANTASIA: The FANTASIA database [195], available on Physionet6, is composed of 120 +min. recordings of ECG, respiration, and blood pressure signals from forty people (twenty +young and twenty old). All signals were acquired at 250 Hz as the subjects remained at rest, +supine, watching Disney’s Fantasia movie; +• Long-Term ST: The LTST database [196], available on Physionet, includes a variety of +ST segment changes for the development of algorithms for the diagnosis of myocardial +ischaemia. This database includes 86 records from 80 subjects, from ambulatory recordings +between 21 and 24 hours, from two and three leads; +• MIT-BIH Arrhythmia: The MIT-BIH Arrhythmia database [287; 301], one of the most used +in ECG-based biometrics research, is available at the Physionet repository. The database +is composed of a total of 48 signals, 30 minutes long excerpts from ambulatory two-lead +recordings. The 47 subjects were selected to obtain a representation of a wide variety of +arrhythmias; +• MIT-BIH Normal Sinus Rhythm: This database is composed of excerpts from 18 subjects, +from the MIT-BIH Arrhythmia database [287; 301], presented above, deemed to be free +from arrhythmias or other abnormalities; +• Physionet 2017 CinC Challenge: This ECG database is available on the Physionet repos- +itory, and was used for the 2017 Computers in Cardiology (CinC) Challenge, consisting +of arrhythmia detection in short single-lead ECG signals. It includes individual 10-60s +recordings from a total of 8528 subjects, acquired at 300 Hz sampling frequency using the +AliveCor KardiaMobile off-the-person acquisition device; +• PTB: The PTB Diagnostic ECG database [49; 230] includes 549 recordings from 290 heal- +thy subjects and individuals with various cardiac conditions (such as myocardial infarction, +dysrhythmia, hypertrophy, or heart failure). It has 1 to 5 recordings per subject, ranging +between 38.4 and 104.2 seconds, from all 12 standard and 3 Frank leads; +• PTB-XL: From the creators of the PTB Diagnostic ECG database, described above, the PTB- +XL database [443; 444] gathers a very large number of ten-second ECG signals (21 837) +from over eighteen thousand subjects in clinical scenarios. The signals are sampled at 500 +Hz (but also available at 100 Hz) and include the twelve standard leads, annotated by up to +two cardiologists considering diagnostic, form, and rhythm; +6FANTASIA database. Available on: https://physionet.org/content/fantasia/1.0.0/. + +52 +Prior Art in Electrocardiogram Biometrics +AHA +CYBHi +DriveDB +ECG-ID +E-HOL 24h +Euro ST-T +LTST +MIT-BIH Arrh. +MIT-BIH NSR +PTB +Physionet CinC 2017 +QT +UofTDB +0 +2 +4 +6 +10 +8 +12 +14 +16 +22 +5 +Number of publications +NUMBER OF PUBLICATIONS THAT USED EACH ECG COLLECTION +0 +3 +28 +26 +36 +0 +2 +2 +0 +12 +PTB-XL +CEBSDB +6 +0 +18 +22 +24 +26 +30 +28 +32 +34 +36 +20 +12 +FANTASIA +4 +Figure 3.1: Currently available ECG collections and the number of surveyed publications that have +used them (adapted from [343]). +• QT: The QT database aims to aid the development of automatic methods of measurement of +QT waveforms [239]. This collection is a compilation of 105 15-minute relevant recording +extracts from other public databases; +• UofTDB: The University of Toronto ECG Database [445] was specifically created for bio- +metrics and addresses several important criteria for a thorough evaluation of biometric per- +formance. The off-the-person ECG signals were captured using dry electrodes at the thumbs +of a total of 1019 subjects. For each subject, the database includes up to six recordings over +a period of six months, in various postures: supine, tripod, exercise, sitting, and standing. +While many researchers opt to use private acquisitions of data for their studies on ECG bio- +metrics, public datasets have been crucial in allowing the appropriate comparison of results across +publications. Nevertheless, if our goal is to increase competitiveness between ECG-based biomet- +rics and more developed traits, we should address some concerns regarding public collections. +Currently, countries like India, China, and the United States, are starting to invest in nationwide +identification systems for their large populations [198], which awakens the need for biometric +systems that can robustly discriminate between several million enrolled subjects. To keep up with +this trend, we need to work towards the creation of public ECG collections with a larger number of +subjects. PTB-XL is impressive in this regard, as it includes signals from over eighteen thousand +subjects. However, it is still quite limited considering the extent and diversity of data per subject, +which are very important aspects for biometrics. +Moreover, researchers can currently choose from small on-the-person datasets that include + +3.2 Related Work +53 +Figure 3.2: General structure of an ECG-based recognition system (from [344]). +health conditions and longer acquisition times (such as the AHA, European ST-T, and the MIT- +BIH Arrhythmia databases), or the off-the-person UofTDB collection with short recordings from +several healthy subjects. This calls for the creation of a public database with a number of sub- +jects similar or superior to UofTDB, with several longer off-the-person recordings (ideally over +one hour), taken over long periods (months to years), during different activities and postures. +ECG-BG, used by Ingale et al. [187], is composed of off-the-person data from 1119 healthy and +unhealthy subjects, but it is still not publicly available. +Finally, it would also be very beneficial to have publicly available collections of signals ac- +quired using recent wearable and seamless technologies, such as the CardioWheel and the Nymi +Band. The highly acceptable acquisition settings offered by such products places, undoubtedly, +new challenges on signal noise and variability, that would be very useful for the development of +robust biometric algorithms. +3.2 +Related Work +Despite being more recent and less developed than the face or fingerprint, the electrocardiogram +(ECG) is quickly growing as a biometric trait, especially due to its inherent liveness and anti- +spoofing capabilities. +A comprehensive review of prior art in ECG-based biometric recognition is available in [343], +published in the scope of this doctoral work, and succinctly summarised in Table A.1. In this +section, we present a summary of the survey, organised in the four common stages of an ECG- +based recognition algorithm: signal denoising, signal preparation, feature extraction, and decision +(see Fig. 3.2). We also offer a discussion on the approaches based on deep learning and the current +challenges and possibilities in the topic. +3.2.1 +Signal denoising +ECG signals are highly susceptible to interference during the acquisition stage [366]. The ampli- +tude of their waveforms may vary depending on the electrode characteristics and placement but, + +54 +Prior Art in Electrocardiogram Biometrics +under ideal conditions (using chest leads in medical settings), the QRS complex only reaches 2−3 +mV, the largest amplitude of the whole cyclic beat [124]. +This means that when the electrodes are placed far from the heart, the signal is weaker and the +noise is more dominant. This can result in different interference types, such as powerline inter- +ference (PLI) from alternating current energy lines, baseline wander from breathing movements, +electrode movement from motion, lead reversal due to electrode mispositioning, or pacemaker +interference [124; 403]. The stage of signal denoising is, thus, of utmost importance for an ECG +biometric system. +On the first initiatives in ECG biometrics, using on-the-person acquisitions, the signal-to- +noise ratio was higher, and noise sources were mainly limited to powerline interference and base- +line wander. Hence, filters, such as bandpass (BPF), lowpass (LPF), highpass (HPF), or notch +(NF), were the first and have been the most frequent option, due to their simplicity and lower +computational cost. Bandpass filters have been most common, with bands between 1 − 40 Hz +[8; 10; 32; 272], 2 − 40 Hz [193; 366; 369], or 2 − 30 Hz [86–88], aiming to keep most useful +individual information of the ECG while attenuating low and high-frequency noise. +Recently, Choudhary and Manikandan [78] proposed the use of the Discrete Cosine Trans- +form (DCT) for simultaneous removal of baseline wander and powerline interference, which +proved more successful than bandpass filters, when compared on simulated scenarios. The Dis- +crete Wavelet Transform (DWT) has also been proposed for denoising of on-the-person signals +[80; 124; 373], as it allows to decompose the signal into several levels, which may be separately +processed to eliminate noise in certain frequency ranges. +When considering off-the-person approaches, wearables, or seamlessly integrated acquisition +settings, it is reasonable to expect a considerable increase in the noise influence, with a lower +signal-to-noise ratio. The ability to capture the ECG signal weakens, so the amplitude of the ECG +components is smaller when compared with chest leads, and movement artefacts are much more +frequent and dominant [278; 289; 394]. +For these, filters have also been widely applied [272; 289], as well as DWT [170]. However, +the enhanced noise content motivated the proposal of new approaches based on line fitting algo- +rithms, such as fitting of polynomial curves and the Savitzky-Golay algorithm [374]. Their use or +combination with moving average or median filters has been shown more successful than filters or +transform denoising [298; 342], likely because noise is widely present across the ECG frequency +range, and such methods avoid restricting their operation to narrow frequency ranges. +Considering all this, it is possible to conclude that the trend in signal denoising has been +the evolution towards methods that can adapt to increasingly unexpected and dominant noise. +Considering the efforts devoted to more acceptable and comfortable acquisition settings, with an +increasing focus on wearables and seamless settings, it is unreasonable to expect this trend would +be reversed in the near future. +While filters appear to be a wise option if the noise is confined to known frequency ranges +outside the ECG frequency range, for on-the-person signals, transforms (especially DCT) have +shown to be good alternatives for denoising without causing distortions [78]. However, when + +3.2 Related Work +55 +the noise is widespread and/or its frequency range is unpredictable (such as with off-the-person +signals), line fitting algorithms such as the Savitzky-Golay filter may be a better option, as they +smooth the signal without making strong assumptions about its noise content. +Nevertheless, research must continue to work towards increasingly robust and adaptable de- +noising methods. Researchers have recently started to use deep learning methodologies (as dis- +cussed further on in subsection 3.2.5), that have shown remarkable robustness to noise and vari- +ability in several pattern recognition applications [163; 484]. These, along with a deep study of +data augmentation, may result in better alternatives to current and future methods devoted to signal +denoising, and should certainly be explored in depth. +3.2.2 +Signal preparation +ECG biometric algorithms frequently resort to the application of several processing operations +over the acquired ECG signal, between denoising and feature extraction. These have the main goal +to prepare the signal for the feature extraction phase, maximise the performance of the system (by +reducing persistent noise and variability), segmenting specific useful parts of the acquired signal, +and/or discarding undesirable or prejudicial parts [191; 278; 342]. +The noise and variability that may remain after the signal denoising stage, which this stage will +aim to attenuate, are generally segment length and alignment inconsistencies, amplitude variations, +heart rate variability, movement artefacts, and contact loss or impedance artefacts [189; 191; 342]. +To fulfil its objective, this stage generally consists of reference point detection, signal segmen- +tation, amplitude normalisation, time normalisation, and/or outlier detection processes. The most +common in the literature methods are fiducial point detection, signal segmentation, and amplitude +normalisation. Below, an overview of these processes is presented. +• Fiducial Point Detection: To aid posterior processes, such as signal segmentation, the prepa- +ration of the signal for recognition can include a step of detection of heartbeat reference +points, designated as fiducials. The majority of the surveyed research works have used this +technique, varying in the methods used. On the other hand, some researchers have opted +to make their algorithms completely non-fiducial, discarding the processes included in this +stage [8; 170; 349]. +The Pan-Tompkins algorithm [328], based on moving-window integration and adap- +tive thresholding, has been the most frequent choice for fiducial detection [272; 303; +327; 390; 446]. +Alternatives include the Discrete Wavelet Transform (DWT) (used in +[124; 125; 326; 373]), the Trahanias algorithm [436] (based on morphological operations +and adaptive thresholding, used in [298; 342]), and the Engelse-Zeelenberg algorithm +[113; 277] (based on differentiation, negative lobe detection, and adaptive thresholding, used +in [272; 276; 342]). Pinto et al. [342] have applied these methods and found Pan-Tompkins +and Engelse-Zeelenberg gave better results for on-the-person signals, while Trahanias per- +formed better in off-the-person settings. + +56 +Prior Art in Electrocardiogram Biometrics +• Signal Segmentation: Signal segmentation is used to limit the signal span for feature extrac- +tion, or to set a fixed size to ease template matching when the feature is the signal itself. +In some cases, the segmentation uses the fiducial point locations and is used to crop QRS +complexes and/or other waveforms [414; 429; 446]. It can also be used to crop the whole +heartbeat (or a majority of it), thus being performed at fixed distances before and after de- +tected R-peaks or QRS complexes [66; 272; 493]. Some research works included signal +segmentation using sliding windows, with or without overlap, regardless of the complete- +ness of the heartbeat cycles inside it [114; 292; 318]. +The alignment and averaging of various signal segments are closely related to the signal +segmentation process. The alignment is generally performed using the R-peaks as a refer- +ence after these are located, or it is performed using cross-correlation [32; 78; 88; 278]. It +usually serves as a way to ensure the template and the collected signal are not affected by +variability, which distorts the personal information the signal contains, and could threaten +the recognition task. +• Amplitude and Time Normalisation: As previously discussed, the electrocardiogram varies +over time with several factors. These include differences in acquisition equipment or in the +interaction of the subject that may cause differences in signal amplitude and DC offset [189], +or heart rate variability that causes significant changes in the duration of the heartbeats and +their waveforms. To mitigate this, some methods include amplitude and time normalisation +techniques. +Amplitude normalisation techniques include the min-max technique [189] (used in[122; +250; 369]), which normalises the signal to the range [0,1], the z − score method [318], +which subtracts the signal and divides it by its standard deviation, or the max-div method +[274; 429], which divides the signal by the maximum amplitude value (generally, the R-peak +amplitude). +Time normalisation techniques aim to reduce the impact of heart rate variability on the +electrocardiogram’s heartbeats. Some methods perform normalisation by simply shrinking +the segmented signal to a predefined length through resampling [250; 274; 368]. As the +heartbeat does not expand uniformly with lower heart rates, Tawfik et al. [429] normalised +only the QT waveform, more prone to heart rate variations, using the Framingham study +formula. This formula computes the linearly corrected QT duration (QTLC), based on the +time between the nearest R-peaks (RR) and the original duration of the waveform (QT), us- +ing QTLC = QT +0.154(1−RR). Fatemian and Hatzinakos [124] went further, segmenting +each ECG heartbeat into its key waveforms (P, QRS, and T), and individually resampling +them, before joining them back together, with regulated intervals between them. By reduc- +ing the effects of heart rate variability and avoiding the typical distortion of the individual +waveforms, this is likely the best technique for time normalisation. However, it requires +the detection of several waveforms’ onset and offset fiducial points, making it potentially +unreliable for off-the-person or seamlessly acquired signals. + +3.2 Related Work +57 +• Outlier Detection: Outlier detection is generally applied to discard false or deflected +heartbeats, segmented from unacceptably noisy signal portions affected by movement or +impedance artefacts or contact loss [342]. Such methods should be able to discriminate +between normal deflections, noise interference, and health-related deflections. DMEAN +was proposed by Lourenço et al. [273] specifically to reject heartbeat outliers. It verifies +the compliance of candidate heartbeats with four rules, regarding the distance to the aver- +age template, the minimum and maximum amplitudes, and the position of the maximum +heartbeat amplitude (which must correspond to the R-peak location). +Louis et al. [272] opted to use Gaussian Mixture Models (GMM) as a supervised method for +outlier detection, after being trained on a set of known clean and desirable heartbeats. Pinto +et al. [342] proposed a clustering algorithm, NCCC, based on normalised cross-correlation +between candidate heartbeats. While GMM, due to the supervised data, can easily be bi- +ased towards certain patterns or subjects seen during training, clustering-based approaches +like NCCC are not susceptible to this issue but can become unreliable for small sets of can- +didate heartbeats. Nevertheless, with the rise of off-the-person, wearable, and seamlessly +integrated acquisition settings, it is expected that robust outlier detection methods will be +increasingly necessary. +3.2.3 +Feature extraction +The stage of feature extraction aims to translate the acquired signal into a representation that +reduces the effects of remaining noise and intrasubject variability and emphasises differences be- +tween subjects. Several feature extraction methods have been proposed for ECG biometrics, which +are generally grouped into three types – fiducial, non-fiducial, or hybrid approaches [288; 292]. +• Fiducial Approaches: Fiducial approaches are those that exclusively use as features mea- +surements of fiducial landmarks of the ECG signal in the time domain. These measurements +vary widely throughout the state-of-the-art, including time intervals, amplitude, widths, and +angles based on the heartbeat waveforms P, Q, R, S, and T, their onset and offset points +[193; 366; 390; 446; 490]. +Nevertheless, these approaches present the significant drawback of requiring the previous +localisation of several fiducial points in the ECG heartbeats (see subsection 3.2.2), which +proves difficult to satisfy when using off-the-person or seamless signals. Hence, fiducial +feature extraction approaches were considerably more frequent in early research works. +• Non-Fiducial Approaches: Non-fiducial approaches are those that use the entirety of the sig- +nal (or segments of it), holistically, to extract features related to the waveform morphology +[274; 288; 292]. These approaches include the use of Fourier, Wavelet, or cosine transforms + +58 +Prior Art in Electrocardiogram Biometrics +[32; 288; 289; 318; 342; 368; 429; 472], autocorrelation coefficients [8; 10; 170; 349], car- +dioid graphs [188; 414], generated tridimensional vectorcardiograms (TVCG) [105], mul- +tiresolution local binary patterns [272], and information-theoretical approaches based on +Lloyd-Max quantisation [87; 88] and Kolmogorov complexity [54]. +Some methods do not perform feature extraction, alternatively using segmented heartbeats, +average ensemble heartbeats, or segments between consecutive R peaks as features [66; 237; +276; 298; 493]. While more applicable to noisier signals, non-fiducial have still to reach the +near-perfect performance reported by earlier works using fiducial features. +• Hybrid Approaches: Hybrid approaches are those that use features from both fiducial and +non-fiducial origins. These are rare among the surveyed literature works and include the +approach from Palaniappan and Krishnan [327], which combined common amplitude, in- +terval and width fiducial features with a non-fiducial QRS complex form factor. Ergin et al. +[114] proposed the fusion of QRS fiducials, with several time-domain, Wavelet transform, +and Power Spectral Density (PSD) features. Also, Dar et al. [96] opted for the extraction of +a total of 46 features from Haar transform and heart-rate-variable R-R intervals. +Through the analysis of the surveyed research works, it is possible to conclude that fiducial +approaches generally contribute more towards a high-performance biometric system, as the use +of specific measurements reduces useless information to a minimum, and allows for feature sets +with fewer dimensions. However, as noise increases, the relevance of robustness overcomes that +of accuracy, and the former can only be offered by non-fiducial methods. The ideal feature extrac- +tion method would be one that combines the conditions for high performance offered by fiducial +approaches with the robustness to noise and variability offered by non-fiducial approaches, per- +haps using deep learning networks and their characteristic robustness to noise and versatile feature +extraction capabilities. +Extracted features may, additionally, suffer dimensionality reduction to improve performance +[445]. Although frequently overlooked, dimensionality reduction has a very important goal in +biometric systems, as the number of features extracted by biometric algorithms can easily become +too high for a time-efficient and reliable recognition process [133]. Thus, dimensionality reduction +aims to select or transform the extracted features, to reduce its number to a more computationally +viable number, while keeping the maximum discriminant power to ensure the system’s recognition +performance [445]. +Dimensionality reduction methods in the surveyed literature range from common methods +such as Linear Discriminant Analysis (LDA) [8; 10; 47; 170; 292] and its Fisher (FLDA) [332] +and Heteroscedastic variants (HLDA) [250], Principal Component Analysis (PCA) [170] and Ker- +nel PCA (KPCA) [170], or Greedy Best-First Search (GBFS) [95; 96], to rarer methods such as +Discrete Cosine Transform (DCT) [349], Wilkes’ lambda stepwise correlation [193], correlation +matrices [42; 43], or bin selection based on symmetric Kullback-Leibler divergence [288; 289]. +The work performed by Plataniotis et al. [349], Agrafioti and Hatzinakos [8], and Hejazi et al. +[170] provides an adequate platform for the comparison of dimensionality reduction algorithms. + +3.2 Related Work +59 +According to their findings, LDA offers better performance than unsupervised techniques such as +PCA and DCT coefficients, despite its supervised nature that requires knowledge of the subjects +prior to the deployment of the biometric system [292]. More recently, other supervised techniques +such as the non-linear KPCA method [170; 326] rendered even better results. Hence, research +should probably focus on more sophisticated dimensionality reduction methods and deep learning +methodologies, which are tunable to provide optimised non-linear dimensionality reduction. +3.2.4 +Decision +Based on the representation of the ECG acquisition, obtained through processes of feature ex- +traction and dimensionality reduction, the decision stage aims to accurately attribute one of the +enrolled identities to the user, in the case of identification tasks, or to accept or reject an identity +claim, for authentication tasks [9; 46; 155]. In the case of identification, the decision stage usually +consists of a classification process while, for authentication, the acceptance or rejection of the +identity claim is generally based on a reference threshold T that is applied to the prediction score. +The decision stage can be based on: +• Classifiers: A classifier can be trained on enrollment templates from a set of subjects, and +then be used to discriminate them, to output an accurate decision when needed. Classi- +fiers are more commonly used for identification tasks. The most common classifiers in +ECG-based biometrics are Support Vector Machines (SVM) [170; 250; 261; 276; 278; 366; +393; 472], mostly using Radial Basis Function (RBF) and Polynomial functions as kernels, +Nearest Neighbour (kNN) classifiers [10; 54; 66; 144; 276; 350; 449; 463], or Multilayer +Perceptrons [144; 188; 327; 447]. +• Metric-based Matching: Some methods are based on the comparison between the currently +acquired trait and the previously acquired templates, stored in the system database. The +comparison is performed based on similarity or dissimilarity metrics. In ECG-Based bio- +metrics, most metric-based matching methods have been based on distance metrics, among +which the most popular was the Euclidean distance [80; 274; 292; 332; 349; 369; 393; 405]. +However, since the Euclidean distance is regarded by some as unreliable in high dimensional +spaces, some researchers have opted to use the cosine [393] or the Mahalanobis distances +[156; 233–235]. Among similarity metrics, literature methods include the correlation coef- +ficient [8; 69; 124; 125; 373; 389; 390], normalised cross-correlation (NCC) [78], Gaussian +log-likelihood [288; 289; 318], and Dynamic Time Warping (DTW) [298; 303; 493]. +In the literature, it is hard to perform a thorough and fair comparison between the algorithms +based on the results reported by the respective authors, as the data used to evaluate such algorithms +is commonly not the same, or is used differently. However, it is important to compare algorithms +to find the advantages and disadvantages of each and find opportunities for improvement. Hence, +to help the comparison of state-of-the-art methods in terms of reported performance, the results +of the surveyed publications that have used the six most common data collections (see Fig. 3.1) + +60 +Prior Art in Electrocardiogram Biometrics +– PTB, ECG-ID, MIT-BIH NSR, MIT-BIH Arrhythmia, UofTDB, and CYBHi – are presented in +Tables 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7, respectively. +SVM and kNN have shown superior performance among traditional classifiers, even in situ- +ations with increased noise and variability. Hence, it is safe to assume that these would be wise +options for new ECG biometric algorithms. However, there is the need for an equally accurate +alternative that would not require re-training with every subject enrollment or update (as SVM +does) or the memory-heavy storage of all subjects’ templates (as kNN does). Recent studies in- +dicate that Deep Learning models could solve these issues, but researchers will need to dedicate +efforts to overcome the challenge of scarce supervised data. +3.2.5 +Deep learning +Deep learning methodologies are quickly revolutionising several fields in pattern recognition, gal- +vanising the machine learning community with outstanding results and unforeseen robustness to +input noise and variability in diverse tasks [163; 164; 242; 440]. It achieved these milestones +mainly due to the flexibility and robustness of convolutional layers for feature learning, the se- +lective memory of recurrent layers connected to their previous instances, and the versatility of +fully-connected layers [242; 483]. Their adaptability to scarce data through techniques such as +data augmentation, fine-tuning, transfer learning, and weakly supervised learning, just add to their +power for pattern recognition applications. +In the topic of ECG biometrics, the study of deep learning is still a pioneering affair. It has, +however, been gathering steam throughout the past four years. Despite a few works which contin- +ued focusing on traditional feature extraction and decision models [37; 252; 358; 433; 450], the +majority of literature methods proposed since 2020 already include some deep learning architec- +ture responsible for the processes of feature extraction, decision, or both. +Initially, Zhang et al. [483] proposed a multiscale CNN that receives, in parallel, selected +autocorrelation coefficients of approximation and detail Wavelet transform coefficient sets of two- +second ECG segments. Eduardo et al. [111] replaced the feature extraction stage using an Autoen- +coder to learn lower-dimensional representations of segmented heartbeats, which were ultimately +fed to a kNN classifier. +However, most researchers aim to integrate several stages into the deep learning model. Sal- +loum and Kuo [371], after signal preprocessing and segmentation, replaced the stages of feature +extraction and decision with an RNN with Long Short-Term Memory (LSTM) and Gated Recur- +rent Unit (GRU). Zhang et al. [484] replaced the stages of feature extraction and classification, by +feeding 2D representations of single-arm ECG signals to a CNN. Luz et al. [284] also integrated +the feature extraction and decision stages, proposing the combined use of two separate CNN, one +receiving segmented heartbeats as input and the other receiving the respective heartbeats’ spectro- +grams, fused at score level. +Labati et al. [238] detected, segmented, and selected QRS complexes from ECG signals, and +concatenated them into a QRS vector that served as input to a unidimensional CNN that fulfilled +the purposes of feature extraction and decision. With a softmax output, the method attained 100% + +3.2 Related Work +61 +Table 3.2: Results of surveyed approaches evaluated with PTB (adapted from [343], ordered by +the number of subjects, NS; works that joined PTB with other databases are not included). +Author +Year +NS +Results +Ingale et al. [187] +2020 +290 +IDR +EER +100% +2% +Ibtehaz et al. [185] +2021 +290 +IDR +EER +99.7% +5.66% +Srivastva et al. [412] +2021 +290 +IDR +99.7% +Thentu et al. [431] +2021 +290 +IDR +99.5% +Chu et al. [79] +2019 +290 +IDR +EER +99.3% +0.59% +Byeon et al. [56] +2020 +290 +IDR +99.1% +Wang et al. [451] +2021 +290 +IDR +98.9% +Hammad et al. [161] +2019 +290 +IDR +98.9% +Pinto and Cardoso [339] +2020 +290 +IDR +97.7% +Jyotishi and Dandapat [209] +2020 +290 +IDR +97.3% +Karimian et al. [210] +2017 +290 +Reliability +97.4% +Pinto and Cardoso [338] +2019 +290 +EER +11.0% +Wang et al. [450] +2020 +248 +IDR +EER +98.2% +2.55% +Huang et al. [180] +2022 +248 +IDR +EER +86.3% +2.26% +Zhang et al. [487] +2019 +234 +IDR +99.5% +Zhang et al. [489] +2021 +234 +IDR +98.8% +Dong et al. [105] +2018 +113 +99 +14 +IDR +IDR +IDR +92.8% +93.3% +98.3% +Safie et al. [369] +2011 +112 +EER +19.2% +Alduwaile and Islam [16] +2020 +100 +IDR +99.9% +Wang et al. [449] +2013 +100 +IDR +99.5% +Pal and Singh [326] +2018 +100 +IDR +97.1% +Wübbeler et al. [463] +2007 +74 +IDR +EER +98.1% +2.8% +Li et al. [251] +2022 +71 +IDR +95.8% +Labati et al. [238] +2019 +52 +IDR +100% +Brás and Pinho [54] +2015 +52 +IDR +99.9% +Coutinho et al. [88] +2013 +51 +IDR +EER +99.9% +0.01% +Plataniotis et al. [349] +2006 +14 +IDR +FAR +100% +0.02% +Waili et al. [446] +2016 +14 +IDR +96% +Zhao et al. [491] +2013 +12 +IDR +96.0% +Ghofrani and Bostani [144] +2010 +12 +IDR +98.6% +Paiva et al. [325] +2017 +10 +IDR +FAR +FRR +97.5% +5.71% +3.44% + +62 +Prior Art in Electrocardiogram Biometrics +Table 3.3: Results of surveyed approaches evaluated with ECG-ID (adapted from [343], ordered +by the number of subjects, NS; works that joined ECG-ID with other databases are not included). +Author +Year +NS +Results +Wang et al. [450] +2020 +90 +IDR +EER +100% +3.3% +Salloum and Kuo [371] +2017 +90 +IDR +100% +Li et al. [252] +2020 +90 +IDR +98.0% +Chu et al. [79] +2019 +90 +IDR +EER +97.8% +2.00% +Wu et al. [462] +2018 +90 +IDR +EER +97.5% +0.52% +Ranjan [359] +2019 +90 +EER +2% +Bento et al. [38] +2020 +90 +IDR +96.9% +Ingale et al. [187] +2020 +90 +IDR +EER +96.7% +2.3% +Zhang et al. [488] +2019 +90 +IDR +EER +96.3% +5.82% +Ibtehaz et al. [185] +2021 +90 +IDR +EER +96.2% +1.29% +Benouis et al. [37] +2021 +90 +IDR +EER +93.3% +3.05% +Jyotishi and Dandapat [209] +2020 +90 +IDR +93.1% +Ivanciu et al. [194] +2021 +90 +IDR +FAR +FRR +Sensitivity +86.5% +13.7% +12.7% +87.3% +Zaghouani et al. [477] +2017 +90 +EER +15% +Dar et al. [96] +2015 +90 +IDR +FAR +FRR +83.88% +16.1% +0.3% +Dar et al. [95] +2015 +90 +IDR +82.3% +Tan and Perkowski [426] +2017 +89 +IDR +100% +Li et al. [251] +2022 +89 +IDR +98.9% +Chun [80] +2016 +89 +EER +5.2% +Kim et al. [216] +2022 +83 +IDR +95.7% + +3.2 Related Work +63 +Table 3.4: Results of surveyed approaches evaluated with MIT-BIH NSR (adapted from [343], +ordered by the number of subjects, NS; works that joined MIT-BIH NSR with other databases are +not included). +Author +Year +NS +Results +Shen et al. [389] +2002 +20 +IDR +100% +Dar et al. [96] +2015 +18 +IDR +EER +100% +0% +Kim and Pyun [215] +2020 +18 +IDR +F-score +100% +1.0 +Ibtehaz et al. [185] +2021 +18 +IDR +EER +99.5% +5.17% +Dar et al. [95] +2015 +18 +IDR +99.4% +Ye et al. [472] +2010 +18 +IDR +FPIR +99.3% +26.9% +Tan and Perkowski [426] +2017 +18 +IDR +98.8% +Li and Narayanan [250] +2010 +18 +IDR +EER +98.3% +0.5% +Ergin et al. [114] +2014 +18 +F-score +0.97% +Zhang et al. [487] +2019 +18 +IDR +95.3% +Zhang et al. [483] +2017 +18 +IDR +95.1% +Zhang et al. [489] +2021 +18 +IDR +93.6% +Li et al. [255] +2020 +18 +IDR +91.4% +N and Jayaraman [303] +2010 +15 +IDR +96% +Palaniappan and Krishnan [327] +2004 +10 +IDR +96.2% +Camara et al. [57] +2017 +10 +IDR +94.8% + +64 +Prior Art in Electrocardiogram Biometrics +Table 3.5: Results of surveyed approaches evaluated with MIT-BIH Arrhythmia (adapted from +[343], ordered by the number of subjects, NS; works that joined MIT-BIH Arrhythmia with other +databases are not included). +Author +Year +NS +Results +Salloum and Kuo [371] +2017 +47 +IDR +EER +100% +3.4% +Ingale et al. [187] +2020 +47 +IDR +EER +100% +4% +Li et al. [252] +2020 +47 +IDR +100% +Tan and Perkowski [426] +2017 +47 +IDR +100% +Kim and Pyun [215] +2020 +47 +IDR +F-score +99.8% +0.99 +Wu et al. [462] +2018 +47 +IDR +EER +99.7% +0.02% +Ye et al. [472] +2010 +47 +IDR +FPIR +99.6% +12.3% +Ibtehaz et al. [185] +2021 +47 +IDR +EER +98.2% +6.36% +Wang et al. [451] +2021 +47 +IDR +97.8% +Jyotishi and Dandapat [209] +2020 +47 +IDR +96.8% +Lee and Kwak [245] +2022 +47 +IDR +96.0% +Dar et al. [96] +2015 +47 +IDR +FAR +FRR +95.9% +4.1% +0.1% +Huang et al. [180] +2022 +47 +IDR +EER +95.7% +0.36% +Chu et al. [79] +2019 +47 +IDR +EER +94.9% +4.74% +Wang et al. [450] +2020 +47 +IDR +EER +94.7% +2.73% +Dar et al. [95] +2015 +47 +IDR +93.1% +Zhang et al. [483] +2017 +47 +IDR +91.1% +Jahiruzzaman and Hossain [197] +2015 +11 +IDR +96.9% +Sasikala and Wahidabanu [373] +2010 +10 +IDR +62.7% +Sufi et al. [413] +2010 +- +MIDR +EER +1% +0.5% + +3.2 Related Work +65 +Table 3.6: Results of surveyed approaches evaluated with UofTDB (ordered by the number of +subjects, NS; works that joined UofTDB with other databases are not included). +Author +Year +NS +Results +Pinto et al. [344] +2019 +1019 +IDR +96.1% +Luz et al. [284] +2018 +1019 +EER (raw) +EER (spect.) +EER (fusion) +16.9% +19.4% +14.3% +Pinto and Cardoso [339] +2020 +1018 +IDR +91.5% +Pinto and Cardoso [338] +2019 +1018 +EER +7.86% +Pinto et al. [345] +2020 +1018 +EER +12.6% +Pinto et al. [347] +2021 +1018 +EER +12.6% +Louis et al. [272] +2016 +1012 +EER +7.89% +Komeili et al. [225] +2017 +82 +EER (sess.) +EER (post.) +6.9% +3.7% +Ciocoiu and Cleju [83, 84] +2019/20 +52 +IDR +EER +95.6% +5.48% +Wang et al. [450] +2020 +46 +IDR +EER +100% +2.17% +Huang et al. [180] +2022 +46 +IDR +EER +87.4% +2.36% +Table 3.7: Results of surveyed approaches evaluated with CYBHi (ordered by the number of +subjects, NS; works that joined CYBHi with other databases are not included). +Author +Year +NS +Results +Pinto and Cardoso [338] +2019 +128 +EER +16.3% +Ingale et al. [187] +2020 +125 +IDR +EER +100% +0.5% +Hammad et al. [161] +2019 +65 +IDR +99.3% +Ciocoiu and Cleju [83, 84] +2019/20 +65 +IDR +EER +95% +8.6% +Belo et al. [36] +2020 +63 +IDR +EER +100% +0.0% +Srivastva et al. [412] +2021 +63 +IDR +99.7% +Wang et al. [451] +2021 +63 +IDR +96.8% +Huang et al. [180] +2022 +63 +IDR +EER +88.5% +1.48% +Jyotishi and Dandapat [209] +2020 +63 +IDR +79.4% +Ibtehaz et al. [185] +2021 +63 +IDR +73.9% +Luz et al. [284] +2018 +61 +EER (raw) +EER (spect.) +EER (fusion) +14.1% +26.4% +12.8% + +66 +Prior Art in Electrocardiogram Biometrics +IDR on the PTB database and, with Hamming distance matching, achieved 2.75% EER with long- +term signals from the E-HOL 24h collection. +Several researchers have also tried to explore two-dimensional representations of ECG signals +in order to use 2D deep learning models. Ciocoiu and Cleju [83, 84] have studied S-Transforms, +Gramian Angular Fields, Phase-Space Trajectories, and Recurrence Plots of segmented heartbeats +on a custom two-dimensional convolutional neural network. They found that S-Transforms offered +the best performance, achieving around 95% identification rates on the UofTDB and CYBHi off- +the-person databases. +Bento et al. [38] explored spectrograms as inputs to a custom 2D CNN architecture and a +DenseNet. The latter achieved the best performance, with almost 97% IDR on ECG-ID and 100% +on FANTASIA. Byeon et al. [56] also studied DenseNets, alongside XCeption and ResNet ar- +chitectures to build ensembles of models receiving spectrograms, log-spectrograms, melspectro- +grams, scalograms, and MFCCs, achieving approximately 99% IDR on the PTB database. +More recently, Srivastva et al. [412] used images of ECG heartbeat plots as inputs to an ensem- +ble of ImageNet-pretrained DenseNet and ResNet models. With this methodology, they achieved +99.7% IDR on PTB and CYBHi. Thentu et al. [431] explored continuous wavelet transform-based +multi-scale representations of ECG heartbeats on various ImageNet pretrained two-dimensional +architectures achieving over 99.5% accuracy on both CEBSDB and PTB databases. +However, beyond the approaches proposed within this doctoral work, no truly end-to-end +methodologies are present in the literature. +Two-dimensional representations are promising, +especially considering the possibility of using pretrained deep models, but the transformations +themselves are still separately optimised processes that may lose important information and limit +achievable performance. +Another promising category of approaches is temporal networks, such as LSTMs, which have +attained interesting results and are a natural match with ECG signals (time series). However, +the aforementioned problem is still valid: if separate processes of denoising, preparation, and/or +feature extraction are added to the pipeline, the model may be limited in the information received +and thus in the performance it can achieve. +3.3 +Open Challenges and Opportunities +Much of the great potential of deep learning for ECG biometrics is still to be explored. Considering +the information presented and discussed throughout this chapter, one can align the main challenges +in ECG biometrics with the corresponding trends in ECG acquisition, thus painting a panorama of +the history and near future of ECG biometrics (see Fig. 3.3). +As illustrated in the aforementioned figure, the use of deep learning is a major research op- +portunity as we move into the future of ECG biometrics. Although deep learning typically brings +significantly increased computational costs to biometric systems, these should be compensated by +considerable boosts in performance and robustness due to the flexibility offered by deep models. + +3.3 Open Challenges and Opportunities +67 +ELECTROCARDIOGRAM BIOMETRICS +Lead I +Lead II +Lead III +1 +2 +3 +45 +6 +RA +LA +LL +RL +aVF +aVR +aVL +MEDICAL +ACQUISITION +HOLTER +SYSTEMS +OFF-THE-PERSON +ACQUISITION +WEARABLES & +SEAMLESS ACQ. +FUTURE +SYSTEMS +EVOLUTION OF ACQUISITION +MAIN TOPICS +Fiducial +Features +Decision +Methods +Long-Term +Variability +Intersubject +Variability +Health Conditions +Activity & Posture +Non-Fiducial +Features +Higher Comfort +& Usability +Efcient Noise +Suppression +Non-Fiducial +Features +Integration in +Everyday Objects +Continuous +Biometrics +Contact Loss +Artifacts +Deep Learning +Improved Multimodal +Biometrics +Contactless/Distance +ECG Acquisition +Biometric Security +Larger Databases +Learning with few data +Figure 3.3: A panorama of ECG biometrics across time: the past, present, and future trends in +ECG acquisition for biometrics and the corresponding research challenges (from [343]). +The integration of all processing stages in a single end-to-end deep model, alongside new +techniques of data augmentation and regularisation, could enable the coordinated optimisation +for individual recognition and lead us to new levels of robustness against noise and variability. +Nevertheless, as we move towards “black-box” deep models, it is important to address the problem +of trustworthiness and transparency through the study of model interpretability and explainability +in ECG biometrics. +The growth of deep learning should further unveil a serious problem in ECG biometrics: the +scarcity of data. This places significant hurdles on the development of accurate and robust biomet- +ric models, which should only worsen with data-hungry deep learning methodologies. As we delve +deeper into deep learning methodologies for ECG biometrics, it is important to build larger and +more complete off-the-person ECG collections. Additionally, researchers should devote further +efforts to data augmentation, siamese architectures, triplet learning methodologies, unsupervised +and self-supervised learning, and other strategies towards the mitigation of the effects of data +scarcity. +Partially linked to data scarcity, another large problem currently plaguing ECG biometrics is +the prevalence of unrealistic and mismatching evaluation settings. The diversity of databases and +data subsets is noticeable throughout this literature review and makes it impossible to adequately +compare different methodologies. The high frequency of random train/test subset splits makes +results unrealistic. Moreover, the rarity of long multi-session data acquisitions makes long-term + +68 +Prior Art in Electrocardiogram Biometrics +performance a hidden problem waiting to be truly unveiled and solved. +Hence, this thesis part focuses on five contributions to these open challenges and opportunities. +Specifically: +• In Chapter 4, we propose the first true end-to-end methodology for ECG-based identifica- +tion, complete with a study on the progressive integration of pipeline stages within the deep +model and various tailored data augmentation strategies; +• In Chapter 5, we adapt the previous model for identity verification and explore identification +vs. triplet loss training for template similarity matching. The proposed methodology is also +benchmarked against state-of-the-art approaches on a restructured evaluation setup for more +realistic results; +• In Chapter 6, we study the effect of long-term variability on the performance of state-of-the- +art approaches using a database of day-long Holter acquisitions, along with the application +of template/model update strategies; +• In Chapter 7, we study the relative importance of ECG waveforms, with a special focus on +the QRS, throughout experimental setups with varying database size and noise/variability +using interpretability tools; +• In Chapter 8, we propose an end-to-end methodology for the recovery of the entire set of +twelve standard leads requiring as input just one single-lead blindly-segmented ECG signal. + +Chapter 4 +End-to-End Models and Augmentation +Strategies for Identification +Foreword on Author Contributions +The research work described in this chapter was conducted entirely by the author of this thesis, under the super- +vision of Jaime S. Cardoso and André Lourenço. The results of this work have been disseminated in the form of +a chapter in a book and an abstract in national conference proceedings: +• J. R. Pinto, J. S. Cardoso, and A. Lourenço, “Deep Neural Networks for Biometric Identification Based on Non- +Intrusive ECG Acquisitions,” in K. V. Arya and R. S. Bhadoria, Eds., The Biometric Computing: Recognition +and Registration, CRC Press, 2019. [344] +• J. R. Pinto, J. S. Cardoso, and A. Lourenço, “Improving ECG-Based Biometric Identification Using End-to-End +Convolutional Networks,” in 24th Portuguese Conference on Pattern Recognition (RECPAD 2018), Oct. 2018. +4.1 +Context and Motivation +The state-of-the-art in ECG-based recognition mostly consists of pipeline algorithms, composed +of separate stages of denoising, signal preparation, feature extraction, and decision, as discussed +in Chapter 3. Even the most recent methods using deep learning techniques still rely on some of +these separate processes. +However, Convolutional Neural Networks (CNNs) possess the tools to integrate all phases of +processing, from acquisition to decision, into a single model. This integration replaces separate, +step-by-step tuning with a holistic optimisation process, synergically adapting the model to attain +the best performance possible. +Furthermore, the flexibility of convolutional and fully-connected layers makes deep networks +able to autonomously learn the most fitted features for the task at hand. Meanwhile, these keep the +ability to generalise and be robust against high variability and noise dominance over the signals +[242; 483]. Hence, it could be the key to improving the inferior performance results verified in +off-the-person ECG biometrics. +69 + +70 +End-to-End Models and Augmentation Strategies for Identification +Figure 4.1: Architecture of the proposed CNN model for ECG-based identification (the number of +neurons on the fully connected layer refers to the entire dataset with 1019 possible identities). +This work aimed to study the full extent of the capabilities of CNNs for biometric identi- +fication using non-intrusive ECG signal acquisitions. A CNN architecture is proposed for the +complete integration of traditional pipeline stages in a single model, for higher accuracy and ro- +bustness in off-the-person settings. To obtain further improved performance, unidimensional data +augmentation strategies are designed specifically for ECG-based biometrics. +4.2 +Methodology +4.2.1 +Model +The proposed convolutional neural network (see Fig. 4.1) integrates all common pipeline stages +into a single, end-to-end model receiving raw five-second ECG segments and delivering the corre- +sponding identity. It follows the typical structure of a convolutional neural network: the first part +includes convolutional and max-pooling layers, and the second part includes one fully-connected +layer. +Convolutional layers hold filter banks to learn the most advantageous representation of the +input signal segment. Pooling quickly reduces the number of parameters, controlling the compu- +tational cost and training time, and improves robustness to small input variations. The proposed +architecture uses Rectified Linear Unit (ReLU) activations, filters’ size 5, stride 1, pooling size 5, +and pooling stride 1. +The feature maps output by the last convolutional layer are concatenated into a single uni- +dimensional vector of features. This serves as input to the fully-connected layer, which weighs +and combines the received features at each neuron. The fully-connected layer is composed of N +neurons (where N is the number of enrolled subjects), with softmax activations. The neuron that +outputs the highest value will correspond to the predicted identity. +Based on a batch of train samples fed to the network, a measure of loss is computed by com- +paring the output of the network with the true labels of the batch. The weights/parameters that +compose the neural network are adjusted to reduce that loss, using an optimiser function. In + +4.2 Methodology +71 +this work, the optimiser Adam [219] was used, with empirically adjusted initial learning rate in +[0.01,0.001], and Sparse Categorical Cross-entropy loss. +To avoid overfitting, the network used dropout. Dropout will avoid learning overly specific +patterns in the training data [231; 411]. They are placed between two layers and act upon the con- +nections between them, setting the corresponding input to zero. In the proposed method, dropouts +are used on the connections between the flattened vector of features and the fully-connected layer, +effectively blocking the access of the classifier to a part of the features, and requiring it to become +less specific to the training set, and more robust to unexpected variability and noise. +4.2.2 +Data augmentation strategies +Data augmentation is used to obtain a more robust classifier. It consists of the application of small +transformations or changes to the train samples while protecting the integrity of the underlying +label of each sample, to simulate larger datasets and ensure the network is robust to such variabil- +ity [71; 231]. Like deep learning in general, data augmentation techniques are significantly more +frequent in 2D networks (for images) than in 1D (signals). +Based on the recent work of [440], and taking into account the unique characteristics of the +ECG signals, seven different types of data augmentation are proposed and explored for 1D convo- +lutional neural networks (see Fig. 4.2). These types are: +• Baseline Wander: simulates a periodic undulation on the signal, by adding a sinusoidal wave +with a frequency near 1 Hz; +• Cropping: takes a contiguous subsegment and resamples it to match the original length. In +the case of ECG signals, this technique simulates slower cardiac frequencies; +• Flip: inverts the signal along the time axis, which causes the inversion of the heartbeat +waveforms and their relative locations; +• Gaussian Noise: introduces Gaussian noise (with mean zero and standard deviation about +ten times lower than the signal amplitude) to cause high-frequency distortions on the signal, +similar to movement artefacts and powerline interference noise; +• Magnitude Scaling: rescales the original train sample by multiplying it by a factor inferior +or superior (but close) to 1; +• Magnitude Warping: similar to the previous technique, it rescales the signal in a non- +uniform fashion, using a sinusoidal wave instead of a fixed factor, so that different parts +of the signal will have their amplitude reduced or increased; +• Random Permutations: divides the signal into N contiguous subsegments with similar +lengths, and their order is randomly changed. This may cause discontinuities in the heart- +beats and their waveforms, simulating sensor faults or abrupt segment terminations. + +72 +End-to-End Models and Augmentation Strategies for Identification +Figure 4.2: Illustration of the effects of the different data augmentation techniques on an exam- +ple five-second ECG segment (for easier visualisation, the original segment was denoised with a +bandpass filter 1–30 Hz and had its amplitude z-score normalised). + +4.3 Experimental Setup +73 +Figure 4.3: Illustration of the progressive phases of integration of the traditional pipeline stages +into the CNN architecture. +4.3 +Experimental Setup +The performance of the proposed convolutional neural network architecture, as previously de- +scribed, was evaluated on off-the-person ECG recordings of the University of Toronto ECG Data- +base (UofTDB) [445]. Besides the entire database of 1019 subjects, two subsets were also used, +with 25 and 100 subjects, to evaluate the performance in smaller datasets. The datasets were +divided with 70% of the data for training and 30% for testing. +The proposed method suffered slight adaptations to allow the study of the progressive inte- +gration of the traditional pipeline stages into the CNN model (see Fig. 4.3). Thus, besides the +proposed end-to-end version that receives raw five-second ECG segments, three other variants +were evaluated. The first receives five-second ECG segments denoised using a 1 − 30 Hz band- +pass filter. The second receives the average of heartbeats detected using the Engelse-Zeelenberg +algorithm and normalised to zero mean and unit variance. The third variant receives DCT fea- +tures extracted from the average heartbeats. The pool size of max-pooling, which was set at 5 for +five-second segments as input, was changed to 3 for ensemble heartbeats, or 2 for DCT features. +The proposed method was compared with a baseline algorithm, adapted from the method +proposed by Pinto et al. [342] using SVM and kNN for decision, and the state-of-the-art algorithm +based on autoencoders proposed by Eduardo et al. [111], and the algorithm based on AC/LDA +features by Matta et al. [292], evaluated in the same conditions. +4.4 +Results and Discussion +With DCT features as input (see Fig. 4.4), the performance of the proposed method is similar to +that of the baseline algorithm for 25 subjects. However, with the increase of subjects on the dataset +(with 100 and 1019 subjects), the proposed algorithm falls behind. This may be caused by the very +concise information that the input carries, fitted for typical pipeline algorithms as the baseline but +not for deep learning. The results of the evaluation using ensemble heartbeats as input support + +74 +End-to-End Models and Augmentation Strategies for Identification +CNN (Proposed) +Baseline SVM +Baseline kNN +85 +90 +95 +100 +Identification Rate (%) +99.2 +99.5 +99.0 +97.0 +98.4 +97.7 +90.6 +95.3 +94.1 +Results with DCT Features as Input +25 subj. +100 subj. +1019 subj. +Figure 4.4: Results of the proposed and baseline algorithms, when using DCT features as input. +this hypothesis (see Fig. 4.5), as the performance increases and approaches that of the baseline +methods, and even surpasses that of the kNN classifier on the two smaller datasets. +Integrating additional stages into the deep learning model allows us to simplify its structure, +and use longer signal segments as inputs (in this case, five seconds). This means an increase in +complexity of the input, which can harm the performance of the network, but also an increase in +available information and variability, which can allow for a more robust model. +The results of the use of denoised and raw five-second segments (see Fig. 4.6) illustrate the +trade-off between signal complexity and the increase of robustness due to extra information and +variability, as the results were similar to those of the CNN receiving ensemble heartbeats. More- +over, in general, the results of the CNN with raw segments surpassed those of the CNN with +denoised segments, which likely result from the benefit of increased variability during training. +Increased variability is, in turn, the goal of data augmentation (D.A.). The aforementioned +techniques were separately tested on the subsets of 25 and 100 subjects (see Fig. 4.7). Most of the +techniques of data augmentation bring improvements to the algorithm’s identification rates. The +exceptions were magnitude scaling in the smallest dataset, cropping and magnitude warping in the +100 subject dataset, and Gaussian noise in both. Likely, this performance decay is the result of a +corruption of the underlying labels with these techniques. +The most promising data augmentation techniques were random permutations (that excelled in +both datasets), baseline wander, and flip. These were evaluated in groups to assess if the combina- +CNN (Proposed) +Baseline SVM +Baseline kNN +85 +90 +95 +100 +Identification Rate (%) +99.0 +99.0 +98.7 +97.6 +97.7 +97.2 +93.1 +94.4 +94.2 +Results with Ensemble Heartbeats as Input +25 subj. +100 subj. +1019 subj. +Figure 4.5: Results of the proposed and baseline algorithms, when using ensemble heartbeats as +input. + +4.4 Results and Discussion +75 +CNN (Denoised) +CNN + D.A. (Denoised) +CNN (Raw) +CNN + D.A. (Raw) +85 +90 +95 +100 +Identification Rate (%) +99.0 +99.1 +99.2 +99.7 +97.4 +98.5 +97.3 +98.7 +92.0 +94.2 +92.7 +96.1 +Results with Denoised or Raw Five-Second Segments as Input +25 subj. +100 subj. +1019 subj. +Figure 4.6: Results of the proposed and baseline algorithms, when using five-second ECG seg- +ments as input, raw or denoised. +tion of two or three techniques would offer performance improvements. The results (see Fig. 4.8) +show that the sole use of random permutations is the best option, although the combinations also +caused an improvement in identification performance. +We compared the proposed and baseline algorithms with state-of-the-art algorithms. As they +were implemented and tested in the same conditions, the algorithms of Eduardo et al. [111] and +Matta et al. [292] can be used for a direct benchmarking (see Fig. 4.9). The proposed method +presents better results than the alternatives and a slightly slower decay with the increase of the +number of subjects, denoting better scalability to larger populations. The state-of-the-art algo- +rithms likely suffer from using nearest neighbour classifiers, prone to overfit, as the results of the +baseline algorithm with kNN were also consistently worse than with SVM. The method of Ed- +uardo et al. [111], despite showing remarkably good results in the denoising of signals during our +experiments (using the entire encoder-decoder), falls short in these conditions. +Finally, the results of the proposed and baseline algorithm can be compared with the results +reported by the most recent prior artworks (see Table 4.1). The IDR of the proposed and baseline +algorithms may pale in comparison with some results reported in some of the considered prior +works, but it is important to consider the evaluation settings. Only Wieclaw et al. [457] used an +off-the-person database, as opposed to the much cleaner signals of on-the-person databases still +used by most researchers. Also, the UofTDB collection enabled the evaluation of the proposed +algorithm with a much larger set of subjects than any other identification method. +None +Rand.Perm. M.Scaling M.Warping B.Wander +Flip +Gauss.Noise Cropping +95 +96 +97 +98 +99 +100 +Identification Rate (%) +99.2 +99.7 +98.9 +99.3 +99.5 +99.4 +99.0 +98.8 +97.3 +98.7 +98.2 +97.3 +98.0 +97.5 +97.2 +97.0 +Results with Data Augmentation Techniques +25 subj. +100 subj. +Figure 4.7: Results of the proposed algorithm receiving raw five-second segments, with each +technique of data augmentation, on the datasets of 25 and 100 subjects. + +76 +End-to-End Models and Augmentation Strategies for Identification +None +Rand.Perm. +R.Perm+B.Wander +R.Perm.+B.Wander+Flip +95 +96 +97 +98 +99 +100 +Identification Rate (%) +99.2 +99.7 +99.8 +99.7 +97.3 +98.7 +98.2 +97.7 +Results with Combinations of Data Augmentation +25 subj. +100 subj. +Figure 4.8: Results of the proposed algorithm, receiving raw five-second segments as input, with +combinations of data augmentation techniques. +Proposed Method +Baseline DCT+SVM +Eduardo (2017) +Matta (2011) +80 +85 +90 +95 +100 +Identification Rate (%) +99.7 +99.5 +97.0 +96.8 +98.7 +98.4 +93.6 +95.5 +96.1 +95.3 +85.0 +90.0 +Benchmarking of CNN with Data Augmentation on Raw 5s Segments +25 subj. +100 subj. +1019 subj. +Figure 4.9: Direct benchmarking between the proposed architecture with the best baseline algo- +rithm and the two implemented state-of-the-art algorithms. +Table 4.1: Comparison of the proposed and baseline algorithms with recent state-of-the-art meth- +ods. +Authors +Brief Description +Dataset +IDR +Proposed Method +Raw segments + CNN +with data augment. +UofTDB +(off-the-person) - 1019 subj. +96.1% +Baseline +DCT features + SVM +UofTDB +(off-the-person) - 1019 sub. +95.3% +Salloum and Kuo [371] +LSTM-GRU RNN +ECG-ID +(on-the-person) - 90 subj. +100% +Zhang et al. [483] +Multiscale CNN +Several +(on-the-person) - 18-47 subj. +93.5% +Wieclaw et al. [457] +Heartbeats + MLP +Private +(off-the-person) - 18 subj. +89.0% +Tan and Perkowski [426] +Fiducials + RF +fused with DWT ++ WDIST kNN +Several +(on-the-person) - 184 subj. +99.5% +Carreiras et al. [66] +Heartbeats + kNN +Private +(on-the-person) - 618 subj. +84.4% +Brás and Pinho [54] +Kolmogorov-based +compression +PTB +(on-the-person) - 52 subj. +99.9% +Wang et al. [449] +Max-pooling of sparse +coding coefficients +PTB +(on-the-person) - 100 subj. +99.5% + +4.5 Summary and Conclusions +77 +However, it is important to recall that deep learning both requires and benefits greatly from +large datasets where each class is represented by a large number of samples. While, as visible in +the results presented here, data augmentation attenuates the prejudicial effects of scarce data, it is +difficult to acquire sufficient ECG signals from each subject to compensate for the increased noise +and variability in off-the-person settings. +In the datasets used, each subject was represented, on average, by just 170 five-second ECG +segments, which is arguably too few to train a convolutional neural network to robustly discrimi- +nate between 1019 individuals. Considering this, with future efforts devoted to adequately dealing +with scarce data, deep learning methodologies could see their potential for ECG biometrics be bet- +ter harnessed and place themselves as clearly better alternatives to traditional pipeline algorithms. +4.5 +Summary and Conclusions +This work proposed a convolutional neural network for biometric identification based on non- +intrusive electrocardiogram acquisitions. The proposed method was evaluated for incremental +integration of traditional ECG biometric pipeline stages, including a complete substitution by +the CNN architecture, that received raw five-second ECG segments and output a decision on the +corresponding identity. +Besides this study, seven data augmentation techniques for unidimensional signals were ex- +plored and their individual and collective impact on the algorithm’s performance was assessed. +The results on the UofTDB database were compared with those of a baseline algorithm and two +promising state-of-the-art methods. +The results show that the total integration of traditional pipeline processes in the CNN ar- +chitecture was successful. The proposed CNN with data augmentation and receiving raw five- +second segments surpassed, in all settings, the baseline and state-of-the-art algorithms in direct +benchmarking. Among other recent state-of-the-art methods, considering the diverse dataset char- +acteristics, the proposed method has also shown promise as an accurate and robust biometric +identification algorithm. + + +Chapter 5 +Triplet Loss and Transfer Learning +for Identity Verification +Foreword on Author Contributions +The research work described in this chapter was conducted entirely by the author of this thesis, under the supervi- +sion of Jaime S. Cardoso. The results of this work have been disseminated in the form of an article in international +conference proceedings: +• J. R. Pinto and J. S. Cardoso, “An End-to-End Convolutional Neural Network for ECG-Based Biometric Au- +thentication,” in 10th IEEE International Conference on Biometrics: Theory, Applications and Systems (BTAS +2019), Sep. 2019. [338] +5.1 +Context and Motivation +The field of ECG biometrics has been steadily evolving from on-the-person signals to off-the- +person acquisition setups. Despite the enhanced usability and comfort, the increased dominance +of noise and variability in off-the-person signals places serious hurdles to the real application of +ECG biometric systems (more detailed information in Chapter 3). +Some researchers have resorted to deep learning in order to fight off noise and variability and +achieve better performance and robustness [111; 284; 344; 483; 484]. However, these still rely on +separate predefined feature transforms and/or noise removal techniques, which are not optimised +for the task at hand and therefore limit the achievable performance. In fact, the work presented in +Chapter 4 shows that end-to-end deep models offer considerable performance benefits in off-the- +person ECG biometric identification, especially when using tailored augmentation techniques. +Building upon the work in Chapter 4, this work studied the use of end-to-end convolutional +neural networks (CNN) for ECG identity verification. The main goal was to discover if dismissing +all separate processes of denoising or preparation in favour of a single integrated model (granted +complete control over the robustness to signal noise and variability) would also improve per- +formance and robustness in the task of identity verification. Besides the use of metric learning +through the triplet loss, this work introduces the technique of weight transfer from a similar model +79 + +80 +Triplet Loss and Transfer Learning for Identity Verification +Conv1 +24@1x5 +ReLu +MaxPool +1x5 +Conv2 +24@1x5 +ReLu +MaxPool +1x5 +Conv3 +36@1x5 +ReLu +MaxPool +1x5 +Conv4 +36@1x5 +ReLu +FC1 +100 +ReLu +Reference +Sample +1x1000 +1st Stored +Template +Kth Stored +Template +Conv1 +MaxPool +Conv2 +MaxPool +Conv3 +MaxPool +Conv4 +FC1 +Current +Segment +... +... +... +... +... +... +... +... +... +Model trained with triplet loss +Trained weights transfer +(TL-CNN) +Euclidean +distance 1 +Euclidean +distance K +Min. +dist. +(score) +... +Positive +Sample +Negative +Sample +Shared weights +Positive +Euclidean +distance +Negative +Euclidean +distance +Triplet +Loss +Conv1 +24@1x5 +ReLu +MaxPool +1x5 +Conv2 +24@1x5 +ReLu +MaxPool +1x5 +Conv3 +36@1x5 +ReLu +MaxPool +1x5 +Conv4 +36@1x5 +ReLu +FC1 +100 +ReLu +FC2 +N +Softmax +Input ECG +Segment +1x1000 +Output +1xN +(Identity +scores) +Trained weights transfer +(IT-CNN) +Identification model +Identity verification model +Figure 5.1: Schemes of the proposed identity verification model, including the weight transfer +between networks for both proposed training methodologies (the input shape 1×1000 refers to the +five-second length of the segments used in this work, 1000 samples at 200 Hz sampling frequency). +trained for identification. This aimed to assess whether parameters optimised for identification +tasks would offer performance benefits in identity verification. +The proposed network and both training methodologies were extensively evaluated on three +ECG collections, which include on-the-person and off-the-person signals with varying signal qual- +ity, multi-session recordings from several subjects, and the influence of emotions, posture, and ex- +ercise. This evaluation included the assessment of the trained model’s applicability to other signal +collections, through cross-database tests using transfer learning and fine-tuning. +5.2 +Methodology +5.2.1 +Model architecture +The proposed method for ECG biometric identity verification is based on a CNN (see Fig. 5.1, +darker grey). All enrolled users have one or more fixed-length ECG segments (templates) stored +in the system, that have been blindly segmented (without requiring any process of reference point +detection) from a recording obtained upon enrollment. +When a user claims to be an enrolled individual, the model receives and processes, simulta- +neously, the K stored templates of the claimed identity and 1 current segment of the user. The + +5.2 Methodology +81 +comparison between the processed current segment and each of the K stored templates allows the +model to output a dissimilarity score, which can be used to accept or reject the identity claim. +After sample-wise normalisation to zero mean and unit variance, the processing of each input +segment or template starts with a succession of convolutional and pooling layers. As visible in +Fig. 5.1, four unidimensional convolutional layers are alternated with three max-pooling layers. +All have 1×5 filters, and the convolution is performed with unit stride and no padding. The first +two convolutional layers hold 24 feature maps, while the last two hold 36. +The second part of the network is composed of a fully-connected layer. The outputs of this +fully-connected layer for each stored template (a) and for the current segment (b) are compared +using normalised Euclidean distance [461] (see Eq. (5.1)), using their variance (Var) so the output +lies in [0,1]. Among the K distances computed, the minimum is output as the final dissimilarity +score for identity verification. +d(a,b) = +Var(a−b) +2(Var(a)+Var(b)). +(5.1) +5.2.2 +Model training +The weights for the identity verification model layers are transferred either from a model trained +for identification or from a model trained using triplet loss (see Fig. 5.1). The training methodol- +ogy of transferring weights from an identification model aimed to take advantage of the training +process of identification deep neural networks and assess how it could benefit a neural network +for identity verification. On the other hand, triplet loss has been recently and successfully used in +biometrics, for identity verification and other similar tasks [73; 74; 102]. +The training process requires specific structural changes to the model, which are illustrated in +Fig. 5.1 and described below. In all cases, during training, the optimiser used was Adam [219] +with an initial learning rate of 0.001, β1 = 0.9, β2 = 0.999, and no decay. Dropout [411] and data +augmentation (random permutations, as in [344]) were used to prevent overfitting. After training, +the weights are transferred to the respective layers on the identity verification model. +5.2.2.1 +Transfer from identification network (IT-CNN) +In the case of identification training (IT-CNN), the model is structured to receive 1 input segment +and contain one additional fully-connected layer (FC2), using softmax activation, that will out- +put N scores. It is trained for identification with data from N identities (following the work of +Pinto et al. [344]). +After receiving a training segment, considering its true label and the network’s output, the +sparse categorical cross-entropy loss [1; 76] is computed and used during training to ultimately +prepare the model to adequately discriminate the subjects. + +82 +Triplet Loss and Transfer Learning for Identity Verification +5.2.2.2 +Triplet loss training (TL-CNN) +To be trained using triplet loss (TL-CNN), the identity verification model, which has K + 1 in- +puts and 1 output, is restructured to receive 3 inputs and offer 2 outputs. The three inputs are +the reference template, a positive template (whose identity is the same as the reference), and a +negative template (of a different identity). The network processes each input and computes the +dissimilarities between the reference and the positive template (p) and between the reference and +the negative template (n). +Using adequate triplets of signal segments, the goal is to minimize p and maximize n. Hence, +the model is trained using triplet loss [73], which can be computed for each triplet of inputs through +the function: +l(p,n) = max(0,α + p−n), +(5.2) +where α controls the margin to be enforced between the scores of positive and negative pairs (in +this work, α = 0.5). This margin eases the choice of an effective threshold for the purpose of +identity verification. +5.3 +Experimental Setup +In this work, one of the main concerns was ensuring the performance results were as realistic as +possible. To achieve this, all databases were split between training subjects and testing subjects, +to ensure the model can be trained and applied to data from two entirely disjoint sets of subjects. +Furthermore, cross-database tests were performed to ensure the model can generalise to other +population samples and acquisition settings. Subject enrollment was limited to realistic durations +(5, 10, 15, or, at most, 30 seconds of the earliest data from each subject). +5.3.1 +Data and reference methods +The three selected databases were UofTDB [445], CYBHi [394], and PTB [49; 146]. UofTDB +(off-the-person, 1019 subjects) was used for most experiments due to its intermediate but realistic +signal quality. The PTB (on-the-person, 290 subjects) and CYBHi (off-the-person, 128 subjects) +databases were used to assess performance in better and worse signal quality settings, respectively. +To match UofTDB, CYBHi and PTB signals were resampled to 200 Hz. For PTB, only Lead I +signals were used. +Three literature methods were used as reference: the AC/LDA method, proposed by Agrafi- +oti et al. [10]; the Autoencoder method, proposed by Eduardo et al. [111]; and the DCT method, +proposed by Pinto et al. [342; 344] (adapted for identity verification, using cosine distance nor- +malised to [0,1] for matching). + +5.4 Results and Discussion +83 +5.3.2 +Evaluation scenarios +The proposed and implemented methods were evaluated across four scenarios, as detailed below, +using the Equal Error Rate (EER, see [343] for more details). Here, each signal segment used as +input for the proposed model was five seconds long (1000 samples at 200 Hz sampling frequency). +In the single-database scenario, the proposed model was evaluated on UofTDB data, and +compared with the aforementioned reference state-of-the-art methods. The last 100 subjects were +reserved for training, while the data from the remaining 919 subjects were used for testing. The +number of enrollment templates varied between 1, 2, 3, or 6 five-second segments. +The varying identity set size scenario aimed to study how the performance is affected by the +number of subjects used to train the model. Instead of the original 100 subjects, training was +performed using the 20, 50, or 150 last subjects of UofTDB, and the remaining 999, 969, or 869 +subjects, respectively, were used for testing. +The cross-database scenario was designed to assess the proposed model’s applicability to sig- +nals from other databases. The proposed model, previously trained on 100 subjects from UofTDB, +was directly tested on data from CYBHi and PTB, without fine-tuning. +At last, in the fine-tuning scenario, the goal was to assess the performance benefits brought by +fine-tuning. As in the cross-database scenario, the proposed model trained on UofTDB data (from +100 subjects) was fine-tuned to CYBHi/PTB data (from 20 subjects). This was compared to the +model directly trained, from scratch, on data from CYBHi or PTB (from 20 subjects, following the +single-database scenario). With 20 subjects reserved for training, the tests on this scenario were +performed for 108 (CYBHi) or 270 (PTB) subjects. +5.4 +Results and Discussion +5.4.1 +Single-database scenario +The results obtained in the single-database scenario are presented in Table 5.1. In all cases, the IT- +CNN model, which used weights trained for identification, attained better results than TL-CNN, +which was trained using triplet loss. With 30 seconds of user enrollment, IT-CNN achieved 7.86% +EER, while TL-CNN offered 9.94% EER in the same circumstances. +When considering shorter enrollment recordings (5 s, 10 s, and 15 s), the performance of both +proposed methods worsens, but always remained below 14% EER. It is noteworthy that IT-CNN +presented a wider advantage over TL-CNN with more enrollment data, which may denote it takes +better advantage of the availability of data. +Among the reference methods, AC/LDA presented the best results in most settings. When +compared with these results, both proposed methods offered consistently lower EER. Considering +the best reference method for each enrollment duration, IT-CNN attained an EER reduction of +around 7−8%, which can be regarded as a significant improvement over the state-of-the-art. + +84 +Triplet Loss and Transfer Learning for Identity Verification +Table 5.1: Single-database scenario: EER results (%) when trained with data from 100 UofTDB +subjects and tested with 919 UofTDB subjects (in italics: proposed methods; in bold: best results). +Enrolment duration +Method +5 s +10 s +15 s +30 s +IT-CNN +13.70 +10.92 +9.52 +7.86 +TL-CNN +13.93 +11.89 +10.90 +9.94 +AC/LDA [10] +30.27 +17.90 +16.55 +15.82 +Autoencoder [111] +21.82 +19.68 +18.84 +17.09 +DCT [342; 344] +23.05 +20.41 +18.55 +17.38 +0.0 +0.5 +1.0 +1.5 +TL-CNN Output +0 +20 +40 +60 +IT-CNN Output +0.0 +0.5 +0 +20 +40 +0.0 +0.5 +1.0 +1.5 +0 +20 +40 +0.0 +0.5 +1.0 +0 +20 +40 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +20 +40 +60 +Figure 5.2: Network outputs for all training samples of five example subjects from the UofTDB +collection (one subject for each row; the average output feature vector is presented as a black line, +and the standard deviation as a grey area). +Among other state-of-the-art works, Luz et al. [284], under similar settings, reported 14.27% +EER with UofTDB data. All IT-CNN and TL-CNN performance results are better, even when +considering only 5 seconds of enrollment (much less than what was used by Luz et al.). +Moreover, Louis et al. [272] reported 7.89% EER, but only using single session data from +1012 UofTDB subjects. Using only data from subjects with more than one session (82 subjects), +Louis et al. reported 10.10% EER, while Komeili et al. [225] reported 6.9% EER. Although the +evaluation settings are different, the proposed method’s results are aligned with these (7.86% for +IT-CNN with 30 s enrollment). +The statistical significance of the results was assessed, repeating the evaluation on one-hundred +random subject data divisions between enrollment and testing (Table 5.2). Overall, the results were +better, as this test is arguably less realistic than the remaining tests performed in this study (a real + +5.4 Results and Discussion +85 +Table 5.2: Single-database scenario: Mean and standard deviation of the EER results (%) obtained +on 100 random data divisions (in italics: proposed methods; in bold: best results). +Enrolment duration +Method +5 s +10 s +15 s +30 s +IT-CNN +11.3 ± 0.14 +9.4 ± 0.12 +8.4 ± 0.14 +7.0 ± 0.14 +TL-CNN +11.6 ± 0.16 +10.3 ± 0.11 +9.7 ± 0.14 +8.7 ± 0.11 +AC/LDA [10] +17.7 ± 0.18 +15.6 ± 0.17 +14.6 ± 0.17 +13.3 ± 0.31 +Autoencoder [111] +18.4 ± 0.17 +16.3 ± 0.14 +15.9 ± 0.16 +13.8 ± 0.12 +DCT [342; 344] +21.2 ± 0.16 +18.6 ± 0.15 +16.4 ± 0.14 +15.5 ± 0.21 +biometric system will always use the very first data of a subject for enrollment). Applying a paired +two-sided t-test to the EER estimates, the results of the proposed methods IT-CNN and TL-CNN +were significantly different in all cases (the differences are statistically significant at the 1% level), +not only from each of the implemented state-of-the-art methods but also between themselves. +Additionally, the outputs of the network for five-second training segments from different sub- +jects were visualised (see Fig. 5.2). These are, effectively, the feature vectors used for the identity +verification decision. It is possible to observe that, despite the blind segmentation and the noise +and variability carried by each five-second segment, the trained network was able to represent +each input segment in a way that maximises similarity with other segments from the same subject. +Although some variability is still present, it is reduced to a manageable level for the biometric +identity verification task, and the differences between the subjects’ output patterns are noticeable +even through a simple visualisation of the plots. +5.4.2 +Varying identity set size scenario +In the varying identity set size scenario, multiple numbers of UofTDB subjects reserved for train- +ing were explored (Fig. 5.3). In all cases, an increase in the number of training subjects resulted +in performance improvements. The best results were obtained with 150 training subjects and 30 +seconds enrollment, with 6.46% EER and 8.71% for IT-CNN and TL-CNN, respectively. Never- +theless, even with just 20 training subjects, IT-CNN offered performance under 10% EER (9.92%, +with 30 s enrollment). +As in the single-database scenario, it was noticeable that the performance advantage of IT- +CNN over TL-CNN was greater when more data was available, either for model training or user +enrollment. For example, the EER difference between IT-CNN and TL-CNN grew from 0.5% to +2.25% when increasing the number of training subjects from 20 to 150 and the enrollment duration +from 5 to 30 s. +Despite this, one could expect the IT-CNN method to perform better than the state-of-the-art, +even under scarce data conditions. Based on these results, when pretrained with only 20 subjects +with 10 s enrollments, IT-CNN should offer an EER lower than 13% on a population of nearly one +thousand individuals. + +86 +Triplet Loss and Transfer Learning for Identity Verification +20 +50 +100 +150 +Number of Training Subjects +6 +8 +10 +12 +14 +16 +EER (%) +IT-CNN +5 s +10 s +15 s +30 s +20 +50 +100 +150 +Number of Training Subjects +10 +12 +14 +16 +EER (%) +TL-CNN +5 s +10 s +15 s +30 s +Figure 5.3: Varying identity set size scenario: EER evolution with number of subjects reserved for +training, for diverse enrollment durations, for the proposed methods IT-CNN and TL-CNN. +5 +10 +15 +30 +Enrollment duration (s) +20 +30 +40 +EER (%) +CYBHi Direct Application +IT-CNN +TL-CNN +DCT +AC/LDA +Autoen. +5 +10 +15 +30 +Enrollment duration (s) +10 +12 +14 +16 +18 +EER (%) +PTB Direct Application +IT-CNN +TL-CNN +DCT +AC/LDA +Autoen. +Figure 5.4: Cross-database scenario: EER for the proposed methods IT-CNN and TL-CNN when +trained with UofTDB data and directly applied to CYBHi or PTB, and comparison with state-of- +the-art methods. + +5.4 Results and Discussion +87 +5 +10 +15 +30 +Enrollment duration (s) +20 +30 +40 +EER (%) +CYBHi Direct Training vs State-of-the-Art +IT-CNN +TL-CNN +DCT +AC/LDA +Autoenc. +5 +10 +15 +30 +Enrollment duration (s) +10 +12 +14 +EER (%) +PTB Direct Training vs State-of-the-Art +IT-CNN +TL-CNN +DCT +AC/LDA +Autoenc. +Figure 5.5: Fine-tuning scenario: EER results for the proposed methods IT-CNN and TL-CNN +when directly trained with CYBHi or PTB data from 20 subjects, and comparison with state-of- +the-art methods. +5 +10 +15 +30 +Enrollment duration (s) +15.0 +17.5 +20.0 +22.5 +25.0 +27.5 +EER (%) +CYBHi Direct Training vs. Fine-Tuning +IT-CNN DT +IT-CNN FT +TL-CNN DT +TL-CNN FT +5 +10 +15 +30 +Enrollment duration (s) +10 +12 +14 +EER (%) +PTB Direct Training vs. Fine-Tuning +IT-CNN DT +IT-CNN FT +TL-CNN DT +TL-CNN FT +Figure 5.6: Fine-tuning scenario: EER results for the proposed methods when (DT) trained, from +scratch, with data from CYBHi or PTB, or when (FT) trained with UofTDB data and fine-tuned +to CYBHi/PTB. + +88 +Triplet Loss and Transfer Learning for Identity Verification +5.4.3 +Cross-database scenario +In the cross-database scenario, the proposed methodologies were directly applied to CYBHi and +PTB data, after training on data from 100 UofTDB subjects (Fig. 5.4). +With CYBHi, IT-CNN offered better performance than TL-CNN when using 30 s enrollment +(16.30% against 17.56% EER). However, with reduced enrollment duration (5 s), TL-CNN per- +formed better (24.66% against 26.89% EER). This reinforces the idea that TL-CNN is better in +scarce data situations, while IT-CNN takes better advantage of the greater availability of data. +With PTB, IT-CNN was, in all cases, the most successful proposed method (13.83% EER with 5 +s enrollment). +Among the state-of-the-art methods, AC/LDA behaved as in the single-database scenario (see +Table 5.1), offering the worst results when using 5 s enrollment, but sharply improving with more +enrollment data, offering the best result with PTB (9.03% EER). DCT presented the best result +with CYBHi (15.40% EER), while IT-CNN offered the second-best result (16.30% EER). Both +proposed methods were, in general, worse than the state-of-the-art with the PTB database. +5.4.4 +Fine-tuning scenario +In the fine-tuning scenario, the model was trained with CYBHi/PTB data and compared with +the state-of-the-art (Fig. 5.5) and when trained with UofTDB data and fine-tuned to CYBHi/PTB +(Fig. 5.6). +Directly trained on CYBHi data, TL-CNN attained 20.04% EER, but it offered 17.56% EER +if trained with UofTDB data, and further improving to 15.37% EER if fine-tuning is performed. +TL-CNN was able to attain better performance than IT-CNN in more difficult settings, once again +indicating that this method may be better fitted for scarcer data or noisier signals. +On PTB, TL-CNN did not offer competitive results. For IT-CNN, fine-tuning (9.06% EER +with 30 s enrollment) improved the results over the direct application, but it was not enough to +significantly improve the results of direct training. Apparently, training with UofTDB data over- +prepared the network for a degree of noise and variability that is not verified on PTB signals, +which ultimately harmed its performance. A hybrid method where, before regular training, the +neural network would be encouraged to mimic the behaviour of traditional methods, could be +beneficial in cross-database settings. +Overall, the proposed methodologies presented more competitive results on CYBHi than on +PTB, likely due to PTB signals’ lesser noise and variability. Thus, while the proposed model has +shown robustness to noise and variability in off-the-person settings, the state-of-the-art methods +are more fitted to cleaner on-the-person signals. +5.5 +Summary and Conclusions +In this work, an end-to-end model, based on a CNN, was proposed for biometric identity verifica- +tion using ECG signals. It was designed to use a set of stored templates of a claimed identity and + +5.5 Summary and Conclusions +89 +an ECG segment of the current user, and output a dissimilarity score used to accept or reject the +identity claim. The model was trained using triplet loss or by transferring weights from a similar +model trained for identification. +The proposed model was successful in improving the performance of state-of-the-art methods, +especially in off-the-person signals, increasingly used in ECG-based biometrics. Using identifica- +tion training has offered better performance than triplet loss when more training and enrollment +data are available and could bring benefits for other tasks or biometric traits. Both methods have +shown the ability to overcome increased noise and variability of off-the-person signals, focusing +on subject-specific signal patterns for accurate identity verification. Nevertheless, further efforts +should be devoted to improving performance and turning the ECG into a reliable biometric trait. + + +Chapter 6 +Long-Term Performance +and Template Update +Foreword on Author Contributions +The research work described in this chapter was conducted in collaboration with Gabriel C. Lopes, under the +supervision of Jaime S. Cardoso. The author of this thesis contributed to this work on the formulation and im- +plementation of the biometric recognition models, the conceptualisation of the template update methodologies, +the preparation and conduction of experiments, the discussion of the results, and the writing of the scientific +publications. +The results of this work have been disseminated as an article in international conference proceedings and an +abstract in national conference proceedings: +• G. Lopes, J. R. Pinto, and J. S. Cardoso, “Don’t You Forget About Me: A Study on Long-Term Performance +in ECG Biometrics,” in IbPRIA 2019: 9th Iberian Conference on Pattern Recognition and Image Analysis, +Jul. 2019. [269] +• G. Lopes, J. R. Pinto, J. S. Cardoso, and A. Rebelo, “Long-Term Performance of a Convolutional Neural Net- +work for ECG-Based Biometrics,” in 25th Portuguese Conference on Pattern Recognition (RECPAD 2019), +Oct. 2019. +6.1 +Context and Motivation +Modern ECG biometric techniques generally report relatively high identification rates and low +verification error, while current off-the-person ECG acquisition techniques contribute towards in- +creased simplicity, usability, and comfort [342; 343]. Nevertheless, as with most alternatives, the +performance decays over time, especially when considering long-term usage [224]. +The natural variability of the input biometric data, the effects of ageing, and variations caused +by the subject’s interaction with the sensor contribute to intrasubject variability [201]. This causes +stored individual templates to quickly lose representativity, resulting in poor recognition perfor- +mance and placing serious challenges on long-term recognition. Thus, long-term biometrics ben- +efits from the frequent update of stored templates to keep up with the variability and ageing of the +users, thus maintaining acceptable performance over time [134]. +91 + +92 +Long-Term Performance and Template Update +Specifically for ECG biometrics, long-term performance and template update remain open +challenges. In the most thorough work yet on this topic, Labati et al. [236] have studied the +performance decay over time on their proposed algorithm for ECG-based authentication, finding +that performance decays significantly even over relatively short periods of time. However, the +focus of this study was limited to authentication and the algorithm proposed by the authors. +In this work, we build upon the study of Labati et al. [236]. We aim to more thoroughly explore +the problem of long-term performance decay in ECG biometrics and how to correctly address it. +Specifically, we extend it to (a) focus on the task of ECG biometric identification, (b) study diverse +state-of-the-art biometric methods, and (c) evaluate how different update techniques may be able +to improve long-term performance. +6.2 +Related Work +In the literature, it is difficult to find a strong and widely accepted rule for template update. Most +methods are based on heuristics and empirically determined thresholds, which are highly depen- +dent on the data and the application scenario. For example, Komeili et al. [224], for authentication, +have set the acceptance threshold equal to the point of zero false acceptance rate, thus ensuring +updates with only genuine samples. +Nevertheless, it is possible to identify some common mechanisms that may vary depending on +different factors: these include the choice of the update criterion (based on thresholds or graphs), +the update periodicity (online or offline), the selection mechanism, and the template update work- +ing mode system (supervised or semi-supervised). The taxonomy of template update (see Fig. 6.1) +divides the existing techniques into two categories: supervised and semi-supervised. +Supervised methods are offline methods in which label attribution is given by a supervisor. +These contain the Clustering subcategory, which includes the MDIST that aims to search for the +templates that minimise the distance among all the samples in the database (i.e., the most similar) +and DEND that aims to search for the templates that exhibit large intraclass variations resort to the +dendrogram (i.e., the most different) [282]. +The second subcategory comprises Editing-based methods, which are independent of the num- +ber of templates and give focus on the whole collected training set T. A subset E ∈ T is generated, +maintaining the classification performance offered by T. The best subsets were obtained by re- +viewing the structure of the data (which needs to be done for each subject) [134; 184]. All the +algorithms (based on k-Nearest Neighbours) must be representative of T and can be roughly de- +scribed as incremental when the E starts empty and grows, or decremental when E starts equal to +T and in each iteration some instances are deleted until some criterion is reached [134]. +Semi-supervised methods merge labelled (in biometrics, these correspond to the initial train- +ing samples) and unlabelled (corresponding to the samples available during system operation) data +to improve the system’s performance. This category comprises the Single Modality (for unimodal +biometric systems) and Multiple Modality (for systems using more than one biometric trait) sub- +categories. The Single Modality subcategory includes the Self-Training approaches such as FIFO + +6.2 Related Work +93 +Template-Based +Adaptive Biometric +Systems +Supervised +Semi-supervised +Clustering +Editing-based +MDIST +DEND +Single Modality +Multimodal +Graph +Approach +Min-Cut +Co-update +Self +Training +Nearest +Neighbour +Selective +Condensing +Edited +Reduced +Penalised +Fixation +MDIST +DEND +Super Template +Figure 6.1: +Dendrogram representing the taxonomy of template update techniques (based +on [364]). +(first-in-first-out), Fixation, Super Template (X composed by N templates x) where new genuine +date is always fused to a common single template [237] updated online during the execution of +continuous verification, Penalised template update method based on the mean of the past ECG’s +and the actual ECG [80] and clock method where the current template is tested against all the +others stored in the database [378]. +Generally, a new unknown trait measurement is used for template update if its score (returned +by the biometric recognition system) is above a set threshold. Hence, the future performance of +the system relies heavily on the chosen threshold value [364]. +The update threshold is commonly estimated using enrollment templates or training data. +When training data are scarce or when using short enrollments, this may lead to some problems. +First, important intrasubject variability information may be missed since only the patterns similar +to the stored templates are used (and all others are discarded). Second, the effectiveness of the +online methods depends on the order of the input data. Third, the methods are vulnerable to large +intraclass variations. At last, since the algorithms normally look for the minimal cost (high scores), +they may get stuck in local maxima and always only use high-confidence data for updating. +Semi-supervised methods also include Graph approaches. These commonly define a graph +where the nodes are either labelled (the identity is known) or unlabelled (unknown identity) data, +and the edges (which can have different weights) are the similarity between those samples [364; +494]. To be considered a graph-based semi-supervised method, it must estimate a function f, +approximate the known Y on the labelled nodes, and include two terms to smooth the graph: a +loss function and a regulariser. These two terms are what define each approach (as can be seen in +Table 6.1) [494], among which the most common in biometrics is min-cut graphs [364]. + +94 +Long-Term Performance and Template Update +Table 6.1: Graph-based template update methods and their respective loss and regulariser func- +tions (based on [494]). +Method +Source +Loss +Regulariser +Min Cut +[44] +∑ +i∈L +(yi −yi|L)2 +1 +2 ∑ +i j +wi j(yi −yj)2 +Gaussian Random Fields +and Harmonic Function +[495] +∑ +i∈L +(fi −yi)2 +f T∆ f +Local and Global Consistency +[479] +n +∑ +i=1 +(fi −yi)2 +D− 1 +2 ∆D +1 +2 +Tikhonov Regularisation +[35] +1 +K ∑ +i +(fi −yi)2 +γ f TS f +Manifold Regularisation +[402] +1 +l +l +∑ +i=1 +V(xi,Yi, f) +γA∥ f∥2 +k +γI∥ f∥2 +I +Graph Kernel from the +Spectrum of Laplacian +[70] +min 1 +2wTW +exp(−σ +2 λ) +Spectral Graph Transducer +[495] +minc(f −γ)TC(f −γ) +f TL f +Local Learning Regularisation +[222] +min 1 +k +k +∑ +i=1 +(yi − fk(xi))2 +γ +k∥ fk∥2 +Considering the topic of template update is still to be adequately addressed in ECG biomet- +rics, this work studies the effect of ECG permanence and variability in long-term identification +performance. Furthermore, it aimed to evaluate the effect of template update techniques, on the +performance of several state-of-the-art methods. +6.3 +Methodology +6.3.1 +Biometric identification methods +To fully and objectively evaluate the effects of ECG variability on the performance of biometric +algorithms, a study was conducted on four literature methods: +• Plataniotis et al. [349] proposed an ECG biometric recognition method using a non-fiducial +approach. Signals are preprocessed using a bandpass filter (0.5 − 40 Hz), followed by fea- +ture extraction with autocorrelation (AC) and dimensionality reduction using discrete cosine +transform (DCT). The fifteen most relevant features were selected, and Euclidean distance +was used for classification; +• Tawfik et al. [429] used a bandpass filter (1 − 40 Hz) in the preprocessing phase. QRS +complexes (the most stable part of ECG) were cut from the signal using a 0.35 second +window. The average ensemble QRS was computed and features were extracted using the +DCT technique (the thirty most relevant features were selected). A multilayer perceptron +(MLP) is used for classification; + +6.3 Methodology +95 +• Belgacem et al. [32] also preprocessed signals with a bandpass filter (1 − 40 Hz). The +QRS complexes were located and cut from the signal, and the average QRS was computed. +The feature extraction resorted to Discrete Wavelet Transform (DWT). From all DWT de- +composition levels, only the most relevant were selected, and a Random Forest is used for +classification. This method was originally proposed for authentication and adapted here for +identification; +• Eduardo et al. [111] used a Finite Impulse Response (5 − 20 Hz) filter for preprocessing. +Heartbeats were cut with a fixed length of [−200,400] ms around each R peak. Outliers +were detected and removed using DMEAN (α = 0.5 and β = 1.5, with Euclidean distance). +For decision, the k-nearest neighbours (kNN) classifier was used with k = 3 and cosine +distance. +Beyond these literature methods, this work also explored the deep learning-based method- +ology presented in Chapter 4 (proposed in Pinto et al. [344]). This method uses an end-to-end +unidimensional convolutional neural network, that receives five-second blindly-segmented z-score +normalised ECG segments to perform biometric identification. The feature extraction part of the +model is composed of four convolutional layers, interleaved with three max-pooling layers, with +filter/pooling size 1 × 5 and ReLU activation units. For classification, the network uses a single +fully-connected layer and softmax activation units. +6.3.2 +Template update methods +FIFO: +First-In-First-Out is the most common strategy and, computationally, is very lightweight. +Here, the database is updated using new samples whose score is above or below a threshold +(whether the score represents similarity or dissimilarity, respectively), or between two threshold +values (discarding previously stored sample) [87; 224]. The score of a new sample can either be +output by a classifier or be a measure of distance or similarity between that sample and the stored +templates [274]. +In this work, the training data were used to search for threshold values. Among all training +samples, 75% were used to train a model, which was used to obtain scores for the remaining data +samples. Comparing the scores with several thresholds, the error at each threshold was analysed +(Fig. 6.2) to find one that simultaneously maximises true positives and minimises false positives. +Fixation: +This method consists of fixing certain templates, allowing only the remaining stored +samples to be updated [157]. In this work, 25, 50, or 75% of the enrollment templates of the +individual are fixed, while the rest of the samples are free to be updated. This ensures some initial, +labelled information of the subjects remains on the system over time. +An adaptation of this technique was explored. Here, n + j × n samples were fixed, where +n ∈ [1,2,3] is the number of fixed initial templates, and j increases over time. In this work, +j ∈ [0,6] increased by one at each testing moment (j ∈ [0,6]), which allowed the system to fix +more and more samples over time, thus storing information on the subject’s variability over time. + +96 +Long-Term Performance and Template Update +Figure 6.2: Illustration of the search for the ideal threshold (the values were chosen near the +intersection, inside the yellow zone). +In a real system with potentially endless use, the parameters n and j should be carefully chosen to +avoid the eventual fixation of the entire template gallery. +Fine-Tuning: +In this technique, the model is briefly optimised with the samples accepted for +update, using the predicted labels. The model retains knowledge of the users’ supervised training +samples, as it was trained using their enrollment samples, but is slightly adapted to the new per- +sonal patterns carried by the new signals. This is explored exclusively for the CNN method [344]. +6.4 +Experimental Setup +6.4.1 +Dataset +For evaluation, the ECG signals used were from the E-HOL-03-0202-003 database1 (most com- +monly designated as E-HOL 24h). This database consists of a study of 202 healthy subjects (only +201 were provided), recorded using three leads at 200 Hz sampling frequency, after an initial rest- +ing supine period of 20 minutes. From the available data of 201 subjects, thirteen were discarded +due to saturation or unacceptable noise (subjects 1043, 9003, 9005, 9020, 9021, 9022, 9025, 9046, +9061, 9064, 9071, 9082 and 9105), similar to what was done by Labati et al. [237]. From each +of the remaining 188 subjects, only the lead most closely resembling Lead I ECG was selected, to +approximate off-the-person settings. +6.4.2 +Experiments +Standard sample wise normalisation was performed following Eq. (6.1) for all methods except +that of Eduardo et al. [111], which required [−1,1] min-max normalisation, Eq. (6.2), where x +1THEW. Available on: http://thew-project.org/Database/E-HOL-03-0202-003.html. + +Search for the optimal update threshold +1.0 +False Positives +0.8 +0.6 +Error +0.4 +0.2 +False Negatives +0.0 +5 +0 +1 +2 +3 +4 +6 +Threshold6.5 Results and Discussion +97 +Figure 6.3: Schema illustrating the use of each E-HOL record for training and testing (in orange - +training segment; in blue - each test segment). +represents the input signal and ˜x the normalised signal. +˜x[n] = x[n]−x +σ(x) +(6.1) +˜x[n] = 2 +� +x[n]−min(x) +max(x)−min(x) +� +−1 +(6.2) +In order to fit the used data, some changes were introduced to the original methods. On the +method from Eduardo et al., the cutoff frequencies of the bandpass filter were changed to 1 and 40 +Hz, to retrieve important information on higher frequencies. Outlier removal was reparametrized +with α = 1.2 and β = 1.5. The autoencoder had the topology [120,60,40,20,40,60,120] and was +trained using the Adam optimiser with a learning rate 0.01. Classification used k = 1. For the +method of Belgacem et al., DWT feature extraction was performed in four decomposition levels, +due to lower data sampling rate, and cd4, cd3, cd2, and cd1 coefficients were used as features. +Data were divided into train and test sets. The training phase used the last 30 seconds (mim- +icking short enrollments on real-life applications) of the first 60 minutes (avoiding unrealistic calm +after the initial resting period) of each subject. Five-second overlap was used to obtain 26 samples +from every 30 seconds of training data. To study performance over time, testing was performed +over seven time points (see Fig. 6.3): one immediately after enrollment, another after one hour, +and regularly until the end of the records. From each point, from 15 minutes of data, thirty 30 s +samples are extracted, and batches are built with one sample from each of the 188 subjects. +6.5 +Results and Discussion +6.5.1 +Handcrafted methodologies +After implementing, for identification, the method proposed by Labati et al. [236] (replicating +their evaluation conditions), it was possible to conclude that the ECG signal is not fully permanent +over 24 h. However, similarly to what was stated by Labati et al., the results are relatively good +over the first two hours (see Fig 6.4a), although permanence was not verified. +The performance results at each test hour, obtained through the weighted average of the corre- +sponding batches, for the state-of-the-art methods can be found in Fig. 6.4b. It was found that the + +→ 30 seconds +fortraining +15 minutes +fortesting +0 +1 +2 +3 +4 +5 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 98 +Long-Term Performance and Template Update +(a) +(b) +Figure 6.4: Identification performance over time corresponding to (a) the Labati et al. method, and +(b) the implemented state-of-the-art methods. +performance is mostly acceptable in the first test point, but performance decays significantly over +time and variability changes considerably over the day. +Moreover, a minimum around the 15th hour occurs independently of the chosen method. Con- +sidering that most of the records start between 8-12 am, after 15 hours the subjects must be sleep- +ing. In this perspective, it appears that the ECG is most different from normal when the subject is +asleep. +Considering the previous results, template update was applied to the methods, in an effort to +avoid performance decay over time. Fig. 6.5 presents the results using the FIFO technique, with +diverse thresholds. +For the methods of Plataniotis et al. and Eduardo et al., the best results were obtained using +two thresholds, respectively, {0.3,0.7} (4.7% accuracy improvement) and {0.1,0.3} (+ 5.7% +accuracy), improving all performance results until the 15th hour. However, for the method of +Belgacem et al., the performance worsens with template update after the first two hours (best +results were obtained when the difference between the highest and second highest scores ∆score ∈ +[0.15,0.3]). The same was verified for the method of Tawfik et al. which, in the first two hours, +offered the best results with ∆score > 0.2. In general, using two thresholds instead of one offered +the best results. +Considering this, it appeared that the Random Forest and MLP classifiers are not suitable for +these kinds of template/model update. This was confirmed after a repetition of the evaluation of +these methods, with kNN replacing the classifiers (see Fig. 6.6). With kNN, the template update +was able to reduce the performance decay over time, improving accuracy, on average, by 7.9% +and 9.2%, respectively, for the methods of Belgacem et al. and Tawfik et al. +As for the Fixation technique, the obtained results were more promising (see Fig. 6.7). This +template update technique brought performance improvements for all methods. The fixation tech- +nique that offered the best results was j×3+3, improving the baseline identification accuracy, on +average, by 10.0%. + +Variability study in Labati conditions +1.0 +- Labati method +Identification Accuracy +0.8 +0.6 +0.4 +0.2 +0.0 +0 +20 +40 +60 +80 +100 +Time (min)Baseline +1.0 +Plataniotis +Tawfik +Identification Accuracy +0.8 +Belgacem +Eduardo +0.6 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)6.5 Results and Discussion +99 +Figure 6.5: Comparison of the FIFO method applied with different thresholds to different identi- +fication methodologies. +Figure 6.6: Results using FIFO update with different thresholds. + +Plataniotis FiFO technigue +1.0 +Baseline +score < 0.7 +Identification Accuracy +0.8 +0.3 < score < 0.7 +0.4 < score < 1 +0.6 +0.2 < score < 1 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Eduardo FiFO technigue +1.0 +Baseline +score < 0.4 +Identification Accuracy +0.8 +0.1 < score < 0.5 +0.2 < score < 0.4 +0.6 +0.1 < score < 0.3 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Tawfik FiFO technique +1.0 +Baseline +0.5 < Ascore < 0.8 +Identification Accuracy +0.8 +0.65 < △score < 0.9 +△score > 0.5 +0.6 +△score > 0.2 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Belgacem FiFO technigue +1.0 +Baseline +0.15 < △score < 0.3 +Identification Accuracy +0.8 +0.1 < △score < 0.4 +△score> 0.2 +0.6 +Ascore > 0.5 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Tawfik with kNN FiFO technique +1.0 +Baseline +5 < Ascore < 7 +Identification Accuracy +0.8 +4 < △score < 8 +△score > 5 +0.6 +△score < 5 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Belgacem with kNN FiFO technique +1.0 +Baseline +0.3 < Ascore < 0.7 +0.8 +Identification Accuracy +0.4 < △score < 0.6 +Ascore > 0.5 +Ascore < 0.5 +0.6 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)100 +Long-Term Performance and Template Update +Figure 6.7: Results using Fixation update (the corresponding value represents the number of sam- +ples that were fixated per subject). +6.5.2 +Deep convolutional network +The results for the implemented end-to-end convolutional neural network are presented in Fig. 6.8. +Model update offered a small improvement in performance in the first test point (91.48% versus +91.15 without update). However, the model experiences sharp performance decay and, after the +fifth hour, the model update is unable to improve identification accuracy. In fact, model update +caused a decrease in identification rate, which is coherent with the findings regarding update with +multilayer perceptron classifiers reported by Lopes et al. [269]. +Different results were obtained when the fully-connected layer of the network was replaced by +a kNN classifier (see Fig. 6.9). Although the results with kNN are slightly worse than those of the +end-to-end network (90.89% vs. 91.15% for the first test point), the template update technique is +more successful and is able to offer performance improvements for almost all test points. +When compared with the results reported by Lopes et al., the CNN (either end-to-end or with +kNN classification) offers the best performance in the first test points after enrollment. However, +it gradually loses that advantage as time passes, even with template update. +Likely, the network will require more data with more variability during the first training phase. +Increasing the training data to thirty minutes or even a few hours per subject would enable the + +Plataniotis Fixation technigue +1.0 +Baseline +1/4 +Identification Accuracy +0.8 +1/2 +3/4 +0.6 +j+1 +j*2 + 2 +j*3 + 3 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Eduardo Fixation technigue +1.0 +Baseline +1/4 +Identification Accuracy +0.8 +1/2 +3/4 +0.6 +j+1 +j*2 + 2 +j*3 + 3 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Tawfik with kNN Fixation technique +1.0 +Identification Accuracy +0.8 +0.6 +Baseline +1/4 +0.4 +1/2 +3/4 +j+1 +0.2 +j*2 ＋ 2 +j* 3 + 3 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)Belgacem with kNN Fixation technique +1.0 +0.8 +Identification Accuracy +0.6 +Baseline +1/4 +0.4 +1/2 +3/4 +0.2 +j+1 +j* 2 + 2 +j*3+3 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)6.6 Summary and Conclusions +101 +Figure 6.8: Performance results over time for the CNN model with fine-tuning-based model up- +date. +Figure 6.9: Performance results over time for the CNN model adapted with kNN decision and +FIFO template update, for several threshold criteria (samples with scores between the presented +values are accepted for update). +network to better learn the common variability patterns of the ECG. This should not only increase +the initial performance, immediately after enrolment but also reduce the performance decay over +time. +6.6 +Summary and Conclusions +This work studied how ECG variability affects the performance of state-of-the-art biometric algo- +rithms, and how template update could mitigate performance decay over time. The results have +shown long-term identification performance in ECG biometrics is generally weak, despite the +promising results often presented in the literature. +Template update techniques proved successful in enhancing the long-term performance of +handcrafted state-of-the-art methods, especially when using template fixation techniques. Addi- +tionally, with a deep learning algorithm, results are better than traditional methods immediately +after enrollment, although it offers slightly worse performance as time progresses. +Generally, one can conclude that further efforts are needed for the study and development of +more advanced techniques. The obtained results in these more realistic settings show that the + +CNN Results +1.0 +Baseline +CNN with fine-tuning update +Identification Accuracy +0.8 +0.6 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)CNN+kNN Results +1.0 +0.8 +0.6 +Baseline +13 < score < 16 +0.4 +12 < score < 18 +14.5 < score < 15.5 +0.2 +14 < score < 15.3 +10 < score < 20 +0.0 +0 +5 +10 +15 +20 +25 +Time (h)102 +Long-Term Performance and Template Update +performance levels commonly reported in the literature would likely not be verified upon real +application. Special focus should be devoted to supervised update techniques, so that ECG-based +biometric systems can offer reliable performances over long periods. + +Chapter 7 +Leveraging Explainability +to Understand ECG Biometrics +Foreword on Author Contributions +The research work described in this chapter was conducted entirely by the author of this thesis, under the supervi- +sion of Jaime S. Cardoso. The results of this work have been disseminated in the form of an article in international +conference proceedings and an abstract in national conference proceedings: +• J. R. Pinto and J. S. Cardoso, “Explaining ECG Biometrics: Is It All In The QRS?,” in International Conference +of the Biometrics Special Interest Group (BIOSIG 2020), Sep. 2020. [339] +• J. R. Pinto and J. S. Cardoso, “xECG: Using Interpretability to Understand Deep ECG Biometrics,” in 27th +Portuguese Conference on Pattern Recognition (RECPAD 2021), Nov. 2021. +7.1 +Context and Motivation +Throughout the past twenty years, research on biometrics based on the electrocardiogram (ECG) +has largely been a success story [343]. After successful proofs-of-concept in cleaner medical +signals (on-the-person), the focus is quickly shifting to acquisitions in more realistic scenarios (off- +the-person). Deep learning approaches [162; 238; 284; 338; 344] have been essential in dealing +with the increased noise and variability in off-the-person settings, despite the performance and +robustness issues that still hinder application in real scenarios. +However, deep learning decisions are obscure: unlike traditional methods based on fiducial +features, we don’t know what information the model uses to distinguish people [107; 372]. One +can assume that the models look mainly to the QRS since it is the most stable part of the ECG in the +face of noise and variability [172; 379]. Several methods have thus focused on QRS complexes +for ECG biometrics [238; 446], but this practice has become uncommon in recent works. This +indicates the true role of this waveform complex in identity discrimination is still to be adequately +recognised. +103 + +104 +Leveraging Explainability to Understand ECG Biometrics +P +Q +R +S +T +Figure 7.1: Illustration of the ECG waveforms on a sample PTB signal segment. +Currently, pattern recognition researchers understand the importance of knowing what specific +information is relevant for their models to reach decisions. Retreating to easily explainable tradi- +tional models (such as decision trees) is often unacceptable due to their performance limitations. +Hence, various interpretability tools are being developed to peek into the inner workings of deep +networks applied to diverse tasks [67; 385; 397]. +This work uses, for the first time in the literature, such interpretability tools on a deep ECG +biometric model, to understand what parts of the ECG are most useful for automatic human iden- +tification. The model is a competitive state-of-the-art method [338; 344] applied for ECG-based +identification in data subsets with diverse signal quality and number of identities. With this, we +aim to assert the importance of the QRS and other waveforms for ECG biometrics and discuss +future possibilities as this topic evolves towards more challenging and realistic scenarios. Addi- +tionally, we propose an intuitive way to visualise interpretations for unidimensional signals. The +code and additional results are available online1. +7.2 +The Electrocardiogram as a Biometric Trait +As presented in Chapter 2, subsection 2.2.2, the ECG is approximately a cyclical repetition of a set +of waveforms (P, Q, R, S, and T) that corresponds to a heartbeat (see Fig. 7.1) [286; 343]. Each of +these waves corresponds to specific phenomena involved in the heart’s contraction and relaxation. +As a measurement of the electrical currents spread across the heart, the ECG signals will +reflect the geometry of this organ. For example, larger hearts, with more cells to depolarise and +repolarise, will result in ECG waveforms with larger amplitudes. Higher or lower basal heart rates +will also result in different signal morphologies. Since heart geometry and basal heart rates vary +across individuals, this intersubject variability is what makes the ECG sufficiently unique to be +used in biometric recognition [172; 441]. +However, the ECG signals are also susceptible to intrasubject variability factors. Noise sources +during acquisition, the short-term and long-term effects of exercise, emotional states, stress, +drowsiness, and fatigue are some of the factors that reflect mainly in the heart rate variability, +changing the morphology of the P-R and S-T segments [10; 379]. These are the sources of un- +certainty that hinder the use of the ECG as a biometric trait. While these are largely controlled +1xECG Github Repository. Available on: https://github.com/jtrpinto/xECG. + +7.3 Methodology +105 +in medical or on-the-person settings (where the subject is at rest, laying down, and signals are +acquired using several high-quality gel electrodes), their effects are dominant for realistic off-the- +person signals (acquired using fewer dry electrodes on the hands, during common daily activi- +ties) [338; 342; 343]. +When compared with the P and T waves, the QRS corresponds to a larger polarisation event +over a shorter period. In practice, this makes the QRS more dominant over noise and intrasubject +variability than the other ECG waveforms [342; 343]. Hence, the QRS is considered more stable +over time and across variable conditions, which makes it better suited for biometric recognition. +Despite this, it is still unclear how much identity information is carried by the QRS complex +compared to the other waveforms, and whether it is enough for an accurate and robust biometric +recognition system. Studies on ECG-based biometric identification have shown it is possible to +distinguish small sets of individuals in on-the-person settings using only the QRS complex or QRS +fiducial amplitude and time measurements [238; 446]. Nevertheless, this practice is becoming +uncommon as research evolves towards realistic off-the-person signals and larger databases. +This denotes that the sole use of the QRS may not be adequate for off-the-person settings, or +the individual information carried by the QRS may not be enough to distinguish individuals in +large populations. This work aimed to address these doubts through a study on the role and rele- +vance of the QRS and the other waveforms in ECG-based biometric identification. Interpretability +tools are used to assess which parts of the ECG are more relevant to the decisions of an end-to-end +identification model [344], with on-the-person and off-the-person signals and data subsets with a +varying number of identities. +7.3 +Methodology +7.3.1 +Biometric identification model +The biometric model for identification followed the architecture proposed by Pinto et al. [344], +which has attained state-of-the-art results in off-the-person settings for both identification and, +later, identity verification [338]. The model (see Fig. 7.2) receives five-second blindly segmented +ECG signals and outputs probabilities for each of the N identities considered. Finding the highest +probability score allows us to assign the respective identity to the input signal. +The model consists of an end-to-end 1D convolutional neural network (CNN) with four con- +volutional layers (with 1 × 5 filters, two layers with 24 followed by two with 36), followed by +ReLU activation. Neighbouring convolutional layers are separated by 1 × 5 max-pooling layers. +The last convolutional layer is followed by two fully-connected layers (100 neurons with ReLU +and N neurons with softmax activation). +7.3.2 +Interpretability tools +To capture the dynamics behind the decisions of the biometric model, four interpretability meth- +ods are applied to the trained model: Occlusion, Saliency, Gradient SHAP, and DeepLIFT. Oc- + +106 +Leveraging Explainability to Understand ECG Biometrics +ID +Conv1D +24@1x5 +ReLU +MaxPool +1x5 +Conv1D +24@1x5 +ReLU +MaxPool +1x5 +Conv1D +36@1x5 +ReLU +MaxPool +1x5 +Conv1D +36@1x5 +ReLU +FullyConn. +100 +ReLU +FullyConn. +N +Softmax +Signal +1x1000 +Figure 7.2: Architecture of the biometric identification model. +clusion and Saliency are two of the simplest interpretability methods, while Gradient SHAP and +DeepLIFT are more sophisticated and powerful. These are implemented in the Captum tool- +box [221] for PyTorch and are described below. +Occlusion +The Occlusion method [478] consists in measuring the influence of hiding a portion +of the input on the output of the model. When hidden, the more relevant input parts will cause +larger changes in the output, and will thus be assigned greater relevance in the explanations offered +by this method. This is the simplest method to interpret a model, although the size of occluded +regions should be carefully defined to obtain meaningful explanations. +Saliency +The Saliency method [401] is based on the gradients of a model given a certain in- +put. Through backpropagation, the gradient of target class scores w.r.t. the input is obtained. +A saliency map is then generated by rearranging the class score derivatives, generating saliency +maps that assign higher relevance to input regions that correspond to higher gradients. Requiring a +single backpropagation pass, this method is a simple and fast way to obtain explanations of model +predictions. +Gradient SHAP +Gradient SHAP [283] is an approach based on game theory which considers +the explanations of a model’s predictions as models themselves. For sophisticated deep learning +models, the explanation models are simplified and interpretable approximations of the respective +models. SHapley Additive exPlanation (SHAP) values, inspired by game theory’s Shapley values, +are computed through the gradient of a random point between a baseline and the input with added +random noise. The SHAP values denote how much a given part of the input raises the probability +for the considered class, and are reportedly better aligned with human intuition and effective in +discriminating among output classes. +DeepLIFT +DeepLIFT (Deep Learning Important FeaTures) [391] performs backpropagation to +track the contributions to the output to the responsible parts of the input. Throughout this process, +it compares the difference in inputs and outputs considering a reference (or baseline) input, as- +signing contribution scores to each neuron of the model. It also allows for the study of negative +contributions: how much a specific part of the input contributes to lower the probability for the +considered class. + +7.4 Experimental Setup +107 +7.3.3 +Visualisation +Decision explanations obtained using interpretability tools are visualised using the multicoloured +line plot feature of Matplotlib [183]. ECG signals are plotted so that the colour of each signal +component represents its relative relevance to the decision. In this case, lighter yellow colours +represent less relevant time samples, whereas more relevant samples assume darker purple colours. +This way, both the ECG morphology and the relevance of each of its components are easily and +intuitively presented. +7.4 +Experimental Setup +The data used for model training and evaluation have been drawn from the Physikalisch- +Technische Bundesanstalt ECG Database (PTB) [49; 146] and the University of Toronto ECG +Database (UofTDB) [445]. The PTB database includes on-the-person (high-quality) 12-lead ECG +signals acquired at 1 kHz from 290 subjects at rest. The UofTDB includes single-lead off-the- +person (more noisy and realistic) data acquired from 1019 subjects. To match the UofTDB, PTB +signals were downsampled to 200 Hz and only Lead I was used. +Five-second segments were blindly extracted (without fiducial detection) from the recordings. +Fifty per cent of those segments (per identity) were used during training and the remaining were +reserved for testing. This provided more challenging test settings than those commonly found in +the literature, but also deliberately avoided the most realistic settings (see [338]), for the sake of +obtaining meaningful interpretations. +To simulate gradually increasing identification difficulty within each database, subsets of N +identities are considered, with N ∈ {2,5,10,20,50,100,200,500,1019}. The identities in each +subset are the first N in lexicographical order. Each subset includes all identities that compose +smaller subsets, so subjects #1 and #2 are the main focus of analysis since these are present in all +subsets. Throughout this paper, TN denotes the subset of UofTDB data from N subjects and PN +denotes the subset of PTB data from N identities. As stated in Table 7.1, P290 was used instead of +P200 to take advantage of the entire PTB dataset. Model training details can be found online at this +project’s repository. +Performance evaluation is based on the True Positive Identification Rate (or accuracy): the +fraction of test samples that are correctly assigned to their true identity by the trained model. +Interpretations are examined through the proposed visualisation method. +7.5 +Results and Discussion +The results of the performance evaluation are presented in Table 7.1. These results roughly follow +the expected patterns considering the use of on-the-person versus off-the-person ECG data. The +model is able to attain high true positive identification rates in both databases when the population + +108 +Leveraging Explainability to Understand ECG Biometrics +Table 7.1: True positive identification rate results (%) on the test data. +Database +Number of Identities +2 +5 +10 +20 +50 +100 +2001 +500 +1019 +PTB +100.0 +100.0 +99.63 +99.50 +98.92 +98.76 +97.73 +- +- +UofTDB +100.0 +97.26 +98.30 +95.46 +93.86 +91.16 +89.70 +91.20 +91.45 +1For PTB, this column corresponds to the entire set of 290 subjects. +is small, but as the set of subjects grows, performance decreases and a wide gap distinguishes the +more challenging off-the-person settings from the more controlled on-the-person settings. +Additionally, one can find some unusual patterns in the performance results. Considering +M > N, one would expect identification performance with subset TN to be higher than with subset +TM. With UofTDB off-the-person data this is not always verified: e.g., from T5 to T10, performance +increases from 97.26% to 98.10%. In these cases, we need to consider that datasets with fewer +identities have fewer data and, thus, more unstable results. Alternatively, the identities added to +TN to create TM may be easier to discriminate (“sheep”, according to the concept of biometric +menagerie [104; 468]) and thus contribute to improving accuracy. However, one should also +regard the substantial regularisation needed to avoid overfitting and the instability during training +as possible causes for these discrepancies. This is a very important insight into the increased +difficulties of using off-the-person data and the need for improved and more robust biometric +models. +Analysing the explanations obtained using the four interpretability tools (examples in Fig. 7.3 +and Fig. 7.4), a trend is verified from smaller to larger identity subsets, consisting on the deviation +from focusing mainly on the QRS complex to the increasing relevance of other parts of the heart- +beats. This is also confirmed when combining the explanations of all heartbeats of each person +into a single average heartbeat (see Fig. 7.5 and Fig. 7.6). +With the cleaner medical signals from PTB, the focus is mostly on the QRS complex, but +information from other waveforms starts to become more and more relevant as more identities are +added. It is noteworthy how, when discriminating PTB subjects #1 and #2 in a two-subject scenario +(see Fig. 7.5), the model still focuses mainly on the QRS, even though subject #2 has a very specific +characteristic, the inverted T-wave, that is arguably their most distinctive feature. This denotes +how, in these cleaner signals, the QRS complex is so stable that the remaining waveforms, more +susceptible to heart rate variability, are largely ignored by the model regardless of any visually +obvious intersubject differences they may present. +With the more realistic off-the-person signals from UofTDB, the QRS retains high importance +but the relevance is more evenly spread among the signal waveforms. In the specific case of subject +#2 (see Fig. 7.6), it is evident that the QRS retains the highest importance for the decision, even +in T1019 (the largest subset). This may denote that, even in these more challenging settings, the +identification models will still give preference to the QRS over other waveforms if it is sufficiently +unique among the considered identities. Nevertheless, in such large sets of identities, the expected +behaviour is that of subject #1 (see Fig. 7.6), since the limited identity information carried by the + +7.5 Results and Discussion +109 +2 id. +Occlusion +Saliency +Gradient SHAP +DeepLIFT +5 id. +10 id. +20 id. +50 id. +100 id. +290 id. +Figure 7.3: Explanations over an example five-second ECG segment from PTB (in each subplot, +the yellow to dark purple colours correspond to increasing time sample relevance and vertical grey +lines denote R-peak locations; signals were filtered for easier visualisation). +2 id. +Occlusion +Saliency +Gradient SHAP +DeepLIFT +5 id. +10 id. +20 id. +50 id. +100 id. +200 id. +500 id. +1019 id. +Figure 7.4: Explanations over an example five-second ECG segment from UofTDB (in each sub- +plot, the yellow to dark purple colours correspond to increasing time sample relevance and vertical +grey lines denote R-peak locations, signals were filtered for easier visualisation). + +110 +Leveraging Explainability to Understand ECG Biometrics +Subject #2 Subject #1 +Figure 7.5: Average explanations over heartbeat waveforms of subjects #1 and #2 on the subsets +of the PTB database (in each subplot, the yellow to dark purple colours correspond to increasing +time sample relevance; signals were filtered for easier visualisation). +Subject #2 Subject #1 +Figure 7.6: Average explanations over heartbeat waveforms of subjects #1 and #2 on the subsets of +the UofTDB database (in each subplot, the yellow to dark purple colours correspond to increasing +time sample relevance; signals were filtered for easier visualisation). + +7.6 Summary and Conclusions +111 +QRS will lead the model to also look to other parts of the signal. +One interesting aspect is the difference between the results with Occlusion versus the other +methods. Occlusion generally grants the QRS complex much more relevance, regardless of the +settings. In the state-of-the-art approaches, the QRS complex is not only a source for identity +features but also frequently used as an easily detectable reference landmark for the location of +other ECG waveforms. This may also be the case in this end-to-end deep model. Although there +are challenging contexts where the QRS may not be the main contributor to the decision, it may +be essential to the deep model as a reference landmark to locate other waveforms in the signal. +Hence, when occluded, it will be the signal component that most impacts the decision, causing the +occlusion method to generally consider it the most relevant. +7.6 +Summary and Conclusions +This work aimed to explain how deep models use ECG signals to distinguish people, using inter- +pretability tools. Overall, the obtained results partially confirm the claim that the QRS is the key +to ECG-based biometrics. With small populations in on-the-person settings, it can alone be used +for reliable recognition. However, as we evolve towards larger populations and off-the-person +settings, other components become relevant in discriminating people, as the models require more +identity information to overcome the hurdles placed by enhanced intrasubject variability. +However, even though relevance is more evenly shared in off-the-person identification in large +sets of identities, the QRS is shown as essential by the occlusion method. It appears that, just like +several literature methods, the implemented end-to-end model learnt to use the QRS as a landmark +for the location of other ECG components in the signal, resulting in large output changes when +the QRS is occluded. Hence, despite the literature claims, one should avoid relying too heavily +on any single part of the ECG, including the QRS complex, since all waveforms carry identity +information that proves increasingly useful in more realistic settings and larger populations. +Beyond these insights, further efforts should be devoted to extending this study and offering +a deeper, more thorough, and more objective analysis of the contribution of each ECG waveform +to the model’s decisions. Obtaining more systematic and complete explanations could create new +opportunities for the use of interpretability tools during model training. Using explanations to +regularise models and promote the focus on the most relevant signal components or the distributed +use of the whole signal (instead of just the QRS) could lead to improved recognition accuracy and +robustness. + + +Chapter 8 +Interlead Conversion of +Electrocardiographic Signals +Foreword on Author Contributions +The research work described in this chapter was conducted in collaboration with Sofia C. Beco, under the super- +vision of Jaime S. Cardoso. The author of this thesis contributed to this work on the formulation, implementation, +and improvement of the interlead conversion methodology, the preparation and conduction of the extended exper- +iments, the discussion of the results, and the writing of the scientific publications. +The results of this work have been disseminated in the form of an extended journal article and a short paper +presented at an international conference: +• S. Beco, J. R. Pinto, and J. S. Cardoso, “Electrocardiogram Lead Conversion from Single-Lead Blindly- +Segmented Signals,” BMC Medical Informatics and Decision Making, 22: 314, 2022. [31] +• S. Beco, J. R. Pinto, and J. S. Cardoso, “Interlead Conversion of Single-Lead Blindly-Segmented Electrocar- +diogram Signals,” in 17th International Conference on Computational Intelligence Methods for Bioinformatics +and Biostatistics (CIBB 2021), Nov. 2021. [30] +8.1 +Context and Motivation +The electrocardiogram (ECG) is the measurement of electrical potentials that make the heart con- +tract and relax as intended. The morphology of the ECG signal depends on the location of the +electrodes used for acquisition: different electrode placement results in different perspectives over +the heart [343]. For medical purposes, the standard configuration acquires the ECG over twelve +leads for more information, but it requires ten electrodes placed on the patient’s arms, legs, and +chest. Using fewer electrodes allows for more comfortable and inexpensive acquisitions, at the +expense of certain leads that could be ideal for a more accurate diagnosis of certain conditions. +To get the best of both worlds, researchers have proposed methods for the automatic interlead +conversion of ECG signals [244; 293; 395; 407; 408]. These transform short ECG segments to +mimic other perspectives, using acquired leads to reconstruct any leads that were not recorded. +However, these methods still present limited applicability, since they typically require multiple +leads as input. Even the most advanced methods [244; 293], that only use one input lead, still +113 + +114 +Interlead Conversion of Electrocardiographic Signals +require the inputs to be single heartbeat segments aligned in time, which makes them dependent +on separate processes and, overall, less flexible and robust. +This chapter presents a study on the feasibility of ECG interlead conversion using short seg- +ments from just one limb lead without any kind of temporal alignment (blindly-segmented). With +such input, the proposed methodology is trained to reconstruct other leads as faithfully as possible. +This aims to open up new possibilities for more comfortable ECG acquisition in clinical scenarios +or wearable devices without giving up the benefits of multi-lead recordings for medical diagnosis. +The proposed methodology, based on deep learning encoder-decoder structures, is explored +for interlead conversion using either lead II or lead I (limb leads) signals as reference, and using +a single shared encoder or an individual encoder for each target lead. Beyond the training and +testing on the widely used PTB database, the conversion models are evaluated on cross-database +scenarios with the INCART and PTB-XL databases. Additionally, the clinical annotations of +the PTB-XL database are also used for a differential performance evaluation in the presence of +medical conditions. The code is available online1. +8.2 +Related Work +At the onset of research on interlead conversion, methodologies commonly required several leads +as reference for robust lead reconstruction. Zhu et al. [496] performed a preliminary study on +the conversion of ambulatory ECG recordings into standard 12-lead ECG signals using lead-field +theory and the least-squares method. Nelwan et al. [306] learned generic and patient-specific +linear regression coefficient templates to reconstruct up to four missing leads with high correlation +results. +Later, Yoshida et al. [474] used 12 lead acquisitions to synthesise additional leads (right ven- +tricular leads V3R, V4R, and V5R and posterior chest leads V7, V8, and V9) which provide +important information for the diagnosis of acute myocardial infarction. Their algorithm was based +on the transfer coefficient estimated from the learning data. +Silva et al. [395] developed three methods for obtaining the Frank leads using the 12 standard +leads as reference: the Kors Quasi-Orthogonal method, the Kors Linear Regression method, and +the Dower Inverse Matrix. The conversion was successful for signals from healthy subjects but +presented limitations on signals from subjects with pathologies. The recent work by Smith et al. +[407] was one of the first to use machine learning techniques for interlead conversion. They used +a focused time-delay neural network (FTDNN), which is well suited for time series prediction. +However, their methodology required seven input leads (all limb leads and V1). +Atoui et al. [22] used ensembles of fully-connected neural networks to learn to synthesise +V1, V3, V4, V5, and V6 heartbeats from three-lead inputs (I, II, and V2). Schreck and Fishberg +[380] performed the first study on the synthesis of the entire set of 12 standard leads and scalar +3-lead derived vectorcardiogram from just three measured leads. Their proposed methodology +used nonlinear optimisation to construct a universal patient transformation matrix. Hansen et al. +1Interlead ECG Conversion Github Repository. Available on: https://github.com/jtrpinto/ecg-conversion. + +8.3 Methodology +115 +[165] applied linear generic and subject-specific transforms to convert recordings from adhesive +patch-type ECG monitors to the standard 12-lead ECG signals. In [435; 438], researchers also +explored personalised statistically determined linear transforms and went on to achieve improved +results. +Lee et al. [243] proposed methods based on linear regression and artificial neural networks +to reconstruct the 12 standard leads from subsets of 35 channels acquired using one single large +patch covering the subject’s chest. Although accurate, the method is arguably incompatible with +scenarios focused on ease of use and patient/user comfort. Similarly, Grande-Fidalgo et al. [154] +used linear regression and fully-connected networks to reconstruct the entire set of twelve stan- +dard leads from a subset of just three input leads. Sohn et al. [408] used long short-term memory +(LSTM) networks to reconstruct the twelve ECG standard leads from a three-lead patch-type de- +vice. Their results show their method was able to correctly retain pathological abnormalities from +medical conditions on the reconstructed signals. +The work of Lee et al. [244] was one of the few that studied the synthesis of standard leads +using only one reference lead. In their study, chest leads (V1 to V6) were synthesised from lead +II using a generative adversarial network (GAN). However, input segments had to be single heart- +beats, aligned according to the R-peaks, which decreases the difficulty of the proposed method but +also its applicability. Matyschik et al. [293] developed patient-specific models to more accurately +reconstruct eleven missing ECG signals from a single available lead of the standard 12-lead sys- +tem. However, the reference lead was either V1, V2, or V3 which, being chest leads, do not enable +the usage in less obtrusive setups which would preferentially use limb leads. +In this work, we explore the more challenging scenario of reconstructing the entire set of +twelve standard leads using only one reference lead. Moreover, the reference signals are blindly- +segmented (without any kind of temporal alignment) and pertain to one of the limb leads to allow +for applications on the least obtrusive setups. Our main goal is to assess whether it is possible to +reconstruct the electrocardiogram signal in such challenging scenarios and discuss the next steps +towards the use of interlead conversion in less obtrusive clinical setups and wearable devices. +8.3 +Methodology +8.3.1 +General overview +The proposed methodology for interlead ECG conversion follows the encoder-decoder structure +typically used for deep image segmentation. The encoder receives an input signal and processes +it to create a compressed representation that retains relevant information for the task at hand. The +decoder receives this representation and processes it so that the output matches the ground-truth as +closely as possible. Here, the input to the encoder is a short ECG segment of one lead (X) and the +ground-truth is the corresponding segment in a different lead (Y). Thus, the encoder is in charge +of selecting the information from X that is needed for Y, and the decoder will use that information +to reconstruct the corresponding lead Y signal. + +116 +Interlead Conversion of Electrocardiographic Signals +8.3.2 +Model architectures +The general encoder-decoder structure allows for diverse specific model architectures. This work +focuses on the U-Net model, a fully convolutional architecture that has found many applications +related to semantic segmentation and can also be adapted for the task of ECG lead conversion. +8.3.2.1 +U-Net +The U-Net was initially proposed by Ronneberger et al. [367] as a tool for biomedical image +segmentation. In this work, the implemented architecture (see Fig. 8.1) receives an input segment +of lead X, which initially goes through a chain of three sequential blocks, each with half the signal +resolution of the previous block. Each block includes two convolutional layers (each followed by +batch normalisation and ReLU activation) and ends with a max-pooling layer. +Between the encoder and the decoder, two convolutional layers compose the latent space or +bottleneck block, which corresponds to the maximum point of information compression. The +decoder mirrors the encoder in its structure, with three similar blocks composed of an upsampling +layer and two transposed convolutional layers. The last transposed convolutional layer outputs a +single-channel signal whose size corresponds to the input segment. The activation function of this +last layer is the hyperbolic tangent for an output signal with amplitudes in [−1,1]. +One aspect of the U-Net which is often cited as the key to its widespread success is the skip- +connection. U-Nets typically include skip-connections between corresponding blocks on the en- +coder and the decoder. This means the feature maps from the encoder blocks are directly routed +to the corresponding decoder blocks, allowing the model to propagate context information from +multiple resolutions between the encoder and the decoder for higher flexibility. +8.3.2.2 +Convolutional autoencoder (AE) +Beyond the aforementioned U-Net architecture, adapted for unidimensional signal inputs, we also +explore a convolutional autoencoder (AE, see Fig. 8.2). Its architecture is very similar to the U- +Net, albeit without skip-connections. As a result, the structure is simplified, when compared to the +U-Net, and the latent representation sent from the encoder to the decoder is smaller. Experiments +with the AE architecture aim to assess if the skip-connections are essential for the task at hand or +if the simplified structure could avoid overfitting and bring performance benefits. +8.3.2.3 +Label refinement network (LRN) +The third architecture explored in this work was based on Label Refinement Network (LRN, see +Fig. 8.3) was originally proposed by Islam et al. [190] for semantic image segmentation. Its archi- +tecture is identical to the aforementioned U-Net. The singularity of the LRN lies in the supervision +strategy: while the U-Net only uses the output of the last decoder block in the reconstruction loss, +the LRN computes the loss at the outputs of every decoder block. This results in supervision at +several resolution levels, leading the decoder to offer a coarse reconstruction right after the first + +8.3 Methodology +117 +1 32 32 +32 32 1 +64 +64 +64 +128 +128 +256 +256 + 512 + 512 +256 +128 +256 +128 +64 +Input +lead X +segment +Output +lead Y +segment +conv 5x5, BatchNorm, ReLU +copy and crop +max pool 5x5 +up-conv 5x5 +conv 1x1, Tanh +(1, 5000) +(1, 5000) +Figure 8.1: Schema of the U-Net architecture. +1 32 32 +32 32 1 +64 +64 +64 +128 +128 +256 +256 +128 +256 +128 +64 +Input +lead X +segment +Output +lead Y +segment +conv 5x5, BatchNorm, ReLU +max pool 5x5 +up-conv 5x5 +conv 1x1, Tanh +(1, 5000) +(1, 5000) +32 +64 +128 +128 +64 +Encoder +Latent Space +Decoder +Figure 8.2: Schema of the convolutional autoencoder (AE) architecture. +block, which should be gradually refined by the subsequent blocks for improved results at higher +resolutions. Experiments with the LRN architecture aim to assess if the multi-level resolution +could bring improved performance to the task of signal lead conversion as they have for semantic +segmentation. +8.3.3 +Shared vs. individual encoders +The conversion of one lead into multiple missing leads requires multiple decoders - each one will +fulfil the task of reconstructing their respective lead based on the compressed latent representation. +In the case of the encoder, however, it is possible to have a single one whose output will be shared +by all decoders or have multiple encoders, each one dedicated to one individual decoder. + +118 +Interlead Conversion of Electrocardiographic Signals +1 32 32 +32 32 1 +64 +64 +64 +128 +128 +256 +256 + 512 + 512 +256 +128 +256 +128 +64 +output 4 +output 3 +output 2 +output 1 +(1, 1000) +(1, 200) +(1, 40) +(1, 8) +conv 5x5, BatchNorm, ReLU +copy and crop +max pool 5x5 +up-conv 5x5 +conv 1x1, Tanh +Input +lead X +segment +(1, 5000) +Output +lead Y +segment +(1, 5000) +Figure 8.3: Schema of the architecture based on label refinement networks (LRN). +In this work, we explore both possibilities for 12-lead reconstruction - using one shared en- +coder connected to all 11 decoders, for all 11 output leads except the one corresponding to the +input, or using one individual encoder for each of the 11 decoders. Using individual encoders +grants more flexibility to each lead conversion process, as each encoder will be able to learn a +unique way to obtain compressed representations and each encoder-decoder pair will work inde- +pendently from all others. On the other hand, using one shared encoder results in a much lighter +and faster algorithm and the added simplicity may contribute to avoiding overfitting. +8.4 +Experimental Setup +8.4.1 +Data +The experiments conducted in this work used mainly the data provided in the PTB Diagnostic ECG +Database [49], available on Physionet [146]. The PTB database includes data from 16 channels, +including all 12 standard leads, sampled at 1 kHz. It contains a total of 549 records from 290 indi- +viduals, with one to five records per subject. Recordings were cropped into segments of 5 s (5000 +samples). A second-order Butterworth bandpass filter with cut-off frequencies fc = [1,40] Hz was +applied to each segment to remove noise while retaining the most useful ECG information. The +amplitudes of the n values of each signal x were then min-max normalised to the interval [−1,1] +following the equation: +xn = 2× xn −xmin +xmax −xmin +−1. +(8.1) + +8.4 Experimental Setup +119 +The data from PTB was divided into train and test sets, with approximately 63%, 7% and 30% +of the segments, respectively, for a total of 7086, 787, and 3509 ECG segments for each set. For +a more thorough and challenging evaluation, subjects are divided between the train/validation and +test sets: the latter had recordings from subjects 1 to 50 while the former had recordings from +subjects 51 to 290. +The INCART database (officially the St. Petersburg INCART 12-lead Arrhythmia Database), +also available on Physionet, was used to test the performance of trained models on cross-database +scenarios. This database contains 75 Holter recordings from 32 subjects undergoing tests for +coronary artery diseases. Each record is 30 minutes long and contains twelve standard leads +sampled at 257 Hz. Recordings from this database were resampled to 1 kHz and processed as +described above for PTB. +The PTB-XL database [443; 444], created by the same team as the PTB, includes 21837 clini- +cal ECG recordings from a total of 18885 patients. Each recording is 10 seconds long, includes all +twelve standard ECG leads, and is originally sampled at 500 Hz. The waveforms were annotated +by up to two cardiologists, who assigned annotations to each record. The 71 possible annota- +tion statements have been clustered into five superclasses: NORM (normal ECG), MI (myocardial +infarction), STTC (ST/T change), CD (conduction disturbance), and HYP (hypertrophy). This +dataset was originally created for the training and evaluation of automatic ECG interpretation al- +gorithms but also shows great promise for the development of lead conversion algorithms. In this +work, we take advantage of expert clinical annotations to study the effect of medical conditions +on the quality of the lead conversion results. From the total of 21837 recordings, we selected +the 16272 that did not have conflicting superclass annotations. From each recording, the first 5 +seconds were cropped, resampled to 1 kHz, and processed as described above for PTB. +8.4.2 +Model training and evaluation +The models were trained using the l1-loss between the model outputs and the corresponding +ground-truth signals as the objective function. The l1 was chosen empirically as it allowed the +model to learn most adequately both the overall morphology of the signals and their finer details. +The Adam optimiser was used with an initial learning rate of 1 × 10−3, over a maximum of 500 +epochs with batch size 32 (shared encoder) or 16 (individual encoder) and early stopping patience +of 50 epochs. +To compare lead conversions with the corresponding measured ground-truth signals, this work +used the following metrics: the average and median Pearson correlation coefficient (r, used in the +majority of the related literature), the average root mean square error (RMSE), and the average +Structural Similarity Index Measure (SSIM). + +120 +Interlead Conversion of Electrocardiographic Signals +Table 8.1: Comparison of encoder-decoder architectures on one-to-one lead conversion. +Model +r (avg.) +r (med.) +U-Net +0.69 +0.78 +Autoencoder +0.67 +0.78 +LRN +0.65 +0.75 +Table 8.2: Average correlation between lead II signals and the remaining leads on the PTB, IN- +CART, and PTB-XL databases. +Average Correlation to Lead II +I +III +aVR +aVL +aVF +V1 +V2 +V3 +V4 +V5 +V6 +PTB +0.45 +0.36 +-0.71 +0.01 +0.77 +-0.34 +-0.20 +0.00 +0.28 +0.72 +0.81 +INCART +0.46 +0.80 +-0.86 +-0.49 +0.95 +-0.45 +-0.19 +0.25 +0.65 +0.82 +0.77 +PTB-XL +0.70 +0.31 +0.25 +-0.82 +0.83 +-0.44 +-0.04 +0.37 +0.68 +0.81 +0.84 +Table 8.3: Test results of the U-Net used for multi-lead conversion from lead II, with shared or +individual encoders. +Shared Encoder +Individual Encoders +Lead +r (avg.) +r (med.) +RMSE +SSIM +r (avg.) +r (med.) +RMSE +SSIM +I +0.67 +0.73 +0.28 +0.28 +0.66 +0.71 +0.29 +0.26 +III +0.56 +0.65 +0.29 +0.63 +0.56 +0.70 +0.29 +0.64 +aVR +0.89 +0.95 +0.12 +0.92 +0.90 +0.95 +0.12 +0.92 +aVL +0.47 +0.58 +0.36 +0.15 +0.47 +0.61 +0.36 +0.16 +aVF +0.81 +0.88 +0.20 +0.64 +0.83 +0.90 +0.19 +0.64 +V1 +0.77 +0.84 +0.20 +0.87 +0.80 +0.86 +0.18 +0.88 +V2 +0.66 +0.72 +0.26 +0.80 +0.67 +0.75 +0.24 +0.81 +V3 +0.56 +0.62 +0.33 +0.65 +0.59 +0.66 +0.31 +0.66 +V4 +0.50 +0.57 +0.36 +0.43 +0.50 +0.58 +0.36 +0.44 +V5 +0.70 +0.77 +0.27 +0.36 +0.74 +0.80 +0.26 +0.40 +V6 +0.79 +0.87 +0.21 +0.49 +0.80 +0.87 +0.21 +0.49 +8.5 +Results and Discussion +8.5.1 +Architecture comparison +To compare the selected architectures, the first experiment entailed the one-to-one lead conversion +from II to I, two of the most used ECG leads for medical purposes (see Table 8.1). According +to the results, the U-Net performs better than both alternatives AE and LRN. Although the AE +achieves the same median r as the U-Net, the average r is lower, meaning that the least successful +results are generally worse with the AE than with the U-Net. +The skip-connections give it the capability to send more information (and at more resolution +levels) from the encoder to the decoders, granting it more flexibility and ultimately better perfor- +mance than the AE. The multi-resolution supervision of the LRN, expected to improve overall +performance, appears to excessively draw the model’s attention away from the details, which re- +sults in worse performance. Following the results of this comparison, subsequent experiments + +8.5 Results and Discussion +121 +Table 8.4: Average correlation between lead I signals and the remaining leads on the PTB, IN- +CART, and PTB-XL databases. +Average Correlation to Lead I +II +III +aVR +aVL +aVF +V1 +V2 +V3 +V4 +V5 +V6 +PTB +0.45 +-0.49 +-0.82 +0.82 +-0.05 +-0.47 +-0.21 +0.03 +0.30 +0.64 +0.68 +INCART +0.46 +0.02 +-0.62 +0.32 +0.26 +-0.36 +0.01 +0.11 +0.35 +0.51 +0.44 +PTB-XL +0.70 +-0.24 +0.77 +-0.86 +0.32 +-0.54 +-0.06 +0.33 +0.63 +0.80 +0.83 +Table 8.5: Test results of the U-Net used for multi-lead conversion from lead I, with shared or +individual encoders. +Shared Encoder +Individual Encoders +Lead +r (avg.) +r (med.) +RMSE +SSIM +r (avg.) +r (med.) +RMSE +SSIM +II +0.49 +0.54 +0.37 +0.17 +0.50 +0.55 +0.36 +0.19 +III +0.44 +0.49 +0.35 +0.53 +0.46 +0.52 +0.35 +0.55 +aVR +0.89 +0.92 +0.14 +0.92 +0.90 +0.93 +0.13 +0.93 +aVL +0.76 +0.84 +0.25 +0.45 +0.77 +0.85 +0.26 +0.46 +aVF +0.26 +0.29 +0.43 +0.28 +0.28 +0.32 +0.42 +0.27 +V1 +0.81 +0.88 +0.18 +0.88 +0.79 +0.88 +0.19 +0.88 +V2 +0.73 +0.80 +0.23 +0.81 +0.70 +0.77 +0.25 +0.80 +V3 +0.67 +0.73 +0.28 +0.70 +0.67 +0.74 +0.29 +0.68 +V4 +0.59 +0.65 +0.33 +0.48 +0.62 +0.71 +0.32 +0.48 +V5 +0.62 +0.73 +0.30 +0.31 +0.64 +0.73 +0.30 +0.28 +V6 +0.66 +0.75 +0.26 +0.39 +0.67 +0.77 +0.26 +0.39 +focus solely on the U-Net architecture. +8.5.2 +One-to-all leads conversion +Not all leads can be converted equally: the correlation between leads depends on their perspectives +of the heart. Table 8.2 presents an overview of the average correlation between lead II and the +remaining eleven standard leads, computed using the PTB, INCART, and PTB-XL test segments. +Specifically for the PTB data, one can observe that some leads such as aVF or aVR are highly +(positively or negatively) correlated with lead II. On the other hand, aVL is almost orthogonal. +Hence, one should expect aVL to be much harder to accurately convert from lead II than aVF or +aVR, since the former shares much less information with lead II than the latter. +This is verified in the results for multi-lead conversion on the PTB database (see Table 8.3). +Conversion from lead II to aVF, aVR, and V6 consistently offer good results, while the conversions +to aVL, lead I, or V4 were overall the least successful. This behaviour is also visible in the example +of Fig. 8.42 where the model is unable to capture the finer details of the signals in lead aVL and +leads V1-V4. The opposite happens in lead III, aVF, V6, and especially aVR, where the model +was consistently able to capture the morphological details of the signals. +2Examples were selected among all test samples to correspond to the median overall r result for each scenario. +Hence, they represent a median result and the methodology should offer better results in half of the occasions. + +122 +Interlead Conversion of Electrocardiographic Signals +PTB - Lead II to I (shared, r=0.613) +Measured +Converted +PTB - Lead II to I (individual, r=0.599) +Measured +Converted +PTB - Lead II to III (shared, r=0.915) +Measured +Converted +PTB - Lead II to III (individual, r=0.93) +Measured +Converted +PTB - Lead II to aVR (shared, r=0.965) +Measured +Converted +PTB - Lead II to aVR (individual, r=0.98) +Measured +Converted +PTB - Lead II to aVL (shared, r=0.754) +Measured +Converted +PTB - Lead II to aVL (individual, r=0.791) +Measured +Converted +PTB - Lead II to aVF (shared, r=0.976) +Measured +Converted +PTB - Lead II to aVF (individual, r=0.988) +Measured +Converted +PTB - Lead II to V1 (shared, r=0.794) +Measured +Converted +PTB - Lead II to V1 (individual, r=0.788) +Measured +Converted +PTB - Lead II to V2 (shared, r=0.583) +Measured +Converted +PTB - Lead II to V2 (individual, r=0.635) +Measured +Converted +PTB - Lead II to V3 (shared, r=0.2) +Measured +Converted +PTB - Lead II to V3 (individual, r=0.357) +Measured +Converted +PTB - Lead II to V4 (shared, r=-0.088) +Measured +Converted +PTB - Lead II to V4 (individual, r=-0.02) +Measured +Converted +PTB - Lead II to V5 (shared, r=0.853) +Measured +Converted +PTB - Lead II to V5 (individual, r=0.881) +Measured +Converted +PTB - Lead II to V6 (shared, r=0.97) +Measured +Converted +PTB - Lead II to V6 (individual, r=0.972) +Measured +Converted +1 +Figure 8.4: Example result of lead II to all conversion on the PTB test dataset (each row depicts +one converted lead; shared encoder on the left; individual encoders on the right). + +8.5 Results and Discussion +123 +PTB - Lead I to II (shared, r=0.53) +Measured +Converted +PTB - Lead I to II (individual, r=0.099) +Measured +Converted +PTB - Lead I to III (shared, r=0.866) +Measured +Converted +PTB - Lead I to III (individual, r=0.859) +Measured +Converted +PTB - Lead I to aVR (shared, r=0.833) +Measured +Converted +PTB - Lead I to aVR (individual, r=0.776) +Measured +Converted +PTB - Lead I to aVL (shared, r=0.956) +Measured +Converted +PTB - Lead I to aVL (individual, r=0.96) +Measured +Converted +PTB - Lead I to aVF (shared, r=0.767) +Measured +Converted +PTB - Lead I to aVF (individual, r=0.309) +Measured +Converted +PTB - Lead I to V1 (shared, r=0.794) +Measured +Converted +PTB - Lead I to V1 (individual, r=0.764) +Measured +Converted +PTB - Lead I to V2 (shared, r=0.617) +Measured +Converted +PTB - Lead I to V2 (individual, r=0.572) +Measured +Converted +PTB - Lead I to V3 (shared, r=0.41) +Measured +Converted +PTB - Lead I to V3 (individual, r=0.152) +Measured +Converted +PTB - Lead I to V4 (shared, r=0.7) +Measured +Converted +PTB - Lead I to V4 (individual, r=0.653) +Measured +Converted +PTB - Lead I to V5 (shared, r=0.748) +Measured +Converted +PTB - Lead I to V5 (individual, r=0.88) +Measured +Converted +PTB - Lead I to V6 (shared, r=0.643) +Measured +Converted +PTB - Lead I to V6 (individual, r=0.626) +Measured +Converted +1 +Figure 8.5: Example result of lead I to all conversion on the PTB test dataset (each row depicts +one converted lead; shared encoder on the left; individual encoders on the right). + +124 +Interlead Conversion of Electrocardiographic Signals +While lead II ECG signals are generally better for medical diagnosis in clinical scenarios, +lead I is becoming increasingly important. The widespread implementation of ECG acquisition +equipment in smartwatches, fitness bands, and other gadgets for daily use allows for the collection +of lead I signals. Combining these growing applications with robust conversion algorithms would +enable the recovery of missing leads on wearables and empower the next generation of robust +continuous health monitoring. +Table 8.4 presents the average correlation between lead I and the remaining eleven standard +leads on the PTB, INCART, and PTB-XL test segments. Like lead II, lead I is more correlated +(positively or negatively) with certain leads, such as aVR, aVL, or V6, while it is almost orthogonal +with aVF or V3. As such, one can observe, in Table 8.5, that the proposed methodology obtains +better performance with aVR and aVL while struggling to convert from lead I to lead aVF. The +same can be observed in Fig. 8.5: for aVR and aVL, the model is able to correctly capture the +target morphology, while the reconstructions of aVF and V3-V6 are largely unsuccessful. +From the example result in Fig. 8.5, one can also identify a shortcoming of the proposed +methodology: the occasional offsets between the baseline of the measured and converted signals. +We suspect this is due to the min-max normalisation of the signals, drawing them into the [−1,1] +amplitude range. Alternatives to this normalisation, such as standard normalisation, should be +further investigated. +Considering the overall results, no lead is perfect for converting all twelve standard leads. +Hence, lead II should be chosen as reference input when aVF or V5-V6 are the most important +leads for the application at hand. Lead I serves better as a reference when aVR, aVL, or V1- +V2 are more important. Otherwise, other leads (such as lead III) should probably be explored. +Nevertheless, the results show it is possible to nicely reconstruct several leads using only one +input lead without temporal alignment. +Using either lead as a reference, there is apparently no considerable or consistent difference +between using one single shared encoder or using an individual encoder for each target lead. It +appears as if the additional flexibility of having multiple encoders is only beneficial up to a point, +and the higher complexity ends up opening the door to overfitting and loss of robustness. As +such, for this application, one should expect a shared encoder to be the best option, considering its +higher simplicity and similar performance. +8.5.3 +Comparison with the state-of-the-art +For a comparison with the state-of-the-art, we implemented the method recently proposed by +Grande-Fidalgo et al. [154] as a baseline. This method is based on a simple fully-connected +model that receives each signal point’s amplitude in three reference leads as inputs and returns the +same point’s amplitude in all twelve leads. Here, we adapt the methodology so it receives signal +point amplitudes from one single lead (leads I or II), to exactly match the evaluation conditions of +the proposed method. +Unlike what has been reported in [154], the baseline was not successful in learning to retrieve +the entire set of leads from just one reference lead. In fact, across all leads, the average test r of this + +8.5 Results and Discussion +125 +Table 8.6: Cross-database test results for INCART conversion from lead II. +Shared Encoder +Individual Encoders +Lead +r (avg.) +r (med.) +RMSE +SSIM +r (avg.) +r (med.) +RMSE +SSIM +I +0.46 +0.51 +0.34 +0.18 +0.44 +0.50 +0.35 +0.16 +III +0.57 +0.63 +0.28 +0.49 +0.57 +0.63 +0.29 +0.45 +aVR +0.91 +0.95 +0.11 +0.92 +0.92 +0.95 +0.11 +0.92 +aVL +0.13 +0.11 +0.41 +0.33 +0.10 +0.09 +0.44 +0.27 +aVF +0.88 +0.93 +0.15 +0.68 +0.92 +0.95 +0.13 +0.71 +V1 +0.63 +0.79 +0.23 +0.82 +0.65 +0.82 +0.23 +0.83 +V2 +0.53 +0.64 +0.26 +0.73 +0.55 +0.69 +0.27 +0.74 +V3 +0.42 +0.51 +0.33 +0.56 +0.42 +0.53 +0.34 +0.55 +V4 +0.52 +0.59 +0.35 +0.28 +0.52 +0.60 +0.35 +0.30 +V5 +0.73 +0.80 +0.25 +0.31 +0.74 +0.80 +0.25 +0.29 +V6 +0.73 +0.83 +0.23 +0.42 +0.72 +0.81 +0.24 +0.39 +Table 8.7: Cross-database test results for INCART conversion from lead I. +Shared Encoder +Individual Encoders +Lead +r (avg.) +r (med.) +RMSE +SSIM +r (avg.) +r (med.) +RMSE +SSIM +II +0.35 +0.37 +0.37 +0.18 +0.36 +0.38 +0.37 +0.19 +III +0.17 +0.19 +0.41 +0.27 +0.17 +0.19 +0.43 +0.28 +aVR +0.65 +0.74 +0.23 +0.81 +0.67 +0.78 +0.22 +0.83 +aVL +0.40 +0.49 +0.32 +0.52 +0.36 +0.46 +0.35 +0.47 +aVF +0.17 +0.15 +0.41 +0.23 +0.17 +0.14 +0.41 +0.22 +V1 +0.55 +0.62 +0.25 +0.78 +0.57 +0.64 +0.24 +0.79 +V2 +0.50 +0.57 +0.27 +0.73 +0.50 +0.56 +0.28 +0.73 +V3 +0.35 +0.35 +0.37 +0.46 +0.36 +0.37 +0.37 +0.44 +V4 +0.27 +0.26 +0.43 +0.16 +0.34 +0.36 +0.41 +0.19 +V5 +0.46 +0.51 +0.36 +0.11 +0.45 +0.53 +0.35 +0.11 +V6 +0.44 +0.49 +0.34 +0.22 +0.45 +0.52 +0.34 +0.21 +method ranged from −0.005 to 0.002, considerably worse than the proposed methodology. One +can assume that, although such a simplistic model presents advantages in terms of lightweight +operation and robustness to overfitting, single-lead information is not enough for it to achieve +reliable interlead conversion. +The fact the baseline method reconstructs signals point-by-point, unable to analyse broader +local context information, makes it hard to reconstruct the signal without already having data from +more than one channel. On the other hand, using convolutional layers allows the proposed method +to use broader local information as context to adequately learn to reconstruct signals using only +one lead as reference. +8.5.4 +Cross-database evaluation +The cross-database tests aimed to assess the behaviour of the proposed methodology in more +diverse scenarios. Here, the models used were the same as in the previous experiments (trained + +126 +Interlead Conversion of Electrocardiographic Signals +Table 8.8: Cross-database test results for PTB-XL conversion from lead II. +Shared Encoder +Individual Encoders +Lead +r (avg.) +r (med.) +RMSE +SSIM +r (avg.) +r (med.) +RMSE +SSIM +I +0.74 +0.80 +0.25 +0.31 +0.72 +0.79 +0.26 +0.29 +III +0.44 +0.50 +0.32 +0.50 +0.45 +0.52 +0.32 +0.50 +aVR +-0.38 +-0.53 +0.72 +0.09 +-0.39 +-0.55 +0.72 +0.09 +aVL +-0.33 +-0.44 +0.64 +0.23 +-0.33 +-0.45 +0.67 +0.17 +aVF +0.83 +0.90 +0.19 +0.61 +0.84 +0.92 +0.19 +0.62 +V1 +0.79 +0.87 +0.17 +0.91 +0.81 +0.89 +0.16 +0.91 +V2 +0.71 +0.79 +0.22 +0.84 +0.72 +0.82 +0.21 +0.85 +V3 +0.61 +0.68 +0.30 +0.66 +0.62 +0.70 +0.30 +0.66 +V4 +0.64 +0.71 +0.31 +0.31 +0.66 +0.74 +0.32 +0.32 +V5 +0.79 +0.86 +0.22 +0.37 +0.80 +0.87 +0.23 +0.39 +V6 +0.85 +0.91 +0.18 +0.58 +0.85 +0.91 +0.18 +0.58 +Table 8.9: Cross-database test results for PTB-XL conversion from lead I. +Shared Encoder +Individual Encoders +Lead +r (avg.) +r (med.) +RMSE +SSIM +r (avg.) +r (med.) +RMSE +SSIM +II +0.60 +0.66 +0.33 +0.23 +0.62 +0.70 +0.31 +0.26 +III +0.31 +0.34 +0.38 +0.43 +0.33 +0.38 +0.37 +0.45 +aVR +-0.61 +-0.76 +0.74 +0.09 +-0.62 +-0.78 +0.74 +0.09 +aVL +-0.63 +-0.75 +0.75 +0.11 +-0.66 +-0.79 +0.76 +0.08 +aVF +0.29 +0.31 +0.41 +0.23 +0.32 +0.36 +0.40 +0.24 +V1 +0.79 +0.86 +0.16 +0.91 +0.81 +0.88 +0.15 +0.91 +V2 +0.71 +0.78 +0.22 +0.84 +0.70 +0.77 +0.23 +0.83 +V3 +0.65 +0.72 +0.29 +0.64 +0.67 +0.76 +0.28 +0.66 +V4 +0.62 +0.72 +0.32 +0.30 +0.69 +0.80 +0.30 +0.32 +V5 +0.76 +0.85 +0.24 +0.35 +0.76 +0.86 +0.25 +0.32 +V6 +0.80 +0.87 +0.20 +0.53 +0.81 +0.89 +0.21 +0.51 +with PTB data), and the evaluation was conducted using data from the INCART and PTB-XL +databases. +For both INCART and PTB-XL, some differences in interlead correlations can be observed +when compared to PTB (see Table 8.2 and Table 8.4). This can be explained due to the different +acquisition setups, especially the positioning of the electrodes, which potentially causes each lead +to offer a different perspective. +For INCART (see Table 8.6 and Table 8.7), the overall quality of the results is inferior to those +with PTB. This is as expected since PTB data was seen by the models during training and the +INCART database is arguably more challenging regarding noise and variability. Despite these +metrics, it is noticeable in Figure 8.6 and Figure 8.7 that both reference leads can offer good +conversion results in some leads, especially with lead II. Using this lead as reference, the proposed +methodology is relatively good at converting most leads except I, V2, and V3. +For PTB-XL (see Table 8.8 and Table 8.9), results are, overall, the worst, although some +leads (namely V4, V5, and V6), due to higher correlation with the reference leads, are better + +8.6 Summary and Conclusions +127 +reconstructed than with the PTB database. In Fig. 8.8 and Fig. 8.9, it is possible to observe that, +despite occasional baseline offset and prevalent noise, both reference leads enable the approximate +reconstruction of most of the set of twelve standard leads. +For either database, differences in acquisition settings and electrode placement result in in- +ferior performance. The ideal solution is to always make sure the acquisition details of training +and inference data match, to ensure optimal performance upon deployment. Nevertheless, the +robustness in cross-database scenarios is a relevant issue that merits further research. +8.5.5 +Influence of medical conditions +As aforementioned, medical conditions may affect differently the various leads of an ECG signal. +While this is the main motivation behind the quest to reconstruct missing leads it may also be one +of the main hurdles. If the medical condition is somehow not evident in the input lead, the algo- +rithm could be led to reconstruct the remaining leads incorrectly without the proper information +on the respective medical condition. +As such, we conducted a differential performance evaluation according to the existence and +type of diagnosed medical conditions on the signals. To do this, we use the expert clinical an- +notations on the PTB-XL database and separate the results by the superclass labelling of each +test sample. The average r results for each converted lead and each superclass are presented in +Table 8.10 (using lead II as reference) and Table 8.11 (using lead I as reference). +Overall, no dominant difference could be observed between the results with normal signals +and the results with signals with medical conditions. Similarly, no specific medical condition +superclass presents considerably different performance results. This is likely due to the presence +of medical conditions on the PTB signals originally used for training the model. Thus, although +the behaviour of the proposed methodology should be expected to vary slightly in the presence of +medical conditions, it should not have a considerable impact on its baseline performance. +8.6 +Summary and Conclusions +This work implemented and compared the performance of three deep learning architectures for +interlead conversion of ECG signals. Unlike the literature, this work focused on the more chal- +lenging scenario of single-lead blindly-segmented inputs from limb leads. The proposed model +was explored on 12-lead acquisitions from three different databases. Ablation studies were con- +ducted on the architectures used for conversion and on the use of a shared encoder vs. individual +encoders. Moreover, the model was evaluated in both single-database and cross-database scenar- +ios, including an experiment on the effect of medical conditions on signal reconstruction and the +study of diagnosis performance with original vs. converted signals. +Despite the considerably more challenging scenario, the proposed methodology based on a U- +Net was capable of obtaining relatively good results. Each reference lead enabled the high-quality +reconstruction of several of the twelve standard ECG leads, in some cases reaching state-of-the-art + +128 +Interlead Conversion of Electrocardiographic Signals +INCART - Lead II to I (shared, r=0.733) +Measured +Converted +INCART - Lead II to I (individual, r=0.569) +Measured +Converted +INCART - Lead II to III (shared, r=0.902) +Measured +Converted +INCART - Lead II to III (individual, r=0.86) +Measured +Converted +INCART - Lead II to aVR (shared, r=0.982) +Measured +Converted +INCART - Lead II to aVR (individual, r=0.989) +Measured +Converted +INCART - Lead II to aVL (shared, r=0.779) +Measured +Converted +INCART - Lead II to aVL (individual, r=0.711) +Measured +Converted +INCART - Lead II to aVF (shared, r=0.985) +Measured +Converted +INCART - Lead II to aVF (individual, r=0.986) +Measured +Converted +INCART - Lead II to V1 (shared, r=0.895) +Measured +Converted +INCART - Lead II to V1 (individual, r=0.916) +Measured +Converted +INCART - Lead II to V2 (shared, r=0.643) +Measured +Converted +INCART - Lead II to V2 (individual, r=0.705) +Measured +Converted +INCART - Lead II to V3 (shared, r=0.166) +Measured +Converted +INCART - Lead II to V3 (individual, r=0.206) +Measured +Converted +INCART - Lead II to V4 (shared, r=0.609) +Measured +Converted +INCART - Lead II to V4 (individual, r=0.156) +Measured +Converted +INCART - Lead II to V5 (shared, r=0.904) +Measured +Converted +INCART - Lead II to V5 (individual, r=0.916) +Measured +Converted +INCART - Lead II to V6 (shared, r=0.935) +Measured +Converted +INCART - Lead II to V6 (individual, r=0.916) +Measured +Converted +1 +Figure 8.6: Example cross-database result of lead II to all conversion on INCART (each row +depicts one converted lead; shared encoder on the left; individual encoders on the right). + +8.6 Summary and Conclusions +129 +INCART - Lead I to II (shared, r=0.52) +Measured +Converted +INCART - Lead I to II (individual, r=0.461) +Measured +Converted +INCART - Lead I to III (shared, r=0.582) +Measured +Converted +INCART - Lead I to III (individual, r=0.755) +Measured +Converted +INCART - Lead I to aVR (shared, r=0.934) +Measured +Converted +INCART - Lead I to aVR (individual, r=0.918) +Measured +Converted +INCART - Lead I to aVL (shared, r=0.915) +Measured +Converted +INCART - Lead I to aVL (individual, r=0.912) +Measured +Converted +INCART - Lead I to aVF (shared, r=0.464) +Measured +Converted +INCART - Lead I to aVF (individual, r=0.614) +Measured +Converted +INCART - Lead I to V1 (shared, r=0.862) +Measured +Converted +INCART - Lead I to V1 (individual, r=0.85) +Measured +Converted +INCART - Lead I to V2 (shared, r=0.79) +Measured +Converted +INCART - Lead I to V2 (individual, r=0.753) +Measured +Converted +INCART - Lead I to V3 (shared, r=0.574) +Measured +Converted +INCART - Lead I to V3 (individual, r=0.412) +Measured +Converted +INCART - Lead I to V4 (shared, r=0.078) +Measured +Converted +INCART - Lead I to V4 (individual, r=0.093) +Measured +Converted +INCART - Lead I to V5 (shared, r=0.366) +Measured +Converted +INCART - Lead I to V5 (individual, r=0.432) +Measured +Converted +INCART - Lead I to V6 (shared, r=0.82) +Measured +Converted +INCART - Lead I to V6 (individual, r=0.75) +Measured +Converted +1 +Figure 8.7: Example cross-database result of lead I to all conversion on INCART (each row depicts +one converted lead; shared encoder on the left; individual encoders on the right). + +130 +Interlead Conversion of Electrocardiographic Signals +PTB-XL - Lead II to I (shared, r=0.18) +Measured +Converted +PTB-XL - Lead II to I (individual, r=0.153) +Measured +Converted +PTB-XL - Lead II to III (shared, r=0.484) +Measured +Converted +PTB-XL - Lead II to III (individual, r=0.772) +Measured +Converted +PTB-XL - Lead II to aVR (shared, r=0.578) +Measured +Converted +PTB-XL - Lead II to aVR (individual, r=0.536) +Measured +Converted +PTB-XL - Lead II to aVL (shared, r=0.147) +Measured +Converted +PTB-XL - Lead II to aVL (individual, r=0.225) +Measured +Converted +PTB-XL - Lead II to aVF (shared, r=0.896) +Measured +Converted +PTB-XL - Lead II to aVF (individual, r=0.957) +Measured +Converted +PTB-XL - Lead II to V1 (shared, r=0.843) +Measured +Converted +PTB-XL - Lead II to V1 (individual, r=0.766) +Measured +Converted +PTB-XL - Lead II to V2 (shared, r=0.763) +Measured +Converted +PTB-XL - Lead II to V2 (individual, r=0.871) +Measured +Converted +PTB-XL - Lead II to V3 (shared, r=0.741) +Measured +Converted +PTB-XL - Lead II to V3 (individual, r=0.816) +Measured +Converted +PTB-XL - Lead II to V4 (shared, r=0.71) +Measured +Converted +PTB-XL - Lead II to V4 (individual, r=0.574) +Measured +Converted +PTB-XL - Lead II to V5 (shared, r=0.874) +Measured +Converted +PTB-XL - Lead II to V5 (individual, r=0.919) +Measured +Converted +PTB-XL - Lead II to V6 (shared, r=0.896) +Measured +Converted +PTB-XL - Lead II to V6 (individual, r=0.903) +Measured +Converted +1 +Figure 8.8: Example cross-database result of lead II to all conversion on PTB-XL (each row +depicts one converted lead; shared encoder on the left; individual encoders on the right). + +8.6 Summary and Conclusions +131 +PTB-XL - Lead I to II (shared, r=0.879) +Measured +Converted +PTB-XL - Lead I to II (individual, r=0.885) +Measured +Converted +PTB-XL - Lead I to III (shared, r=0.708) +Measured +Converted +PTB-XL - Lead I to III (individual, r=0.677) +Measured +Converted +PTB-XL - Lead I to aVR (shared, r=-0.491) +Measured +Converted +PTB-XL - Lead I to aVR (individual, r=-0.446) +Measured +Converted +PTB-XL - Lead I to aVL (shared, r=-0.378) +Measured +Converted +PTB-XL - Lead I to aVL (individual, r=-0.477) +Measured +Converted +PTB-XL - Lead I to aVF (shared, r=0.856) +Measured +Converted +PTB-XL - Lead I to aVF (individual, r=0.765) +Measured +Converted +PTB-XL - Lead I to V1 (shared, r=0.57) +Measured +Converted +PTB-XL - Lead I to V1 (individual, r=0.707) +Measured +Converted +PTB-XL - Lead I to V2 (shared, r=0.578) +Measured +Converted +PTB-XL - Lead I to V2 (individual, r=0.507) +Measured +Converted +PTB-XL - Lead I to V3 (shared, r=0.785) +Measured +Converted +PTB-XL - Lead I to V3 (individual, r=0.866) +Measured +Converted +PTB-XL - Lead I to V4 (shared, r=0.768) +Measured +Converted +PTB-XL - Lead I to V4 (individual, r=0.743) +Measured +Converted +PTB-XL - Lead I to V5 (shared, r=0.976) +Measured +Converted +PTB-XL - Lead I to V5 (individual, r=0.893) +Measured +Converted +PTB-XL - Lead I to V6 (shared, r=0.969) +Measured +Converted +PTB-XL - Lead I to V6 (individual, r=0.98) +Measured +Converted +1 +Figure 8.9: Example cross-database result of lead I to all conversion on PTB-XL (each row depicts +one converted lead; shared encoder on the left; individual encoders on the right). + +132 +Interlead Conversion of Electrocardiographic Signals +Table 8.10: Average correlation results for PTB-XL conversion from lead II according to medical +condition class (using the U-Net with a shared encoder). +Converted leads +Class +I +III +aVR +aVL +aVF +V1 +V2 +V3 +V4 +V5 +V6 +NORM +0.79 +0.42 +-0.39 +-0.32 +0.86 +0.83 +0.76 +0.64 +0.70 +0.85 +0.90 +MI +0.65 +0.47 +-0.37 +-0.38 +0.76 +0.75 +0.65 +0.56 +0.54 +0.68 +0.77 +STTC +0.71 +0.41 +-0.41 +-0.37 +0.80 +0.79 +0.69 +0.58 +0.59 +0.78 +0.84 +CD +0.65 +0.60 +-0.31 +-0.28 +0.84 +0.59 +0.54 +0.58 +0.58 +0.63 +0.71 +HYP +0.75 +0.42 +-0.40 +-0.24 +0.80 +0.85 +0.70 +0.56 +0.61 +0.81 +0.87 +Table 8.11: Average correlation results for PTB-XL conversion from lead I according to medical +condition class (using the U-Net with a shared encoder). +Converted leads +Class +II +III +aVR +aVL +aVF +V1 +V2 +V3 +V4 +V5 +V6 +NORM +0.73 +-0.23 +-0.35 +-0.66 +0.23 +0.79 +0.73 +0.68 +0.69 +0.84 +0.85 +MI +0.50 +-0.15 +-0.26 +-0.57 +0.036 +0.67 +0.58 +0.47 +0.38 +0.56 +0.61 +STTC +0.57 +-0.20 +-0.36 +-0.64 +0.00 +0.74 +0.64 +0.55 +0.48 +0.71 +0.76 +CD +0.40 +-0.31 +-0.11 +-0.48 +-0.06 +0.35 +0.38 +0.42 +0.39 +0.52 +0.54 +HYP +0.65 +-0.29 +-0.37 +-0.60 +0.07 +0.78 +0.65 +0.58 +0.59 +0.78 +0.82 +level performance. Both lead I and II appear to be especially suitable for certain sets of leads, and +could be used on specific target applications that focus on those. +In the cross-database scenario, despite the acquisition setup differences, results were promising +especially with the INCART database. Finally, the analysis of the influence of medical conditions +has shown no considerable effect of pathologies on the performance of the proposed methodology. +However, a state-of-the-art methodology for automatic diagnosis revealed lower accuracy when +using reconstructed signals, a problem that should be addressed in future research. +Although the results are promising, further efforts should be devoted towards the improvement +of the methodologies for interlead conversion using single-lead blindly-segmented inputs. Namely, +the pre-processing and normalisation of the signals, as well as the robustness to diverse acquisition +setups, should be the target of further research. Additionally, task-oriented objective functions +should be explored to ensure useful signal information is kept and avoid performance losses in +subsequent diagnoses. +With some consolidation, the proposed methodology could be the key to better cardiac health +monitoring in wearable devices and less obtrusive clinical scenarios. Taking the example of emer- +gency rooms, if we can retrieve all twelve leads (or the most important among these) from Lead +I signals, then patients will only need two electrodes placed on the wrists to have their ECG col- +lected, instead of the full set of 10 electrodes on wrists, ankles, and chest. This is a meaningful +step towards higher comfort and usability for patients in clinical settings or users in other scenarios +involving the monitoring of ECG signals. Additionally, albeit outside the scope of this work, this +methodology could also be applicable to other multi-channel signals where the different channels +correspond to different perspectives over the same physiological phenomenon. + +Part III +Face Biometrics +133 + + +Chapter 9 +Prior Art in Face Biometrics +9.1 +Data +There are several publicly available databases for research purposes, to develop and benchmark +face biometric recognition algorithms. Considering the face is one of the most developed biometric +traits, the databases available are some of the largest and most complete, thoughtfully structured +for deep and adequate evaluation of recognition algorithms. Table 9.1 compiles some relevant in- +formation about the most important databases currently available, which are also described below: +• CASIA NIR-VIS: Also known as CBSR NIR, the CASIA NIR-VIS 2.0 database was cre- +ated by the Institute of Automation of the Chinese Academy of Sciences (CASIA). It in- +cludes pairs of mugshots and corresponding NIR images from 725 people, acquired over +four recording sessions, from 2007 to 2010 [253]; +• CASIA WebFace: This database was created and made available by the Institute of Au- +tomation of the Chinese Academy of Sciences (CASIA). It includes almost five hundred +thousand images from more than ten thousand identities collected to support the research in +unconstrained face recognition; +• CelebA: This database results from a previous one, CelebFaces+, which has been enriched +with fiducial and attribute annotations. It includes over two hundred thousand pictures from +over ten thousand celebrities, with five fiducial locations, and forty binary attributes per +image [267]; +• COX Face: The COX Face database was designed to study recognition across still images +and videos. Thus, it includes still images from one thousand subjects in a controlled envi- +ronment with high quality, and surveillance videos from the subjects in unconstrained and +low-quality settings [181]; +135 + +136 +Prior Art in Face Biometrics +Table 9.1: Details on the main face recognition databases that are currently available. +Database +Spectrum +Subjects +Images +Videos +Resolution +Unconstr. +CASIA NIR-VIS [253] +Vis. + NIR +725 +17 580 +None +640x480 + +CASIA WebFace +Visible +10 575 +494 414 +None +250x250 +CelebA [267] +Visible +10 177 +202 599 +None +- +COX Face [181] +Visible +1000 +1000 +3000 +- + +CSIST Lab1 [466] +Vis. + NIR +50 +1000 +None +100x80 + +CSIST Lab2 [466] +Vis. + NIR +50 +2000 +None +200x200 + +FERET +Visible +1199 +14 126 +None +512x768 +IJB-C +Visible +3531 +138 836 +11 779 +- +IMDb-Face [448] +Visible +59 000 +1 700 000 +None +- +LFW [177; 178] +Visible +5749 +13 233 +None +250x250 +MegaFace [305] +Visible +672 057 +4 753 520 +None +- +MS-Celeb +Visible +99 892 +8 200 000 +None +- +PaSC [39] +Visible +293 +9376 +2802 +- +PolyU [480] +NIR +335 +34 000 +None +768x576 + +UMD Faces [25] +Visible +8277 +367 888 +None +- +UMD Videos [25] +Visible +3107 +None +22 075 +- +VGGFace [331] +Visible +2622 +2 600 000 +None +Diverse +VGGFace2 [58] +Visible +9131 +3 310 000 +None +Diverse +Yale Face +Visible +15 +165 +None +320x243 + +YouTube Faces [460] +Visible +1595 +None +3495 +- +• CSIST Lab1: The CSIST database was developed by the Chung-Shan Institute of Science +and Technology, with images from volunteers at the Harbin Institute of Technology of Shen- +zhen. The Lab1 dataset contains ten visible light and ten NIR images from each of fifty +subjects [466]; +• CSIST Lab2: The CSIST Lab2 dataset is part of the CSIST database, and includes twenty +visible light and twenty NIR images from fifty volunteers, with natural lighting and artificial +lighting [466]; +• FERET: The Facial Recognition Technology (FERET) Program was sponsored by the US +Department of Defence, and the FERET database is distributed by the National Institute of +Standards and Technology (NIST). The database, with more than fourteen thousand face +images collected between 1993 and 1996, has the goal to support the development of new +techniques for automatic recognition of faces; + +9.1 Data +137 +• IJB-C: The IARPA Janus Benchmark-C (IJB-C) is a database resulting from a group of +challenges created by NIST addressing verification, identification, detection, clustering, and +processing of videos. It includes over one hundred and thirty-eight thousand images and +eleven thousand face videos from over three thousand identities; +• IMDb-Face: This dataset includes approximately 1.7 million images with faces from fifty- +nine thousand celebrities on the IMDb movie database. According to the authors, efforts +have been devoted to ensuring this database is cleaner than most other large available +databases, making it better for training robust algorithms [448]; +• LFW: The Labelled Faces in the Wild dataset was one of the first databases of unconstrained +face images, and includes over thirteen thousand face images of almost six thousand identi- +ties [177; 178]; +• MegaFace: The MegaFace collection includes an average of 7 unconstrained face photos +(between 2 and 2469) for each of almost seven hundred thousand identities. It is the sur- +veyed database with the most identities, and the second with most total images [305]; +• MS-Celeb: The Microsoft Celeb is a dataset of over eight million images obtained online. +It includes faces from nearly one hundred thousand celebrities in unconstrained settings, +aiming to accelerate research in face recognition with large target sets; +• PaSC: The Point and Shoot Face Recognition Challenge (PaSC) dataset was created by +NIST to encourage the development of face recognition algorithms that are more robust +to very unconstrained settings and inexpensive camera technology. It includes over nine +thousand images and almost three thousand videos with faces from almost three hundred +identities with different distances to the camera, perspective, and locations [39]; +• PolyU: This database was created by the Biometric Research Centre (UGC/CRC) at The +Hong Kong Polytechnic University, using their own real-time NIR face capture device. It +includes approximately thirty-four thousand NIR face images from over three hundred sub- +jects [480]; +• UMD Faces: This dataset was created at the University of Maryland (UMD) and contains +almost four hundred thousand face images from over eight thousand subjects. Each image +includes human-curated face bounding boxes and annotations on pose, gender, and twenty- +one keypoints [25]; +• UMD Videos: This dataset is similar to UMD Faces, only it includes over three million video +frames from twenty-two thousand videos with over three thousand identities. The frames +are also annotated with pose, keypoints, and gender information [25]; +• VGGFace: This database was created by the Visual Geometry Group (VGG) of the Univer- +sity of Oxford, UK. The group developed a method for minimally-supervised online image + +138 +Prior Art in Face Biometrics +Face +Detection +Feature +Extraction +Recognition +Image +Identity +Figure 9.1: Stages of a biometric recognition algorithm based on face images (based on [28]). +collection, which enabled the creation of this large database, with over two million faces in +unconstrained conditions [331]; +• VGGFace2: The VGGFace database was later extended to create the VGGFace2 database, +which includes more than three million unconstrained face images from almost ten thousand +identities [58]; +• Yale Face: The Yale Face database includes eleven images from each of fifteen subjects. +Although not an unconstrained database, it includes annotations on certain expressions and +configurations simulated by the subjects, which can be useful in training models for other +tasks such as emotion recognition; +• YouTube Faces: This dataset is composed exclusively of faces on YouTube videos. It in- +cludes over three thousand videos with over one thousand identities. The videos range from +48 to 6070 frames (average of 181 frames per video) [460]. +These databases already cover most bases and offer a good starting point for the study and +development of strong biometric algorithms. Nevertheless, it is important to acknowledge the +growth of heterogeneous approaches in facial recognition, and the subsequent need for databases +of face images acquired in different modalities (e. g., visual spectrum vs. NIR). These databases +are still too small and too controlled for the development of robust algorithms. Moreover, it would +be useful to have larger databases focused on more specific applications (e. g., face images and +videos of car drivers). +9.2 +Related Work +According to Barnouti et al. [28], face recognition can be decomposed into three processes: face +detection, feature extraction, and face recognition (see Fig. 9.1). Below, we delve into the state- +of-the-art in each of these processes. Due to the currently common practice of joining feature +extraction and face recognition into a single model using deep learning, these two processes are +jointly discussed. +9.2.1 +Face detection +Given an image or video stream, the process of face detection has the goal of locating and extract- +ing all human faces visible in the received input. It is an extremely important process not only + +9.2 Related Work +139 +Figure 9.2: Examples of face detection in unconstrained settings (images from the FDDB data- +base [203], ground-truths in green and predictions in red). +for face-based biometric recognition, but also for face tracking and person re-identification across +surveillance cameras, recognition of expressions and emotions, and analysis of soft-biometrics +such as gender or age [232; 476]. +Some earlier, simpler methods relied on skin colour for face detection on images [89; 220]. +These presented the advantage of being orientation-invariant, as it would serve to detect a face even +if it did not present a frontal pose. However, they fail to consider the great variety of skin colours, +both due to natural differences between individuals, and due to diverse illumination conditions. +More sophisticated algorithms have been proposed, including the method by Sirovich and +Kirby [406] based on eigenvectors from large face image datasets, the method by Viola and Jones +[442] which uses cascades of Haar transform filters selected using AdaBoost, or the method by +Dalal and Triggs [92] which uses histograms of intensity gradients from image regions and their +orientation. These commonly offer very fast detection, adequate for real-time systems, but often +fail on non-frontal face detection and faces of very diverse scales. +Like most pattern recognition tasks, traditional methods from earlier literature have been re- +cently replaced with deep learning algorithms. These offer more robust and accurate face detec- +tions, especially for non-frontal face detection, making better use of very large datasets currently +available. +Some of these datasets currently offer a public benchmark for fair and direct comparison with +state-of-the-art methods. These include the WIDER face dataset [469] and the Face Detection +Database (FDDB) [203]. These benchmarks are currently largely dominated by deep learning +approaches. +The FDDB benchmark is currently dominated by the S3FD, the DeepIR, and the RSA algo- +rithms. S3FD [486] is based on a single deep network specifically fitted to better detect small +faces. The DeepIR method [416] uses a Faster Region-based CNN (RCNN) adapted with fea- +ture concatenation, multiscale training, and hard negative mining for more robust detections. The +RSA algorithm [264] is based on a convolutional network with a recurrent strategy for detection +at different scales. +The best scores on the WIDER Face benchmark belong to the AInnoface and the RetinaFace +algorithms. The RetinaFace algorithm [99] is a single-stage pixelwise face detector that takes +advantage of extra-supervised and self-supervised multitask learning. AInnoface [481] is based + +140 +Prior Art in Face Biometrics +on RetinaFace with two-step classification and regression, an IoU loss function, improved data +augmentation, and several other structural network changes. +These sophisticated algorithms perform accurate and robust detection for faces in different +poses and at different scales. But they are still computationally heavy and require GPUs for +real-time operation in images with VGA resolution. Having overcome most challenges in uncon- +strained face detection, research should now focus on making algorithms faster and more efficient. +9.2.2 +Feature extraction and recognition +Having extracted the detected faces from the input, face-based recognition systems need to extract +appropriate features from those faces to accurately decide on their identities. Wang and Deng +[452], in their survey of deep learning face recognition, have pointed out how the field of face bio- +metrics has moved from traditional machine learning approaches to deep learning (see Fig. 9.3). +However, the best results were only obtained when the development of sophisticated tailored ob- +jective functions began (see Fig. 9.4). +As such, Wang and Deng [452] divide approaches into four categories: holistic learning, local +handcrafted, shallow learning, and deep learning methods. Below, the most relevant examples of +each category are presented, along with their advantages and shortcomings. +Holistic learning approaches are those that use the whole face image to obtain representa- +tions that ease the process of face recognition. The Eigenface method [406], described for face +detection, is one of these methods, along with the Fisherface method [34], which is similar to +Eigenface, but uses the Fisher Linear Discriminant Analysis (FLDA, instead of PCA) for dimen- +sionality reduction. Such methods are simple and fast, but lack robustness to several variability +factors. +Eventually, researchers started to explore methods that extracted features from regions of the +face image. These methods mostly used Gabor filters and Local Binary Patterns for feature extrac- +tion based on intensity gradients and image edges [12; 262]. Methods like these and the Elastic +Bunch Graph Matching [459] were able to improve recognition accuracy, but not to make it high +enough for real use. +To improve accuracy and robustness to pose variations, researchers proposed learning-based +methods, that used available data to learn the best features. The first method to use shallow learning +was proposed by Cao et al. [59], using gradient filtering after facial landmark alignment and +clustering methods to learn encodings for better recognition. +But truly high accuracy and robustness in face recognition were only attained with the rise +of deep learning. The first models were convolutional neural networks with conventional archi- +tectures, such as DeepFace [423], VGG-Face [331], or VGG-Face2 [58]. Over time, researchers +started to focus on adapting the networks for specific details of face recognition, such as custom +loss functions that force increased intersubject separability. This resulted in improved methods +such as DeepID [417], L2-Softmax [360], and ArcFace [98]. +Both Facenet and DeepID are among the top five non-commercial methods in the LFW bench- +mark, with 99.63% and 97.45% accuracy, respectively. However, as discussed by Wang and Deng + +9.2 Related Work +141 +Figure 9.3: Evolution of face recognition approaches, from holistic to deep learning (from [452]). +Figure 9.4: Recent history of face recognition, from deep learning to tailored objective functions (from [452]). + +Representation +Deepface +%L682%) +Genericend-to-end learning repeated elementarylayersByBackProp +LEDFDFV +PCANet +Edges/ +Spatial pooling +K-Means +Sparsecoding +Classifier +Shallow +Filtering +/Histogram +LeamedMetric +Gabor,LBP +Fisher Vector +learning +(LFW>70%) +Specialized learning components +Eigenface +(LFW-60%) +EBGM +Gabor +LBP +LGBP +HD-LBP +G +Edges +Spatial pooling +Classifier/ +Local +Filtering +/Histogram +LearnedMetric ++"Messi" +handcraft +Handcrafteddesign +Eigenface +Fisherface +Bayes +Laplacianface2DPCA +SRC +CRC +Metric Learning +Holistic +Classifier +Learmed Metric ++"Messi" +learning +Pixel +1991 +1997 +2010 +2014 +TimevMFloss +(weight and feature +Fairloss +normalization) +(largemargin) +DeepID2 +L-softmax +Normface +A-softmax +Cosface +RegularFace +(contrastive loss) +(feature +(large margin) +normalization) +(large margin) +(large margin) +(largemargin) +DeepID +DeepID3 +TSE +L2softmax +CoColoss +Range loss +(feature +Arcface +AdaptiveFace +(sofumax) +(contrastiveloss) +(triplet loss) +normalization) +normalization) +(largemargin) +(large margin) +Deepface +DeepID2+ +FaceNet +VGGface +TPE +Centerloss +Center +AMSloss +Adacos +Marginal loss +(softmax) +(contrastive loss) +(triplet loss) +(triplet+softmax) +(triplet loss) +(center loss) +invariantloss +(centerloss) +(largemargin) +(largemargin) +2014 +2015 +2016 +2017 +2018 +2019 +2020 +米 +Negative +A +Anchor +LEARNING +Negative +Anchor +9 +Positive +Positive +W +Softmaxloss +Contrastive loss +Triplet loss +Center loss +Feature and weight normalization +Largemarginloss142 +Prior Art in Face Biometrics +[452], the best results yet haven’t been achieved with traditional deep learning losses such as soft- +max or even triplet loss, but with tailored objective functions specifically designed to make the +most of available data for the task of face recognition. According to results reported by Wang and +Deng [452], both L2-softmax and ArcFace offer even better performance in the LFW benchmark, +with 99.78% and 99.83% accuracy, respectively. In fact, ArcFace, using the tailored ArcLoss +objective function, is still widely recognised as the state-of-the-art approach for face recognition. +These methods and results show the high potential offered by adapted deep learning networks +for face recognition. +However, these present the same problem as deep learning approaches +for face detection: high computational cost. These models are generally very complex, and re- +searchers should devote efforts to making them more efficient. Furthermore, video-based bench- +marks should be used more to evaluate methods also based on their timeliness. Finally, as stated +by Arya et al. [21], research in visible or infrared spectra may be reaching its limits, and the future +may be based on multispectral imaging. +Overall, face biometric recognition is already a thoroughly developed topic, unlike other bio- +metric characteristics (such as the electrocardiogram). Joint efforts dedicated by several inter- +national research groups throughout multiple decades have brought this topic to a stage of high +maturity that enables real applications. Proof of that is the currently endless variety of applica- +tions of face biometrics in our day-to-day routine, from unlocking our phones to border control, +including opening bank accounts remotely. +9.2.3 +Presentation attack detection +Just like any other biometric solution or access control system, face biometric systems are prone +to attacks. These systems commonly guard goods or information whose value entices attackers +to try to fool it into granting them access. In face biometrics, one of the main ways to do this is +through presentation attacks: here, an attacker presents to the sensor fake or altered samples of the +biometric trait that contains identity information from an authorised person [386]. +Presentation attack detection (PAD) algorithms aim to automatically recognise when captured +biometric samples have been faked or altered in such a way, preventing a biometric system from +granting access to an attacker. They consist of binary classifiers that distinguish between presen- +tation attacks or bona fide samples. Presentation attacks involve presentation attack instruments +(PAI) which can be printed photographs, digital screens, paper masks, or even tridimensional sili- +cone masks: each of these types of PAI is called a PAI species (PAISp) [192]. +Earlier PAD approaches focused on one single PAISp (i.e., trained and tested only with one +type of PAI), a scenario that can be designated as one-attack. This can naturally lead to overly +optimistic results that may not be verified in real-life applications, since attackers are perpetually +working on new and improved PAI species and face biometric systems will expectedly be faced +with more than one during application. +Another (more challenging) scenario is the one where multiple PAISp are used for training +and a set of unknown PAISp are used to evaluate performance. This is called unseen-attack and +typically enables performance results much more closely related to what would be verified in real + +9.2 Related Work +143 +applications. A PAD algorithm that is able to generalise well from the seen PAISp to the unseen +PAISp and offer good performance in the unseen-attack scenario is much more likely to perform +well when faced with novel PAISp once deployed. +As with several other pattern recognition tasks, deep learning architectures have been widely +applied in PAD, especially in recent years [40; 334]. These have enabled reaching improved +results that allude to the possibility of real implementation. Nevertheless, the evaluation scenarios +typically resemble the one-attack scenario and, thus, raise serious doubts about the realism of the +reported results. In fact, several works on PAD have doubled down on this issue and called for the +widespread adoption of unseen-attack scenarios for more realistic evaluation [132; 365; 384]. +The issue of performance degradation in the presence of unseen attacks is currently the biggest +challenge in face PAD. As such, some have addressed it using one-class classification or anomaly +detection strategies [20; 126; 143; 315; 333; 334; 465] achieving meaningful breakthroughs and +fostering the robustness of face PAD to unknown attacks. Others defend that the true solution lies +in the use of domain adaptation and interpretability to better control model behaviour and lead +PAD systems to achieve true generalising capabilities [335; 385; 386]. +9.2.4 +Robustness and trustworthiness +Faced with the current state of face recognition, some would say face recognition is a solved +topic. Advances in deep learning architectures, tailored loss functions, and massive online-sourced +databases have enabled the topic of face biometrics to achieve near-perfect performance metrics. +This is true even for edge scenarios, on challenging datasets with significant pose, illumination, +and environment variability. +However, diverse challenges remain to be solved or have surged over the recent years due +to developments in face recognition or society in general. Here, we focus on two of the most +pressing problems: trustworthiness and robustness. The first relates to the significant opacity of +deep learning-based state-of-the-art approaches. The second is linked to the difficulty of current +methodologies to recognise faces under occlusions, especially masks. Below, we delve deeper into +each of these challenges. +9.2.4.1 +Face recognition in a masked society +The ongoing Covid-19 pandemic has had a meaningful negative impact on face recognition sys- +tems [152]. The widespread (and generally mandated) use of face masks covering the nose, mouth, +and chin regions of the face has been reported to significantly degrade the accuracy of existing face +recognition solutions [93; 204; 313; 314]. +Before the pandemic, research on the robustness to occlusions in face recognition was fairly +common. However, it was also rather limited to small occlusions like sunglasses or scarves [324; +409], which do not typically hide as much information as a face mask (see Fig. 9.5). One should +easily understand how unprepared the existing face recognition solutions were for this new global +paradigm. + +144 +Prior Art in Face Biometrics +Figure 9.5: Example of how a mask can significantly occlude a face and limit the information that +can be used by a face recognition algorithm (from [93]). +Since the dawn of these challenging circumstances, multiple authors have studied in detail +the effects of wearing masks on face recognition [152]. Two of the most relevant studies were +conducted by the National Institute of Standards and Technology (NIST), focusing on pre-Covid- +19 [313] and post-Covid-19 algorithms [314]. These studies were part of the ongoing Face Re- +cognition Vendor Test (FRVT), an independent and thorough benchmark of face recognition solu- +tions, and the results indicate that competitive algorithms, which fail to authenticate less than 1% +of probes in typical circumstances, failed up to 50% more frequently in the presence of synthetic +masks. +Beyond these studies, some have studied the effect of real masks in academic and commer- +cial solutions. Damer et al. [93] has found that the FMR100 score of the state-of-the-art method +SphereFace can increase from 0.065% to 27.35% when evaluated with real masked face images +from twenty-four participants. Wearing masks has also been reported to significantly impact verifi- +cation performance of human operators [94], face image quality estimation [136] and presentation +attack detection performance [120]. +Early works on this topic included the automatic detection of face masks in images [268; 355] +which, however, do not effectively solve the problem. Later, Li et al. [256] proposed an attention- +based method to train a model on the periocular face region, avoiding the mask areas and achieving +promising results. +Most other solutions have focused on the adaptation of existing pretrained face recognition +networks, likely due to the limited amount and diversity of data currently available for masked +face recognition. Anwar and Raychowdhury [19] studied the benefits of synthesising masks on the +training data. [140] augmented existing datasets with realistic masked images using a generative +adversarial network (GAN) specifically tailored to retain identity information. +More recently, Boutros et al. [52] proposed an efficient solution to be integrated on top of +existing face recognition models, which attempts to unmask the embedding produced by the back- +bone. The unmasking module is based on a neural network trained with a self-restraining triplet +loss which prioritises more affected genuine pairs. +Huber et al. [182] used a template-level knowledge distillation approach to approximate em- +beddings produced from masked and unmasked images. The teacher model is trained with Elas- + +9.2 Related Work +145 +USER +MODELS +BIOMETRIC +DATA +INTERPRETABILITY +& EXPLAINABILITY +BETTER BEHAVIOR +UNDERSTANDING +DEEPER +PERFORMANCE +EVALUATION +EXPLANATION-BASED +OBJECTIVE FUNCTIONS +TRANSPARENT & +CLEAR DECISION +EXPLANATIONS +TRAINING +FEEDBACK +PROMOTE DESIRED +BEHAVIORS +PROVIDE SIMPLE AND COMPLETE +DECISION EXPLANATIONS +DECISIONS +Figure 9.6: Illustration of how interpretability/explainability can be used to understand and im- +prove a biometric model (from [311]). +ticFace loss, which is also used to ensure the resulting embeddings retain identity information. +Similarly, Li et al. [248] used knowledge distillation combined with image-level face completion. +Despite all these recent efforts and meaningful strides, there are still plenty of hurdles to over- +come. The rise of masks in our society uncovered the feeble nature of face recognition systems +in the face of extensive occlusions, and researchers are organising to face the challenge (e.g., +through large competitions in masked face recognition [50; 100]). Nevertheless, a definitive so- +lution should only be achievable with strong concerted efforts on improving available databases, +adapting existing algorithms, and designing novel training strategies. +9.2.4.2 +Interpretability in face biometrics +The growing use of increasingly sophisticated and elusive deep learning models has been spark- +ing the need for strategies to better understand their inner workings. This is also true for several +biometric applications. Namely, as face recognition systems permeate further into more criti- +cal applications such as border control or law enforcement, trustworthiness and transparency are +paramount [3; 17; 137; 352]. +In recent years, multiple studies have found that biometric systems (especially those based +on face) generally present significant demographic biases which have been able to survive largely +unnoticed. For example, lower false match scores among men (when compared to women) have +been linked to a significantly larger variety of facial hair styles. Similarly, the variety of hairstyles, +face morphology, and makeup styles have also been cited as possible causes behind gendered +differences in face recognition accuracy [14; 15; 356]. +If a biometric model suffers from bias, traditional metrics or visualisation methods will not be +able to detect it or explain that these features played a prominent role in the decision. Having more +thorough mechanisms of understanding the behaviour of biometric models, for instance through +interpretability, could be the key to unveiling and avoiding such biases and ensuring fair treatment +at all times (see Fig. 9.6). + +146 +Prior Art in Face Biometrics +The topic of interpretability in biometrics is still in its early stages, although some researchers +have already delved into the study of transparency for face biometrics. Some of these focused on +attention mechanisms [6; 206; 387], which allow models to focus on relevant areas that can be +adjusted to avoid certain undesirable behaviours. +One of the first approaches for interpretable face recognition was proposed by Yin et al. [473], +using feature and spatial activation diversity losses. These were used to promote, respectively, +filter robustness against occlusions and the inclusion of semantic information. Williford et al. +[458] explored a new way to obtain explanations by combining triplets and an inpainting game. +Using subtree excitation backpropagation and density-based input sampling for the explanation, +model interpretability is promoted and saliency maps can be built to support explanations. Liu +et al. [263] used adversarial training for heterogeneous face recognition, explicitly promoting the +introduction of semantic and interpretable information in the model’s latent space. +As for face PAD, most works so far are limited to the simple application of explainability +tools (typically GradCAM), or the use of t-distributed stochastic neighbour embedding (t-SNE) +as a way to interpret the way features lead to certain decisions [121; 207; 336; 454; 455; 475]. +Beyond these works, some authors proposed the application of auxiliary supervision techniques to +promote interpretability [265; 266; 470]. +Beyond face recognition and presentation attack detection, Seibold et al. [382] studied face +morphing attack detection using focused layer-wise relevance propagation (FLRP) as a way to +explain decisions to humans. Similarly, Xu et al. [467] proposed the use of FLRP to explain de- +cisions of their deepfake detection method. Alongside a method to simulate face aging, Genovese +et al. [141] proposed the cross-GAN filter similarity index (CGFSI) that can be used to explain the +behaviour of face GANs. +Despite these first works on interpretability for face biometrics, there is still a long way to +go. The effect of causality on the production of explanations and the possibility for multiple +simultaneous explanations are still open topics. Likewise, the production of semantic and textual +explanations (beyond simply visual ones) is yet to be addressed. All of these issues remark the +relevance of calling for transparency in face biometrics and adopting the new technologies in +interpretability for this topic. +9.3 +Open Challenges and Opportunities +As discussed throughout this chapter, face recognition is an intensively and thoroughly researched +topic with praised results and a significant impact in the real world. Through the study of increas- +ingly sophisticated deep learning methodologies and the design of tailored objective functions, +face recognition was able to conquer a prominent place in our society with robust and reliable +commercial solutions. +Nevertheless, it is also clear that some problems remain largely unsolved. Nowadays, those +are mainly related to robustness in masked face recognition scenarios, the trustworthiness of so- +phisticated deep learning-based solutions, and the detection of presentation attacks using unseen + +9.3 Open Challenges and Opportunities +147 +species. These are the topics which currently pose the greatest threats to face recognition applica- +tions and, thus, the most promising research opportunities. +Hence, this thesis part focuses on two contributions to these open challenges and opportunities. +Specifically: +• In Chapter 10, we propose two methodologies based on triplet and contrastive learning +strategies, combined with ArcFace and mean squared error losses to promote similarity be- +tween masked and unmasked image embeddings and close the performance gap on masked +face recognition; +• In Chapter 11, we study the use of interpretability to better understand the decisions of deep +learning models in face PAD, in order to motivate the broader application of interpretability +and explainability for more transparent and trustworthy biometrics. + + +Chapter 10 +Masked Face Recognition +Foreword on Author Contributions +The research work described in this chapter was conducted in collaboration with Pedro C. Neto, Fadi Boutros, +Mohsen Saffari, and Naser Damer, under the supervision of Jaime S. Cardoso and Ana F. Sequeira. The au- +thor of this thesis contributed to this work on the conceptualisation of the training strategies, the discussion and +comparison of the results, and the preparation of the scientific publications. +The results of this work have been disseminated in three articles in international conference proceedings: +• P. C. Neto, F. Boutros, J. R. Pinto, N. Damer, A. F. Sequeira, J. S. Cardoso, “FocusFace: Multi-task Con- +trastive Learning for Masked Face Recognition,” in Workshop on Face and Gesture Analysis for COVID-19 +(FG4COVID19), Dec. 2021. [308] +• P. C. Neto, F. Boutros, J. R. Pinto, M. Saffari, N. Damer, A. F. Sequeira, and J. S. Cardoso, “My Eyes Are Up +Here: Promoting Focus on Uncovered Regions in Masked Face Recognition,” in International Conference of +the Biometrics Special Interest Group (BIOSIG 2021), Sep. 2021. [309] +• F. Boutros, N. Damer, J. Kolf, K. Raja, F. Kirchbuchner, R. Ramachandra, A. Kuijper, P. Fang, C. Zhang, F. +Wang, D. M. Martin, N. Aginako, B. Sierra, M. Nieto, M. E. Erakin, U. Demir, H. Ekenel, A. Kataoka, K. +Ichikawa, S. Kubo, J. Zhang, M. He, D. Han, S. Shan, K. Grm, V. Struc, S. Seneviratne, N. Kasthuriarachchi, +S. Rasnayaka, P. C. Neto, A. F. Sequeira, J. R. Pinto, M. Saffari, and J. S. Cardoso, “MFR 2021: Masked Face +Recognition Competition,” in International Joint Conference on Biometrics (IJCB 2021), Aug. 2021. [50] +10.1 +Context and Motivation +Face recognition is one of the most advanced biometric traits. The adoption of sophisticated deep +learning architectures and tailored loss functions led to the achievement of very high accuracies. +This enabled the widespread adoption of face recognition solutions, from automated border control +to personal biometric applications [153; 280]. +However, as detailed in Chapter 9, the Covid-19 pandemic and the global mask mandates +created yet another hurdle to face recognition solutions. The occlusion of information by the use +of a mask has been reported to result in meaningful performance degradation in state-of-the-art +face recognition solutions [93; 204; 313; 314]. +149 + +150 +Masked Face Recognition +Unmasked Anchor + Masked Negative +Masked Positive +Unmasked Anchor +Masked Negative +Masked Positive +Learning +Figure 10.1: Expected effect of the original triplet loss on the output embedding space. +Recent works have focused mainly on training existing architectures or fine-tuning pretrained +models with relatively small datasets of synthetic masked images [19; 140]. Some have proposed +more sophisticated approaches for avoiding information that could be occluded by a face mask [52; +182; 248]. However, it is clear that further efforts are needed in the improvement of available +databases, the adaptation of existing algorithms, and the design of novel training strategies in +order to close the performance gap. +This chapter presents a work partially framed within the Masked Face Recognition (MFR +2021) challenge hosted by the 2021 International Joint Conference on Biometrics (IJCB). Here, +we propose two learning strategies based on the combination of triplet and contrastive losses with +mean squared error loss to promote both identity information capture and similarity between em- +beddings of masked and unmasked images. With this, we aimed to adapt face recognition state- +of-the-art deep learning methodologies to close the performance gap between face recognition in +generic unconstrained scenarios and in the presence of face masks. +10.2 +Methodology +Two approaches were explored to close the performance gap between unmasked and masked face +recognition. The first one is an adaptation of the triplet loss training strategy [309]. The second +one is a multi-task contrastive learning combination of cross-entropy (CE), ArcFace, and mean +squared error (MSE) objective functions [308]. Both aim to influence the behaviour of the models +through the respective loss functions by leading them to steer clear of information that can be +occluded by the use of face masks. +10.2.1 +Adapted triplet loss +The triplet loss [72] has been frequently used in general pattern recognition tasks, including the +particular case of face recognition models. Instead of instance-based learning, this methodology +organises training data into triplets composed of an anchor (xA, which serves as a reference), +a positive (xP, which shares the same identity as the anchor), and a negative sample (xN, of a +different identity). Each input corresponds to a model embedding representation in the learned +output space: yA, yP, and yN, respectively. With these, the triplet loss follows the equation: +l = max(0,α −d(yA,yN)+d(yA,yP)), +(10.1) + +10.2 Methodology +151 +Unmasked Anchor + Masked Negative +Masked Positive +Unmasked Anchor +Masked Negative +Masked Positive +Learning + Masked Anchor +Masked Anchor + +Figure 10.2: Expected effect of the proposed adapted triplet loss on the output embedding space. +where d(a,b) denotes the Euclidean distance between two embeddings a and b, and α is a tunable +margin parameter. By optimising towards the minimisation of this loss, the model is effectively +led to bring closer the representations y which share the same identities and draw apart those that +do not, in the learned output space (see Fig. 10.1). +The triplet loss will lead to embeddings of the same identity being clustered together in the +output space. However, they are still allowed some variability as long as it verifies the α margin +to the negative samples. In the scenario of masked face recognition, adding a mask to an image +can be enough to result in a large difference in the embedding within the margin allowance. This +is, however, undesirable. A method that is truly robust to face masks should be able to completely +ignore masks and, ideally, offer the exact same outputs for a face image and the exact same image +but with a face mask. Only thus do we have complete certainty that the model is robust to such +occlusions. +As such, we adapt the triplet loss learning strategy to promote such behaviour. Alongside the +typical members of a triplet (yA, yP, and yN), the model also receives a version of the anchor with +a synthetically added face mask (yAm) covering the mouth, chin, and nose according to health and +safety guidelines. The mean squared error (MSE) between yA and yAm is added to the aforemen- +tioned triplet loss formulation to further promote the similarity between the anchor and the masked +anchor: +ladapted = max(0,α −d(yA,yN)+d(yA,yP))+MSE(yAm,yA). +(10.2) +With this, we aim to lead the trained model to avoid the regions of the face that are commonly +occluded by masks and thus retain performance levels when performing inference on masked face +images (see Fig. 10.2). +10.2.2 +Multi-task contrastive learning +Building upon the idea of the adapted triplet loss, we propose a multi-task approach for masked +face recognition combining multiple objective functions. The proposed methodology, illustrated +in Fig. 10.3, receives pairs of masked and unmasked images and is composed of two symbiotic + +152 +Masked Face Recognition +shared +weights +MSE loss +Masked? +Identity +Masked? +Identity +CE loss +ArcFace loss +ArcFace loss +CE loss +total +loss +backpropagation +backpropagation +Figure 10.3: Schema of the proposed multi-task contrastive learning approach. +parts: one makes the network aware if a face mask exists in the input image, and the other uses +mask awareness for higher stability in identity learning among masked and unmasked images. +The first part consists of the detection of masks in the input image. Embedding features from +a common backbone (which can be a pretrained face recognition network) are delivered to a fully- +connected module which performs binary classification (masked or unmasked). This module is +optimised by minimising the cross-entropy objective function: +LCE = − 1 +N ∑ +i∈N +log +eyit +∑k∈n eyik , +(10.3) +where N represents the number of samples in the mini-batch, n is the class number, and yit repre- +sents the output of the module for the sample i and target class t. This strategy leads the model to +become aware of the presence of a mask in the input image and to include this information in the +output embedding. +The second part focuses on the identity recognition task. For improved results, we take the +ArcFace approach [98], widely considered the state-of-the-art in face recognition. ArcFace learns +identity by minimising the following loss function: +Larc = − 1 +N +N +∑ +i=1 +log +es(cos(θit+m)) +es(cos(θit+m)) +∑n +j=1, j̸=t escosθij , +(10.4) +where m represents the embedding distance margin, s denotes the scale, and θit the angle between +features (xi) and weights (Wt) for sample i and target class t. Both m and s are tunable hyperparam- +eters of the loss. ArcFace loss explicitly promotes higher intraclass similarity and diversity among +inter-class samples. Also, thanks to the l2 normalisation of both the weights and the feature vector, +the loss becomes equal to the geodesic distance margin penalty in a normalised hypersphere. +The global loss is the combination of ArcFace and CE losses computed for both masked and +unmasked inputs as well as the MSE between the embeddings of each input. As with the afore- + +10.3 Experiments and Results +153 +described adapted triplet loss approach, the MSE is intended to reinforce the similarity between +the latent representations of masked and unmasked inputs throughout the training process, thus +further promoting robustness to this kind of face occlusions. +10.3 +Experiments and Results +10.3.1 +Adapted triplet loss +10.3.1.1 +Experimental setup +The development of the adapted triplet loss methodology was partly motivated by the Masked Face +Recognition (MFR) competition hosted at the 2021 International Joint Conference on Biometrics +(IJCB). As such, the developed methodology is mainly evaluated using the official MFR dataset +(MFRC-21) [50]. This dataset is composed of face images of 47 subjects acquired over three +different days. The first day is considered the reference, while the second and third days are +considered probe sessions. Each session encompassed the acquisition of three videos, two with +face masks and one without, using a webcam. In total, 470 non-masked images and 940 masked +images are available as references, and 940 non-masked images and 1880 masked images are +available as probes. +The training methodology is applied to a ResNet-50 backbone architecture [167] that is used +to extract features from masked/unmasked face images. This backbone was trained using cross- +entropy loss and stochastic gradient descent for 150 thousand epochs with a batch size of 400. The +initial learning rate was set as 0.1, decreased by a factor of 10 whenever validation accuracy began +to decay. After convergence, the backbone was fine-tuned either with the original triplet loss or +the adapted triplet loss, with α empirically set to 0.2. Triplets were randomly generated during +training, without using any mining strategy, across a total of 65 thousand epochs. +Training used synthetic masked face images. The VGGFace2 dataset [58], composed of over +3.3 million face images from more than nine thousand identities, was adapted to include masked +faces. The NIST Dlib C++ toolkit [314] is used to obtain sixty-eight face landmarks which are +then used to generate masks appropriately fitted to the mouth/nose region, as detailed in [218; 314], +allowing for some controlled variability regarding mask shape and colour. No face alignment was +performed, and input images to the model were shaped 224×224×3. +The trained models were evaluated in a biometric verification task in two scenarios: U-M, +where the reference is an unmasked image and the probe is a masked image, and M-M, where +both the reference and probe are masked face images. The performance is reported through the +false non-match rates (FNMR), the FMR100 and FMR10, which are the lowest FNMR for a false +match rate (FMR) < 1.0% and < 10.0%, respectively. Additionally, the equal error rate (EER) +and the area under the receiver operating characteristic curve (AUC) results are also reported. The +genuine mean (GMean) and impostors mean (IMean) scores, which represent the mean distances +between the mated and non-mated embedding pairs, were also computed. + +154 +Masked Face Recognition +Table 10.1: Results with the adapted triplet loss on synthetic masked face data (SMFD). +Method +GMean +IMean +AUC +EER +FMR100 +FMR10 +VGG Face [58; 381] +0.505 +0.325 +0.951 +11.8% +38.2% +13.5% +CE Loss +0.528 +0.426 +0.941 +13.2% +38.5% +21.5% +CE + TL +0.601 +0.320 +0.977 +7.8% +28.9% +11.9% +CE + Adapted TL +0.596 +0.319 +0.985 +6.2% +18.5% +4.1% +10.3.1.2 +Results and discussion +The proposed adapted triplet loss methodology was evaluated on two distinct datasets: one with +synthetic masks (SMFD) and the other with real masked face images (RMFD). A detailed step- +wise ablation study is used to understand the behaviour of the models and the impact of the pro- +posed triplet loss adaptation in masked face recognition. +Table 10.1 presents the results on SMFD data. The proposed methodology, combining cross- +entropy pretraining, triplet loss, and MSE loss offers the best results across all performance met- +rics. As visible in the referenced table, the triplet loss training enables considerable performance +gains over the CE baseline, outperforming the literature VGG Face model [58; 381] trained on +unmasked face images. However, it is the MSE loss between masked and unmasked face images +that allows the model to achieve promising results, especially in FMR100 and FMR10. +The results on real masked data are presented in Table 10.2, including experiments on the U-M +and M-M scenarios. It can be observed that, in general, the performance results are considerably +inferior to those on SMFD, likely due to the models being trained with synthetic masks and tested +with real masks. However, the proposed adapted triplet loss attained superior performance across +all metrics. +It is noteworthy that M-M results, in general, do not differ considerably or consistently from +U-M ones. This could be a result of the optimisation process of the adapted triplet loss, which +leads the model to minimise the distance between masked and unmasked face embedding pairs. +However, it is expected that some applications, such as border control, could consist of the com- +parison of masked probes (face images captured in loco) with unmasked references (passport +Table 10.2: Results with the adapted triplet loss on real masked face data (RMFD). +Method +Mode +GMean +IMean +AUC +EER +FMR100 +FMR10 +VGG Face [58; 381] +U-M +M-M +0.523 +0.616 +0.426 +0.461 +0.769 +0.847 +29.419% +23.552% +90.587% +68.979% +58.959% +38.159% +CE Loss +U-M +M-M +0.610 +0.702 +0.475 +0.503 +0.931 +0.936 +11.687% +9.002% +32.041% +16.628% +12.852% +8.791% +CE + TL +U-M +M-M +0.647 +0.699 +0.396 +0.414 +0.943% +0.945% +11.213 +10.806 +34.744% +26.457% +11.874% +11.249% +CE + Adapted TL +U-M +M-M +0.649 +0.699 +0.383 +0.390 +0.957 +0.959 +9.799% +9.292% +28.252% +23,507% +9.678% +9.035% + +10.3 Experiments and Results +155 +Figure 10.4: Explanations obtained for each trained model with the Smooth Grad-CAM++ ex- +plainability tool (top row images were computed from the cross-entropy model - CE; middle row +images were computed from the triplet loss model - CE + TL; bottom row images were computed +from the adapted triplet loss model - CE + Adapted TL). +photographs). +Alongside the quantitative results presented above, the behaviour of the models trained with +the proposed and alternative approaches was also evaluated through explainability. Fig. 10.4 +presents six example images for which the Smooth Grad-CAM++ method [322] was used to as- +sess the relevance of the input pixels for the output embedding features. Relevance maps were +computed for each embedding feature and then averaged. +It can be observed that the first method, trained only with cross-entropy loss, was already +largely able to ignore mask regions in the figures, even when masks are not present. However, it +also commonly uses the region of the chin to construct the output embeddings, which does not +happen with the remaining two methods. When comparing the explanations of the triplet loss and +the adapted triplet loss, it can be observed that the latter typically considers more information from +wider regions of the face, thus capturing more information that could be useful for more robust +decisions. +10.3.2 +Multi-task contrastive learning +10.3.2.1 +Experimental setup +After the proposal of the adapted triplet loss, the multi-task contrastive learning methodology +aimed to be a more thorough study of masked face recognition. The multi-task contrastive learn- +ing methodology was explored for two backbone architectures, ResNet-100 and ResNet-50, both +widely used in the face recognition literature [167]. Following the example of Boutros et al. [51], + +156 +Masked Face Recognition +Figure 10.5: Example of masked face images generated to validate the multi-task contrastive +learning approach: images of the two individuals before and after mask generation. The first +example is correctly masked, while the second illustrates a scenario of incorrect masking. +the parameter s is set to 64 for the ResNet-100 and 30 for the ResNet-50, with m = 0.5. Train- +ing used the Stochastic Gradient Descent (SGD) optimiser with an initial learning rate of 0.1, a +momentum of 0.9, and weight decay of 5×10−4. +Following the example of several literature works [18; 98; 179; 295], this research used the +MS1MV2 dataset [98]. The MS1MV2 dataset is based on the MS-Celeb-1M dataset [158] and is +composed of 5.8 million images from 85 thousand identities. In this work, this dataset has been +augmented, offline, with face masks by generating one image with a face mask for each image in +the original dataset. +Face masks were synthesised using the MaskTheFace open source tool [19], which includes +five mask options (N95, KN95, surgical, cloth, and gas masks). For each image in the MS1MV2 +dataset, one of the first four mask types was randomly selected for the process of synthesising +masked face images, and the MaskTheFace tool took care of correctly reshaping and rotating the +mask templates to match the detected face landmarks in each image. +The described process was similarly followed for the images in the Labeled Faces in the Wild +(LFW) dataset [178], used for model validation during training (see Fig. 10.5 for some examples). +Although trained and validated on synthetic data, the method was evaluated on the MFR dataset, +composed of real masked face images, just as the adapted triplet loss methodology. Also, like with +the adapted triplet loss, we explore the evaluation scenarios U-M (with unmasked references and +masked probes) and M-M (with masked references and probes). +10.3.2.2 +Results and discussion +This section presents the results achieved by the multi-task contrastive loss approach. Compar- +isons focus mainly on the adapted triplet loss methodology, alternative approaches submitted to +the MFR competition, and multiple ablation studies. +The conducted ablation studies explored smaller backbone models, the use of pretrained net- +works, and diverse image selection approaches for the contrastive learning module. According to +the results presented in Table 10.3, four of the explored variants were capable of outperforming +the official MFR competition baseline [50] under the U-M scenario. Two of them were capable of +outperforming the baseline under the M-M scenario as well. + +10.3 Experiments and Results +157 +Table 10.3: Ablation results with the multi-task contrastive learning approach on the test dataset +(original selection stands for the use of the masked version of the unmasked image on the second +run of the network, whereas random selection denotes the process of choosing a random masked +image from the same subject). +Model +Selection +Mode +GMean +IMean +AUC +EER +FMR100 +FMR10 +ResNet-50 +Original +U-M +M-M +0.561 +0.625 +0.333 +0.341 +0.988 +0.983 +5.377% +5.346% +7.285% +6.546% +4.519% +4.642% +ResNet-100 +(Pretrained) +Original +U-M +M-M +0.571 +0.627 +0.357 +0.365 +0.987 +0.983 +4.917% +5.005% +5.986% +6.109% +3.922% +3.882% +ResNet-100 +(Pretrained) +Random +U-M +M-M +0.567 +0.625 +0.354 +0.361 +0.984 +0.984 +5.371% +5.581% +5.986% +6.184% +4.991% +5.056% +ResNet-100 +Original +U-M +M-M +0.624 +0.675 +0.373 +0.383 +0.992 +0.992 +4.594% +4.329% +5.750% +5.509% +2.582% +2.836% +ResNet-100 +Random +U-M +M-M +0.621 +0.671 +0.371 +0.382 +0.988 +0.991 +5.164% +4.757% +5.917% +5.695% +4.197% +3.668% +Table 10.4: Comparison of the modules of the proposed approach by the total number of parame- +ters (includes information regarding the use of each module for inference). +Module +No. Parameters +Inference +ResNet-100 w/o Embedding Layer +52 309 568 +Yes +Face Recognition Embedding Layer +12 846 080 +Yes +Mask Detection Embedding Layer +802 880 +No +ArcFace Layer +43 899 904 +No +Mask Detection Fully-Connected Layer +66 +No +Total (Training w/o pretraining) +109 858 498 +- +Total (Training w/ pretraining) +57 548 930 +- +Total (Inference) +65 155 648 +- +The non-pretrained ResNet-100 backbone model offered the best overall performance, espe- +cially when using the original pair selection process where the masked image is the masked version +of the unmasked image. Nevertheless, the results with pretrained models reveal that the proposed +method can effectively make existing models able to recognise masked faces with reasonable ac- +curacy and reduced training effort. +In fact, even though accuracy is important, time is also a key factor with such processing-heavy +algorithms. Most of the MFR competition submissions using a ResNet-100 backbone required the +training of more than 65 million parameters in the backbone model and up to 44 million parameters +in the ArcFace layer. Using pretrained weights for the entire backbone (except for the last layer), +the number of trainable parameters is reduced by approximately 47.6% while still outperforming +the baseline algorithm (see Table 10.4). + +158 +Masked Face Recognition +Table 10.5: Comparison of the FMR100 results of the methods presented in the MFR compe- +tition [50], the official baseline, the adapted triplet loss, and the multi-task contrastive learning +approach. +FMR100 +Method +U-M +M-M +Baseline [50] +6.009% +5.925 % +A1_Simple [50] +5.538% +5.771% +VIPLFACE-M [50] +5.681% +5.759% +MaskedArcFace [50] +5.687% +5.825% +Adapted Triplet Loss [309] +28.252% +23.507% +Multi-Task Contrastive Learning +5.750% +5.509% +In general, one can observe that better results were obtained when the masked image corre- +sponds to the unmasked image (original pair selection), likely as this better allowed to reinforce +in the model the behaviour of avoiding information that could be occluded by a mask. As for +the U-M and M-M scenarios, performance differences were not relevant nor consistent for any of +the models. As such, the proposed method is able to outperform the challenge baseline in both +scenarios. +A comparison with other submissions to the MFR 2021 competition is presented in Ta- +ble 10.5, for the U-M and M-M scenarios. The compared methods (A1_Simple, VIPLFACE-M, +and MaskedArcFace) were selected based on the competition results, the chosen loss functions +(ArcFace loss), the input and feature vector sizes (112 × 112 × 3 and 512, respectively), and the +dataset used (MS1MV2). Additionally, the proposed method is also compared to the challenge +baseline and the adapted triplet loss method [309]. +As presented in Table 10.5, it is possible to see that, despite presenting very similar perfor- +mance between the U-M and M-M scenarios, the proposed multi-task contrastive learning method- +ology is only able to outperform the selected competition submissions on the M-M scenario. How- +ever, the difference between the methods’ performances is relatively small and the contrastive ap- +proach was able to consistently outperform both the official baseline and the adapted triplet loss +methodology in both scenarios. +In the U-M scenario, a possible explanation for the advantage of A1_Simple is that it uses a +larger backbone model than ours, with roughly 34% more parameters. As for VIPLFACE-M, the +proponents used a different (improved) mask synthesis technique, which could be responsible for +the performance benefit. MaskedArcFace appears to also apply synthetic masks to the test set. +As for the M-M scenario, the proposed contrastive solution was able to outperform all alternative +methods by a margin of at least 0.250% FMR100. +Thanks to the multi-task nature of the proposed methodology, it is possible to leverage the +second embedding to perform simultaneous mask detection. Fig. 10.6 shows the ROC curve results +of the proposed model on the mask detection task, evaluated on the augmented LFW dataset. +The model achieved a perfect face detection score, which could be exaggerated by the relative + +10.4 Summary and Conclusions +159 +Figure 10.6: Receiver operating characteristic (ROC) curve for mask detection on the LFW dataset +with simulated masks. +simplicity of the validation data, but nevertheless show the effectiveness of the proposed multi- +task methodology. +Overall, these results showcase the capabilities of the proposed methodology, not only versus +the official MFR 2021 competition baseline and the submitted algorithms, but also against the pro- +posed adapted triplet loss. Moreover, the ability to achieve competitive results with largely frozen +pretrained backbones enables obtaining multi-task architectures with up to 80% fewer trainable +parameters. +10.4 +Summary and Conclusions +The work presented in this chapter addressed the challenge of masked face recognition, made +relevant by the worldwide face mask mandates born out of the recent Covid-19 pandemic. Despite +the extensive work on the robustness to occlusions in face biometrics, the presence of a mask +hiding the nose, chin, and mouth has resulted in significant performance decay. +Here, two approaches were presented to mitigate this issue. The first is an adaptation of the +triplet loss by combining it with the mean squared error (MSE) loss, which will minimise the +distance between embeddings from masked and unmasked versions of a face image. The second +is a multi-task contrastive learning approach that aims to make the model aware of the presence +of a mask and uses it alongside the ArcFace and MSE losses to achieve a more robust recognition +model. Both approaches aim to lead a biometric model into offering the same templates for a face +image, whether or not it contains a mask, effectively leading the model to avoid information from +the face regions that could be occluded by a mask. + +ROCcurveforfacemaskdetection +1.0 +0.8 +Rate +True Positive +0.6 +0.4 +0.2 +LFW +(area = 1.00) +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate160 +Masked Face Recognition +Results show that the combination of different losses and the promotion of embedding similar- +ity between masked and unmasked versions of a face image are successful at improving masked +face recognition performance. Experiments with an explainability tool also show the models are +indeed led to avoid regions that could be occluded by masks and to better use information from +visible face regions. +However, the second proposed approach, based on a multi-task contrastive learning strategy, +is the one which offered the best results. By combining the ArcFace loss (widely praised in face +recognition literature) with the MSE loss (which showed promise in the adapted triplet loss results) +and a strategy to make the network aware of the presence of masks, it was able to outperform the +official baseline and the best algorithm submissions in the MFR 2021 challenge. +Although the results are promising, further efforts should be devoted to the improvement of +the contrastive learning approach, the preprocessing stage of adding synthetic masks to test data, +and the study of larger and more accurate backbone architectures. The topic of masked face +recognition would also benefit heavily from larger and improved databases, with real masks of +diverse types worn by larger sets of individuals. + +Chapter 11 +Interpretability for Face Biometrics +Foreword on Author Contributions +The research work described in this chapter was conducted in collaboration with Ana F. Sequeira, Wilson Silva, +and Tiago Gonçalves, under the supervision of Jaime S. Cardoso. The author of this thesis contributed to this +work on the formulation and implementation of the face presentation attack detection framework, the preparation +and conduction of experiments, the discussion of the results, and the writing of the scientific publications. +The results of this work have been disseminated as a journal article, an article in international conference pro- +ceedings, and an abstract in national conference proceedings: +• A. F. Sequeira, T. Gonçalves, W. Silva, J. R. Pinto, and J. S. Cardoso, “An Exploratory Study of Interpretability +for Face Presentation Attack Detection,” IET Biometrics, 10 (4), 2021. [386] +• A. F. Sequeira, W. Silva, J. R. Pinto, T. Gonçalves, and J. S. Cardoso, “Interpretable Biometrics: Should We +Rethink How Presentation Attack Detection is Evaluated?,” in 8th International Workshop on Biometrics and +Forensics (IWBF 2020), Apr. 2020. [385] +• W. Silva, J. R. Pinto, T. Gonçalves, A. F. Sequeira, and Jaime S. Cardoso, “Explainable Artificial Intelligence +for Face Presentation Attack Detection,” in 26th Portuguese Conference on Pattern Recognition (RECPAD +2020), Oct. 2020. +11.1 +Context and Motivation +Like plenty of other artificial intelligence fields, biometrics has been witnessing a steadily in- +creasing dominance of deep learning-based approaches. This is largely due to the availability of +unprecedentedly large datasets and major computational gains offered by powerful graphics pro- +cessing units (GPU) [212; 242]. As discussed in Chapter 9, the largest downside of such a trend is +the lack of transparency of the resulting algorithms, which has been increasingly criticised by the +research community [107; 174; 372]. +Beyond simple decision accuracy-based metrics, there are several other aspects of a model that +may be useful (and even to understand during its development and evaluation to avoid undesirable +future consequences. One great example of this was described by Lapuschkin et al. [241], who +observed that for the detection of the class “horse” by a deep neural network, the model assigned +relevance to the bottom left corner of the images. A careful inspection revealed the presence of a +161 + +162 +Interpretability for Face Biometrics +copyright tag in that location, meaning the model was relying on this tag for the decision instead +of using meaningful image information. +For the example of a face presentation attack detection (PAD) model, it may be important for +a model to verify certain properties that may not be immediately obvious. First, the model should +look for the same information in a given sample whether or not that sample has been seen during +training. Second, a presentation attack sample should be processed similarly by a model whether +or not it was trained to detect that specific attack. Third, the behaviour of the model should be +coherent (similar) for different samples with the same predicted label. At last, the model’s choice +of information within a sample should be meaningful (as in, a human would likely look to the +same regions to provide the same decision). +These ideal behaviours may not be entirely objective and consensual. However, they are as- +pects of deep learning models that cannot be measured by traditional metrics. The ability to peek +deeper into the inner workings of biometric models is the true advantage behind integrating inter- +pretability in their development and evaluation. +The exploratory study presented in this chapter focused on the face PAD task to illustrate how +interpretability could be integrated with biometrics to reach the aforementioned goal. As such, +this work did not aim to push forward the state-of-the-art in face PAD, but to push forward the +almost nonexistent field of the explainability analysis in biometrics. An end-to-end CNN was im- +plemented to perform face PAD, and Grad-CAM was used to explain its decisions, delving deeper +into the behaviour of the model and assessing whether or not it verifies the desirable properties +discussed above. +11.2 +Methodology +11.2.1 +Implemented face PAD network +A PAD method receives a biometric trait measurement as input and returns a prediction of whether +that measurement belongs to a live individual (referred to as a bona fide sample) or a spoof attempt +to intrude the system (in this case referred to as a presentation attack). +In this work, the model used for PAD is an end-to-end convolutional neural network (CNN). +As an end-to-end CNN, the model was granted the flexibility to freely learn the most appropriate +features for the task at hand. This provides the most interesting context for interpretability studies +focused on gaining insight into the inner workings of a classifier. A relatively simple architecture +was chosen (see Fig. 11.1), as the emphasis of this work was to study the interpretability of the +face PAD model and not necessarily to surpass the PAD state-of-the-art. +The input to the network is a 224×224 RGB image and the output is a bidimensional softmax +layer providing two probability scores. The network is composed of four convolutional layers, +with three max-pooling layers interposed between them, and three fully-connected layers. The +four convolutional layers are composed of 32, 32, 64, and 64 filters, respectively, with size 3×3, +unit stride, and padding. The max-pooling is performed in 2 × 2 regions with stride 2. The + +11.3 Experimental Setup +163 +Conv2D +MaxPool2D +Conv2D +MaxPool2D +Conv2D +MaxPool2D +Conv2D +Dense +Dense +Dense +32@3x3 +ReLU +2x2 +32@3x3 +ReLU +2x2 +2x2 +64@3x3 +ReLU +64@3x3 +ReLU +100 +ReLU +100 +ReLU +2 +Softmax +224x224x3 +(RGB) +Bona Fide +or +Attack +Figure 11.1: Architecture of the PAD end-to-end deep model used in this work. +dense layers are composed of 100, 100, and 2 neurons, respectively. All convolutional and fully- +connected layers are followed by rectified linear unit (ReLU) activations, except for the last dense +layer, which is followed by softmax activation. +11.2.2 +Interpretability method +The Gradient-weighted Class Activation Mapping (Grad-CAM) [383] method is inspired by the +Class Activation Mapping (CAM) [492]. CAM was introduced for the identification of discrim- +inative regions in CNNs without fully-connected layers and restricted to the architectures that +perform global average pooling over convolutional maps immediately before prediction. Grad- +CAM is a generalisation of CAM in the sense that it was designed to be used with any type of +CNN architecture. +Grad-CAM consists of the combination of feature maps using the gradient signal. The gradient +information flows to the last convolutional layer of the model, thus assigning different importance +values to each neuron according to a particular decision of interest. This interpretability tool +enables the generation of explanations for any layer of the network. Additionally, it is possible to +obtain explanations per class, allowing the analysis of the model predictions at a class-level. +11.3 +Experimental Setup +11.3.1 +Data +This work used bona fide and presentation attack images extracted from the ROSE-Youtu Face +Liveness Detection Dataset [249]. This dataset is composed of 3497 videos of twenty subjects, +including attack videos of seven different PAI species (see Table 11.1). From each video, frames +were extracted every five seconds and faces were detected on each frame using an MTCNN [482]. +Face regions were cropped, resized to 224 × 224, and normalised to [0,1]. The samples from +subjects {2,3,4,5,6} were reserved for testing, while the data from the remaining fifteen subjects +were used for training and validation. + +164 +Interpretability for Face Biometrics +Table 11.1: PAI species in the ROSE Youtu DB and the respective number of extracted samples. +PAI Species +No. Frames +- +Genuine (bona fide) +2794 +#1 +Still printed paper +1136 +#2 +Quivering printed paper +1188 +#3 +Video of a Lenovo LCD display +923 +#4 +Video of a Mac LCD display +1113 +#5 +Paper mask with two eyes and mouth cropped out +608 +#6 +Paper mask without cropping +1194 +#7 +Paper mask with the upper part cut in the middle +1162 +11.3.2 +Implementation details +The PAD end-to-end model was trained on the extracted ROSE Youtu data using the Adam opti- +miser, with an initial learning rate of 10−4 for a maximum of 150 epochs with batch size 8. Early +stopping was used, monitoring the validation loss, with patience of 20 epochs. For regularisation, +dropout (0.5) was used between each pair of consecutive dense layers. Horizontal flips, rotations +with a range of 20 degrees, and width and height shifts with a range of 0.2 were used for data +augmentation. +Explainability experiments were performed using the Grad-CAM [383] implementation pro- +vided by the Keras Visualisation Toolkit [227] for Python, which is a library that enables visual- +ising and debugging a trained Keras model. It supports the visualisation of class activation maps, +saliency maps, and activation maximisation. In the obtained maps, each pixel is assigned a rele- +vance value that corresponds to a specific colour from blue (less relevant to the decision) to yellow +(more relevant). +11.3.3 +Experimental scenarios and evaluation +The explored scenarios follow those discussed in Chapter 9: +• In one-attack, the PAD model is trained and tested with bona fide samples and only one type +of attack/PAD species (PAISp). Therefore, the only type of attack shown to the network +during the test phase was already seen in the training step. The expression One-Attack#i, +used throughout the remainder of this chapter, denotes the respective model was trained and +tested with bona fide samples and presentation attack samples of type i; +• In unseen-attack, the PAD model is trained with all but one PAISp and tested with the +remaining PAISp, besides the bona fide samples in the train and test steps. Therefore, during +the test phase, the network is evaluated only with the type of PAD attack that was not present +in the training step (the unseen attack). This scenario enables a more thorough evaluation +of the generalisation capabilities of the PAD model. The expression Unseen-Attack#i, used +throughout the remainder of this chapter, denotes the respective model was tested with bona + +11.4 Conducted Studies on PAD Interpretability +165 +fide samples and presentation attack samples of type i and trained with bona fide samples +and the remaining types of attacks (i.e., trained with j ∈ {1,...,7}\i). +Evaluation results are reported using standardised metrics commonly used by the literature +in PAD. The Bona fide Presentation Classification Error Rate (BPCER) measures the proportion +of bona fide samples erroneously classified as attacks and the Attack Presentation Classification +Error Rate (APCER) indicates the proportion of presentation attacks wrongly classified as bona +fide [192]. The Equal Error Rate (EER) is the error at the operation point where the APCER and +BPCER present the same value. +11.4 +Conducted Studies on PAD Interpretability +11.4.1 +Representation of a model’s explanations +Beyond qualitative visualisation of explanations, this work entails a quantitative analysis consist- +ing of the comparison between the explanations obtained with bona fide or presentation attack +samples and taking into account the different evaluation scenarios (one-attack or unseen-attack). +Bona fide samples are present in both training and testing in any scenario, hence these can be +tested and explained in any framework One-Attack#i or Unseen-Attack#i. +The presentation attack samples belong to a specific type of attack. Considering the one-attack +evaluation scenario, an attack sample of type #i can only be tested for the respective One-Attack#i +scenario. It is not meaningful to test this sample with the models of One-Attack#j (with j ̸= i) +because then this would be an Unseen-Attack#i scenario. +As such, let I = {I1,...,In} and Exi = {Exi +1 ,...,Exi +n } be a set of images and the respective set of +explanations. For each image Ik (for k = 1,...,n) there is a corresponding explanation Exi +k . Note +that x = o or x = u whether the explanation refers to a classification result within the one-attack +or unseen-attack scenario, respectively, and i = 1,...,7 is the type of attack that defines the model +used for testing. Thus, each explanation is obtained with a specific model determined by the +evaluation framework and the attack used in testing. +11.4.2 +Semantic representation of explanations +One of the objectives of this work was to measure the variability of the explanations for both bona +fide and presentation attack samples in the two different evaluation scenarios. To do this, one needs +a suitable representation of the produced explanations, so that it becomes possible to quantify how +much two explanations differ from each other. In this work, this comparison is performed in a +semantic context. An illustrative example of how the approach used to perform a quantitative +comparison between explanations is depicted in Fig.11.2. +As stated before, the explanations are generated using Grad-CAM, which highlights the image +regions that maximise the predicted class. Since Grad-CAM typically produces blobby and coarse +explanations that fail to preserve finer details, it was decided to multiply the saliency maps by + +166 +Interpretability for Face Biometrics +Ik +Image +of type +bona fde +or attack #i +Ek +xj +Explanation * Image +Ek +yh +Explanation * Image +Model xj +(scenario x, attack j) +Grad-CAM +Model yh +(scenario y, attack h) +Grad-CAM +Facenet +Embedding +Embedding +Facenet +Euclidean +distance +d(Ek +xj, Ek +yh) +Figure 11.2: Example of the approach used to quantify the difference between two explanations. +the respective images. However, this space is still not ideal for image comparison, as it would be +highly impacted by the spatial location of important features. +To overcome this issue, and inspired by what is being done in image retrieval [173; 398] and +concept-based interpretability [145] to find similar images, the learned features computed by a +pre-trained CNN were used as the space to measure the distance between two explanations. This +follows the finding of Zhang et al. [485] that the Euclidean distance in the activation space of final +layers is an effective similarity metric. +Since this work focused on face images, FaceNet [381], a face-specific network pretrained +on the VGGFace2 dataset [58], was used for the extraction of the deep features. This CNN was +trained using a triplet loss, aiming to optimise the embedding space and ensure that the FaceNet +could learn a function that correctly maps the face images to a compact Euclidean space where +distances directly correspond to a measure of face similarity. +To take advantage of these FaceNet properties and achieve meaningful mappings of the Grad- +CAM explanations, we start by multiplying the original images by their respective Grad-CAM +explanations. We then input the resulting image into FaceNet, and extract the features generated +in the penultimate layer. All the Euclidean distances reported in this paper are computed in this +semantic space. +11.4.3 +Comparison of explanations across different scenarios +One could hypothesise that, for a robust PAD model, explanations for the same sample should be +similar whether or not the model is trained to detect that specific attack. Verifying this property +means we are in the presence of a PAD algorithm with thorough generalisation capabilities. In this +section, we describe a study that addresses the assessment of this property. +For bona fide images, Fig. 11.3 illustrates the process to compare the explanations, having an +image Ik, the evaluation framework x (either one-attack or unseen-Attack), and fixing as reference +the explanation obtained by the model for the PAISp #i in both frameworks. +As for presentation attack images, Fig. 11.4 illustrates the process to compare explanations, +having a presentation attack image Ik of type Attack #i. In this case, the comparison is made by + +11.4 Conducted Studies on PAD Interpretability +167 +Ik +Bona fde +Ek +xi +Ek +xj +Ek +xj +... +j = {1, ..., 7}\i +dk +xj +dk +xj +... +All distances +{dk +xj} = d(Ek +xi, Ek +xj) +for j = {1, ..., 7}\i +j = {1, ..., 7}\i +dk +xi = mean{dk +xj} +Figure 11.3: Comparison of explanations for a bona fide sample Ik, on the evaluation scenario x, +and fixing Attack #i. +Ik +Attack type #i +Ek +oi +Ek +xj +Ek +xj +... +j = {1, ..., 7}\i +dk +xj +dk +xj +... +All distances +{dk +xj} = d(Ek +oi, Ek +xj) +for j = {1, ..., 7}\i +j = {1, ..., 7}\i +dk +xi = mean{dk +xj} +Figure 11.4: Comparison of explanations for a presentation attack sample Ik, on the evaluation +scenario x, and fixing Attack #i. +fixing as reference the explanation of the result of the classification in the one-attack framework +obtained by the model for the Attack #i. Thus, the evaluation framework is x = o. As mentioned +before, this is done in order to have a more stable benchmark for comparison, since in the one- +attack scenario the model is trained and tested with the same type of attack. +So, for each image Ik (either bona fide or presentation attack), evaluation framework x, and +using the model regarding attack #i, a set of six total values {dx j +k : j ∈ {1,...,7}\i} is obtained. It +results of the comparison between the explanation Exi +k (always Eoi +k in the presentation attack case) +and the explanation Ex j +k (for j ∈ {1,...,7}\i). +The distance measurements between explanations {dx j +k } provide a quantitative measure of the +variability of the explanations produced by the different models when processing the same image. +Averaging these values will provide the ¯dxi +k values for comparison, given by: +¯dxi +k = 1 +6 ∑ +j +dx j +k . +(11.1) +For sake of clarity, Table 11.2 presents these distance values and their correspondence to the +images and scenarios. The values ¯dxi +k obtained for each image Ik are unique for a presentation +attack sample and multiple (one for each i ∈ {1,...,7}) for a bona fide sample. They can be used +for other quantitative interpretability endeavours, including to obtain image average and attack +average distances: +• The Image Average (Iµ) provides a quantitative measure of the variability, across all models, +of the explanations produced by each model under the evaluation framework defined by xi +(for x = o or x = u and i = 1,...,7) regarding one image Ik. The Iµ for the bona fide is given + +168 +Interpretability for Face Biometrics +Table 11.2: Overview of the strategy to compare explanations for a sample across different evalu- +ation scenarios. +Sample Class +Scenario +Comparisons +Bona Fide +One-Attack +{ ¯dxi +k : ¯dxi +k for i = 1,...,7} +Unseen-Attack +Presentation attack +One-Attack +¯dxi +k +(type #i) +Unseen-Attack +by Eq. (11.2) and for the presentation attacks is given by Eq. (11.3). +for a bona fide sample: +Iµx +k = 1 +7 +7 +∑ +i=1 +¯dxi +k +(11.2) +for an attack sample (type #i): +Iµx +k = ¯dxi +k +(11.3) +• The Attack Average (Aµ) provides a quantitative measure of the variability, across all sam- +ples, of the explanations produced by the model under one evaluation scenario defined by +xi (for x = o or x = u and i = 1,...,7). Consider the values ¯dxi +k as defined in Table 11.2 and +being n ad m the number of bona fide and attack samples, respectively. The Aµ for the bona +fide is given by Eq. (11.4) and for the presentation attacks is given by Eq. (11.5). +for a bona fide sample: +Aµxi = 1 +n +n +∑ +i=1 +¯dxi +k , for i = 1,...,7 +(11.4) +for an attack sample (type #i): +Aµxi = 1 +m +m +∑ +i=1 +¯dxi +k , for i = 1,...,7 +(11.5) +11.4.4 +Interclass comparison in the unseen-attack scenario +This study investigates the interclass comparison between explanations obtained using the models +in the unseen-attack framework. In other words, it investigated the variability of the explanations +between bona fide and presentation attack samples. To achieve the desired goal, the explanations +obtained from the classification of each image, with the different models trained in the Unseen- +Attack scenario, are compared in a pairwise manner. This comparison is performed for all images +within each class. +By using the unseen-attack models it is possible to test the robustness of the models to the +variability in the attacks present in the training and testing steps. Recall that a model resulting +from unseen-attack#i is trained with attacks j for j ∈ {1,...,7}\i. +For a bona fide sample Ik, the process to obtain the comparison of all explanations is illustrated +in Fig. 11.5. It shows one example of how to obtain the pairwise distances Dk given by: Dk = {d jh +k : +d jh +k = d(Eu j +k ,Euh +k ), with j,h ∈ {1,...,7} and j ̸= h}. Then the values in Dk are averaged and ¯dk is +obtained for image Ik. A global value is obtained averaging all these values, ¯dBF, as given by + +11.4 Conducted Studies on PAD Interpretability +169 +Ik +Bona fde +{Ek +uj} +Ek +1 +Ek +e +... +j = {1, ..., 7} +All pairwise distances +{dk +jh} = d(Ek +uj, Ek +uh) +for j,h = {1, ..., 7}, j ≠ h +dk = mean{dk +jh} +#{Ek +e} = 7 +Ek +2 +d12 +d2e +d1e +Figure 11.5: Pairwise comparison of explanations produced by the models in unseen-attack sce- +narios, for a bona fide sample Ik. +Ik +Attack type #i +{Ek +uj} +Ek +1 +Ek +e +... +j = {1, ..., 7}\i +All pairwise distances +{dk +jh} = d(Ek +uj, Ek +uh) +for j,h = {1, ..., 7}\i, j ≠ h +dk = mean{dk +jh} +#{Ek +e} = 6 +Ek +2 +d12 +d2e +d1e +Figure 11.6: Pairwise comparison of explanations produced by the models in unseen-attack sce- +narios, for the presentation attack sample Ik of type #i. +Eq. (11.6): +¯dBF = 1 +n ∑ +k +¯dk, +(11.6) +for each bona fide image Ik, k = 1,...,n. +As for a presentation attack image Ik (of type #i), Fig. 11.6 shows the process to obtain the +comparison between the explanations obtained for the models Unseen-Attack#j for j ∈ {1,...,7}\ +i. The figure shows one example, regarding image Ik of type Attack#i, on how to obtain the +pairwise distances Dk = {d jh +k : d jh +k = d(Eu j +k ,Euh +k ), with j,h ∈ {1,...,7}\i} and j ̸= h}. The values +in Dk are averaged and ¯dk is obtained for image Ik. A global value is obtained averaging all these +values, ¯dPA, as given by Eq. (11.7): +¯dPA = 1 +m ∑ +k +¯dk, +(11.7) +for each presentation attack image Ik, k = 1,...,m, of type #i. +11.4.5 +Intraclass comparison across different samples +One other property that a robust PAD solution should verify is that explanations should be similar +for different samples with the same label. In other words, the explanations should reveal that +the model looks for the same regions of the face even when the images are very different, since +the ground-truth label is the same. This means the algorithm is coherent in its decisions and +has effectively pinpointed the features that allow for accurate and stable detection of presentation +attacks. + +170 +Interpretability for Face Biometrics +It is important to investigate the coherence of explanations for the bona fide samples, in order +to understand if the considered PAD solution knows indeed what is a “real face”. However, it +is also important to understand how much the explanations vary for presentation attack samples, +so we can assess if our algorithm is indeed capable of generalising well to the multiple possible +attack species. +This analysis can be performed by measuring how much the explanations for the models’ +decisions are affected by variations in the types of presentation attacks known in the learning +phase. In this work, this comparison is done by comparing features extracted from the explanations +(using the FaceNet model described above) and not in a pixel-to-pixel manner, making it possible +to compare the explanations obtained for different samples. +11.5 +Results and Discussion +11.5.1 +Performance of the face PAD algorithm +The main focus of this work was interpreting the decisions of a face PAD model to illustrate and +motivate the application of interpretability to biometrics. Achieving improved face PAD results +versus the state-of-the-art was not a priority. Nevertheless, one should not aim to interpret a model +that lacks PAD abilities by design, as this will impair and bias the drawn conclusions. As such, +the performance results of the implemented model in the one-attack and unseen-attack scenarios +are presented in Table 11.3. +The results were generally inferior in the unseen-attack scenario when compared to the one- +attack scenario, as expected given the considerably higher difficulty of the former scenario and the +insight found in face PAD literature [384]. One reason for such results may be the large variability +between the different PAI species found in the ROSE Youtu dataset. Hence, to obtain a truly robust +PAD algorithm, the test set should always be composed of PAI species which have not been seen +by the algorithm during training [132]. +Attack #7, an upper face mask with eyes cropped out, is a good example of the implemented +model’s difficulty to generalise. Although the results are acceptable in the one-attack scenario, +the error rates are considerably higher in the unseen-attack scenario. Some exceptions can also be +found, such as Attack #1 (full face printed photo), for which both EER and APCER are actually +lower in the unseen-attack scenario vs. one-attack. For this specific PAI species, the model is +probably learning most of the needed features from various other paper-based attack types in the +unseen-attack scenario, taking advantage of the greater availability of data despite the absence of +Attack #1 samples in the training. +11.5.2 +Comparison of explanations across different scenarios +11.5.2.1 +Image average (Iµ) +One of the objectives of this work was to study the behaviour of the PAD models under varying +data diversity in the training dataset. For an ideally robust PAD model, the same explanation + +11.5 Results and Discussion +171 +Table 11.3: Performance of the PAD models in the one-attack and unseen-attack evaluation sce- +narios (EER, APCER, and BPCER in %; APCER and BPCER calculated for a threshold of 0.5). +Attack +One-Attack +Unseen-Attack +EER +APCER +BPCER +EER +APCER +BPCER +1 +7.29 +12.15 +3.06 +5.90 +6.94 +4.90 +2 +3.62 +6.67 +1.35 +5.55 +3.00 +10.65 +3 +2.79 +8.37 +0.12 +10.38 +26.29 +4.28 +4 +12.66 +30.38 +1.84 +25.34 +45.73 +3.92 +5 +1.61 +1.61 +1.59 +4.84 +3.55 +7.10 +6 +4.46 +5.10 +1.10 +10.19 +12.74 +7.71 +7 +0.73 +5.23 +0.00 +15.49 +34.31 +7.71 +should be obtained for a given bona fide image, regardless of the PAI species seen during the +training stage. However, in reality, models are often sensitive to variations in the training data, +both for bona fide and presentation attack samples. +Fig. 11.7 presents the mean results (µ(Iµ)) and respective standard deviation (σ(Iµ)) of the +Image average (Iµ) experiments for bona fide (BF) and presentation attack (PA) samples across the +one-attack (OA) and unseen-attack (UA) scenarios. These results show that intraclass variability +is higher in the one-attack scenario, as indicated by higher µ(Iµ) values. +The Iµ values also show higher variability in one-attack, according to the σ(Iµ) results. This +suggests that the models in the unseen-attack scenario are better able to generalise in the recog- +nition of bona fide samples, since they see a wider variety of attacks during training. For the PA +samples, the variability results also suggest the models are more robust when a wider diversity of +attacks is available during training. +Fig. 11.8 and Fig. 11.9 show an example of a PA sample presenting a higher µ(Iµ) in the +one-attack scenario when compared to unseen-attack. These results further support the idea that a +model trained with more than one PAI species is able to learn better patterns and become a more +robust model with better generalisation capabilities. +11.5.2.2 +Attack average (Aµ) +The mean and standard deviation of the attack average results (respectively, µ(Aµ) and σ(Aµ)) +across the one-attack and unseen-attack scenarios for bona fide and presentation attack samples +are presented in Fig. 11.10. +Similar to the Iµ results presented before, the Aµ results show higher variability in the one- +attack scenario. This suggests once again that a training setup that integrates more than one PAI +species may be more successful at promoting the learning of more coherent features for the bona +fide class. The same is also verified for presentation attack samples, as the mean distance in the +unseen-attack scenario is inferior to that in one-attack. +Since each PAI species contains intrinsic specificities, one could hypothesise that grouping +multiple of them into a single class “presentation attack” would confuse the model and result in + +172 +Interpretability for Face Biometrics +Figure 11.7: Image Average mean and standard deviation (StD) results for bona fide (BF) and +presentation attack (PA) samples in the comparison across the one-attack (OA) and unseen-attack +(UA) scenarios. +lesser robustness. Yet, the results indicate that the models used to detect them seem to benefit from +the integration of more attacks during the training phase. +It is interesting to observe the variability for presentation attack samples across the different +attacks in the unseen-attack scenario: although, in general, the values are inferior to those in +the one-attack scenario, some specific attacks present much lower results than others. This may +denote that differences (and similarities) between PAI species seen in training and evaluation lead +to models that are much more sensitive to unseen PAI species in the testing phase. +In particular, in the presentation attack graphs in Fig. 11.10, Unseen-Attack#5, consisting of +paper masks with eyes and mouth cut off, presents the lowest µ(Aµ). This may result from the fact +that, in this scenario, the model has seen multiple print-based PAI species (like complete photos, + +"Image Average" Mean Intraclass Analysis: +One-Attack vs. Unseen-Attack +BFIAMeanOABFIAUAPAIAMeanOAiOAjPAIAMeanOAiUAj +1.2000 +1.0000 +0.8000 +0.6000 +0.4000 +0.2000 +0.0000"Image Average" StD Intraclass Analysis: +One-Attack vs. Unseen-Attack +BF IA StD OA BF IA StD UA PA IA StD OAiOAjPA IA StD OAiUAj +0.6000 +0.5000 +0.4000 +0.3000 +0.2000 +0.1000 +0.000011.5 Results and Discussion +173 +Attack #5 +One-Attack 1 +One-Attack 2 +One-Attack 3 +One-Attack 5 +One-Attack 4 +One-Attack 6 +One-Attack 7 +Figure 11.8: Comparison of explanations in intraclass one-attack for an example PA sample of +type 5 presenting high Iµ value (obtained when the One-Attack#5 is compared against the One- +Attacks{#1,...,#7}\#5). +Attack #5 +Unseen-Attack 1 +Unseen-Attack 2 +Unseen-Attack 3 +One-Attack 5 +Unseen-Attack 4 +Unseen-Attack 6 +Unseen-Attack 7 +Figure 11.9: Comparison of explanations in intraclass unseen-attack for an example PA sample of +type 5 presenting low Iµ value (obtained when the One-Attack#5 is compared against the Unseen- +Attacks{#1,...,#7}\#5). +complete paper masks, and half paper masks) which help the model prepare for PAI species #5. +The Unseen-Attack#7 (consisting of upper-half face paper masks) presents a higher result since +these samples combine skin and paper in the facial area and may be more difficult to learn from +the PAI species seen during training. + +174 +Interpretability for Face Biometrics +Figure 11.10: Mean Aµ results for bona fide and presentation attack samples in the one-attack +(OA) and unseen-attack (UA, i = 1,...,7) scenarios and respective mean value. + +Bona Fide"Attack Average"Mean Values: +One-Attackvs.Unseen-Attack +0.8000 +0.7000 +0.6000 +0.5000 +0.4000 +0.3000 +0.2000 +0.1000 +0.0000 +AttackAverage OA +AttackAverage UA +Attack 1 +1 Attack 2 +Attack 3 +Attack4 +Attack 5 +Attack 6 +Attack 7 +IMeanPresentation Attack"Attack Average" Mean Values: +One-Attack vs. Unseen-Attack +0.8000 +0.7000 +0.6000 +0.5000 +0.4000 +0.3000 +0.2000 +0.1000 +0.0000 +AttackAverage OAi OAj +AttackAverage_OAi_UAj +Attack 1 +Attack 2 +■Attack 3 +1Attack 4 +Attack 5 +Attack 6 +Attack 7 +IMean11.5 Results and Discussion +175 +Bona Fide +Unseen-Attack 1 +Unseen-Attack 2 +Unseen-Attack 3 +Unseen-Attack 4 +Unseen-Attack 5 +Unseen-Attack 6 +Unseen-Attack 7 +Figure 11.11: Explanations for an example bona fide samples with pairwise distance close to the +obtained average ( ¯dBF = 0.54). +Attack #2 +Unseen-Attack 1 +Unseen-Attack 3 +Unseen-Attack 4 +Unseen-Attack 5 +Unseen-Attack 6 +Unseen-Attack 7 +Figure 11.12: Explanations for an example presentation attack sample of type #2 with pairwise +distance close to the obtained average ( ¯dPA = 0.52). +11.5.3 +Interclass comparison in the unseen-attack scenario +This experiment aimed to quantify the variability of the explanations between bona fide and pre- +sentation attack samples. For the bona fide samples, a result of ¯dBF = 0.54 was obtained, by +averaging the value ¯dk of all bona fide images (Ik, for k = 1,...,n) obtained from a pairwise com- +parison of the explanations of all models in the unseen attack scenario. The associated standard +deviation is 0.13. As for the presentation attack samples, a result of ¯dPA = 0.52 was obtained by +averaging the value ¯dk of all attack images (Ik, for k = 1,...,m) obtained from a pairwise compari- +son of the explanations of Unseen-Attack#j for j ∈ {1,...,7}\i models. The associated standard + +176 +Interpretability for Face Biometrics +Bona Fide +Unseen-Attack 1 +Unseen-Attack 2 +Unseen-Attack 3 +Unseen-Attack 4 +Unseen-Attack 5 +Unseen-Attack 6 +Unseen-Attack 7 +Figure 11.13: Explanations for an example bona fide sample with pairwise distance above the +obtained average ( ¯dBF = 0.54). +deviation is 0.14. +The obtained results are very similar and the standard deviation values are not only relatively +high but similar for both sample classes. The similarity of the values does allude to any definitive +comparative conclusion, but it does motivate further investigation. Fig. 11.11 and Fig. 11.12 depict +examples of images whose pairwise distance results ¯dk are close to the respective mean values +¯dBF and ¯dPA. A visual inspection allows us to conclude there is, in fact, considerable variability +between explanations. Good examples of this are the Unseen-Attacks#4,#6 in comparison to +#2,#3,#5 in Fig. 11.11, and the Unseen-Attacks#3,#4 in comparison to #1,#6) in Fig. 11.12. +Nevertheless, despite the observed variability, in both types of samples there are certain regions +of the images that are consistently used by all models to make their decisions. This may denote +that, despite the variety of training conditions and the resulting noise found in the explanations, the +model is generally able to pinpoint some regions of the face that correspond to the real underlying +label information. This idea is verified when the pairwise distance is above the median values (see +Fig. 11.13 and Fig. 11.14). +Naturally, this rationale based on subjective visual evaluations is limited, but this is a result of +the current unavailability of a ground-truth for what is a good or meaningful explanation. These +are still muddy unexplored grounds that require further research into the combination of inter- +pretability methods and human expert knowledge in the field of biometrics. +11.5.4 +Intraclass comparison across different samples +The intraclass comparison experiment aimed to analyse an expected property of a robust and sound +PAD solution: explanations for different samples of the same class should be similar. If true, this +means the model learned to behave in a coherent manner, looking for the same regions in the +images to reach the same decisions. The results of this experiment are presented in Fig. 11.15. + +11.6 Summary and Conclusions +177 +Attack #7 +Unseen-Attack 1 +Unseen-Attack 2 +Unseen-Attack 3 +Unseen-Attack 4 +Unseen-Attack 5 +Unseen-Attack 6 +Figure 11.14: Explanations for an example presentation attack sample of type #7 with pairwise +distance above the obtained average ( ¯dPA = 0.52). +One can observe that the one-attack scenario (both for bona fide and presentation attack sam- +ples) leads to overall higher variability in explanations. This confirms the idea, brought up previ- +ously in this section, that a greater variety of PAI species in the training phase will result in models +that are more robust and consistent in the information they use for their decisions. +11.6 +Summary and Conclusions +This study consisted of an analysis of the explanations produced for a face PAD model explored +in different scenarios and settings. Both the standard one-attack and the more challenging unseen- +attack scenarios were considered in this exploratory study. Additionally, intraclass and interclass +experiments were also conducted to evaluate certain desirable properties of a robust and effective +PAD solution. +Focusing on the intraclass comparison of explanations’ variability, it was possible to verify +that the one-attack framework led to an increased mean distance value for both bona fide and +presentation attack samples. This denotes that the presence of more attacks during training has +a positive effect on the generalisation capabilities of the models, despite one-attack performance +metrics being generally better than those in unseen-attack. +As for the interclass comparison of explanations, it was found that both classes exhibit similar +levels of variability. Further analysis of bona fide and presentation attack samples has shown that, +despite the wide variability in explanations, the models were generally able to pinpoint important +regions of the face images that correspond to the true underlying labels. +Overall, this exploratory study illustrates the deeper level of insight that can be obtained when +biometric studies are combined with interpretability. If one is to overcome the elusive nature of + +178 +Interpretability for Face Biometrics +Figure 11.15: Bona fide and presentation attack intraclass comparison mean and standard deviation +results for the one-attack and unseen-attack (i = 1,...,7) scenarios and respective overall results. +sophisticated deep learning models and move towards more transparent approaches, it is important +that evaluation frameworks allow the assessment of the quality and comparison of explanations. +It is clear that there is still much to do. Regardless of the biometric task at hand, it may be +very difficult to find a consensual definition of a “good” explanation, or what behaviours an ideal +model should exhibit. According to Phillips and Przybocki [335], an explanation must describe +how the system came to its conclusion and an “accurate” explanation (whatever that may mean) +does not imply that a system provided the correct answer. +As such, the major open challenge is for the biometrics community to evolve from established +evaluation metrics based on decision accuracy to novel explanation-based performance metrics. +This work took one step forward in this direction. However, improving objectivity by combin- +ing subjective but knowledgeable opinions from several experts is essential to consolidate inter- +pretability and thus enable more meaningful performance analysis on biometrics. Additionally, +further efforts should be devoted to investigating the integration of the explanations on the training +itself as a regularisation method to guide models through the learning of more meaningful features. + +BonafideIntraclassComparison +0.8 +0.7 +0.6 +0.5 +LLL +0.4 +0.3 +0.2 +0.1 +0.0 +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Attack1Attack2Attack3Attack4Attack5Attack6Attack7 +Mean +OneAttack_BF +UnseenAttackBF +PresentationAttackIntraclassComparison +0.9 +0.8 +0.7 +0.6 +LLL +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Mean +StD +Attack1Attack2Attack3Attack4Attack5Attack6Attack7 +Mean +OneAttackPA +UnseenAttackPAPart IV +Wellbeing Monitoring +179 + + +Chapter 12 +Emotion Valence +Classification in the Wild +Foreword on Author Contributions +The research work described in this chapter was conducted within the Easy Ride project in collaboration with +Tiago Gonçalves, Carolina Pinto, and Luís Sanhudo, under the supervision of Jaime S. Cardoso, Pedro Carvalho, +Joaquim Fonseca, and Filipe Gonçalves. The author of this thesis contributed to this work on the conceptuali- +sation and implementation of the video module, the evaluation of the unimodal and multimodal algorithms, the +preparation of submissions to the EmotiW 2020 AV Group sub-challenge, and the preparation of the scientific +publication. +The results of this work were disseminated in the form of an article in international conference proceedings: +• J. R. Pinto, T. Gonçalves, C. Pinto, L. Sanhudo, J. Fonseca, F. Gonçalves, P. Carvalho, and J. S. Cardoso, +“Audiovisual Classification of Group Emotion Valence Using Activity Recognition Networks,” in Fourth IEEE +International Conference on Image Processing, Applications and Systems (IPAS 2020), Dec. 2020. [346] +This work was awarded the Best Session Paper Award at the aforementioned international conference. +12.1 +Context and Motivation +Emotion recognition is a fast-growing research topic, due to its potential for enhanced human- +computer interfaces and automatic services that immediately respond to the emotions of the user or +client [131]. Horror videogames that adapt the gameplay and sound effects based on the player’s +fear, as well as autonomous vehicles that adapt the travel experience based on the occupants’ +emotions, are only two of the endless innovations attainable through emotion recognition [294]. +State-of-the-art methods for emotion recognition are mainly based on facial expressions, and +important hurdles have been overcome in this field [131; 294]. Group-level emotion is a fairly +uncharted research topic that extends the analysis to the emotional state displayed by a group of +people as a whole [425]. While there are several challenges in individual emotion recognition, +approaches for group emotion recognition also need to deal with the variety of emotions, their +valence, and arousal levels, that can differ among members of the same group. This topic was the +focus of the EmotiW 2020 [101] sub-challenge that motivated this work. The scarce data and the +181 + +182 +Emotion Valence Classification in the Wild +difficulty in obtaining annotations is the reason why few have addressed this topic [159; 415; 456], +and why current approaches still offer low accuracy levels. +The task of group emotion recognition shares some similarities with the recognition of human +activity based on the video. Unlike the former, the latter boasts several large and thoroughly +labelled datasets, such as the Kinetics [65] or the ActivityNet [169], even when restricting to data +focused on groups rather than on individuals. These larger sets of available data have allowed +for the development of very robust and high-performing algorithms, such as the I3D [65], the +SlowFast networks [127], or the stagNet [354]. +While methods based on visual information compose most of the literature, some works dis- +cuss the advantages of including additional sources of information, especially audio [85; 214; +258; 453]. Specifically, it has been shown that using audio complements some of the flaws of +video-based recognition [214], despite offering subpar accuracy results when in a unimodal re- +cognition system. These results have confirmed the advantages of combining audio information +with a strong method for video-based recognition. +This work explores the novel application of inflated convolutional neural networks (CNN) to +classify emotion valence at the group level in videos. The network uses weights pretrained for +activity recognition, to take advantage of the greater availability of data to boost performance +on our target task. We also study the use of audio for improved performance, through score- +level fusion, with a Bi-LSTM network receiving spectral features. Throughout the experiments, +we assess the performance of the proposed method for multimodal and unimodal classification, +analyse its behaviour in different scenarios, and compare it directly with the EmotiW 2020 sub- +challenge official baseline. +12.2 +Methodology +12.2.1 +General overview +The proposed algorithm is composed of three modules: a video-based emotion recognition model, +an audio-based emotion recognition model, and a multimodal fusion module (see Fig. 12.1). Video +and audio-based emotion recognition modules are trained independently, while the fusion module, +based on a multiclass SVM receives the softmax scores provided by the other two. Thus, the +proposed method consists of a pipeline that relies on late audio-video fusion, at the score level, +using a multi-class SVM emotion recognition classifier. +12.2.2 +Video-based emotion recognition +The video-based emotion recognition module is based on an inflated bidimensional (2D) convolu- +tional neural network (CNN), similar to I3D [65], the state-of-the-art in activity recognition. The +model is an end-to-end network: it receives frames extracted from a video, ordered and concate- +nated over a time dimension, and returns class probabilities for that video. The architecture of the +network follows the structure of a ResNet-50 (see Fig. 12.2), proposed by He et al. [167], whose + +12.2 Methodology +183 +Frame +Extraction +Audio +Extraction +(pretrained for activity recognition) +Sliding Window +Feature Extraction +Bi-LSTM +Fusion +SVM +Class Probabilities +Class Probabilities +Class +Probabilities +Video +Figure 12.1: Illustration of the structure of the proposed method for audiovisual group emotion +recognition (the proposed methodology for group emotion recognition processes a video in two +streams: one processes concatenated video frames using an inflated ResNet-50 pretrained on a +large activity recognition dataset, and the other extracts sliding window features from the audio +and processes them using a bi-directional long short-term memory (Bi-LSTM) network; a support +vector machine (SVM) classifier receives class probabilities from each stream and returns a final +class prediction). +name stands for residual networks. The shortcut connections that perform identity mapping on +each residual learning block enable the stable training of models with more convolutional layers, +resulting in deeper representations of the input data. +The inflated ResNet-50 consists of a bidimensional ResNet-50 model where the convolutional +filters and layers have been converted into 3D. This allows them to process several frames simul- +taneously as a single input. Downsampling operation before the first block of each type enables +learning multi-resolution features. This model has been pretrained1 to discriminate between 339 +activity classes on the Multi-Moments In Time database [300]. To offer probability outputs for +each of the three group-level emotion valence classes, the last fully-connected layer of this net- +work is replaced by a three-neuron fully-connected layer, followed by softmax activation, trained +on the EmotiW 2020 sub-challenge train dataset. +12.2.3 +Audio-based emotion recognition +The audio-based recognition module (see Fig. 12.1) is composed of two main processes: fea- +ture extraction on sliding windows, and a Bi-LSTM recognition model. Audio features were +extracted using pyAudioAnalysis2 which contains an off-the-shelf feature set with 34 available +features, including signal zero-crossing rate, signal energy, entropy of energy, spectral centroid, +spectral spread, spectral entropy, spectral flux, spectral roll-off, mel-frequency cepstral coeffi- +cients (MFCC), chroma vector, and chroma deviation. All these features are extracted over sliding +windows of 25 milliseconds with a time step of 10 milliseconds. +The features are received by a Bi-LSTM model with local attention that returns the class prob- +abilities for the respective audio (see Fig. 12.3), adapted from [296]. Its weighted-pooling strategy +enables the focus on the specific sound parts which contain strong emotional characteristics, con- +trolled by an attention function trained simultaneously with the Bi-LSTM model. +1Multi-Moments in Time models. Available at: https://github.com/zhoubolei/moments_models. +2pyAudioAnalysis. Available at: https://github.com/tyiannak/pyAudioAnalysis. + +184 +Emotion Valence Classification in the Wild +Conv3D 64@7x7x7 +BatchNorm + ReLU +Block A +Block B +Block C +Block D +AvgPool +Fully-Connected 3 +Softmax +Block A +Block A +Block B +Block B +Block B +Block C +Block C +Block C +Block C +Block C +Block D +Block D +Block D +Conv3D 64@1x1x1 +Block A +BatchNorm +Conv3D 64@3x3x3 +BatchNorm +Conv3D 256@1x1x1 +BatchNorm + ReLU +Conv3D 128@1x1x1 +Block B +BatchNorm +Conv3D 128@3x3x3 +BatchNorm +Conv3D 512@1x1x1 +BatchNorm + ReLU +Conv3D 256@1x1x1 +Block C +BatchNorm +Conv3D 256@3x3x3 +BatchNorm +Conv3D 1024@1x1x1 +BatchNorm + ReLU +Conv3D 512@1x1x1 +Block D +BatchNorm +Conv3D 512@3x3x3 +BatchNorm +Conv3D 2048@1x1x1 +BatchNorm + ReLU +Figure 12.2: Structure of the video-based group emotion recognition module, based on an inflated +ResNet-50. +12.2.4 +Score-level ensemble +The softmax scores obtained by both video and audio-based emotion recognition models are then +concatenated in a feature vector, composed of six class probability values. This vector is then +given to a multi-class SVM classifier, which combines the separate audio and video predictions +into a single decision. + +12.3 Experimental Setup +185 +FullyConnected 512 +Recurrent Layer +Sliding Window +Attention Model +Softmax +Feature Extraction +Weighted Pooling +FullyConnected 512 +Recurrent Layer +Feature Extraction +FullyConnected 512 +Recurrent Layer +Feature Extraction +... +... +... +ReLU +ReLU +ReLU +FullyConnected 512 +FullyConnected 512 +FullyConnected 512 +... +ReLU +ReLU +ReLU +... +... +Figure 12.3: Structure of the audio-based group emotion recognition module, based on a Bi-LSTM +network [296]. +12.3 +Experimental Setup +12.3.1 +Data +All the experiments were conducted on an adapted version of the “Video-level Group AFfect” +(VGAF) dataset [388] for the EmotiW 2020 AV Group-level sub-challenge [101]. The VGAF +is a video-based database that contains labels for emotion and cohesion. The data was collected +from the YouTube platform and consists of videos under the creative commons license (CC0) and +present keywords that correspond to the range of emotions and cohesion. +Since the number of individuals per video is variable, and the groups on each video can also +present a varying number of persons over time, the videos have been divided so that each video +clip has always the same number of persons per frame. Each VGAF clip was manually labelled +by different annotators for emotion and cohesion and every annotator was informed of the basic +concepts of emotion and cohesion. Only videos with mutual consensus were kept in the final +database. The labels for group emotion are related to emotion valence (i.e., positive, neutral, and +negative) whereas the group cohesion labels are in the range [0−3], being 0 the state of very low +cohesion (dominance over the group members) and 3 the state of very high cohesion. +For the EmotiW 2020 AV Group-Level Emotion sub-challenge, the task was the classification +of group emotion. The VGAF dataset videos were divided into five-second videos: 2661 for the +train, 766 for the validation, and 756 for the test. Each video (except those in the test set) is +accompanied by a discrete group emotion valence ground-truth label. In the training dataset, there + +186 +Emotion Valence Classification in the Wild +are 802 positive videos, 923 neutral videos, and 936 negative videos. In the validation set, there +are 302 positive videos, 280 neutral videos, and 184 negative videos. +12.3.2 +Baseline algorithm +The baseline algorithm is the audiovisual group-level emotion recognition sub-challenge base- +line [388] of the EmotiW 2020 Grand Challenge. This method is composed of two streams, for +audio and video data processing, fused at the feature level. The video stream is a pretrained Incep- +tion V3 network that separately processes frames extracted from a video. The extracted features +are combined using a long short-term memory (LSTM) network. The audio stream is composed of +a fully-connected network that receives OpenSMILE [117] features extracted from the audio. The +outputs from the video and audio streams are concatenated and used by a fully-connected layer to +offer two outputs: the probabilities for the three emotion valence classes and an emotion cohesion +value on the [0−3] range. +12.3.3 +Preprocessing +The videos on the EmotiW 2020 AV Group-level Emotion sub-challenge were subject to prepro- +cessing before being used by the proposed method. For each five-second video, ten frames were +extracted, thus resulting in 2 frames per second. This is an adaptation from the original model +pretrained on the Multi-Moments In Time database (MMIT), which worked at 5 frames per sec- +ond. We found that reducing the frame rate did not harm performance and sped up the recognition +process. Before being used on the audio-based recognition module, the audio was extracted from +each file by converting them (originally in the MP4 format) to audio files (in the WAV format). +Regarding audio, we noticed that after feature extraction some of the generated features could as- +sume non-number values. For training purposes, we removed these samples and trained only with +valid ones. For inference during validation/test, we replaced non-number values with zero. +12.3.4 +Training +The audio-based recognition module was trained from scratch3. The weights were randomly ini- +tialised, and the model was trained over a maximum of 200 epochs, with a batch size of 128, +categorical-cross-entropy as the loss function, and using the Adam optimiser with an initial learn- +ing rate of 10−2. To prevent overfitting, we used dropout and early-stopping with a patience value +of 15 epochs. +The video-based recognition module, pretrained on the MMIT database, was adapted to output +probabilities for each of the three valence classes in group emotion recognition. This was achieved +by replacing the last fully-connected layer with a new one, with three neurons. +Since this layer needs to be trained, all weights of the network have been frozen (except those +of this layer). The network was briefly fine-tuned until convergence over a maximum of 250 +epochs, with batch size 32, using the Adam optimiser with an initial learning rate of 10−5. +3Audio submodule code adapted from: https://github.com/RayanWang/Speech_emotion_recognition_BLSTM. + +12.4 Results and Discussion +187 +When training the audio-based module and the video-based module the hyperparameters have +been selected empirically, to maximise performance in the validation set. The hyperparameters +for the fusion module (e.g., the regularisation parameter “C”, the kernel, or the polynomial degree +of the kernel) were found through a grid search. Once the optimal hyperparameters were found, +the train and validation set were combined to make a “full train” set, and thus take full advantage +of all available labelled data for better performance in the test set. +12.3.5 +Experiments +In this work, the aim is not only to assess the proposed method’s performance for group-level +emotion recognition but also to examine its behaviour in several conditions. +We use the accuracy metric and confusion matrices to examine the overall performance of the +method and also analyse its class-wise accuracy. The performances of the multimodal method +and its audio and video-based modules, separately, are evaluated in both the validation set (with +available ground-truth labels) and the test set (accuracy values delivered by the EmotiW 2020 +sub-challenge organisation upon request). The performance is compared with the official sub- +challenge baseline, following the results reported in [388]. +The performance is also evaluated according to the number of people in the video. Since the +number of people in each video is not included, we use the MTCNN method [482] on each frame +of each video, and infer the group size based on the average number of detected faces: a group +with less than five detected faces are considered small (total of 502 videos on the validation set), +otherwise, it is considered a large group (264 videos on the validation set). With this, we aim to +evaluate the difficulties associated with recognising emotion in large groups, where cohesion is +likely to be generally lower. +12.4 +Results and Discussion +The performance results of the proposed method, and the comparison with the official sub- +challenge baseline, on the validation and test sets, is presented, respectively, in Table 12.1 and +Table 12.2. +On the validation set, the performance offered by the proposed multimodal method is superior +to the baseline. The accuracy attained by the video-only approach is close to that offered by the +multimodal method, over 62%. This is evidence of the advantages of using pretrained networks (in +this case, transferred from the task of human activity recognition). The audio-only approach offers +considerably lower performance (47%) than the audio-only baseline (50%), which indicates the +use of OpenSMILE features and fully-connected networks may be better fitted for group emotion +recognition based on audio. +From the validation to the test set (Table 12.2), the official video-only baseline suffers a sharp +performance decay (from 52% to 42% accuracy), which is also felt with the proposed video-only +approach (albeit not as dramatic, from 62% to 59% accuracy). Fusing with audio on a multimodal +approach reduced that decrease in the case of the official baseline (from 50% to 48% accuracy), + +188 +Emotion Valence Classification in the Wild +Table 12.1: Accuracy (%) of the proposed method on the validation set (A - only audio; V - only +video; A+V - multimodal). +Method +Accuracy +Overall +Positive +Neutral +Negative +Proposed (A) +47.19 +19.93 +69.64 +57.61 +Proposed (V) +62.40 +69.20 +52.50 +66.30 +Proposed (A+V) +61.83 +58.13 +61.78 +67.93 +Baseline (A) +50.23 +- +- +- +Baseline (V) +52.09 +- +- +- +Baseline (A+V) +50.23 +- +- +- +Table 12.2: Accuracy (%) of the proposed method on the test set (A - only audio; V - only video; +A+V - multimodal). +Method +Accuracy +Overall +Positive +Neutral +Negative +Proposed (V) +58.86 +55.76 +57.93 +63.04 +Proposed (A+V) +65.74 +54.38 +77.99 +60.00 +Baseline (V) +42.00 +- +- +- +Baseline (A+V) +47.88 +45.00 +10.00 +70.00 +Table 12.3: Confusion matrix of audio-based recognition on the validation set. +Predicted Class +True Class +Positive +Neutral +Negative +Positive +60 +139 +102 +Neutral +35 +195 +50 +Negative +24 +54 +106 +Table 12.4: Confusion matrix of video-based recognition on the validation set. +Predicted Class +True Class +Positive +Neutral +Negative +Positive +209 +59 +34 +Neutral +98 +147 +35 +Negative +44 +18 +122 +Table 12.5: Confusion matrix of multimodal recognition on the validation set. +Predicted Class +True Class +Positive +Neutral +Negative +Positive +175 +99 +27 +Neutral +78 +173 +29 +Negative +27 +32 +125 +and even reversed it in the case of the proposed method (from 62% to 66% accuracy). This +confirms the idea present in the literature that, while audio alone is not suitable for recognition, it +offers additional information that is essential for the robustness and accuracy of the method. + +12.4 Results and Discussion +189 +Table 12.6: Accuracy (%) on the validation set for videos of small groups vs. large groups (A - +only audio; V - only video; A+V - multimodal). +Method +Group Size +N < 5 +N ≥ 5 +Proposed (A) +48.61 +44.49 +Proposed (V) +66.33 +54.92 +Proposed (A+V) +65.74 +54.37 +Analysing the class-wise accuracies and the confusion matrices (Table 12.3, Table 12.4, and +Table 12.5), one can notice that video is, overall, the best modality to recognise emotions. The +advantages of using video rely mainly on the “extreme” classes, positive and negative, which +denote visual information is more advantageous to recognise strong group emotions. The proposed +audio-only approach attains very poor accuracy in the positive class. +Since the positive class is the minority class in the training dataset, the results of the audio-only +approach may partially be explained by this slight class imbalance. However, as mentioned before, +the video-only approach does not verify this, which is fortunate when combining both approaches +into the multimodal proposed method. Using both modalities slightly decreases the accuracy of +positive and negative videos, when compared with the video-only approach, but takes advantage +of the audio information to considerably improve accuracy on neutral videos and achieve overall +better performance. +At last, the results of the group size study are presented in Table 12.6. In both audio and +video-only approaches, as well as the multimodal method, the recognition performance is higher +in smaller groups. The performance values should serve as a rough reference, since the process of +face detection may present errors, and the number of faces may not accurately describe the number +of people in the video’s group (which may include occluded faces or people facing the opposite +direction of the camera). +Nevertheless, the performance differences are considerable and show expected behaviour: it +should be harder for larger groups to consistently show the same emotion than smaller groups. +Hence, emotion cohesion should be higher, on average, for smaller groups, and thus the certainty +of the algorithms when recognising the emotion valence. This could perhaps be addressed using +hierarchical methodologies (from individual-level to group-level) as used in current group activity +recognition approaches (discussed in the related work section). +Through an analysis of some videos where the proposed model failed, a pattern emerged. +While there are certain videos where the error was evident, there are several examples where it +is very difficult to notice that an apparently neutral scene displays, in fact, a positive or negative +group emotion. +Some examples are shown in Fig. 12.4. On the top left, is a short video of a calm conversation +on a TV show that is labelled as positive. On the top right, is a negative emotion video that the +proposed method classified as positive, since the video only covers the moment before the boy be- +ing bullied started crying. On the bottom left, a conference presentation that the method classified + +190 +Emotion Valence Classification in the Wild +Figure 12.4: Some examples of validation set videos where the model offered unsuccessful pre- +dictions (top left: label positive, predicted neutral; top right: label negative, predicted positive; +bottom left: label positive, predicted neutral; bottom right: label negative, predicted neutral). +as neutral since the positive ground-truth emotion could only be verified by facial expressions. On +the bottom right, a conversation is deemed neutral by the proposed method, where only a closer +inspection of the audio shows that it is, in fact, part of a protest or a similar confrontation. +Although the information about the underlying ground-truth labels is indeed present in these +example videos, it is hidden in contextual clues, expressions in small faces, or the content of con- +versations. Exploring ways to integrate these aspects into the recognition of group-level emotion +could be the way to avoid the most common mistakes of the proposed method, and ultimately +achieve better overall performance and robustness. +12.5 +Summary and Conclusions +In this work we proposed a novel method for the automatic recognition of group emotion that uses +a late-fusion multimodal approach, combining scores from both video and audio-based emotion +recognition models that are used to feed a multiclass SVM that returns a final class probability. +This method showed significant improvement against the baseline, confirming that the use of ac- +quired knowledge from activity recognition is useful for group-emotion recognition and that the +joint utilisation of audio and video benefits the learning of the model. +On the other hand, taking into account the maximum accuracy value, we believe that there is +still room for further improvements. Further efforts should be devoted to the study of the links +between the tasks of activity recognition and emotion recognition, especially at the group level. +Approaches for abnormal behaviour recognition through video anomaly detection, such as [23], +could offer meaningful improvements in the automatic distinction between positive and negative + +mornino +BLEND +lasvegas +NEWBOOK"ONCEYOUGOPERSIAN..."OUTNOW! +morning +BLENDIMTERMATIOMAL +AGILEANDLEANSOT +1111 +qile +IndiaGOVEG12.5 Summary and Conclusions +191 +emotions. Also, this method could benefit from OpenSMILE features in the audio-based emotion +recognition module, the development of multimodal approaches that are based on early-fusion +(e.g., input or intermediate-layer levels), and the design of a “fully” end-to-end network that re- +ceives both video and audio as input and learns the relevant features for the classification task (e.g., +through regularisation methods such as loss functions with different terms and weights). + + +Chapter 13 +Activity and Violence +Recognition in Shared Vehicles +Foreword on Author Contributions +The research work described in this chapter was conducted in collaboration with Carolina Pinto, Afonso Sousa, +and Leonardo Capozzi, under the supervision of Jaime S. Cardoso and Pedro Carvalho. The author of this thesis +contributed to this work on the reformulation and implementation of the audio module, the conceptualisation and +implementation of the cascade strategy, the preparation of data and the experimental setup, the evaluation of the +parallel and cascade algorithms, the discussion of the results, and the preparation of the scientific publication. +The results of this work were disseminated in the form of an article in international conference proceedings: +• J. R. Pinto, P. Carvalho, C. Pinto, A. Sousa, L. G. Capozzi, and J. S. Cardoso, “Streamlining Action Recognition +in Autonomous Shared Vehicles with an Audiovisual Cascade Strategy,” in 17th International Conference on +Computer Vision Theory and Applications (VISAPP), Feb. 2022. [348] +13.1 +Context and Motivation +Human action or activity recognition is a vibrant and challenging research topic. Being able to +recognise actions automatically is game-changing and often crucial for several industries, includ- +ing the scenario of shared autonomous vehicles. Without a driver responsible for the vehicle’s and +occupant’s security and integrity, it falls upon automatic recognition systems to monitor passenger +well-being and actions, and eventually recognise harmful behaviours or even violence [23]. How- +ever, the wide range of possible actions that can be portrayed, the variability in the way different +individuals portray the same actions, the heterogeneity of sensors and the type of information +captured and the influence of external factors still pose significant hurdles to this task. +Despite all the above-mentioned challenges, the topic of action recognition has thrived by fol- +lowing a very recognisable recipe for success. As in plenty of other pattern recognition tasks, +the state-of-the-art gradually evolved towards larger and more sophisticated models based on deep +learning methodologies [65; 127; 354]. These have achieved increasingly higher accuracy thanks +193 + +194 +Activity and Violence Recognition in Shared Vehicles +to a growing number of massive databases typically using public video data gathered through on- +line sourcing, such as Kinetics [65], Multi-Moments in Time (MMIT) [300], or ActivityNet [169]. +This also means most research in action recognition is based on visual information (images or +video). This is the case of the I3D [65], the methodology currently deemed the state-of-the-art +in this topic. In fact, I3D goes further beyond simple visual spatial information by adopting a +two-stream approach, including optical flow for temporal action encoding. Other approaches have +explored recurrent networks for the same purpose [175; 226; 330], but have seldom managed to +reach the accuracy level offered by the I3D method. +Despite the meaningful strides brought by such sophisticated methods and large databases, +some limitations can be observed. On the one hand, the general nature of the data sourced to train +and evaluate the state-of-the-art models lead to overly general results that may not be verified in +more specific scenarios, such as in-vehicle passenger monitoring. On the other hand, hefty models +based on visual information and optical flow (such as I3D) may offer very high accuracy, but +their complexity does not allow for real-time applications in inexpensive limited hardware, such +as embedded devices. +This work proposes a set of changes to the state-of-the-art I3D method to bring it closer to real +applicability in edge computing scenarios: in this case, we focus on action recognition and vio- +lence detection in shared autonomous vehicles. First, inspired by [346], the current work discards +the time-consuming optical flow component of I3D and introduces a lightweight model for action +recognition with audio. Despite being less frequently used than video, audio is considered one of +the most promising options for a multimodal system for action recognition [85; 214; 258]. This +way, we obtain a simpler methodology that can use both video and audio modalities for a greater +variety of information. Then, as each modality is likely to contribute differently to the recognition +of each action, we propose a cascade strategy based on confidence score thresholding. This strat- +egy allows a simplification of the multimodal pipeline by using only one (primary) modality as +often as possible; the two modalities are used together only when the primary one is not enough +for sufficiently confident predictions. Hence, it is possible to attain significant time and computing +energy savings without overlooking classification accuracy. +This chapter is organised as follows: beyond this introduction, a description of the proposed +multimodal methodology and cascade strategy is presented in section 13.2; the experimental setup +is detailed in section 13.3; section 13.4 presents and discusses the obtained results; and the con- +clusions drawn from this work are presented in section 13.5. +13.2 +Methodology +13.2.1 +Multimodal pipeline +The baseline consists of a multimodal pipeline for activity recognition based on an audio-visual +module previously proposed for group emotion recognition [346]. The pipeline is composed of +three submodules (as illustrated in Fig. 13.1): the visual submodule, which processes visual data; + +13.2 Methodology +195 +VISUAL +SUBMODULE +AUDIO +SUBMODULE +Frames +Sound +Video +yvisual +yaudio +^ +^ +FUSION +SUBMODULE +yjoint +^ +Figure 13.1: Diagram of the full multimodal pipeline for activity recognition. +the audio submodule, which processes sound data; and the fusion submodule, which combines +individual decisions from the previous two submodules into joint multimodal classifications. The +specific structures of each of these submodules are described below. +13.2.1.1 +Visual submodule +As in [346], the visual submodule is based on an inflated ResNet-50 [167] using pretrained weights +for the Multi-Moments in Time (MMIT) activity recognition database [300]. Using an inflated +ResNet-50 ensures optimal performance by following the successful example of the state-of-the- +art I3D method [65]. Using model weights pretrained on the large MMIT database allows us to +transfer deeper and more general knowledge to our narrower task of activity recognition inside +vehicles. +The inflated ResNet-50 model (see Fig. 13.2) is composed of seventeen residual blocks, each +including three 3D convolutional layers with 64 to 2048 filters, batch normalisation and ReLU +activation. Downsampling at each block allows the model to capture important information at dif- +ferent levels of resolution. After an average pooling layer, the last fully-connected layer, followed +by a softmax activation function, offers probability scores for each of the N considered activity +labels. +13.2.1.2 +Audio submodule +The audio submodule consists of a simple network based on a bi-directional long short-term mem- +ory (LSTM) model. These are known for their ability to encode temporal information, important +for audio-related topics, and have been previously successful for tasks such as group emotion +valence recognition [346] or speech-based sentiment analysis [296]. + +196 +Activity and Violence Recognition in Shared Vehicles +Frames +Residual Block A +yvisual +^ +Batch Norm + ReLU +3x +Residual Block B +4x +Residual Block C +6x +Residual Block D +4x +Average Pooling +Softmax +Fully-Connected (N) +Conv3D (64@7x7x7) +Figure 13.2: Diagram of the visual submodule (more details on the ResNet-50 and the residual +blocks in [167]). +Unlike the visual submodule, which largely follows the method proposed in [346] to approach +the state-of-the-art performance of I3D, the audio submodule was reformulated. In the aforemen- +tioned work, the audio Bi-LSTM model received a set of cepstral, frequency, and energy hand- +crafted features extracted from each signal window. Moreover, it included multiple convolutional +layers with 512 filters each and an attention mechanism after the LSTM layer. In this work, we +design a streamlined and faster audio submodule. +The simplified and lighter Bi-LSTM model (see Fig. 13.3), with less trainable parameters, re- +ceives a raw audio signal divided into 100 ms windows with 50 ms overlap, without any preceding +process of feature extraction. Each window is processed by three convolutional layers (with 16, +32, and 64 1 × 5 filters, respectively, stride 1, and padding 2), each followed by ReLU activation +and max-pooling (with pooling size 5). A Bi-LSTM layer receives features from the convolu- +tional part for each window, and its output for the last window is sent to a fully-connected layer +for classification (with N neurons, one for each activity class, followed by softmax activation). +In section 13.4, we analyse the advantages of using the proposed audio submodule vs. the one +in [346]. +13.2.1.3 +Fusion submodule +The aforedescribed visual and audio submodules output their respective sets of class probability +predictions for a given video. To combine the two separate sets of predictions for each task into a +single audio-visual multimodal classification, the fusion submodule is used. + +13.2 Methodology +197 +Audio +Sliding window +Conv1D (16@1x5) +Conv1D (32@1x5) +Conv1D (64@1x5) +Bi-LSTM Layer +Conv1D (16@1x5) +Conv1D (32@1x5) +Conv1D (64@1x5) +Bi-LSTM Layer +Fully-Conn. (N) +yaudio +^ +... +... +... +... +ReLU, MaxPool (1x5) +ReLU, MaxPool (1x5) +ReLU, MaxPool (1x5) +ReLU, MaxPool (1x5) +ReLU, MaxPool (1x5) +ReLU, MaxPool (1x5) +Softmax +Figure 13.3: Diagram of the audio submodule. +The fusion submodule is composed of a simple support vector machine (SVM) classifier. This +classifier receives the probability score sets from the two previous submodules concatenated as a +single unidimensional feature vector. The SVM model is trained to use these probability sets to +output a joint class prediction for the respective video. +13.2.2 +Cascade strategy +In the multimodal pipeline described above, all submodules are used for each instance (video) that +needs to be classified. This means that regardless of the difficulty of a given video or the activity +portrayed, both visual and sound data are always processed, resulting in two sets of unimodal class +predictions which are then combined into a set of multimodal class probabilities. +Given the considerable complexity of both the residual-network-based visual submodule and +the Bi-LSTM-based audio submodule, this multimodal pipeline is arguably too heavy for the target +application. This is especially true considering, as observed in [346], that different classes may +benefit much more from one of the modalities and thus not need the other one. Hence, we design +a cascade strategy to explore the possibility of using just one of the modalities and “turning off” +the remaining two submodules as often as possible. This aimed to achieve improved processing +times and energy usage, offering an alternative or complement to model compression strategies. +In the proposed cascade strategy, one of the data modalities (visual or audio) is selected as +the “primary” modality and, as such, the corresponding submodule is always used to offer a start- +ing prediction. The probability score offered for the predicted class is considered a “confidence +score”: a measure of how confident the primary submodule is in the prediction it provided. If the +confidence score is above a specific confidence threshold T ∈ [0,1], the remaining modules remain + +198 +Activity and Violence Recognition in Shared Vehicles +unused, and the primary submodule predictions are considered final. However, if the aforemen- +tioned condition is not verified, the secondary submodule is called to offer additional information +for more confident predictions, which are then combined into a single multimodal prediction (just +like the original multimodal pipeline). +The performance benefits of such a strategy are intimately related to the defined confidence +threshold. If T is too high, most of the instances will use both data sources, thus retaining the +accuracy offered by the parallel pipeline but reaping very few benefits related to complexity or +processing time. Conversely, if T is too low, most instances will be classified using only the +primary submodule, which may result in heavily impacted accuracy, despite the complexity bene- +fits of the simplified pipeline. Section 13.4 includes thorough experimental results on the impact +of the confidence score on the accuracy and processing requirements of the pipeline for activity +classification. +13.3 +Experimental Setup +13.3.1 +Databases +For generic scenarios, this work used the Multi-Moments in Time (MMIT) database [300], made +available by the creators upon request. The MMIT database includes a total of 1 035 862 videos, +split between a training set (1 025 862 videos) and a validation set (10 000 videos). These cor- +respond to a total of 339 classes, describing the main activity verified in each video. From those +classes, only those related to the target scenario of in-vehicle passenger monitoring were included. +This resulted in a subset of twenty-one classes: fighting/attacking, punching, pushing, sitting, +sleeping, coughing, singing, speaking, discussing, pulling, slapping, hugging, kissing, reading, +telephoning, studying, socialising, resting, celebrating, laughing, and eating. Train and test divi- +sions use the official predefined MMIT dataset splits. +For the in-vehicle scenario, a private dataset was used. The dataset includes a total of 490 +videos of the back seat of a car occupied by one or two passengers (see example frames in +Fig. 13.4). Videos are acquired using a fish-eye camera to capture most of the interior of the +car and microphones to acquire sound data. Each video includes annotations for forty-two action +classes: “entering”, “leaving”, “buckle on/off”, “turning head”, “lay down”, “sleeping”, “stretch- +ing”, “changing seats”, “changing clothes”, “reading”, “use mobile phone”, “making a call”, “pos- +ing”, “waving hand”, “drinking”, “eating”, “singing”, “pick up item”, “come closer”, “handshak- +ing”, “talking”, “dancing”, “finger-pointing”, “leaning forward”, “tickling”, “hugging”, “kissing”, +“elbowing”, “provocate”, “pushing”, “protecting oneself”, “stealing”, “screaming”, “pulling”, “ar- +guing”, “grabbing”, “touching (sexual harassment)”, “slapping”, “punching”, “strangling”, “fight- +ing”, and “threatening with weapon”. Videos are randomly drawn into the train dataset (70%) or +the test dataset (30%). + +13.3 Experimental Setup +199 +Figure 13.4: Example frames from the in-vehicle dataset, depicting normal activities (top row) +and violence between passengers (bottom row) (grey areas were used to protect the subjects’ +identities). +13.3.2 +Data preprocessing +A total of 10 frames, evenly spaced, was extracted from each video in the MMIT selected data +subset (each with about 5 sec). These frames were concatenated over a third dimension follow- +ing their temporal order to serve as input to the visual submodule. Two seconds of audio were +extracted from each video and normalised to 16 kHz sampling frequency to serve as input to the +audio submodule. +For the in-vehicle dataset, each video can have multiple labels (the passengers portray different +actions over the course of each acquisition). As such, each video period labelled as one of the 42 +classes, is divided into two-second long individual samples. From each of these, 8 frames are + +200 +Activity and Violence Recognition in Shared Vehicles +Table 13.1: Summary of the accuracy (%) results obtained in the various experimental scenarios. +Scenario +Unimodal +Multimodal +Cascade +Audio +Visual +Parallel +Audio-First +Video-First +Generic (21 classes) +41.65 +52.96 +55.12 +55.30 +55.12 +In-Vehicle (42 classes) +44.42 +35.96 +43.26 +46.05 +43.47 +In-Vehicle (3 classes) +64.10 +61.87 +66.61 +68.88 +66.77 +extracted, resized and cropped into 224×224 squares, and concatenated over a third dimension to +be used by the visual submodule, and the corresponding audio is resampled to 16 kHz to be used +by the audio submodule. +In the specific scenario of in-vehicle violence recognition, the aforementioned forty-two +classes of the in-vehicle dataset have been clustered into three classes: normal car usage (from +‘entering’ to ‘pick up item’, in order of appearance), normal interactions (from ‘come closer’ to +‘kissing’), and violence (from ‘elbowing’ to ‘threatening with weapon’). +13.3.3 +Model training +The inflated ResNet-50 model used on the visual submodule uses the official pretrained weights +from the MMIT database. Given that it was pretrained on the same database used in the labora- +tory experiments, this work took full advantage of this by setting most of the parameters of the +network as non-trainable. The only parameters that were trained are those of the fully-connected +layers which correspond to the classification of the selected twenty-one categories. This layer was +optimised for a maximum of 250 epochs according to categorical cross-entropy loss, with batch +size 32, using the Adam optimiser with an initial learning rate of 10−4. For regularisation, dropout +with a probability of 0.5 is used before the fully-connected layer. For the in-vehicle scenario, +the training process is identical to the one described above. However, since the nature of the in- +vehicle video data is substantially different from MMIT, the pretrained weights are also trained +(not frozen) alongside the fully-connected layer for classification. +For both the generic and in-vehicle scenarios, the audio submodule was trained for a maximum +of 200 epochs with batch size 64 and early-stopping patience of 25 epochs. The optimisation was +performed using Adam with an initial learning rate of 10−4 and cross-entropy loss. +13.4 +Results +After training the methodologies previously described, including the proposed cascade strategy, +an overview of the obtained accuracy results is presented in Table 13.1. It is clear the overall +better performance of the proposed cascade pipeline, particularly with the audio submodule as +the first block. Subsections 13.4.1 and 13.4.2 offer a deeper discussion of the results for different +configurations of the cascade pipeline for the more generic scenarios and the specific case of in- +vehicle monitoring respectively. + +13.4 Results +201 +1 +2 +3 +5 +7 +10 +Rank +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Accuracy +Rank Accuracies (MMIT) +Video only +Audio only +Parallel Fusion +Cascade Video-First Fusion +Cascade Audio-First Fusion +Figure 13.5: Rank accuracy results for the 21 selected classes from the MMIT database. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Confidence Threshold +0.42 +0.44 +0.46 +0.48 +0.50 +0.52 +0.54 +Accuracy +Fusion Accuracy After Confidence Threshold (MMIT) +Video-first +Audio-first +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Confidence Threshold +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ratio that required second trait +Fusion Frequency After Confidence Threshold (MMIT) +Video-first +Audio-first +Figure 13.6: Cascade results for the 21 selected classes from the MMIT database (overall classi- +fication accuracy, on the left, and fraction of instances that need the secondary modality, on the +right, for different confidence thresholds). +13.4.1 +Generic scenarios +For the laboratory experiments using the selected data from the MMIT database, the full parallel +multimodal pipeline explored in this work offered 55.12% accuracy. However, when considering +the proposed cascade strategy based on confidence score thresholds, it was possible to achieve +an improved accuracy score of 55.30% (see Fig. 13.5). Beyond this relatively small accuracy +improvement, the largest benefit of the proposed cascade algorithm is related to processing time. +As presented in Fig. 13.6, the best accuracy of 55.30% is achieved with an audio-first cascade +with a confidence score threshold T = 0.5. This means it is possible to avoid the visual and fusion +submodules for approximately 51% of all instances without performance losses. +An analysis of size, number of parameters, and average run time per instance for each sub- +module (see Table 13.2) shows that the visual model is the heftiest among the three submodules. +Hence, being able to bypass it on more than half of the instances translates into significant time +savings: while the full multimodal pipeline takes, on average, 85.9 ms to predict an instance’s + +202 +Activity and Violence Recognition in Shared Vehicles +Table 13.2: Summary of the size, total number of parameters, and average run times per instance +of the three pipeline submodules for the in-lab scenario (run times were computed using a NVidia +GeForce GTX 1080 GPU, with the exception of the fusion submodule, computed on an Intel +i7-8565U CPU). +Submodule +Size (MB) +Params. +Run Time (ms) +Visual +176 +46.2 M +77.18 +Audio +1.70 +220 K +8.30 +Fusion +1.94 +- +0.38 +activity label, the proposed cascade can do it in only 46.3 ms, on average, without accuracy losses. +This brings us closer to real applications using inexpensive hardware in the target in-vehicle sce- +nario. +As visible in Table 13.2, the proposed audio submodule has a total size of 1.70 MB, approx- +imately 230 thousand parameters, and an average GPU run time of 8.30 ms per instance. Con- +versely, the audio submodule used in [346], on this task of activity recognition, has a total size of +3.94 MB, approximately 1.026 million parameters, and an average GPU run time of 1666 ms per +instance (due to the CPU-based handcrafted feature extraction process). Despite the significant +reduction in run time, size, and complexity, the proposed audio submodule performed similarly +vs. the alternative (41.65% and 42.01% accuracy, respectively). +13.4.2 +In-vehicle scenario +The results on the data from the target in-vehicle scenario largely follow those discussed above +for the laboratory experiments. On the 42-class activity recognition task, an audio-first cascade +strategy achieved the best performance (46.05% accuracy) versus the full multimodal pipeline +(43.26%) and the best unimodal submodule (44.42%). Similar accuracy improvements are verified +up to rank 10 (see Fig. 13.7). With a confidence score threshold of T = 0.3, this cascade strategy +is able to avoid the visual submodule for approximately 74.1% of the instances (see Fig. 13.8). +Considering the average run times presented in the previous subsection, this means the cascade +is able to offer activity predictions in 28.4 ms, on average, while offering considerably higher +accuracy than the full multimodal pipeline (which would take 85.9 ms). +For the three-class violence recognition task, the results follow the same trend, albeit with +higher accuracy scores for all submodules and fusion strategies. The proposed cascade strategy +with audio as the primary modality was able to attain 68.88% accuracy, considerably better than +the 66.61% offered by the full multimodal pipeline. This accuracy corresponds to T = 0.8, which +enabled avoiding the visual submodule for 35% of all instances (see Fig. 13.9). While this value is +lower than those reported for the previous experiments, it still translates into average time savings +of 27.2 ms per instance (58.7 ms for the cascade vs. 85.9 ms for the full pipeline), accompanied +by a considerable improvement in accuracy. + +13.5 Summary and Conclusions +203 +1 +2 +3 +5 +7 +10 +Rank +0.4 +0.5 +0.6 +0.7 +0.8 +Accuracy +Rank Accuracies (In-Vehicle) +Video only +Audio only +Parallel Fusion +Cascade Video-First Fusion +Cascade Audio-First Fusion +Figure 13.7: Rank accuracy results for the in-vehicle scenario with 42 classes. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Confidence Threshold +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +Accuracy +Fusion Accuracy After Confidence Threshold (In-Vehicle) +Video-first +Audio-first +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Confidence Threshold +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ratio that required second trait +Fusion Frequency After Confidence Threshold (In-Vehicle) +Video-first +Audio-first +Figure 13.8: Cascade results in the in-vehicle scenario with 42 classes (overall classification ac- +curacy, on the left, and fraction of instances that need the secondary modality, on the right, for +different confidence thresholds). +13.5 +Summary and Conclusions +This work explored a different strategy for the recognition of human activities, focusing on the +scenario of autonomous shared vehicles. In addition to the inherent difficulties of automatically +recognising human actions using audio-visual data, this specific scenario poses specific constraints +regarding available hardware and energy consumption. +Inspired by state-of-the-art multimodal approaches, the main contributions are two-fold: +a lighter-weight deep-learning based audio processing submodule; and a cascade processing +pipeline. The proposed audio processing module demonstrated state-of-the-art performance while +presenting lesser memory requirements and computational demands. With the submodules imple- +mented, different configurations were tested for the cascade strategy to assess which one provides +the best performance, taking into account two critical axes: accuracy and computational perfor- +mance. Results show that by using audio as the first processing block, it was possible to obtain + +204 +Activity and Violence Recognition in Shared Vehicles +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Confidence Threshold +0.62 +0.63 +0.64 +0.65 +0.66 +0.67 +0.68 +0.69 +Accuracy +Fusion Accuracy After Confidence Threshold (In-Vehicle - 3 classes) +Video-first +Audio-first +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Confidence Threshold +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ratio that required second trait +Fusion Frequency After Confidence Threshold (In-Vehicle - 3 classes) +Video-first +Audio-first +Figure 13.9: Cascade results in the in-vehicle scenario with 3 classes (overall classification ac- +curacy, on the left, and fraction of instances that need the secondary modality, on the right, for +different confidence thresholds). +an accuracy score higher than the state-of-the-art, along with a significant reduction in process- +ing/inference time. +The obtained results are interesting and reveal a high potential for further improvement. Modi- +fications to the individual processing submodules could contribute to even higher accuracies while +further reducing computational weight. The latter may benefit from a combination with model +compression and acceleration techniques, such as quantisation, avoiding likely losses in accuracy +due to compression. +The proposed strategy demonstrated benefits from cascading the processing modules. Other +early modules could bring other benefits by filtering out incoming audio-visual data, without rele- +vant content (e. g., without people present or without movement/sound). + +Part V +Broader Topics on Biometrics and +Pattern Recognition +205 + + +Chapter 14 +Learning Template Security +on End-to-End Biometric Models +Foreword on Author Contributions +The research work described in this chapter was conducted entirely by the author of this thesis, under the su- +pervision of Jaime S. Cardoso and Miguel V. Correia. The results of this work have been disseminated in the +form of a journal article, an article in international conference proceedings, and an abstract in national conference +proceedings: +• J. R. Pinto, M. V. Correia, and J. S. Cardoso, “Secure Triplet Loss: Achieving Cancelability and Non- +Linkability in End-to-End Deep Biometrics,” IEEE Transactions on Biometrics, Behavior, and Identity Science, +3 (2): 180–189, 2021. [347] +• J. R. Pinto, J. S. Cardoso, and M. V. Correia, “Secure Triplet Loss for End-to-End Deep Biometrics,” in 8th +International Workshop on Biometrics and Forensics (IWBF 2020), Apr. 2020. [345] +• J. R. Pinto, M. V. Correia, and J. S. Cardoso, “Achieving Cancellability in End-to-End Deep Biometrics with +the Secure Triplet Loss,” in 26th Portuguese Conference on Pattern Recognition (RECPAD 2020), Oct. 2020. +This work was awarded the Computers Journal Best Paper Award at the 2020 International Workshop on Biomet- +rics and Forensics and the Best PhD Student Live Presentation, awarded by the jury, at the 2021 NIS Workshop. +14.1 +Context and Motivation +It is easy to change our keys or passwords when a traditional authentication system is compro- +mised, but it is very hard to change our compromised biometric characteristics. This is the reason +why it is paramount that biometric templates are kept secure [186; 304]. This is not easily achiev- +able since, unlike password-based systems, biometric comparison is not binary and must also +account for the natural intrasubject biometric variability [200; 304]. +While several methods have been proposed to protect biometric templates, most require spe- +cific feature extraction or additional processes based on salting, biohashing, or cryptographic pro- +tection [200; 304]. Even those proposed for deep learning biometric methods [329; 424] are inte- +grated into end-to-end models, thus creating hurdles that often limit the achievable performance. +207 + +208 +Learning Template Security on End-to-End Biometric Models +Hence, this chapter presents the Secure Triplet Loss, a reformulation of the well-known triplet +loss that enables training end-to-end deep learning models to obtain secure biometric templates. +Binding keys with the input within the model and considering key divergences in the objective +function enables learning template cancelability. A component based on Kullback-Leibler di- +vergence or distance statistics measures and actively promotes unlinkability. Thus, the proposed +method aims to allow taking full advantage of the capabilities of end-to-end deep networks while +still ensuring the security of the stored biometric data. +A detailed presentation of the Secure Triplet Loss can be found throughout the next sections. +This training methodology was thoroughly evaluated for the task of identity verification, consid- +ering the template security properties of cancelability, unlinkability, non-invertibility, and secrecy +leakage. It was thoroughly evaluated for both ECG and face, to confirm its solidity and flexibility, +using the off-the-person University of Toronto ECG database (UofTDB) [445] and the uncon- +strained YouTube Faces [460] database. +Moreover, it was studied in two scenarios: (a) training a model “from scratch” (initialised +with random parameters), or (b) adapting an existing end-to-end biometric model to make it se- +cure (taking advantage of pretrained weights and fine-tuning with the proposed method). The +Secure Triplet Loss was also compared with the original triplet loss and competitive state-of-the- +art approaches based on Bloom Filters [149] and Homomorphic Encryption [109]. The code used +in this work is available online1. +14.2 +Related Work +Beyond accounting for natural biometric characteristic variability, biometric data protection meth- +ods need to verify template cancelability, non-invertibility, and unlinkability. Cancelability (or +revokability) means the templates can be easily and effectively rendered useless if they be- +come compromised, generally through the change of a personal key that is bound with the tem- +plate [353; 427]. +Non-invertibility requires the transformation from biometric samples to templates to be as +close to irreversible as possible. Thus, if the template is compromised, the original biometric +sample cannot be reliably recovered or approximated [200; 304]. Finally, template unlinkability +means it is difficult to assess if compromised templates from different biometric systems belong +to the same identity [149]. +One of the first template protection methods was the fuzzy commitment scheme proposed +by Juels and Wattenberg [208], using cryptography and error-correcting codes for template can- +celability. Later, Teoh et al. [430] proposed BioHashing, an adaptation of the hashing process +commonly applied to passwords to deal with fingerprint variability. A similar approach has been +proposed by Sutcu et al. [418]. +More recently, Rathgeb et al. [362, 363] proposed the Bloom filter approach for alignment- +free template cancelability and irreversibility. This approach was later adapted by Gomez-Barrero +1Secure Triplet Loss Github repository. Available on: https://github.com/jtrpinto/SecureTL. + +14.3 The Secure Triplet Loss +209 +et al. [149, 151] to ensure template unlinkability, and by Drozdowski et al. [108] for higher com- +putational efficiency. Raja et al. [357] proposed a highly efficient method using neighbourhood- +preserving manifolds and hashing for biometric template protection in smartphones. +Among cryptography-based methods, homomorphic encryption (HE) approaches are particu- +larly promising as HE allows arithmetic operations on the encrypted domain [45]. This allows the +biometric comparison to be fully conducted on the encrypted domain, ensuring data security [109]. +Fully HE approaches, that allow for unlimited operations in the encrypted domain, most notably +include Gentry’s [142], Brakerski’s [53], and Fan-Vercauteren’s [118] schemes. +HE has been successfully applied for biometric template protection in face [45; 109; 223], +signature [148], and even multibiometric recognition [150]. However, with HE the protection of +templates remains the responsibility of a separate process that should, ideally, be harmoniously +integrated within the feature extraction algorithm. +Using deep learning, Pandey et al. [329] proposed a template protection scheme that receives +features from a convolutional neural network (CNN), quantises them, and applies hashing to obtain +exact comparison despite the variability. Later, Talreja et al. [424] used forward error control +(FEC) decoding and hashing to protect biometric features extracted by deep neural networks. +While these are applied to deep learning, they still require separate protection and comparison +schemes. Hence, they are inadequate for recent state-of-the-art biometric recognition methods, +which largely rely on end-to-end deep learning models for significantly improved performance. +Considering this, this work proposes the Secure Triplet Loss, a reformulation of the triplet +loss [72] that promotes cancelability and unlinkability in end-to-end biometric models. More +importantly, it aims to achieve this while avoiding additional protection processes and decreases +in performance relative to the original triplet loss. +14.3 +The Secure Triplet Loss +14.3.1 +Original triplet loss +The triplet loss [72] has been widely used in deep learning to train networks to accurately deter- +mine whether or not two samples belong to the same class [73; 74; 338]. During training, such +networks receive three inputs (a triplet), in parallel: one is the anchor (xA, the reference with iden- +tity iA), the second is the positive sample (xP, with identity iP = iA), and the third is the negative +sample (xN, with identity iN ̸= iA). In biometrics, triplets are groups of three biometric samples +(images or signals): the anchor and positive inputs correspond to the same individual, unlike the +negative input. +For each input, the network will output a representation: e. g., for the anchor, yA = f(xA). The +three representations are then compared using a measure of distance or dissimilarity d(y1,y2), and +the network is optimised through the minimisation of the triplet loss function: +lTL = max[0,α +d(yA,yP)−d(yA,yN)], +(14.1) + +210 +Learning Template Security on End-to-End Biometric Models +Figure 14.1: Comparison between the model training schemes of the original triplet loss and the +proposed Secure Triplet Loss method [345]. +which leads representations of the same class to be more similar than those of different classes, +minimising d(yA,yP) and maximising d(yA,yN). The loss also aims to enforce a minimum margin +α > 0 between the two distances. +This is a generally successful strategy when training neural networks for biometric verification +(assessing if the identities of a biometric template and a biometric query match). However, it does +not address the important issue of security in biometrics, especially the topics of cancelability and +unlinkability. +14.3.2 +Learning cancelability +In its initial formulation, proposed in [345], the Secure Triplet Loss modifies the original triplet +loss to make the final sample representations cancelable (as illustrated in Fig. 14.1). Besides the +triplet inputs (xA, xP, and xN), the network also receives two different keys (k1, k2) that are bound +with the inputs by the network itself. +Unlike the original triplet loss, xP and xN are processed by the network twice. First, they are +combined with k1 and then with k2. The anchor xA is only bound with k1. Thus, five representations +are obtained: yA = f(xA,k1), yP1 = f(xP,k1), yP2 = f(xP,k2), yN1 = f(xN,k1), yN2 = f(xN,k2). +From these, four distances are computed: dSP = d(yA,yP1) (with matching identities and keys), +dDP = d(yA,yP2) (with matching identities but different keys), dSN = d(yA,yN1) (with different +identities but matching keys), and dDN = d(yA,yN2) (with non-matching identities and keys). +The objective is to minimise dSP, when both the identities and the keys match, and maximise +the remaining three distances (see Fig. 14.2). Hence, the loss is computed through: +lSTL = max(0,α +dSP −dn), +(14.2) +where dn results from the combination of all three distances to be maximised. Here, we consider +dn = min({dSN,dDP,dDN}), with the three distances to be maximised being considered equally + +14.3 The Secure Triplet Loss +211 +Figure 14.2: Illustration of the expected results when training with the proposed Secure Triplet +Loss (during training, the Secure Triplet Loss promotes the proximity between yA and yP1, which +match in identity and key, and larger distance to the three negative samples with a margin α). +relevant. This results in: +lSTL = max[0,α +dSP −min({dSN,dDP,dDN})]. +(14.3) +As with triplet loss, α enforces a margin between positive and negative distances. In this case, +the loss involves four distances, since it also takes into account whether or not the keys match. By +minimising the loss in Eq. (14.3), the network learns to deal with the intrasubject and intersubject +variability of the biometric characteristic. More importantly, it learns to recognise when the keys +do not match, even if the identity is the same. Hence, if the stored templates become compromised, +they can easily be invalidated through a key change. However, as reported in [345], lSTL fails to +promote unlinkability. +14.3.3 +Promoting unlinkability +Unlinkability can be achieved by combining the original formulation of the Secure Triplet Loss, +lSTL, with a component that quantifies linkability in the representations output by the network +during training, lL. Thus, the proposed reformulation of the Secure Triplet Loss, as presented +in [347], follows the equation: +lSTL2 = γlSTL +(1−γ)lL. +(14.4) +The lSTL component is the original Secure Triplet Loss in Eq. (14.3), focused on biometric +performance and template cancelability, following the formulation in Eq. (14.3). The parameter +γ ∈ [0,1] balances the lSTL and lL loss components. The lL component is focused on measur- +ing template linkability. To achieve unlinkability, one has to ensure similar distance values are +obtained when keys don’t match (regardless of whether or not the templates are from the same +identity). Hence, dDP and dDN should assume similar values. This can be achieved using the +Kullback-Leibler divergence (KLD), computed over each batch. This agrees with the reference + +212 +Learning Template Security on End-to-End Biometric Models +linkability metric, which is also inspired by the KLD. In this case, this part of the loss becomes: +lL = DKL(PdDP||PdDN) = ∑PdDP log +�PdDP +PdDN +� +, +(14.5) +where PdDP and PdDN are the probability density functions for the distributions of dDP and dDN, +respectively. To obtain these distributions and their respective probability density functions, this +part of the loss cannot be computed over each triplet, instead being computed over each batch of +triplets. For brevity, this formulation of the Secure Triplet Loss with Kullback-Leibler divergence- +based linkability is from now on designated as SecureTL w/KLD. +Alternatively, one can avoid estimating these distributions and the computation of the +Kullback-Leibler divergence using simple statistics to promote linkability. If we consider µ(dDP) +and σ(dDP) as the mean and standard deviation, respectively, of the distances dDP on a given batch, +and likewise µ(dDN) and σ(dDN) for the distances dDN on the same batch, then we can reformulate +lL = |µ(dDP)− µ(dDN)|+|σ(dDP)−σ(dDN)|. +(14.6) +This should lead the model to offer embeddings that result in similar distance scores when the keys +do not match, regardless of whether or not the identities match, thus avoiding template linkability. +Throughout the remainder of this chapter, for brevity, the formulation of the Secure Triplet Loss +with this statistics-based linkability module is designated as SecureTL w/SL. +14.4 +Experimental Setup +The proposed methodology for learning secure biometric models was explored for the two bio- +metric characteristics addressed within this doctoral work: the ECG and the face. This section +presents the details of the models, the data, and the conducted experiments. +For either characteristic, keys have been randomly generated for each triplet, consisting of +unidimensional arrays with 100 binary values. Each key is processed after generation to verify +unit l2 norm. For SecureTL w/KLD and SecureTL w/SL, the parameter γ that controls the balance +between the Secure Triplet Loss and the linkability component was set to 0.9: this value has +overall been able to offer good template unlinkability without considerably harming the validation +performance and cancelability. +14.4.1 +ECG identity verification +14.4.1.1 +Data +The ECG data used comes from the University of Toronto ECG Database (UofTDB) [445]. This +database includes recordings from 1019 subjects over up to six sessions and five different posi- +tions. The signals are off-the-person (less obtrusive and more comfortable for realistic biometric +applications) and have been acquired at 200 Hz using dry electrodes on the pointer fingers. Each +recording is generally 2 to 5 seconds long. + +14.4 Experimental Setup +213 +Figure 14.3: Architecture of the models used for ECG and face identity verification (x denotes +an input biometric sample, k a key, and y a biometric template; the structure of the face model +before concatenation with k follows precisely the structure of the Inception-ResNet-v1, which is +presented in higher detail in [420]). +Data from the last 100 identities were used for training, while the data from the remaining +919 subjects have been reserved for testing. From these 919, one has been discarded for only +having a total of 30 seconds of data. Triplets have been generated by selecting an anchor from the +first 30 s of data from a subject and positive and negative samples from the remaining data of the +same or another identity, respectively. From the 100 training identities, 100 000 triplets have been +generated, with 20% being used for validation. A total of 10 000 triplets have been generated for +testing. Each of the three samples in a triplet is a blindly-segmented five-second raw ECG sample, +normalised to zero mean and unit variance. +14.4.1.2 +Model +The model for ECG identity verification (see Fig. 14.3) is adapted from the end-to-end architec- +tures proposed in [338; 344] and described in Chapter 4 and Chapter 5. The model is composed +of four unidimensional convolutional layers (with 16, 16, 32, and 32 filters, respectively, with size +1 × 5, unit stride, and zero padding), each followed by ReLU activation and max-pooling (with +1×3 kernels and stride 3). The model ends with two fully-connected layers, each with 100 units +and followed by ReLU activation. +Once trained, this model receives a 5 second long raw ECG segment (1000 samples long +at 200 Hz sampling frequency) and outputs an embedding or template that can be compared to +a reference through the Euclidean distance (during training) or through the normalised Euclidean +distance [461] (with the trained model, to obtain dissimilarity scores in the [0,1] range). In the case +of Secure Triplet Loss, the feature vector s(x) (the flattened feature maps from the last max-pooling +layer) is concatenated with the key array k, and both are bound together by the fully-connected +layers to make the final secure template f(x,k). +The model was trained using the Adam [219] optimiser, with an initial learning rate of 0.0001 +and l2 weight regularisation with λ = 0.001. The training lasted a maximum of 250 epochs, with + +214 +Learning Template Security on End-to-End Biometric Models +batch size 32, with early stopping based on validation loss with patience of 25 epochs. +14.4.2 +Face identity verification +14.4.2.1 +Data +To fine-tune and evaluate the model, images from the YouTube Faces database [460] were used. +This database is composed of frames from 3425 YouTube videos, depicting a total of 1595 sub- +jects (up to six videos of each subject). Each video corresponds to between 48 and 6070 frames. +This work used the aligned images provided on the database, which resulted from face detection, +cropping, and alignment. +Each face image has been reduced to 70% height and width and resized to 160×160 to match +the input dimensions of the model. Ten random triplets have been generated for each of the first +500 subjects on the database for a total of 5000 training triplets, of which 1000 have been used for +validation. Ten random triplets have also been generated from each of the remaining identities, +reserved for testing, resulting in a total of 10 950 test triplets. Whenever possible, the anchor and +positive samples corresponded to different videos of the same identity. +14.4.2.2 +Model +The model for face identity verification (see Fig. 14.3) is based on the Inception-ResNet [420]. +This network has been pretrained2 for identification on the VGGFace2 dataset [58] and offered an +accuracy of 99.63% on the Labelled Faces in the Wild (LFW) dataset and 95.12% on the YouTube +Faces database [381]. The original fully-connected layer has been replaced with two new fully- +connected layers, each with 100 units and followed by ReLU activation. For the Secure Triplet +Loss, the first of these layers receives the feature vector s(x) from the first part of the model, +concatenated with the key k. The second outputs the template y(x,k). +All layers on the model have been frozen, to take advantage of the pretrained parameters. The +exceptions are the last convolutional block and the fully-connected layers that come, respectively, +before and after the average pooling operation. The last convolutional block is fine-tuned to allow +for small adjustments during training, while the fully-connected layers are newly created and thus +require training. The model was trained for a maximum of 250 epochs at batch size 32, with +early stopping based on validation EER with a patience of 25 epochs. As with the ECG model, +the Adam optimiser was used with an initial learning rate of0.0001 and l2 regularisation with +λ = 0.001. +14.4.3 +Evaluation frameworks and metrics +The experiments have been designed to quantify the performance of the models trained with the +original and Secure Triplet Loss formulations, not only considering verification accuracy but also +biometric security. +2FaceNet Pytorch Package. Available on: https://github.com/timesler/facenet-pytorch. + +14.4 Experimental Setup +215 +14.4.3.1 +Verification performance +The verification performance is quantified through the measurement of false match rates (FMR) +and false non-match rates (FNMR) over the range of possible decision thresholds (for these mod- +els, t ∈ [0,1]). These values are presented in FMR vs. FNMR plots and detection error trade- +off (DET) curves and used to compute the equal error rate (EER), corresponding to the error +where FMRV = FNMR, and the FNMR@FMR = 0.01%. +14.4.3.2 +Cancelability +Avoiding additional processes such as biohashing or template encryption, the proposed Secure +Triplet Loss integrates cancelability into the single output of the system, the template y(x,k), +and is reflected in the distance measure d between two templates. Although the proposed loss is +designed to promote cancelability, this property may not necessarily be achieved. +Hence, the experiments with the Secure Triplet Loss include the measurement of cancelability +error. The plots of false match vs. false non-match rates over the dissimilarity/distance scores +include both the false match rate based on identity (when identities don’t match, denoted as FMRV) +and the false match rate based on cancelability (when keys don’t match, denoted as FMRC). The +false non-match rate (FNMR) values are the same for identity and cancelability since they refer +to situations when both identity and keys match. The value of cancelability false accept rate at the +operation point that corresponds to the verification EER, FMRC@EER, is also computed. +14.4.3.3 +Unlinkability +The template unlinkability analysis followed the method described by Gomez-Barrero et al. [149]. +The test samples were paired into mated (different biometric samples from the same identity with +different keys) and non-mated instances (different identities and keys). These have been used to +compute p(d|Hm) and p(d|Hnm): the probability density functions of the distance score d given +the instances are, respectively, mated (hypothesis Hm) or non-mated (hypothesis Hnm). From the +likelihood ratio LR(d) = p(d|Hm)/p(d|Hnm), D↔(d) is computed through +D↔(d) = +� +� +� +� +� +0, +if LR(d) ≤ 1 +2 +�� +1+e−(LR(d)−1)�−1 +− 1 +2 +� +, +if LR(d) > 1 +(14.7) +which allows to compute the Dsys +↔ linkability metric with +Dsys +↔ = +� dmax +dmin +D↔(d)· p(d|Hm)dd. +(14.8) +The Dsys +↔ is considered the main metric to quantify template linkability. A biometric system +verifying perfect template unlinkability, which is highly desirable, will assume Dsys +↔ = 0. A bio- +metric system creating entirely linkable templates will verify Dsys +↔ = 1. + +216 +Learning Template Security on End-to-End Biometric Models +Table 14.1: Summary of the test results for ECG identity verification. +Performance +Cancelability +Linkability +Method +EER (%) +FNMR @FMRV = 0.1% +FMRC@ EER +Dsys +↔ +Triplet Loss +12.56 +0.9033 +- +- +BF [149] +15.76 +0.9242 +0.0075 +0.234 +HE [109] +12.49 +0.9573 +0.0806 +0.002 +SecureTL +11.36 +0.8362 +0.0035 +0.288 +SecureTL w/KLD +13.58 +0.8700 +0.0 +0.005 +SecureTL w/SL +13.33 +0.9458 +0.0 +0.004 +14.4.3.4 +Non-invertibility and secrecy leakage +Other aspects of template security offered by the proposed method were evaluated, namely non- +invertibility and secrecy leakage. Non-invertibility is measured through the privacy leakage rate, +which can be computed through the expression: +H(X|Y) +H(X) = 1− I(X;Y) +H(X) , +(14.9) +where X is the input biometric, Y is the output of the model, H(X) denotes the entropy of X, +H(X|Y) denotes the conditional entropy of X given Y, and I(X;Y) denotes the mutual information +between X and Y. The privacy leakage rate, in the range [0,1], should be as close to 1 as possible: +obtaining information on X should be impossible even when one has all knowledge of Y. The +secrecy leakage measures the mutual information between the stored template Y and the key K, +through the expression I(Y;K). The keys are public, unlike the templates, so they should reveal as +little information as possible on the templates. Hence, the secrecy leakage should be close to zero. +These require the computation of some information theoretical measures, such as entropy and +mutual information. This is very difficult in biometrics, due to the high dimensionality of the in- +puts and the feature sets, as well as their variability. In this work, entropy and mutual information +were estimated using a Python implementation3 of the methods proposed in [228] and in [229], +respectively, for continuous multivariate data. These methods, based on nearest neighbour statis- +tics, were shown to be more accurate than the alternatives [106]. Since the processing cost of such +estimations grows exponentially with the size of the dataset, a subset of 1000 test anchors has been +used for this test. +14.5 +Results and Discussion +A general overview of the results obtained is presented in Table 14.1 and Table 14.2, respectively +for ECG and face identity verification. The following subsections discuss the results on verification +performance, cancelability, and unlinkability, and the comparison with state-of-the-art alternatives. +3Paul Brodersen’s Entropy Estimators. Available on: https://github.com/paulbrodersen/entropy_estimators. + +14.5 Results and Discussion +217 +Table 14.2: Summary of the test results for face identity verification. +Performance +Cancelability +Linkability +Method +EER (%) +FNMR @FMRV = 0.1% +FMRC@ EER +Dsys +↔ +Triplet Loss +13.99 +0.8496 +- +- +BF [149] +17.07 +0.9103 +0.0396 +0.245 +HE [109] +15.06 +0.8312 +0.0371 +0.001 +SecureTL +13.61 +0.8314 +0.0966 +0.399 +SecureTL w/KLD +15.93 +0.8586 +0.0089 +0.132 +SecureTL w/SL +15.15 +0.8771 +0.0182 +0.070 +14.5.1 +Verification performance +On ECG identity verification, the baseline method trained with triplet loss offered 12.56% +EER. This is similar to the results presented in the work that first proposed this end-to-end +model [338; 344]. As presented in Table 14.1 and in the receiver-operating characteristic (ROC) +curves in Fig. 14.4, the first formulation of the Secure Triplet Loss (SecureTL), without consider- +ing linkability, attained 11.36% EER, which is an improvement in performance over the original +triplet loss despite the inclusion of a cancelability module. +The linkability-focused reformulations of the Secure Triplet Loss, which use the Kullback- +Leibler divergence (SecureTL w/KLD) or distance statistics (SecureTL w/SL), led the model to +attain, respectively, 13.58% and 13.33% EER. These results show that a small performance gap +should be expected when considering both cancelability and linkability in the triplet loss. Recall- +ing the performance improvements with the SecureTL formulation, it can be hypothesised that the +performance decrease in SecureTL w/KLD and SecureTL w/SL is caused by measuring linkability +in a separate loss module computed batch-by-batch. It is likely that, if linkability was better inte- +grated into the Secure Triplet Loss, as was cancelability, then the performance gap would remain +closed. +Nevertheless, the model trained with any of the proposed loss formulations still offers con- +siderably better performance than the state-of-the-art methods. The best state-of-the-art method +evaluated in the same conditions (in [338]) offered 21.82% EER vs. 13.58% attained by SecureTL +w/KLD and 13.33% achieved by SecureTL w/SL. This denotes that the proposed method, while +presenting a small performance gap with the linkability loss module, still retains most of the per- +formance advantages associated with deep end-to-end models. +For face identity verification, the performance results are presented in Table 14.2 and in the +ROC curves in Fig. 14.5. The model trained with the triplet loss attained 13.99% EER, which +seems adequate given the difficulty of the evaluation settings: YouTube Faces provides a challeng- +ing framework for evaluation (noted by the 95.12% accuracy achieved by the Inception-ResNet +model on this database vs. 99.63% on the LFW database), disjoint subsets of identities are used +for training/validation and testing, and each identity is only represented by a single template for +each comparison (the gallery size is 1). +In harmony with the results on ECG, the model trained with the Secure Triplet Loss without + +218 +Learning Template Security on End-to-End Biometric Models +10 +3 +10 +2 +10 +1 +100 +FMR +10 +3 +10 +2 +10 +1 +100 +FNMR +DET Curves (ECG) +Triplet Loss +SecureTL +SecureTL w/KLD +SecureTL w/SL +Bloom Filters +Homomorphic Encryption +Figure 14.4: Detection Error Tradeoff (DET) curves for the ECG identity verification model when +trained with the original triplet loss vs. the proposed formulations of the Secure Triplet Loss. +10 +3 +10 +2 +10 +1 +100 +FMR +10 +3 +10 +2 +10 +1 +100 +FNMR +DET Curves (Face) +Triplet Loss +SecureTL +SecureTL w/KLD +SecureTL w/SL +Bloom Filters +Homomorphic Encryption +Figure 14.5: Detection Error Tradeoff (DET) curves for the face identity verification model when +trained with the original triplet loss vs. the proposed formulations of the Secure Triplet Loss. + +14.5 Results and Discussion +219 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +Original Triplet Loss (ECG) +FNMR +FMR +EER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +SecureTL (ECG) +FNMR +FMRV +FMRC +EER +FMRC@EER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +SecureTL w/KLD (ECG) +FNMR +FMRV +FMRC +EER +FMRC@EER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +SecureTL w/SL (ECG) +FNMR +FMRV +FMRC +EER +FMRC@EER +Figure 14.6: False match rate (FMR) and false non-match rate (FNMR) curves w.r.t. the dis- +tance comparison threshold t, for ECG identity verification with triplet loss and the proposed +Secure Triplet Loss formulations (the latter include both FMRP, relative to verification error, and +FMRC, relative to cancelability error, as well as the FMRC that corresponds to the EER point, +FMRC@EER). + +220 +Learning Template Security on End-to-End Biometric Models +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +Original Triplet Loss (Face) +FNMR +FMR +EER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +SecureTL (Face) +FNMR +FMRV +FMRC +EER +FMRC@EER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +SecureTL w/KLD (Face) +FNMR +FMRV +FMRC +EER +FMRC@EER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Threshold (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error Rate +SecureTL w/SL (Face) +FNMR +FMRV +FMRC +EER +FMRC@EER +Figure 14.7: False match rate (FMR) and false non-match rate (FNMR) curves w.r.t. the dis- +tance comparison threshold t, for face identity verification with triplet loss and the proposed Se- +cure Triplet Loss formulations (the latter include both FMRP, relative to verification error, and +FMRC, relative to cancelability error, as well as the FMRC that corresponds to the EER point, +FMRC@EER). + +14.5 Results and Discussion +221 +a linkability component offered a small improvement in verification performance (13.61% EER). +Likewise, the addition of a linkability-measuring term to the loss leads to a 2% increase in EER. +This confirms the aforementioned belief that the separate linkability loss term is affecting perfor- +mance and improvements could be achieved by integrating it into the Secure Triplet Loss in a more +cohesive way. +Overall, the verification performance results denote that it is possible to adequately train or +fine-tune an end-to-end model with the proposed loss formulations. With either biometric charac- +teristic, the performance difference between using KLD and distance statistics is not appreciable, +which denotes these formulations may each be fitted for specific settings or used interchangeably. +14.5.2 +Cancelability evaluation +As aforementioned, by integrating identity verification and template cancelability into a single +comparison score, template cancelability is not necessarily ensured. Hence, the results of false +match rates based on cancelability (FMRC) are presented, in Fig. 14.6 and Fig. 14.7, alongside the +false match rates based on verification (FMRV) and the common false non-match rates (FNMR). +In all cases, the FMRC is lower than FMRV at and around the EER operation point. In most +cases, FMRC at this point is very small and is lower than or equal to FMRV for all operation +points, which is highly desirable. As presented in Table 14.1 and Table 14.2, cancelability error +is significantly lower in the ECG models. As shown by the results, SecureTL w/KLD and w/SL +appear to be better at promoting cancelability than the original secure loss formulation, denoting +that the linkability loss term could have a positive effect on cancelability. +Considering these results and the increased difficulty experienced while fine-tuning the face +models, one can conclude that the proposed Secure Triplet Loss is likely better fitted for training +models from scratch than to adapting previously trained models to become secure. Nevertheless, +the cancelability results, especially with the SecureTL w/KLD and SecureTL w/SL, are encourag- +ing in either case. +14.5.3 +Unlinkability evaluation +The results of the linkability analysis following the framework established in [149] are presented in +Fig. 14.8. In both cases, the original formulation of the Secure Triplet Loss presents relatively high +Dsys +↔ (0.288 for ECG and 0.399 for face). However, the result with ECG is better than the equivalent +reported earlier in [345] (0.67). This results from the fact that linkability was not promoted by this +loss during model training: hence, the model may achieve adequate unlinkability, but that would +be accidental. +In the case of the proposed SecureTL w/KLD and SecureTL w/SL, linkability is actively pro- +moted during training through the loss. The effects of this loss reformulation are clear: the prob- +ability density functions of mated and non-mated are more superposed, which indicates that it +would be more difficult, as desired, to distinguish identities in pairs of templates where the keys +do not match. + +222 +Learning Template Security on End-to-End Biometric Models +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance score (d) +0 +1 +2 +3 +4 +5 +6 +Probability density (p) +SecureTL (ECG) +p(d|Hm) +p(d|Hnm) +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance score (d) +0 +1 +2 +3 +4 +5 +6 +Probability density (p) +SecureTL (Face) +p(d|Hm) +p(d|Hnm) +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance score (d) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Probability density (p) +SecureTL w/KLD (ECG) +p(d|Hm) +p(d|Hnm) +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance score (d) +0 +1 +2 +3 +4 +Probability density (p) +SecureTL w/KLD (Face) +p(d|Hm) +p(d|Hnm) +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance score (d) +0 +1 +2 +3 +Probability density (p) +SecureTL w/SL (ECG) +p(d|Hm) +p(d|Hnm) +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance score (d) +0 +1 +2 +3 +4 +5 +Probability density (p) +SecureTL w/SL (Face) +p(d|Hm) +p(d|Hnm) +D +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +D +(d) +Figure 14.8: Template linkability analysis for the ECG and face identity verification models (fol- +lowing the procedure proposed by Gomez-Barrero et al. [149]). +With ECG, Dsys +↔ assumes the values 0.005 for SecureTL w/KLD and 0.004 for SecureTL w/SL. +With face, it assumes 0.132 for SecureTL w/KLD and 0.070 for SecureTL w/SL. All of these can +be considered semi to fully-unlinkable. Just as with cancelability, the proposed method seems +more adequate for training models from scratch than for fine-tuning existing biometric models. +Additionally, using KLD appears to offer some advantages in linkability for ECG verification, but +that should be weighted with the increased instability this alternative has shown during training, +relative to SecureTL w/SL, especially in face verification. +14.5.4 +Non-invertibility and secrecy leakage +Regarding other security metrics, the privacy leakage rate was estimated as 1 for the model trained +with any of the losses. This indicates that it is highly difficult for an attacker to recover the original + +14.6 Summary and Conclusions +223 +biometric measurements x based on compromised templates y output by the model. This could +be a result of using end-to-end deep learning models: recent research indicates that optimised +deep models compress the inputs retaining only the information needed for the task [434]. This +means perfect non-invertibility can be achieved without carefully handcrafted feature extraction +algorithms. +Similarly, all losses led the model to offer a perfect secrecy leakage rate of 0, which denotes +that the public keys used to make the templates cancelable reveal no information on them. These +results on non-invertibility and secrecy leakage do not show a superiority of the proposed loss +formulations over the original triplet loss but emphasise the meaningful advantages of using end- +to-end deep learning models for secure biometrics. +14.5.5 +Comparison with state-of-the-art approaches +The proposed method was compared with two state-of-the-art approaches: Bloom Filters (BF) +and Homomorphic Encryption (HE), as described in [149] and [109], respectively. To provide a +fair and direct comparison between the template protection schemes, the features given to those +methods were those output by the triplet loss baseline model. +The results are presented in Table 14.1, Table 14.2, Fig. 14.4, and Fig. 14.5. Both with face and +ECG, the proposed method outperformed BF in EER, cancelability, and linkability. HE offered the +best linkability results, at the cost of poor cancelability. Additionally, HE took significantly longer +for biometric comparison than any of the alternatives, which may grant it limited real applicability. +Although the error results are relatively high, the Secure Triplet Loss is competitive vs. the +state-of-the-art alternatives, especially on cancelability and linkability. Moreover, improved re- +sults are expected when the Secure Triplet Loss is used on more accurate biometric models. +14.5.6 +Effects of varying γ +Fig. 14.9 presents the EER and Dsys +↔ results obtained when varying the γ parameter which balances +the original secure triplet loss formulation and the template linkability component. As shown, +lower γ values (γ < 0.7) lead to higher EER with either SecureTL w/KLD or SecureTL w/SL, +since template unlinkability takes precedence over verification accuracy on the loss that guides +model training. For γ ≥ 0.7, lower EER results are obtained, albeit with a slight increase in +template linkability (Dsys +↔ ), especially for γ > 0.9. Results may vary in other application scenarios +depending on their specificities, but 0.7 < γ < 0.95 should offer the highest likelihood of success. +14.6 +Summary and Conclusions +This work presented the Secure Triplet Loss, a methodology focused on training end-to-end deep +biometric models, without any additional processes, to verify template cancelability, unlinkability, +and non-invertibility. The results on ECG and face identity verification show that the proposed + +224 +Learning Template Security on End-to-End Biometric Models +0.30 +0.50 +0.70 +0.80 +0.90 +0.95 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Results when varying +EER (SecureTL w/KLD) +Dsys (SecureTL w/KLD) +EER (SecureTL w/SL) +Dsys (SecureTL w/SL) +Figure 14.9: Results with the proposed loss when varying the γ parameter. +method is not only able to fulfil this purpose, but also to adapt pretrained biometric models to +offer secure templates, with competitive performance results. +However, there is still room for improvement. Further efforts should be devoted to design- +ing ways to better integrate linkability in the Secure Triplet Loss, in order to avoid performance +decreases. A scheme where linkability would be measured triplet-by-triplet (instead of batch-by- +batch), similarly to cancelability, should lead to improved performance using the Secure Triplet +Loss. +This would also enable the formulation of triplet mining approaches for the proposed +method. Nevertheless, the Secure Triplet Loss is, overall, a suitable and flexible general scheme +for template protection in end-to-end deep biometrics. + +Chapter 15 +Self-Supervised Learning +with Sequential Data +Foreword on Author Contributions +The research work described in this chapter was conducted entirely by the author of this thesis, under the supervi- +sion of Jaime S. Cardoso. The results of this work have been disseminated in the form of an article in international +conference proceedings: +• J. R. Pinto and J. S. Cardoso, “Self-Learning with Stochastic Triplet Loss,” in International Joint Conference +on Neural Networks (IJCNN 2020), Jul. 2020. [340] +15.1 +Context and Motivation +In recent years, deep learning algorithms have offered improved performance over handcrafted +methodologies in several pattern recognition tasks. These commonly take advantage of convolu- +tional layers, which enable the autonomous learning of the most relevant features for the task at +hand, and use fully-connected layers for more intricate decision boundaries [242]. However, these +improvements come with an important drawback: the need for labelled data. +Most tasks where such performance breakthroughs have been achieved are those where re- +searchers have plenty of labelled data at their disposal. The ImageNet dataset enabled the training +of deeper models for better performance in the detection and recognition of objects. Similarly, da- +tasets such as VGGFace [331] and VGGFace2 [58] helped in the development of improved models +for biometric recognition based on face images. +However, for some pattern recognition problems, supervised data is scarce. In most of these +cases, even though available data is plenty, the annotation process is cumbersome and/or expen- +sive. This is very frequent in automatic medical image diagnosis tasks, where several imaging +exams are usually available but lack specific annotations which typically would need to be offered +by experts. +Another exemplary application is video surveillance. Given the current ubiquity of surveil- +lance cameras, the availability of data is not a problem. However, the annotation of individuals on +225 + +226 +Self-Supervised Learning with Sequential Data +the recordings is a long and expensive endeavour. This limits the performance one can attain in +these tasks since deeper models will be harder to train. +Yet another key application is continuous biometrics with electrocardiogram (ECG) sig- +nals [343]. Deep learning has offered improved performance through increased robustness to +signal noise and variability [344]. However, in scenarios with off-the-person signals (acquired +during normal activity using few dry electrodes on the fingers and palms), the performance still +fails to match the use of cleaner on-the-person signals (acquired on medical-grade settings from +subjects at rest, using several wet electrodes on the chest and limbs) [284; 338; 483]. This is ver- +ified because current off-the-person signal datasets are too few and too small to train larger and +deeper models. +Self-learning (SL) has emerged as a promising approach to learn from unlabelled data. Con- +trarily to unsupervised learning, SL uses contextual or prior information to automatically define +labels and tasks, which then support the learning. Generally, self-learning methods are focused +on specific applications, including visual representation learning [128; 147], action classifica- +tion [129], or human motion capture [439], thus including details that restrict their use to those +specific tasks. +On the other hand, several self-learning methods use simple low-level tasks for the training, +such as ranking samples [254] or recovering masked parts of an image [437], that do not nec- +essarily guarantee that the learned parameters can be useful for high-level tasks. For example, +features that prove adequate for the approximate reconstruction of biometric samples may not be +good enough to discriminate their identities. Hence, there is currently a need for a more general +and capable self-learning methodology. +In this work, we propose a novel formulation of the triplet loss [72] for self-supervised learning +with unlabelled data, unrestricted to specific problems. The triplet loss is particularly suited for +this task since it does not require absolute labels for the samples. As samples are combined into +triplets, only their relative label information is required. The proposed formulation finds its key +application in sequential data scenarios, where mild assumptions about the data acquisition process +enable the adoption of a stochastic adaptation of the triplet loss to trigger and sustain the learning. +In the experimental work, the proposed methodology is successfully applied to off-the-person +ECG-based biometric identity verification tasks, using signals from the University of Toronto ECG +Database (UofTDB) [445], and to unconstrained face identity verification, using the YouTube +Faces dataset [460]. Specific stress experiments were conducted on the ECG-based identity veri- +fication task to evaluate the behaviour of the proposed methodology in strain conditions. +15.2 +The Stochastic Triplet Loss +The triplet loss [72] uses triplets of data samples to train a network to accurately assess if two +samples belong to the same class. Each triplet is composed by an anchor xa with identity ia and +two other samples, one positive (xp) and one negative (xn), where ip = ia ̸= in. The three samples +are processed, in parallel, by the same network, which returns a learned representation of each + +15.2 The Stochastic Triplet Loss +227 +of them (ya, yp, and yn). A measure of distance d is then used to compare the anchor-positive +(d+ = d(ya,yp)) and anchor-negative (d− = d(ya,yn)) pairs of representations, which are used in +the computation of the triplet loss. The triplet loss for a single triplet can be defined as: +L (xa,xp,xn) = max(0,α +d+ −d−). +(15.1) +During training, the goal to decrease the triplet loss will lead the model to adjust its weights +to obtain a final representation which brings samples of the same class closer together (reducing +d+), and samples of different classes further apart (increasing d−). Here, the margin parameter α +will contribute to enforce a minimum distance margin between the samples of different classes. +In the standard triplet loss, it is certain that xp is sampled from the same class as xa and xn is +sampled from a class different from the class of xa. This can be rewritten as P(Iia(ip) = 1) = 1 +and P(Iia(in) = 0) = 1, where IA(x) is the indicator function. +With unlabelled samples, there is an uncertainty associated with the generation of the triplets, +arising from the possibility of errors during the selection of the positive and negative samples. +We generalise the previous assumption by modelling Iia(ip) as a random variable following a +Bernoulli distribution with parameter β, i. e., P(Iia(ip) = 1) = β. Similarly, Iia(in) is assumed to +follow a Bernoulli distribution with parameter γ, i. e., P(Iia(in) = 0) = γ. +Assuming the independence of Iia(ip) and Iia(in), and conditioned on the true identity of the +observations in the triplet, the triplet loss follows a multinoulli distribution, with: +L (xa,xp,xn) = +� +� +� +� +� +� +� +� +� +� +� +� +� +max(0,α +d+ −d−), +with probability βγ +max(0,α +d− −d−), +with probability (1−β)γ +max(0,α +d+ −d+), +with probability β(1−γ) +max(0,α +d− −d+), +with probability (1−β)(1−γ) +(15.2) +On average, the middle terms in Eq. (15.2) do not contribute to the learning. The last term in +the equation negatively impacts the learning process. In practice, one would need more data/time +to learn under the noisy sampling of the triplets. The parameters β and γ guide the training of the +model through the triplet loss, and their values depend on the specificities of the task and the data. +In ideal conditions, these should be as close as possible to 1 to approximate the original triplet +loss in supervised settings. Lower values would work against the purpose of the triplet loss and +diminish its training effectiveness (or, equivalently, increase the difficulty of the training). +The proposed self-learning methodology can be used to train models with unsupervised data. +During triplet generation, after the selection of an anchor sample, one can randomly draw one +sample from the dataset to serve as the negative sample. Assuming a balanced dataset with C +classes will give γ = 1−1/C. If C is large, the probability of errors in negative sample selection +p(ia = in) will be very low (e. g., 0.1 for C = 10 or 0.01 for C = 100), and so will be their impact on +the training process. More importantly, in practice, prior knowledge allows us to adopt a sampling +strategy with a much higher probability of success. + +228 +Self-Supervised Learning with Sequential Data +The positive sample can be obtained through the transformation of the anchor according to +xp = f(xa). The transformation f should be carefully defined in order to change the anchor ac- +cording to an expected range of intraclass variability but without degrading the underlying label +information carried by the sample. The probability β will depend on the degree to which f com- +plies with this need. Similarly to the negative sample selection, prior knowledge of the data may +be useful to maximise the probability of success. For example, when dealing with sequential data, +choosing a positive sample closer in time to the anchor will increase the probability of both sam- +ples sharing the same label. However, the anchor and the positive sample will likely be more +similar, which will restrict the model’s robustness to intraclass variability. Hence, one should find +a trade-off between ensuring intraclass variability and maximising the probability of success in +positive sample generation. +An approximation of the expected value of the loss in Eq. (15.2) can be computed under some +simplified conditions. Assuming a setting with two classes C1 and C2, with a probability density +functions p1(x) and p2(x), respectively. If xa is sampled from either of the distributions, it results +in pa(xa) = π p1(xa) + (1 − π)p2(xa), with 0 ≤ π ≤ 1. Setting x = [x′ +a x′ +p x′ +n]′ with a probability +density function p(x) = p(xa,xp,xn) assumed to be equal to +p(x) = p(xa,xp,xn) += pa(xa)pp(xp|xa)pn(xn|xa) += π p1(xa)p1(xp)p2(xn)+(1−π)p2(xa)p2(xp)p1(xn) +, +(15.3) +the triplet loss between xa, xp, xn can be described using the Euclidean distance function as +Ex∼pL (xa,xp,xn) with +L (xa,xp,xn) = max +� +0,α +||r(xa)−r(xp)||2 −||r(xa)−r(xn)||2� +, +(15.4) +where r(x) is the learned representation of x. +In the presence of the assumed noise model in the sampling process of the triplets (xa,xb,xc), +the probability density function becomes +g(x) = βγ pa(xa)pp(xp)pn(xn) ++(1−β)γ pa(xa)pn(xp)pn(xn) ++β(1−γ)pa(xa)pp(xp)pp(xn) ++(1−β)(1−γ)pa(xa)pn(xp)pp(xn) +. +(15.5) + +15.3 Application Scenarios +229 +With this, the triplet loss becomes: +Ex∼gL (xa,xp,xn) = βγEx∼pL (xa,xp,xn) ++(1−β)γEx∼h1L (xa,xp,xn) ++β(1−γ)Ex∼h2L (xa,xp,xn) ++(1−β)(1−γ)Ex∼pL (xa,xn,xp) +, +(15.6) +with: +h1(xa,xp,xn) = π p1(xa)p2(xp)p2(xn)+(1−π)p2(xa)p1(xp)p1(xn) +(15.7) +and: +h2(xa,xp,xn) = π p1(xa)p1(xp)p1(xn)+(1−π)p2(xa)p2(xp)p2(xn). +(15.8) +Noting that the expected value of the gradient of the loss L (xa,xp,xn) is zero under h1 and h2 +(since xp and xn are sampled from the same distribution and the loss is symmetric), the impact of +those two cases in a gradient-based learning scheme is small. The total loss is then: +βγ max(0,α +||ya −yp||2 −||ya −yn||2)+ +(1−β)(1−γ)max(0,α +||ya −yn||2 −||ya −yp||2) +, +(15.9) +under the p(x) probability density function, where y = r(x). +Finally, this loss can be compacted to: +max +� +0,βγ(α +||ya −yp||2 −||ya −yn||2) +� ++ +max +� +0,(1−β)(1−γ)(α +||ya −yn||2 −||ya −yp||2) +�. +(15.10) +In section 15.3, example applications of this methodology are presented for the tasks of +electrocardiogram-based biometric identity verification and face identity verification. +15.3 +Application Scenarios +The proposed method can be used to train models relying solely on unsupervised data. On clas- +sification tasks, the negative sample can be generated through the random selection of a sample +in the dataset. Assuming a large number of balanced classes, errors in negative sample selection +should be rare. For the positive samples, the function f(x) that generates them based on an anchor +can be a data augmentation procedure. This should be carefully adjusted to cover the expected +intraclass noise and variability while retaining the information pertaining to the underlying image +label, which can be difficult. +Alternatively, when training with sequential data, the triplet generation can forgo the data +augmentation procedures. In these situations, depending on the acquisition context or protocol, + +230 +Self-Supervised Learning with Sequential Data +the temporal distance or proximity between data can be used to infer the identity of the subjects. +A sample that is very close in time to the anchor can safely be used as xp. Similarly, a sample that +is sufficiently distant in time to the anchor can be assumed to belong to a different user, and thus +used as xn. Some knowledge of the domain and the acquisition settings can be used to adjust the +distance between xa, xp, and xn to maximise β and γ. +Both aforementioned alternatives (entirely unsupervised or using sequential data) were ex- +plored for the applications described below, through the experiments described in section 15.4. +15.3.1 +ECG identity verification +Deep learning models have previously shown improved robustness to off-the-person noise and +variability in electrocardiogram-based biometrics [338; 344]. +However, to train such models +and match the performances reported for cleaner on-the-person signals, one would need large +databases of off-the-person acquisitions, which are currently unavailable [343]. In such circum- +stances, a pretrained network would often be the natural option in computer vision tasks. However, +these too are currently nonexistent for unidimensional physiological signals such as the electro- +cardiogram. +The integration of ECG sensors in everyday objects, e. g. using the CardioWheel steering +wheel cover [279] for shared vehicles or similar solutions for shared bicycles or scooters, enables +the continuous acquisition of data from several subjects over long periods. This large amount of +collected data could be used to train deeper and more sophisticated models. However, this data is +commonly unlabelled, as the identity of the users at the moment of acquisition cannot be easily +verified. +The proposed methodology for self-learning can be applied to train models for ECG-based +identity verification using such data. As aforementioned, perturbations based on data augmenta- +tion procedures can be applied to the anchor to generate a positive sample. Thus, the four most +successful data augmentation procedures proposed by Pinto et al. [344] were implemented. For +each triplet, one of these was randomly selected to generate a positive sample from the anchor: +• Cropping: a smaller contiguous segment is taken from the anchor sample and resampled to +match the anchor’s length, to simulate slower heart rates; +• Baseline Wander: a periodic undulation, with a frequency near 1 Hz, is added to the anchor +segment to simulate breathing movement artefacts; +• Gaussian Noise: Gaussian noise is added to the anchor signal, simulating high-frequency +distortions similar to the electromyogram (EMG) and powerline interference; +• Random Permutation: the anchor is divided into N subsegments, which are shuffled to gen- +erate a different sample that simulates discontinuities or sensor faults. +With continuous ECG recordings, it is possible to avoid errors in positive and negative sample +selection. Having separate recordings for each person, positive samples are obtained through the + +15.3 Application Scenarios +231 +CONV +POOL +CONV +POOL +CONV +POOL +CONV +FC +CONV +POOL +CONV +POOL +CONV +POOL +CONV +FC +TEMPLATE +QUERY +16@1x5 +ReLU +16@1x5 +ReLU +32@1x5 +ReLU +32@1x5 +ReLU +1x5 +1x5 +1x5 +100 +ReLU +d +score +shared weights +Figure 15.1: Architecture of the ECG identity verification model that was trained with the pro- +posed methodology. +selection of a segment of the anchor’s recording. The negative sample is obtained from a different +recording. In this case, there should be no errors in positive sample selection. Although there +can be several recordings for the same person, errors in negative sample selection should be rare +considering the large number of identities in the dataset and the balanced number of recordings +per identity. +15.3.2 +Face identity verification +More face data are available now than ever before, especially from surveillance feeds or public +videos shared on online social media platforms. However, as with ECG-based biometrics, the +labelling of faces in acquired datasets is a tedious and lengthy task. Some researchers have taken +advantage of online videos to build large datasets for face recognition, such as the YouTube Faces +dataset from Wolf et al. [460]. However, these datasets are limited by the number of annotations +available. +The proposed self-learning method can be used to train models for face verification without +labelled data. In this case, common image data augmentation based on rotations, width and height +shifts, and horizontal flips were used as the transformation function f(x) that generates a positive +sample xp based on an anchor xa. +Having short videos, a random detected face from the same recording as the anchor can serve +as a positive sample, while a negative sample can be drawn from a different recording. With +some knowledge of the recordings, we minimise the probability of errors in positive and negative +sample selection. Specifically, we know the YouTube Faces data consists of frames from short +video recordings, with several people, with no more than one person per frame. Hence, although +the short recordings lack much intrasubject trait variability, selecting triplets in the aforementioned +way avoids errors in positive and negative sample selection. +CONV +POOL +CONV +POOL +CONV +POOL +CONV +FC +CONV +POOL +CONV +POOL +CONV +POOL +CONV +FC +TEMPLATE +QUERY +16@3x3 +ReLU +16@3x3 +ReLU +32@3x3 +ReLU +32@3x3 +ReLU +2x2 +2x2 +2x2 +100 +ReLU +d +score +shared weights +FC +FC +1000 +ReLU +POOL +CONV +POOL +CONV +POOL +CONV +POOL +CONV +64@3x3 +ReLU +64@3x3 +ReLU +2x2 +2x2 +Figure 15.2: Architecture of the face identity verification model that was trained with the proposed +methodology. + +232 +Self-Supervised Learning with Sequential Data +15.4 +Experimental Setup +15.4.1 +Data +15.4.1.1 +ECG data +The data used to train and evaluate the model is from the University of Toronto ECG Database +(UofTDB) [445]. This database includes data from 1019 subjects, acquired at 200 Hz using dry +metallic button electrodes, held by the subjects in contact with one finger of each hand. Each +recording is 2 − 5 minutes long, and each subject has recordings for up to five different postures +(supine, tripod, exercise, standing, and sitting) on up to six sessions over a period of six months. +The data was divided for model training and evaluation as done by Pinto et al. [338]. The last +100 subjects (from subject 921 to subject 1020) were reserved for model training. The data from +the remaining 918 subjects were used for evaluation. One subject (8) was discarded for having too +few data. From the 918 subjects reserved for evaluation, the first 30 seconds of the first recording +were used for enrollment, while the remaining data were used for testing. This aimed to mimic a +realistic context with scarce supervised data as expected in real ECG-based biometric applications. +15.4.1.2 +Face data +For face identity verification, data from the YouTube Faces database [460] were used. This data- +base contains frames from 3425 videos of 1595 subjects, sourced from YouTube. Each video is +48 to 6070 frames long, and there are up to six videos of each subject. This work used the aligned +images provided on the database, which resulted from face detection, cropping, and alignment. +The first 150 subjects (in alphabetical order) were used to build the dataset used in this work: +the first 100 subjects were reserved for training and validation, while the data from the remaining +subjects were used for testing. Triplets were generated using this data subset, after resizing the +images to 224×224, as detailed below in the experiments’ description. +15.4.2 +Models +The self-supervised training method proposed in this work was explored for ECG-based identity +verification using an adapted version of the end-to-end network proposed by Pinto et al. [338] (see +Fig. 15.1). The model receives two z-score normalised five-second raw ECG segments (a stored +template and a query sample) and returns a measure of dissimilarity related to their identity. +The network is composed of four convolutional layers followed by a dense layer. A max- +pooling layer (pooling size 1×5) follows each of the first three convolutional layers. The convo- +lutional layers have 16, 16, 32, and 32 filters, respectively, with unit stride, without padding. The +dense layer is composed of 100 units. All convolutional and dense layers are followed by ReLU +activation. +For face identity verification, the model is a simple convolutional neural network (see +Fig. 15.2), which receives two 224 × 224 RGB face images, normalised to [0,1] intensities, and +outputs a measure of their dissimilarity. It is composed of six convolutional layers interposed with + +15.4 Experimental Setup +233 +five max-pooling layers (pooling size 2×2) and followed by two dense layers. The convolutional +layers have 16, 16, 32, 32, 64, and 64 filters, respectively, with size 3 × 3, unit stride, without +padding. The dense layers are composed of 1000 and 100 units, respectively. All convolutional +and dense layers are followed by ReLU activation. +Both models were trained using the Adam optimiser, with an initial learning rate of 0.0001. As +in [338], the Euclidean distance was used as distance measure d during training, while for identity +verification this was replaced by the normalised Euclidean distance for scores in [0,1]. The triplet +loss margin was set as α = 1.0. A maximum of 200 epochs was given, with batches of 12 triplets, +along with early stopping with a patience of 10 epochs. Dropout was used before each dense +layer, with rates of 0.5 and 0.2 for the ECG and the face models, respectively. L2 regularisation +(λ = 0.01) was used for the convolutional layers in both models. +15.4.3 +Evaluation metrics +The evaluation metrics used are the False Acceptance Rate (FAR), the False Rejection Rate (FRR), +the Equal Error Rate (EER), and the Receiver-Operating Characteristic (ROC) curve [343]. The +FAR measures the rate at which impostors meet a given acceptance threshold and are falsely +granted access. The FRR measures the rate at which genuine users are incorrectly denied access +due to their scores not meeting the given threshold. The EER corresponds to the error at the +operation threshold where FAR and FRR have equal values. The ROC curve plots the values of +1−FRR versus FAR for the possible range of threshold values. +15.4.4 +Experiments’ description +15.4.4.1 +Without supervision +In this experiment, the models were trained with triplets whose negative samples are drawn ran- +domly from the entire respective dataset. The positive samples are created through the application +of data augmentation procedure to the respective anchor samples. To train the ECG identity ver- +ification model, 100 000 triplets were generated for the training, of which 10% were used for +validation during training, and 10 000 triplets were generated for evaluation. For the face model, +10 000 triplets were generated for the training, of which 20% were used for validation, and 5000 +triplets were generated for testing. +Naturally, depending on the dataset used for training, the probability of error in the random +selection of a negative sample will vary. In datasets with fewer classes, the probability of randomly +selecting a negative sample whose class matches that of the anchor is greater than in datasets with +more classes. Hence, the aforedescribed experiment with the ECG identity verification model +was repeated, but giving the selection of a negative sample a probability pe = 1 − γ of returning +a sample from the anchor’s identity. This probability of error was linked to a simulated number +of subjects Ns, with pe = 1/Ns and Ns = {2,5,10,20,50,100,200,500,1000}. This enabled the +assessment of how a balanced dataset with fewer classes could impact the training process and the +effect on the final identity verification performance. + +234 +Self-Supervised Learning with Sequential Data +15.4.4.2 +Using recordings +This experiment used the recordings of the UofTDB database and the video recordings of YouTube +Faces as a way to infer the identity of the samples through the temporal proximity between them, +using prior knowledge to minimise triplet generation errors. Here, the positive sample is drawn +from the same recording as the anchor, while the negative sample is drawn from a different record- +ing. As each subject can have several recordings, there is an error associated with the selection +of the negative sample, which can accidentally be selected from a different recording of the same +subject. The number of generated ECG and face triplets used for training, validation, and testing, +was the same as mentioned in 15.4.4.1. +When training the network with longer recordings spanning several users, as described in +section 15.2, the possible errors are different. Although the positive sample is selected in the +temporal vicinity of the anchor, it can belong to a different identity. The negative sample, despite +the distance from the anchor, can accidentally belong to the same user as the anchor. Hence, +additional experiments were conducted where the positive and negative sample selection processes +failed purposely with probability p = {0.05,0.1,0.2,0.3,0.5,0.7}, to assess the effect of such +errors in the final model performance. +15.5 +Results and Discussion +15.5.1 +ECG identity verification +The baseline results correspond to the identity verification model trained with supervised data. The +equal error rate of 12.56% represents a small improvement over the corresponding result reported +in [338]. Considering the evaluation data and conditions were the same, this method also offered +significantly better results than the state-of-the-art methods implemented and tested in [338]: the +Autoencoder-based solution proposed by Eduardo et al. [111], the AC/LDA method proposed by +Agrafioti et al. [10], and the DCT approach proposed by Pinto et al. [342; 344]. +The performance results of the model trained with the two unsupervised training approaches +are presented in Fig. 15.3, in comparison with the baseline results. The Equal Error Rate values +were 12.56%, 19.19%, and 12.70%, for supervised, entirely unsupervised, and recording-based +training, respectively. The difference between the performance with entirely unsupervised training +and the performance with recording-based training denotes the data augmentation procedures have +not been able to completely mimic the variability of the signals, and could perhaps be improved +using optimised data augmentation [90; 260]. Despite the worse performance attained with the +entirely unsupervised training approach, all of these methods offered better performance than the +handcrafted methods evaluated in the same settings in [338], among which the best result was +21.82% EER. + +15.5 Results and Discussion +235 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +FAR +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1-FRR +Supervised +Unsupervised +Recordings +Figure 15.3: Comparison of the Receiver-Operating Characteristic curves on ECG identity verifi- +cation for supervised training, unsupervised training, and recording-based supervision. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +FAR +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1-FRR +Supervised +Unsupervised +Recordings +Figure 15.4: Comparison of the Receiver-Operating Characteristic curves on face verification for +supervised training, unsupervised training, and recording-based supervision. +15.5.2 +Face identity verification +As with the ECG-based identity verification task, the model trained with supervised data was +used as a baseline for comparison of results in face identity verification. The performance offered +by the baseline was 18.45% EER. This is considerably higher than the state-of-the-art, which is +explained by the relative simplicity of the implemented model and the relatively small dataset +used. Nevertheless, the goal of this work was not to overcome or match the state-of-the-art in face +verification but to illustrate how the proposed self-learning methodology can be applied to face +biometrics with small performance losses relative to a supervised baseline in similar conditions. + +236 +Self-Supervised Learning with Sequential Data +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +FAR +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1-FRR +2 +5 +10 +20 +50 +100 +200 +500 +1000 +Figure 15.5: Receiver-Operating Characteristic curve for negative selection error based on the +number of database subjects. +The results with the proposed methodology are illustrated in Fig. 15.4. When using entirely +unsupervised data, the proposed method offered 22.81% EER, a 4.36% increase relative to the +use of supervised data. With recording-based triplet generation, the model offered 19.77% EER, +a 1.32% increase. These small performance losses when forgoing labels during training show +that the model can learn without supervision using the stochastic triplet loss, as verified above for +ECG biometrics. Besides these two applications, one should expect the proposed self-learning +methodology to be successfully applied to similar problems. +15.5.3 +Stress experiments +As discussed in subsection 15.4.4, the success of the unsupervised triplet generation technique +depends on the number of identities (classes) in the database. Hence, an experiment was performed +on ECG identity verification to simulate the variation of the number of identities on the dataset, +inducing errors in the negative sample selection with the respective probability. The results (see +Fig. 15.5) show that, although the performance worsens with fewer subjects, the errors have a +very small effect for datasets with more than 20 subjects. In fact, with 50 subjects or more, +the performance results stabilised around 20% EER. Hence, a dataset with 50 classes should be +enough to adequately apply this method with better performance than handcrafted state-of-the-art +approaches. +For the training based on temporal proximity between samples, both the selection of positive +samples and the selection of negative samples may fail. Hence, enforcing a probability of each +error in the recording-based training experiments allows the study of the impact of such errors +on the model’s performance. The increase of either positive or negative sample selection error +probabilities leads to a decrease in performance (see Fig. 15.6 and Fig. 15.7). However, that + +15.5 Results and Discussion +237 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +FAR +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1-FRR +5% +10% +20% +30% +50% +70% +Figure 15.6: Receiver-Operating Characteristic curves for varying positive sample selection error +probability. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +FAR +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1-FRR +5% +10% +20% +30% +50% +70% +Figure 15.7: Receiver-Operating Characteristic curves for varying negative sample selection error +probability. +decrease is relatively small unless the probabilities of error are over 50%. This means that some +knowledge of the typical usage times and patterns during data acquisition would be enough to +adjust the process of positive and negative sample selection and ensure the best results. +Results could be further improved using more enrollment data (see Fig. 15.8). As studied by +Pinto et al. [338], instead of the simple one-vs-one comparisons performed in the aforedescribed +experiments, which correspond to five-second enrollments, the query template can be compared +with each of several enrollment templates from each person, and only the minimum score is con- + +238 +Self-Supervised Learning with Sequential Data +5 +10 +15 +20 +25 +30 +Enrollment (s) +10 +12 +14 +16 +18 +EER +Supervised +Unsupervised +Recordings +Figure 15.8: Equal Error Rate (EER) results when using more enrollment data from each subject. +sidered. ECG identity verification performance with the proposed unsupervised and recording- +based training approaches reached 14.55% and 9.89%, respectively, when using thirty-second +enrollments. +15.6 +Summary and Conclusions +This work proposed a novel formulation of the triplet loss for self-supervised learning with un- +labelled data. This method considers the uncertainty associated with the triplet generation in +unsupervised settings and maximises the probability of success using prior knowledge. +In harmony with the goals of this thesis, the proposed methodology was applied to the task +of ECG-based biometric identity verification, using transformations based on data augmentation +or the temporal proximity between samples to generate valid triplets. The method offered better +performance than handcrafted state-of-the-art methods, especially when using temporal proximity +between samples, with performance results similar to supervised training. +This pattern was also confirmed on the task of unconstrained face identity verification. Train- +ing with entirely unsupervised data using the proposed triplet loss formulation resulted in just a +small performance loss when compared with the use of supervised data. When generating triplets +based on video streams, this loss was considerably smaller. +It should be noted that the proposed method can be influenced by imbalanced classes and +errors in the unsupervised triplet generation. However, the results of the stress experiments show +its robustness is sufficient to avoid considerable impacts on performance in most cases. Thus, +this method would, according to the presented results, be a viable training option in multiclass +classification problems where only unlabelled data are available, especially with sequential data. + +Part VI +Epilogue +239 + + +Chapter 16 +Summary and Conclusions +The doctoral work presented throughout this thesis had the main goal of advancing the integration +of biometric recognition and wellbeing monitoring solutions, especially focused on the case of +intelligent vehicles. This goal was carefully targeted through two specific scenarios. The first was +personalised wellbeing monitoring systems using biometrics, mainly focused on advancing ECG +and face biometric solutions for improved driver assistance systems. The second was occupant +monitoring for autonomous shared vehicles, through the development of solutions for efficient +and robust group-based monitoring of passengers’ wellbeing inside fully autonomous vehicles. +Always with this major goal in mind, several novel and meaningful contributions were +achieved, as introduced in Chapter 1 and described in the various chapters of this document. The +main contributions, resulting completely or mostly from the work of the author of this thesis, were: +• The most comprehensive state-of-the-art survey on ECG biometrics to date. This work +encompassed one hundred and twenty-five literature methods, a thorough description of the +fundamentals behind the ECG as a biometric characteristic, and a critical overview of the +evolution of this topic, including extensive considerations on the current research challenges +and future opportunities. At the time of writing this thesis, the resulting survey article had +been cited in the literature over one hundred and fifty times1; +• The first truly end-to-end approach for ECG-based biometrics, proposed for both identifi- +cation and identity verification tasks, including the development of tailored data augmenta- +tion strategies and studies on fine-tuning and transfer learning. Meaningful and significant +performance improvements were achieved in challenging and realistic evaluation scenarios +using large off-the-person databases. At the time of writing this thesis, the resulting book +chapter and international conference article had been cited in the literature over forty times; +• A pioneering study on the relevance of ECG waveforms for the biometric identification +task through explainability. End-to-end deep networks trained for biometric identification +and analysed using multiple explainability tools offered new insights on the importance of +1João R. Pinto’s Google Scholar. Available on: https://scholar.google.com/citations?user=hhF9Q8kAAAAJ. +241 + +242 +Summary and Conclusions +the different parts of the ECG signal, especially the QRS complex, when distinguishing +identities in diverse scenarios; +• A multimodal approach for audiovisual recognition of emotion valence in groups of individ- +uals. The proposed approach, based on adapted state-of-the-art sound and video recognition +modules, presented promising results in regards to both accuracy and efficiency in the re- +cognition of emotional states of groups of people; +• A cascade strategy to streamline multimodal audiovisual activity recognition in time- +sensitive scenarios. When evaluated for multiclass activity recognition and violence de- +tection, including inside real vehicles, the proposed approach was able to offer improved +accuracy with up to 66% reduction in runtime; +• The first approach to achieve template security in end-to-end biometric models. Aiming to +combine the benefits of end-to-end deep learning with the security of template cancelability +and unlinkability, the proposed Secure Triplet Loss was able to outperform the state-of-the- +art alternatives and offer minimal performance gaps without requiring separate encryption +processes or protection schemes. At the time of writing this thesis, the resulting journal +article and international conference article had been cited in the literature fifteen times; +• An adaptation of the triplet loss to allow for fully unsupervised learning. Taking advantage +of multiclass balance and the natural structure of sequential data, the proposed approach +was able to considerably close the gap to supervised performance levels, even in the more +challenging evaluation scenarios. +The secondary contributions of this doctoral work, resulting from collaborations within the +scope of this thesis which benefitted partially from the work of the author, were: +• The first study on long-term performance evolution of multiple state-of-the-art ECG bio- +metric methodologies. This study highlighted the problem of intrasubject variability in ECG +biometrics, even over relatively short time periods, and proposed multiple template/model +update strategies to mitigate its negative effects on identification performance; +• An approach for recovering the full set of twelve standard ECG leads relying only on short +single-lead blindly-segmented recordings. The proposed approach has offered promising +results in a considerably more challenging evaluation scenario than those found in the liter- +ature, and paves the way towards robust and efficient methods for retrieving missing leads +in more comfortable acquisition setups; +• Two novel strategies to reduce the performance gap in masked face recognition. Among +the proposed approaches, the one based on multi-task contrastive learning outperformed the +alternative methods and illustrated the benefits of promoting the similarity between latent +masked and unmasked face image representations. At the time of writing this thesis, the +resulting three international conference articles had been cited in the literature over fifty +times; + +Summary and Conclusions +243 +• A pioneering study on interpretability for face biometrics, through the presentation attack +detection task. This exploratory experiment illustrated how interpretability tools could be +used to achieve a deeper understanding of the behaviour of biometric models and motivated +their use in the next generation of biometric evaluation strategies. +Through these contributions, this doctoral project touched on several important topics related +to each individual research area and the thesis theme as a whole. In ECG biometrics, the work +addressed the topics of end-to-end models, transfer learning, data augmentation, long-term perfor- +mance, template update, interpretability, and interlead conversion. In face biometrics, the topics +of masked face recognition, presentation attack detection, and interpretability were covered. In +wellbeing monitoring, this work focused on multimodal fusion, group emotion recognition, vio- +lence detection, in-vehicle scenarios, and optimisation/efficiency. Additionally, the broader topics +of biometric template security and self-supervised learning were also addressed. +Considering the achievements of this work, one can conclude that, although the ideal of truly +personalised wellbeing monitoring is yet to be achieved, meaningful and valuable strides have been +successfully taken to reach it. As such, a strong framework is now built to support future work +towards the central goal of tightly integrating biometric recognition and wellbeing monitoring in +a multimodal, seamless, continuous, and realistic way. +This conclusion is supported by the reception of this research within the scientific commu- +nity. The work described in this thesis has directly resulted in twenty-four scientific publications, +including five articles in peer-reviewed journals and eleven articles in the proceedings of inter- +national conferences. This increases to thirty-eight total publications if one also considers other +minor contributions to other topics in biometrics, computer vision, and pattern recognition, which +have not been addressed in this document. These had been welcomed by the scientific community +with over three hundred citations by the time this thesis was written. +This doctoral work has been published in multiple major journals and conferences in the in- +ternational biometrics community. These include the IEEE Transactions in Biometrics, Behavior +and Identity Science, IET Biometrics, the International Joint Conference on Biometrics (IJCB), +the International Conference on Biometrics: Theory, Applications and Systems (BTAS), and the +International Conference of the Biometrics Special Interest Group (BIOSIG). Additionally, part of +the work has also been published in reputed venues related to general machine learning, computer +vision, or pattern recognition research, such as IEEE Access or the International Joint Conference +on Neural Networks (IJCNN). Part of the results of this work also contributed to the AUTOMO- +TIVE, Easy Ride, and Aurora research projects. +In recognition of the value of the achieved contributions, this doctoral research and its author +have also been the recipients of multiple awards. These are the EAB Max Snijder Award at the +2022 European Biometrics Awards organised by the European Association for Biometrics (EAB), +the Computers Journal Best Paper Award at the 2020 International Workshop on Biometrics and +Forensics, the Best Session Paper Award at the 2020 IEEE International Conference on Image +Processing, Applications and Systems (IPAS), and the Jury’s Best Presentation Award at the 2021 +NIS Workshop organised by INESC TEC. + + +Chapter 17 +Future Work Considerations +Despite the results achieved in the doctoral work described throughout this thesis, plenty is yet to +be done to achieve the full symbiotic integration of biometric recognition and wellbeing monitor- +ing. The successful integration of these two tasks in real scenarios is a challenging endeavour in +itself. Nevertheless, plenty of opportunities are yet to be explored in the topics of ECG biomet- +rics, face biometrics, and wellbeing monitoring focused on this doctoral work, which would be +essential to achieve our major objective. +When considering the current state of ECG biometrics, it is hard to disagree with the notion +that data is the main problem to be tackled. Many would argue that face biometrics is more evolved +than ECG biometrics because the ECG is more deeply affected by noise and variability. But this +is misleading, since the face suffers (and heavily so in truly unconstrained scenarios) from most of +the same factors that affect the ECG, including emotions, exercise, drowsiness, and medical con- +ditions. Even the loss of information due to heavy noise or sensor contact losses in off-the-person +ECG is analogous to common occlusions that can deeply limit the available information in uncon- +strained face images. Face biometrics is more developed thanks to the unprecedented magnitude +of data available to train increasingly sophisticated and robust models. The number of subjects, +the unconstrained nature of the data, the diversity of scenarios, the variety of acquisition sessions, +and the comprehensiveness of current benchmark data in ECG biometrics pale in comparison to +what can be found for face biometrics. +As such, researchers should dedicate special efforts to creating larger (and better) datasets for +ECG biometrics. Increased number of identities, longer recordings, more sessions across wider +periods of time, and more realistic off-the-person scenarios are only some of the aspects that +need to be verified by new datasets. Since the ECG could likely serve society better as part of +multimodal solutions, new datasets could focus on the simultaneous (and continuous) acquisition +of other traits alongside the ECG, especially face video. While such datasets remain unavailable, +it would be interesting to find new ways to mitigate the effects of data scarcity. This includes +the development of sophisticated pretrained models that could be fine-tuned to multiple tasks, the +study of learnable 1D to 2D transformations to take advantage of image pretrained models, and +the development of tailored solutions for unsupervised and self-supervised learning. At last, it is +245 + +246 +Future Work Considerations +also important to standardise the way performance is evaluated in ECG biometrics, through the +definition of complete and realistic benchmark datasets to assess and compare the performance of +state-of-the-art methods. +Conversely, in face biometrics, data is plenty and models are considerably more accurate. +However, as we could witness from the results of this doctoral research, there is still much to +do regarding the robustness of face recognition in realistic scenarios. The advent of face masks +revealed the feeble nature of current state-of-the-art approaches to unexpectedly drastic scenarios +and reignited the old topic of occlusions in face biometrics. In fact, after over two years of heavy +and dedicated research in masked face recognition, there is still a considerable performance gap +vs. unmasked scenarios. Masks may soon leave our society, but the possibility for such severe +occlusions to be witnessed once again (in the shape of masks or any other object) is enough to +warrant further research. As such, the successful example of multi-task contrastive learning should +be followed with the development of more sophisticated multi-objective schemes for learning to +ignore masked face regions. Moreover, the creation of larger datasets of real masked face images +and videos is also a key step towards closing the performance gap in masked face recognition. +Additionally, given that face biometrics is a much more developed topic than ECG biometrics, +interpretability is also a much greater opportunity in this topic. Face recognition solutions have +permeated countless aspects of our society, including deeply sensitive applications such as border +control or urban surveillance. This situation starts to raise doubts about the trustworthiness of such +algorithms. Interpretability is essential to offer the deeper level of understanding needed to avoid +biases, lead models away from undesirable behaviours, and ultimately make the general public +more comfortable with the use of such technologies. As such, it is important that more attention +is devoted to the topic of interpretability in biometrics (especially face biometrics), through the +redefinition of evaluation standards and the development of approaches with interpretability and +explainability as one of the main goals (and not just an afterthought). +On the topic of wellbeing monitoring, two major problems are evident from the results of the +work conducted during this doctoral project. The first is performance in specific scenarios such as +in-vehicle monitoring. Current models present promising results in general purpose scenarios, for +which data has become plenty through online sourcing, but are still relatively weak for recognition +in specific scenarios. These, naturally, should not be overlooked and researchers should dedicate +efforts to better take advantage of generalistic models for specific scenarios, e.g., through more +sophisticated methods for transfer learning and domain adaptation. The second problem is subject +dependency. This is a well-documented problem in the literature and, while not as evident in the +group-level emotion recognition work presented in this thesis, has been encountered and discussed +frequently in emotion and drowsiness recognition works linked to this doctoral work and the AU- +TOMOTIVE project. It is troubling how performance can be affected when wellbeing monitoring +models are applied to data from individuals they have never seen before. If wellbeing monitoring +is to be applied in real scenarios, this problem should be tackled head-on through dedicated ro- +bustness studies and the reformulation of evaluation setups to ensure realistic subject-independent +results. + +Future Work Considerations +247 +After the aforementioned individual topics are addressed, the integration of biometrics and +wellbeing, combined with the integration of multiple data sources, raises new problems in terms +of efficiency and data security. Since these solutions would be deployed to edge scenarios, it is +important to consider performance costs and develop models with efficiency as one of the main +priorities. Multi-task and multimodal integrated solutions, minimising separate processing of in- +formation, should be chosen as frequently as possible. At last, since such solutions would deal +with highly intimate data carrying identity and health information, it is paramount to continue the +efforts on learnable template security to ensure all data is stored in a thoroughly protected way. + + +Part VII +Appendices +249 + + +Appendix A +ECG Biometrics Literature Methods +This appendix presents an overview of the methods proposed for ECG-based biometrics in the +literature, from the beginning of the topic in 1999 till today, in Table A.1. This table includes +a straightforward description of the most important aspects of ECG biometric methodologies: +the denoising methods, the signal preparation pipeline, the feature extraction and dimensionality +reduction schemes, and the decision models/strategies. Moreover, it includes the performance re- +ported in the respective publications, alongside information on the evaluation scenario correspond- +ing to such results: the dataset, the number of subjects, and whether the data has been acquired in +off-the-person scenarios. +This survey of state-of-the-art approaches began with the research work in [337], initially +covering sixty-five literature methods. It was later expanded during this doctoral work to cover +ninety-three publications in [343], which has since been cited nearly one hundred and fifty times +by the scientific community. In this thesis, it presently covers a total of one hundred and twenty- +five state-of-the-art methods for ECG biometrics. +This unprecedentedly comprehensive collection of information on ECG biometric methodolo- +gies aimed mainly to empower the easy pinpointing of open challenges and research opportunities. +Hence, this work, as published before in [343], is thought to have helped accelerate the develop- +ment of novel solutions in ECG biometrics and is expected to continue doing so in the near future. +The information gathered in this appendix has also been used to thoroughly describe the evolution +and current landscape of ECG biometrics in Chapter 3. +251 + +252 +ECG Biometrics Literature Methods +Table A.1: Summary of the surveyed state-of-the-art unimodal methods proposed for ECG biometrics (ordered by year of publication and first author +name, DR – Dimensionality Reduction, NS – Number of Subjects, OP – Off-the-Person). +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Biel et al. +[42; 43] +1999 +2001 +- +Siemens Megacart +ECG Processing +10 Lead I fiducial +features / PCA +SIMCA +Private +20 +No +IDR +100% +Kyoso +et al. [235] +2000 HPF 0.06 Hz + +LPF 60 Hz + +NF 50 Hz +Beat segment. and +fiducial detection +PQ and QT times +Mahalanobis +distance + LDA +Private +3 +No +IDR +99.5% +Kyoso +et al. +[233; 234] +2001 HPF 0.06 Hz + +LPF 60 Hz + +NF 50 Hz +Second-order +derivative fiducial +detect. +QRS duration and QT +time +Mahalanobis +distance + LDA +Private +9 +No +IDR +94.2% +Shen et al. +[389] +2002 - +- +RQ, RS, ST, QS time, +QT time, RS slope, +QRS area +Correlation + +DBNN +MIT NSR +20 +No +IDR +100% +Palaniappan +et al. [327] +2004 LPF 30 Hz +Adapted +Pan-Tompkins +fiducial detect. +R, QR, RS, QRS +width, R-R, beat form +factor +MLP; SFA +MIT NSR +10 +No +IDR: +MLP +SFA +96.2% +83.6% +Israel et al. +[193] +2005 BPF 2–40 Hz +R detection, +heartbeat +segmentation and +alignment by +R-peaks +RQ, RS, RP, RL, RP’, +RT, RS’, RT’, P and T +widths, ST, PQ, PT, +LQ, ST’ / LDA +Contingency +matrix majority +voting +Private +49 +No +IDR: +Anxty. +Norm. +97% +98% +Saechia +et al. [368] +2005 - +PQRST heart rate +normaliz., P, QRS, +and T +segmentation +Fourier transform of +PQRST (whole), P, +QRS, and T +Neural +Networks +- +- +No +FRR: +Whole +Apart +17.1% +2.85% +Plataniotis +et al. [349] +2006 BPF 0.5–40 Hz +Fixed-length +window blind +segmentation +Autocorrelation +coefficients / DCT +Norm. +Euclidean dist. ++ Gaussian +LLR +PTB +14 +No +IDR +FAR +100% +0.02% + +ECG Biometrics Literature Methods +253 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Zhang +et al. [490] +2006 - +R-peak and QRS +detection, +heartbeat +segmentation +Amplitudes, +durations, intervals, +levels, & areas / PCA +Bayes- +minimum-error- +rate +Private +(leads I, II, +V1, and V2) +502 +No +IDR +L.I +L.II +L.V1 +L.V2 +85.3% +92.0% +95.2% +97.4% +Molina +et al. [298] +2007 Savitzky-Golay +Trahanias R +detection, R-R +segmentation +R-R segments +DTW path + +kNN +Private +10 +Yes +EER +2% +Wübbeler +et al. [463] +2007 Moving Median ++ LPF 75 Hz +R-peak detection +by thresholding +2D QRS (combination +of leads I, II, and III) +Temporal +derivatives dist. ++ kNN +PTB +74 +No +IDR +EER +98.1% +2.8% +Agrafioti +et al. [8] +2008 BPF 1–40 Hz +Fixed-length +window blind +segmentation +Normalized +autocorrelation / DCT +or LDA +Correlation + +kNN +PTB + MIT +NSR +27 +No +IDR: +DCT +LDA +96.3% +100% +Chan et al. +[69] +2008 NF 60 Hz +Fid. detection with +backward diff., +alignment and +outlier rej. by +cross-corr. +Signal-averaged ECG +PRD, CC, +WDIST + kNN +Private +50 +Yes +IDR: +PRD +CC +WD +70% +80% +89% +Irvine et al. +[189] +2008 BPF 0.05–60 +Hz +Beat segm. and +alignment by +R-peaks using AC, +amplitude +normalization +Covariance matrix +eigenvectors / PCA +kNN +Private +39 +No +IDR +100% +Boumbarov +et al. [47] +2009 HPF 0.05 Hz + +DWT soft +thresholding +HMM-GMM +PQRST +segmentation +Cardiac cycle vector +matrix / PCA and +LDA +RBF NN +Private +9 +No +IDR +83.3% +Fang et al. +[122] +2009 BPF 2–50 Hz +R detection, 5-beat +average, amplitude +normalization +Avg. beat phase space +portrait +Correlation; +Mutual nearest +pt. dist. + kNN +Private (one +or three +leads) +100 +No +IDR: +1 l. +3 l. +93% +99% + +254 +ECG Biometrics Literature Methods +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Fatemian +et al. [124] +2009 DWT 3rd scale ++ Mov. Average +Discarding outlier +beats, DWT QRS, +T, & P delineation +Heart-rate normalized +heartbeat construction +Correlation + +kNN +PTB + MIT +NSR +27 +No +IDR +99.6% +Guennoun +et al. [156] +2009 LPF 30 Hz +- +Fiducial amplitude +and time feat. / +Physiological-state- +indepen. feature +select. +Mahalanobis +dist. + Thresh. +and Voting +Private +16 +No +FRR +FAR +0.01% +0% +Coutinho +et al. [86] +2010 BPF 2–30 Hz +Beat segment. and +alignment, 10-beat +avg. +Uniformly quantized +avg. beats +Ziv-Merhav +relative entropy ++ kNN +Private +19 +Yes +IDR +99.5% +Fatemian +et al. [125] +2010 DWT 3rd scale +DWT beat +delineation, +P-wave time norm. +Avg. ensemble +heartbeat +Correlation + +kNN +Private +21 +No +IDR +EER +95.4% +3.3% +Ghofrani +et al. [144] +2010 BPF 0.5–150 +Hz + NF 50 Hz +- +AR; PSD; Lyapunov; +Approximation +Entropy; Higuchi +Fractal Dim.; Shannon +Entropy +kNN; MLP; +PNN +PTB +12 +No +IDR: +AR +ApEn +Hig. +Lya. +Sha. +98.6% +94.3% +87.4% +96.7% +92.8% +Li et al. +[250] +2010 - +Beat segment., +amplitude and time +norm. +Hermite poly. +expansion; Cepstral +features / HLDA +SVM + +GMM-UBM +fusion +MIT NSR +18 +No +IDR +EER +98.3% +0.5% +Murthy +et al. [303] +2010 BPF +Pan-Tompkins +P, T, ST, PR, QRS and +QT intervals / FLDA +DTW + kNN +MIT NSR +15 +No +IDR +96% +Odinaka +et al. [318] +2010 HPF 0.5 Hz + +LPF 500 Hz + +NF 60 Hz +R detect., beat +normaliz. and +Hamming seg. +Log-STFT +spectrogram / Bin +selection +Gaussian +models LLR +Private +269 +No +IDR +EER +99% +0.37% +Sasikala +et al. [373] +2010 Median Filters ++ DWT +QRS detection w/ +Daubechies DWT, +P and T detection +Fiducial amplitudes +and differences +Correlation +MIT Arrh. +10 +No +IDR +62.7% + +ECG Biometrics Literature Methods +255 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Tawfiq +et al. [429] +2010 HPF 1 Hz + +LPF 40 Hz +R-alignment, +amplit. norm., +QRS segment. +QRS DCT coefficients +Neural +Networks +Private +22 +No +IDR +99.1% +Ye et al. +[472] +2010 BPF +Beat segment. with +R location +annotations +Daubechies DWT / +ICA +RBF SVM +MIT Arrh. +MIT NSR1 +MIT LT +MIT NSR2 +47 +18 +65 +18 +No +No +No +No +IDR +IDR +IDR +IDR +99.6% +99.3% +98.1% +97.5% +Coutinho +et al. [87] +2011 BPF 2–30 Hz +Beat segment., +alignment, 10-beat +avg. +User-tuned +Lloyd-Max quantised +avg. beat +Ziv-Merhav +cross parsing +similarity + +kNN +Private +19 +Yes +EER +0.36% +Lourenço +et al. [274] +2011 BPF 0.5–30 Hz +Engelse- +Zeelenberg, beat +segment., +amplitude and time +normaliz. +Avg. normalized beat +Euclidean dist. ++ kNN +Private +16 +Yes +IDR +EER +94.3% +13% +Matta et al. +[292] +2011 BPF +Fixed-length +window blind +segmentation +Autocorrelation coeff. +/ LDA +Euclidean dist. ++ kNN +Private +10 +No +IDR +TPIR3 +75% +99% +Safie et al. +[369] +2011 BPF 2–40 Hz +ECGPUWAVE +fiducial detect., +P-R & P-T quality +check, 5-beat avg. +Pulse Active Ratio +Euclidean dist. ++ kNN +PTB +(healthy or +w/ +arrhythmias) +112 +No +EER: +Heal. +Arrh. +9.98% +19.2% +Shen et al. +[390] +2011 BPF 1–50 Hz +Pan-Tompkins, +heartbeat +segmentation +Amplitudes, +durations, slopes, +angles, and QRS area +/ LDA +Correlation + +kNN +Private +168 +Yes +IDR +98% +Sufi et al. +[414] +2011 - +P, QRS, and T +segmentation, +cardioid 2D loop +generation +Cardioid graph +centroid, extremas, +area, and perimeter +Straight line +and percentage +dist. + kNN +MIT Arrh. +- +No +MIDR +FAR +FRR +1% +0.5% +0.5% + +256 +ECG Biometrics Literature Methods +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Agrafioti +et al. [10] +2012 BPF 1–40 Hz +Fixed-length +window blind +segmentation +Autocorrelation coeff. +/ LDA +Euclidean dist. ++ kNN +Private +42 +No +EER +3.96% +Belgacem +et al. [32] +2012 BPF 1–40 Hz +Amplitude +normal., beat +segmentation and +R-peak alignment +Avg. beat Daubechies +DWT +Random Forest +MIT Arrh. + +ST-T + MIT +NSR + PTB ++ Private +80 +No +FAR +FRR +0.60% +0.58% +Lourenço +et al. [276] +2012 BPF 1–30 Hz +Steep-slope R +detection, beat +segment. +Segmented heartbeats +kNN +Private +32 +Yes +EER +9.39% +Singh et al. +[405] +2012 - +QRS, P, and T +delineation +Interval, angle, and +amplitude fid. feat. +Euclidean dist. ++ kNN +MIT Arrh. + +ST-T + MIT +NSR + QT +73 +No +EER +10.8% +Belgacem +et al. [33] +2013 BPF 1–40 Hz +Beat segment., +amplitude +normalization, +100-beat avg. +Avg. beat Daubechies +DWT +Random Forest +MIT Arrh. + +ST-T + MIT +NSR + PTB ++ Private +80 +No +IDR +FAR +FRR +100% +0.63% +0.66% +Coutinho +et al. [88] +2013 BPF 2–30 Hz +Beat segm. and +alignment, 10-beat +mean wave +Fid. latency and +amplitude from mean +waveform +subsampling +Euclidean dist. ++ kNN +PTB +Private +51 +26 +No +IDR +EER +IDR +EER +99.9% +0.01% +99.6% +0.70% +BPF 2–30 Hz +Beat segm. and +alignment, 10-beat +mean wave +User-tuned +Lloyd-Max quantized +heartbeats +Ziv-Merhav +cross-pars. +similarity + +kNN +PTB +Private +51 +26 +No +IDR +EER +IDR +EER +99.4% +0.13% +99.9% +0.29% +Labati +et al. [236] +2013 HPF 0.5 Hz + +NF 50 Hz +R detection and +QRS segment., +rejection of +low-correl. seg. +QRS segment set +templates +Cross-corr. +similarity mat. ++ kNN +E-HOL 24h +185 +No +EER +5.36% + +ECG Biometrics Literature Methods +257 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Matos +et al. [288] +2013 HPF 0.5 Hz + +NF 50 and 150 +Hz +R detection, beat +segment., +Hamming +segmentation +STFT spectrogram + +Spectral zoom / Bin +selection +Gaussian +models LLR +Private +27 +No +EER +10% +Silva et al. +[393] +2013 BPF 5–20 Hz +R detection and +beat segmentation +Mean and median +ensemble beats +Euclidean and +cosine dist. + +kNN and SVM +Private +63 +Yes +EER: +kNN +SVM +0.99% +9.10% +Wang et al. +[449] +2013 - +Sliding-window +segmentation +Max-pooling +representation +elements +kNN +PTB +100 +No +IDR +99.5% +Zhao et al. +[491] +2013 HPF and DWT +soft-thresh. +Beat segment. and +normaliz., quality +check +EEMD main IMFs +and their PSD / PCA +kNN +ST-Change +LTST +PTB +15 +18 +12 +No +No +No +IDR +IDR +IDR +98.0% +95.8% +96.0% +Ergin et al. +[114] +2014 - +Segm. 2 s sliding +windows +QRS fid., time +domain, wavelet trans. +and PSD +C4.5 and +Bayesian +Network +MIT NSR +18 +No +F-s.: +C4.5 +Bay. +0.97% +0.96% +Iqbal et al. +[188] +2014 - +QRS detection and +segment. +QRS cardioid graph +coord. +MLP +Private +30 +No +IDR +96.4% +Labati +et al. [237] +2014 HPF 0.5 Hz + +NF 50 Hz +R detection, QRS +segment. +QRS segments +Cross-corr. +simil. kNN +E-HOL 24h +185 +No +EER +5.36% +Lin et al. +[261] +2014 - +Time-delay space +reconst., Chaos +theory feature +extract. +Corr. dimension +Lyapunov exp. +SVM +Private +26 +Yes +IDR +81.7% +Lourenço +et al. [278] +2014 - +QRS detection, +DMEAN +Mean ensemble beats +SVM +Private +63 +Yes +EER +2.5% +Matos +et al. [289] +2014 LPF 50 Hz +Slope sum + +thresh. for R det., +beat segm. +STFT window +features / +Kullback-Leibler +LLR + kNN +Private +10 +Yes +IDR +EER +100% +14% + +258 +ECG Biometrics Literature Methods +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Pathoumvanh +et al. [332] +2014 BPF 0.4–40 Hz +R detection and +hearbeat +segmentation +CWT / FLDA +Euclidean dist. ++ kNN +Private +(normal + +increased +HRV) +10 +No +IDR: +Norm. +HRV +97% +80% +Zhou et al. +[493] +2014 BPF 0.5–40 Hz +R detection, +interval vs. +amplitude plot +Signal between 3 +consec. R peaks +DTW path + +kNN +Private +20 +No +HTER 1.45% +Brás et al. +[54] +2015 NF 50 Hz + +Moving Avg. + +LPF 40 Hz +Amplitude norm., +SAX conversion +Kolmogorov-based +normalised rel. +compression +kNN +PTB +52 +No +IDR +99.9% +Choudhary +et al. [78] +2015 DCT +FOGD peak det., +peak correction, +heartbeat +segmentation +Avg. ensemble beat +RMSE, PRD, +NCC, +WWPRD, +WDIST + kNN +MIT Arrh. + +STC + QT + +MIT NSR + +SLP +127 +No +FAR +FRR +5.8% +11.6% +Dar et al. +[95] +2015 Poly. line fitting +Local-maxima R +det., QRS +segmentation +Haar Transform / +GBFS +kNN +MIT Arrh. +MIT NSR +ECG-ID +47 +18 +90 +No +No +No +IDR +IDR +IDR +93.1% +99.4% +83.2% +Dar et al. +[96] +2015 Poly. line fitting +Local-maxima R +det., QRS +segmentation +Haar Transform and +HRV / GBFS +Random Forest +MIT Arrh. +MIT NSR +ECG-ID +47 +18 +90 +No +No +No +IDR +FAR +IDR +EER +IDR +FAR +95.9% +4.1% +100% +0% +83.9% +16.1% +Jahiruzzaman +and +Hossain +[197] +2015 BPF 0.5–45 Hz +None +CWT and Chaotic +Encryption +Identification of +unique CE +sequen. +MIT Arrh. +11 +No +IDR +96.9% +Carreiras +et al. [66] +2016 BPF 5–20 Hz +QRS det., beat seg. +and align., +DMEAN +Segmented heartbeats +kNN +Private +618 +No +EER +MIDR +9.01% +15.6% + +ECG Biometrics Literature Methods +259 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Chun et al. +[80] +2016 DWT 3rd scale +Pan-Tompkins, +heartbeat +segmentation +Guided filtering avg. +beat / PCA +DTW or +Euclidean dist. ++ kNN +ECG-ID +89 +No +EER: +DTW +Eucl. +5.2% +2.4% +Hejazi +et al. [170] +2016 DWT 3rd scale +Fixed-length +window blind +segmentation +Autocorrelation coeff. +/ KPCA +SVM +Private +52 +Yes +IDR +FAR +FRR +76.3% +3.5% +4.83% +Louis et al. +[272] +2016 BPF 1–40 Hz +Pan-Tompkins, +beat segment. & +alignment +1D multi-res. LBP +Bagging +UofTDB +1012 Yes +EER +7.89% +Porée et al. +[350] +2016 LPF 45 Hz +Pan-Tompkins, +beat segment. +10 beat avg. ensemble +Discrimination +coeff./kNN +Private +14 +No +IDR +100% +Rezgui +et al. [366] +2016 BPF 2–40 Hz +ECGPUWAVE +QRS detection, +segmentation +Amplitudes, areas, +intervals and fid. +slopes +SVM +MIT NSR + +Arrh. +No +48 +IDR +98.8% +Waili et al. +[446] +2016 HPF 0.05 Hz + +LPF 40 Hz +Pan-Tompkins, +signal mean +subtract. norm. +12 QRS fid. +amplitudes +MLP +PTB +14 +No +IDR +96% +Camara +et al. [57] +2017 BPF 0.67–45 +Hz +None +Walsh-Hadamard +features, outliers +rejected +kNN +MIT NSR +10 +No +IDR +94.8% +Eduardo +et al. [111] +2017 BPF 5–20 Hz +Beat cropping, +DMEAN outlier +detection +Fully-connected +autoencoder +representations +kNN +Private +709 +No +MIDR 0.91% +Islam et al. +[191] +2017 BPF 0.25–40 +Hz +Curvature QRS +detect., beat segm., +time norm., AC +outlier reject. +Avg. ensemble +heartbeats / PCA +Euclidean dist. +Private +112 +Yes +EER +10.5% +Karimian +et al. [210] +2017 BPF 1–40 Hz +Pan-Tompkins, +heartbeat segm., +heart rate QT time +normalization +DWT, Maximal +Overlap DWT, DCT, +Normalize Convolute +Norm. encrypted by +IOMBA +Key matching +PTB +BioSec +290 +13 +No +Yes +Rel. +Rel. +97.4% +94.7% + +260 +ECG Biometrics Literature Methods +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Komeili +et al. [225] +2017 BPF 0.5–40 Hz +Heartbeat segm. +and outlier +removal, z-score +norm. +CWT, STFT, AC, +max., st. dev., kurtosis +and skewness / MSFS +SVM +UofTDB +(different +session or +posture) +82 +Yes +EER: +Sess. +Post. +6.9% +3.7% +Paiva et al. +[325] +2017 - +Pan-Tompkins, +LPF for Q, S, T +detection +Fiducial distances ST, +RT, and QT +SVM +PTB +10 +No +IDR +FAR +FRR +97.5% +5.71% +3.44% +Pinto et al. +[342] +2017 Savitzky-Golay ++ moving avg. +Trahanias, +heartbeat +segmentation, +z-score, NCCC +DCT coefficients +SVM +Private +6 +Yes +IDR +EER +94.9% +2.66% +Salloum +and Kuo +[371] +2017 - +Pan-Tompkins, +beat segm., z-score +norm. +Sequences of +appended heartbeats +RNN with +LSTM/GRU +ECG-ID +MIT Arrh. +90 +47 +No +IDR +EER +100% +0% +Tan et al. +[426] +2017 BPF 2–50 Hz +Pan-Tompkins, +P–QRS–T +segmentation +Temporal, amplitude, +and angle fid. + DWT +coefficients +Random Forests ++ WDIST kNN +Private +ECG-ID +MIT Arrh. +MIT NSR +Combined +30 +89 +47 +18 +184 +Yes +No +No +No +No +IDR +IDR +IDR +IDR +IDR +99.4% +100% +100% +98.8% +99.5% +Wieclaw +et al. [457] +2017 BPF 4–35 Hz +Hamilton R detect., +outlier rejection +Individual heartbeats +MLP +Private +18 +Yes +IDR +89% +Zaghouani +et al. [477] +2017 Median filter +Fixed-length +window segm. +AC / DCT, feature +security locking +Norm. +Euclidean dist. +ECG-ID +90 +No +EER +15% +Zhang +et al. [483] +2017 BPF 2–50 Hz +Normalisation, 2 s +blind +segmentation, +DWT +Autocorrelation, +component selection +Multiresolution +1D CNN +CEBSDB +WECG +FANTASIA +MIT NSR +STDB +MIT Arrh. +AFDB +VFDB +20 +22 +40 +18 +28 +47 +23 +22 +No +IDR +99.0% +94.5% +97.2% +95.1% +90.3% +91.1% +93.9% +86.6% + +ECG Biometrics Literature Methods +261 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Zhang +et al. [484] +2017 None +Heartbeat detection +and segmentation +Cardioid-like 2D +representations +2D CNN +Private +10 +Yes +IDR +98.4% +Dong et al. +[105] +2018 - +Construction of 3D +VCG with 12-lead +ECG +Banks of state and +errors from 2D VCG +Minimum L1 +norm of bank of +errors +PTB +(healthy and +ill subjects) +14 +99 +113 +No +IDR: +Healt. +Ill +All +98.3% +93.3% +92.8% +Guven +et al. [160] +2018 HPF 0.5 Hz + +LPF 150 Hz + +mov. avg. +Z-score norm., 5 s +window +segmentation +AC, DCT, cepstral, +and QRS features +Euclidean dist. ++ kNN +Private +30 +45 +60 +Yes +IDR +IDR +IDR +100% +100% +98.3% +Kim and +Lim [217] +2018 - +Min-max norm. +Pan-Tompkins, +beat ensemble +Haar Wavelet +Transform +Fuzzy +membership +ANN +- +73 +No +FRR +FAR +1.68% +5.84% +Lee et al. +[246] +2018 BPF 0.3–35 Hz, +6th order +polynomial line +fitting +R & T detection, +R-R segmentation, +resampling +R-R segments, +including two or three +heartbeats (hb.) +Cosine, +euclidean, +manhattan +dists., & CC +Private +55 +No +IDR: +2 hb. +3 hb. +89.9% +93.3% +Luz et al. +[284] +2018 BPF 0.5–40 Hz +Pan-Tompkins, +beat segmentation, +z-score +normalisation, +outlier removal +Raw 1D heartbeats +and 2D spectrogram +representations +1D CNN + 2D +CNN +CYBHi +UofTDB +61 +1019 +Yes +Yes +EER: +Raw +Spect. +Fusion +Raw +Spect. +Fusion +14.1% +26.4% +12.8% +16.9% +19.4% +14.3% +Pal et al. +[326] +2018 HPF 1 Hz, NF +50 Hz, LPF 40 +Hz +DWT fiducial det., +P-QRS-T +segmentation +Interval, amplitude, +angle, and area +fiducial features / +KPCA +Euclidean +distance +PTB +100 +No +IDR +97.1% +Wu et al. +[462] +2018 BPF 1–40 Hz +Pan-Tompkins, +beat segmentation +1D CNN for feat. +extraction and outlier +removal +Attention-based +LSTM +ECG-ID +MIT Arrh. +90 +47 +No +No +IDR +EER +IDR +EER +97.5% +0.52% +99.7% +0.02% + +262 +ECG Biometrics Literature Methods +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Carvalho +et al. [68] +2019 LPF 30 Hz, 1st +order derivative +Lloyd-Max +quantisation, 10 s +segmentation +Extended-alphabet +Finite-Context Model +(xaFCM) +Normalised +Relative +Compression +(NRC) +Private +25 +No +IDR +89.3% +Chu et al. +[79] +2019 RNN-based +noise removal +Threshold beat +detection, +segmentation, and +concatenation +Parallel Multiscale 1D +ResNet +Template +similarity or +fully-connected +layer +ECG-ID +PTB +MIT Arrh. +90 +290 +47 +No +No +No +IDR +EER +IDR +EER +IDR +EER +97.8% +2.00% +99.3% +0.59% +94.9% +4.74% +Ciocoiu +and Cleju +[83, 84] +2019 +2020 +BPF 1–40 Hz +DTW R-peak +detection, beat +segmentation +S-Transform, Gramian +Angular Fields, +Phase-Space +Trajectories, or +Recurrence Plots +2D CNN +UofTDB +CYBHi +52 +65 +Yes +Yes +IDR +EER +IDR +EER +95.6% +5.48% +95% +8.6% +Hammad +et al. [161] +2019 - +2 s blind +segmentation, +z-score norm. +None +1D Res-Net +with Attention +PTB +CYBHi +290 +65 +No +Yes +IDR +98.9% +99.3% +Labati +et al. [238] +2019 HPF 0.5 Hz, NF +R detection, QRS +cropping and +concatenation, +normalisation +1D CNN +Softmax layer +or template +similarity +PTB +E-HOL 24h +52 +92 +No +No +IDR +EER +100% +2.15% +Pinto et al. +[344] +2019 None +5 s blind +segmentation, +z-score norm. +None +1D CNN +UofTDB +1019 Yes +IDR +96.1% +Pinto and +Cardoso +[338] +2019 None +5 s blind +segmentation, +z-score norm. +1D CNN +Template +similarity +UofTDB +PTB +CYBHi +1018 +290 +128 +Yes +No +Yes +EER +7.86% +11.0% +16.3% +Ranjan +[359] +2019 BPF 1–20 Hz +Pan-Tompkins, +QRS cropping, +normalisation +Nine QRS +concatenated in 2D +2D CNN +ECG-ID +90 +No +EER +2% + +ECG Biometrics Literature Methods +263 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Zhang +et al. [487] +2019 BPF 1–40 Hz +Blind segm., +z-score norm. +Residual 1D CNN +kNN + majority +voting +PTB +CEBSDB +MIT NSR +234 +20 +18 +No +No +No +IDR +98.7% +99.9% +92.9% +BPF 1–40 Hz +Blind segm., +z-score norm. +Residual 1D CNN +SVM + +majority voting +PTB +CEBSDB +MIT NSR +234 +20 +18 +No +No +No +IDR +99.5% +100% +95.3% +Zhang +et al. [488] +2019 DWT hard +thresholding +3 s blind +segmentation +Recurrence plots +GoogLeNet +CNN +ECG-ID +90 +No +IDR +EER +96.3% +5.82% +Alduwaile +and Islam +[16] +2020 BPF +QRS detection, +beat segm. +1D CNN +Softmax layer +PTB +100 +No +IDR +99.9% +Belo et al. +[36] +2020 Moving average +Blind +segmentation, max. +normalisation +Blind segment +RNN +FANTASIA +MIT NSR + +Arrh. + LT +CYBHi +20 +72 +63 +No +No +Yes +IDR +EER +IDR +EER +IDR +EER +100% +0.02% +92.7% +1.25% +63.5% +4.3% +Moving average +Blind +segmentation, max. +normalisation +Blind segment + +extracted heartbeat +Temporal +Dual-stream +CNN +FANTASIA +MIT NSR + +Arrh. + LT +CYBHi +20 +72 +63 +No +No +Yes +IDR +EER +IDR +EER +IDR +EER +99.1% +0.02% +96.4% +0.08% +100% +0.0% +Bento et al. +[38] +2020 Hann window +filter, mov. avg. +Normalisation +Spectrogram +2D CNN +FANTASIA +ECG-ID +40 +90 +No +No +IDR +99.4% +94.2% +Hann window +filter, mov. avg. +Normalisation +Spectrogram +DenseNet +FANTASIA +ECG-ID +40 +90 +No +No +IDR +99.8% +96.9% +Byeon +et al. [56] +2020 Convolution- +based +denoising +R-peak detection, +beat segmentation +Log-spectrogram, +melspectrogram, +spectrogram, MFCC, +and scalogram +Xception, +ResNet, +DenseNet +ensembles +PTB +290 +No +IDR: +Xcep. +ResN. +Dens. +99.1% +99.0% +99.0% + +264 +ECG Biometrics Literature Methods +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Ingale +et al. [187] +2020 BPF 1–40 Hz +R-peak detection, +beat segmentation +Thirty amplitude and +time fiducial features +or DWT +Euclidean +distance or +DTW +PTB +MIT Arrh. +CEBSDB +CYBHi +ECG-ID +Private +290 +47 +20 +125 +90 +1119 +No +No +No +Yes +No +Yes +IDR +EER +IDR +EER +IDR +EER +IDR +EER +IDR +EER +IDR +EER +100% +2% +100% +4% +100% +0% +100% +0.5% +96.7% +2.3% +100% +1.2% +Jyotishi +and +Dandapat +[209] +2020 HPF 0.5 Hz + +NF 50 Hz + NF +100 Hz +Min-max +normalisation +100 ms sliding +windows +LSTM +PTB +MIT Arrh. +ECG-ID +CYBHi +290 +47 +90 +63 +No +No +No +Yes +IDR +97.3% +96.8% +93.1% +79.4% +Kim and +Pyun [215] +2020 Derivative filter, +moving avg. +Min-max +normalisation, beat +segmentation +Sequence of +heartbeats +Bi-LSTM +MIT NSR +MIT Arrh. +18 +47 +No +No +IDR +F-s. +IDR +F-s. +100% +1.0 +99.8% +0.99 +Lehmann +and +Buschek +[247] +2020 BPF 3–45 Hz +Engelse- +Zeelenberg, beat +segmentation, +outlier removal +QRS fiducial +statistics, feat. +selection with +correlation matrix +RF +SVM +MLP +Private +20 +No +EER +16.7% +25.7% +17.7% +Li et al. +[252] +2020 BPF 1–40 Hz +Pan-Tompkins, +beat segmentation +and avg. ensemble +GNMF +Sparse +representation- +based +matching +ECG-ID +MIT Arrh. +90 +47 +No +No +IDR +98.0% +100% +Li et al. +[255] +2020 BPF 2–50 Hz +Normalisation, +R-peak detection, +beat segmentation, +outlier detection +Segmented heartbeats +Cascaded 1D +CNN +FANTASIA +CEBSDB +MIT NSR +STDB +AFDB +40 +20 +18 +28 +23 +No +No +No +No +No +IDR +99.3% +93.1% +91.4% +92.7% +89.7% + +ECG Biometrics Literature Methods +265 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Pinto et al. +[345] +2020 None +5 s blind +segmentation, +z-score norm. +1D CNN +Template +similarity +UofTDB +1018 Yes +EER +12.6% +Pinto and +Cardoso +[339] +2020 None +5 s blind +segmentation, +z-score norm. +None +1D CNN +PTB +UofTDB +290 +1018 +No +Yes +IDR +97.7% +91.5% +Randazzo +et al. [358] +2020 - +Heartbeat +segmentation +AC/DCT +MLP +Private +5 +Yes +IDR +99.1% +Tirado- +Martin +et al. [433] +2020 BPF 1–35 Hz +R detection, QRS +segmentation +Differentiated QRS +MLP +Private +55 +No +EER +2.69% +Wang et al. +[450] +2020 BPF 1–40 Hz +Pan-Tompkins, +beat segmentation +Multi-Scale +Differential Features +fusion of 1D +Multi-Resolution +LBPs +(MSDF-1DMRLBP) +Euclidean +distance +MIT Arrh. +ECG-ID +PTB +UofTDB +47 +90 +248 +46 +No +No +No +Yes +IDR +EER +IDR +EER +IDR +EER +IDR +EER +94.7% +2.73% +100% +3.3% +98.2% +2.55% +100% +2.17% +Benouis +et al. [37] +2021 Savitzky-Golay +filtering +Pan-Tompkins, +beat segmentation +1D Local Difference +Patterns (1D LDP) +kNN, SVM, +PNN +ECG-ID +90 +No +IDR +EER +93.3% +3.05% +Ibtehaz +et al. [185] +2021 None +U-Net based +R-peak detection, +beat segmentation +1D CNN +Softmax layer +(identification) +or siamese +architecture +(verification) +MIT Arrh. +ECG-ID +PTB +MIT NSR +CYBHi +47 +90 +290 +18 +63 +No +No +No +No +Yes +IDR +EER +IDR +EER +IDR +EER +IDR +EER +IDR +98.2% +6.36% +96.2% +1.29% +99.7% +5.66% +99.5% +5.17% +73.9% + +266 +ECG Biometrics Literature Methods +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Ivanciu +et al. [194] +2021 - +Christov R-peak +detection, beat +segmentation +Heartbeat plot images +Siamese 2D +CNN +ECG-ID +90 +No +IDR +FAR +FRR +Sens. +86.5% +13.7% +12.7% +87.3% +Pinto et al. +[347] +2021 None +5 s blind +segmentation, +z-score norm. +1D CNN +Template +similarity +UofTDB +1018 Yes +EER +12.6% +Srivastva +et al. [412] +2021 BPF 0.5–30 Hz +Pan-Tompkins, +beat segmentation, +z-score norm. +2D heartbeat image +ImageNet +pretrained +DenseNet and +ResNet models +ensemble +PTB +CYBHi +290 +63 +No +Yes +IDR +99.7% +99.7% +Tai et al. +[422] +2021 BPF 1–40 Hz +Normalisation, +R-peak detection, +fixed-window +segmentation +CNN trained with +N-Pair Loss +Euclidean +distance +ECG-ID + +E-HOL 24h ++ YSYW + +MIT NSR +2293 No +AUC +97.0% +Thentu +et al. [431] +2021 - +Pan-Tompkins, +beat segmentation +Multi-scale CWT +representation +Various +ImageNet +pre-trained 2D +networks +CEBSDB +PTB +20 +290 +No +No +IDR +99.9% +99.5% +Tirado- +Martin and +Sanchez- +Reillo +[432] +2021 BPF 1–35 Hz +R-peak detection, +QRS segmentation +CNN +LSTM + +softmax +fully-connected +layer +Private +105 +No +IDR +EER +94.6% +0.04% +Wang et al. +[451] +2021 BPF 1–40 Hz +Pan-Tompkins, +heartbeat +segmentation +Segmented heartbeats +and their STFT +Locality- +preserving +semantic space +learning +MIT Arrh. +PTB +CYBHi +47 +290 +63 +No +No +Yes +IDR +97.8% +98.9% +96.8% +Zhang +et al. [489] +2021 BPF 1–40 Hz +Blind +segmentation, +z-score +normalisation +CNN +SVM +PTB +CEBSDB +MIT NSR +234 +20 +18 +No +No +No +IDR +98.8% +99.7% +93.6% + +ECG Biometrics Literature Methods +267 +Author +Year Denoising +Preparation +Features/DR +Decision +Dataset +NS +OP +Results +Huang +et al. [180] +2022 DWT hard- +thresholding +denoising +Pan-Tompkins, +heartbeat +segmentation, +normalisation +AC/DCT and Wavelet +LBP histogram matrix +factorisation +Euclidean +distance +matching +MIT Arrh. +PTB +CYBHi +UofTDB +47 +248 +63 +46 +No +No +Yes +Yes +IDR +EER +IDR +EER +IDR +EER +IDR +EER +95.7% +0.36% +86.3% +2.26% +88.5% +1.48% +87.4% +2.36% +Kim et al. +[216] +2022 - +Pan-Tompkins, +beat segmentation +Segmented heartbeats +AlexNet CNN +with heart-rate +based data +augmentation +ECG-ID +83 +No +IDR +95.7% +Lee and +Kwak +[245] +2022 - +- +ICA convolutional +network with 2D +scalogram +representations +SVM +Private +MIT Arrh. +95 +47 +No +No +IDR +98.5% +96.0% +Li et al. +[251] +2022 DWT adaptive +soft- +thresholding +Second-order diff. +R-peak detection, +beat segmentation +Uniform Manifold +Approximation and +Projection (UMAP), +Stacked Extremely +Randomised Trees +(ETs) +XGBoost +ECG-ID +PTB +89 +71 +No +No +IDR +98.9% +95.8% + + +Referências +[1] M. Abadi, A. Agarwal, P. 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In IEEE EMBS Asian- +Pacific Conference on Biomedical Engineering, pages 48–49, Kyoto, Japan, Oct. 2003. +doi: 10.1109/APBME.2003.1302577. + diff --git a/ytE1T4oBgHgl3EQfQwPe/content/tmp_files/load_file.txt b/ytE1T4oBgHgl3EQfQwPe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..099dceb064b29ac3efa9038a82186e8856269b89 --- /dev/null +++ b/ytE1T4oBgHgl3EQfQwPe/content/tmp_files/load_file.txt @@ -0,0 +1,24287 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf,len=24286 +page_content='Seamless Multimodal Biometrics for Continuous Personalised Wellbeing Monitoring João Tiago Ribeiro Pinto Doctoral Programme in Electrical and Computer Engineering Supervisor: Professor Jaime dos Santos Cardoso Co-Supervisor: Professor Miguel Velhote Correia December 21, 2022 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='03045v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='CV] 8 Jan 2023 © João Tiago Ribeiro Pinto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Seamless Multimodal Biometrics for Continuous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Personalised Wellbeing Monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='João Tiago Ribeiro Pinto ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Doctoral Programme in Electrical and Computer Engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Approved in public examination by the Jury: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='President: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Professor Luís Miguel Pinho de Almeida ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Referee: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Professor Hugo Filipe Silveira Gamboa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Referee: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Professor Armando José Formoso de Pinho ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Referee: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Professor Jaime dos Santos Cardoso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Referee: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Professor Ana Maria Rodrigues de Sousa Faria de Mendonça ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Referee: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Professor Luís Filipe Pinto de Almeida Teixeira ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='December 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2022 Resumo A perceção através da inteligência artificial está cada vez mais presente nas nossas vidas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Os veículos não são exceção, uma vez que sistemas avançados de assistência ao condutor auxiliam no cumprimento de limites de velocidade, na manutenção dentro das faixas e na prevenção de acidentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Num futuro próximo, o reconhecimento de padrões terá um paper ainda mais prepon- derante nos veículos, uma vez que o automóvel autónomo necessitará de meios automáticos para compreender o que acontece ao seu redor (e no seu interior) para agir de forma adequada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Em reconhecimento de padrões, a biometria oferece aplicações promissoras para veículos, do acesso keyless à personalização automática de opções de condução com base no condutor recon- hecido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' De igual modo, as tecnologias de reconhecimento de bem-estar têm atraído atenção pela possibilidade de reconhecer atividade, emoções, sonolência ou stress em condutores e passageiros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' No entanto, estes dois tópicos são diametralmente opostos, uma vez que o reconhecimento de bem- estar usa a variabilidade intra-sujeito, enquanto a biometria se baseia na variabilidade inter-sujeito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Apesar das diferenças, a biometria e o reconhecimento de bem-estar poderiam (e deveriam) co-existir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' O reconhecimento contínuo de identidade em dados adquiridos de forma impercetível poderiam ser usados para personalizar modelos de reconhecimento de bem-estar e obter melhor desempenho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Estes modelos personalizados poderiam ser a chave para meios mais robustos de monitorizar sonolência e atenção em condutores e evitar acidentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Num sentido mais amplo, estes poderiam ser aplicados a todos os ocupantes, abrindo o caminho em direção ao reconheci- mento eficaz de atividade, emoções, conforto e até episódios de violência em veículos autónomos partilhados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Este doutoramento focou-se em avançar o tópico de perceção em veículos através do es- tudo de novas metodologias de visão computacional e reconhecimento de padrões para biome- tria e reconhecimento de bem-estar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' O foco principal foi na biometria com electrocardiograma (ECG), um traço reconhecido pelo seu potencial em monitorização impercetível de condutores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Esforços foram dedicados à obtenção de desempenho melhorado em identificação e verificação de identidade em cenários off-the-person, reconhecidos pelo elevado teor de ruído e variabilidade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Aqui, foram propostas soluções deep learning end-to-end e analisados tópicos importantes como o desempenho cross-database e a longo prazo, a importância relativa das ondas através da inter- pretabilidade, e a conversão entre canais.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A biometria com face, um complemento natural ao ECG em cenários impercetíveis, foi tam- bém estudada nesta tese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Os desafios em reconhecimento de faces com máscaras e na interpretabil- idade em biometria foram abordados com o intuito de avançar para algoritmos mais transparentes, confiáveis e robustos a oclusões significativas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Dentro do tópico de reconhecimento de bem-estar, foram propostas soluções melhoradas para o reconhecimento multimodal de emoções em grupos de pessoas e de atividade/violência dentro de veículos partilhados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Por fim, foram propostos ainda uma forma inovadora de aprender segurança de templates em modelos end-to-end, evitando pro- cessos adicionais de encriptação, e um método auto-supervisionado adaptado a dados sequenciais, para garantir segurança de dados e desempenho otimizado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' i ii Segundo os resultados deste trabalho, é possível concluir que o ideal de reconhecimento per- sonalizado de bem-estar está ainda por atingir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' No entanto, este trabalho construiu uma base sólida para suportar trabalho futuro em direção à integração da biometria com o reconhecimento de bem- estar de forma multimodal, impercetível, contínua e realista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Em geral, este doutoramento levou a múltiplas contribuições para os tópicos de biometria e reconhecimento de bem-estar, resultando diretamente em vinte e quatro publicações científicas em fóruns de renome em biometria e recon- hecimento de padrões.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A sua qualidade e impacto foram reconhecidas pela comunidade científica com mais de trezentas citações e múltiplos prémios, incluindo o prémio EAB Max Snijder 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Palavras-chave: Aprendizagem Computacional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Atividade;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Áudio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Biometria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Electrocardio- grama;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Emoção;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Face;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Monitorização de Bem-Estar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reconhecimento de Padrões;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Processamento de Sinal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Veículos Autónomos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Vídeo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Visão Computacional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Abstract Artificially intelligent perception is increasingly present in the lives of every one of us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Vehicles are no exception, as advanced driver assistance systems (ADAS) help us comply with speed limits, keep within the lanes, and avoid accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the near future, pattern recognition will have an even stronger role in vehicles, as self-driving cars will require automated ways to understand what is happening around (and within) them and act accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Within pattern recognition, biometrics offer promising applications in vehicles, from keyless access control to the automatic personalisation of driving and environmental conditions based on the recognised driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Similarly, wellbeing monitoring technologies have long attracted attention to the possibility of recognising activity, emotions, sleepiness, or stress from drivers and pas- sengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, these two topics are starkly opposed, since wellbeing recognition relies on intrasubject variability while biometrics thrives on intersubject variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite their differences, biometric recognition and wellbeing monitoring could (and should) coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Continuous identity recognition from seamlessly acquired data could be used to person- alise wellbeing monitoring models and attain improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These personalised models could be the key to more robust ways of monitoring drivers’ drowsiness and attention and avoid- ing accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In a broader sense, they could be applied to all vehicle occupants, paving the way towards the accurate recognition of activity, emotions, comfort, and even violence episodes in shared autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This doctoral work focused on advancing in-vehicle sensing through the research of novel computer vision and pattern recognition methodologies for both biometrics and wellbeing moni- toring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The main focus has been on electrocardiogram (ECG) biometrics, a trait well-known for its potential for seamless driver monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Major efforts were devoted to achieving improved performance in identification and identity verification in off-the-person scenarios, well-known for increased noise and variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Here, end-to-end deep learning ECG biometric solutions were proposed and important topics were addressed such as cross-database and long-term performance, waveform relevance through explainability, and interlead conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Face biometrics, a natural complement to the ECG in seamless unconstrained scenarios, was also studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The open challenges of masked face recognition and interpretability in biometrics were tackled in an effort to evolve towards algorithms that are more transparent, trust- worthy, and robust to significant occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Within the topic of wellbeing monitoring, improved solutions to multimodal emotion recognition in groups of people and activity/violence recognition in in-vehicle scenarios were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' At last, we also proposed a novel way to learn template security within end-to-end models, dismissing additional separate encryption processes, and a self-supervised learning approach tailored to sequential data, in order to ensure data security and optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Following the results of this work, one can conclude that truly personalised wellbeing is yet to be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, this work has built a strong framework to support future work towards the goal of integrating biometric recognition and wellbeing monitoring in a multimodal, seamless, iii iv continuous, and realistic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Overall, this doctoral work led to numerous contributions to biomet- rics and wellbeing monitoring in general, resulting directly in twenty-four scientific publications in major biometrics and pattern recognition venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Its quality and impact have been recognised by the scientific community with over three hundred citations and multiple awards, including the EAB Max Snijder Award 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Keywords: Activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Audio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Autonomous Vehicles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Biometrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Computer Vision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Electrocardio- gram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Emotion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Face;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Machine Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pattern Recognition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Signal Processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Video;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Wellbe- ing Monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Acknowledgements Doutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Finalmente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Volvidos cinco anos desde o início do meu percurso em investigação, a es- crita deste documento dá-me a oportunidade de reviver os sucessos, fracassos, ideias, desafios, pessoas e momentos que marcaram este meu doutoramento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Foi uma excelente aventura, mas não foi fácil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As vozes do impostor syndrome fizeram-me duvidar se realmente teria a inteligên- cia e a capacidade suficientes para completar um doutoramento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Foi inevitável, por vezes, tentar comparar com os doutoramentos de outros, o que frequentemente resultou em desilusão.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mas fui aprendendo a evitar os obstáculos no caminho e a compreender os avisos de quem já passara por eles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Diziam que um doutoramento não é um sprint, mas sim uma maratona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Que não re- quer capacidades extraordinárias, mas sim resiliência, motivação e determinação.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Duvidei, mas depois deste longo percurso reconheço a veracidade dessa afirmação.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Podemos duvidar da nossa inteligência e capacidades, mas o importante é continuar, e insistir, e procurar até encontrar a meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Com tempo percebi que cada doutoramento é único, e cada um tem o seu caminho a traçar, com desafios e dificuldades específicos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A comparação com os doutoramentos dos outros, apesar de inevitável, será sempre incompleta e injusta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Aprendi ainda que um doutoramento completo deverá ir bem para além da investigação científica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Não devemos ser apenas “máquinas de fazer artigos”, mas sim procurar explorar todas as outras vertentes de um investigador completo, como o ensino, a mentoria, as colaborações, e a organização de eventos científicos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' É natural que rapidamente esgotemos as horas do dia (e a nossa energia) quando nos dividimos entre tantas atividades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Foram frequentes as longas noites de trabalho e os fins-de-semana que não o foram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mas considero que encontrei o verdadeiro caminho para um bom doutoramento, e fico feliz por ter decidido segui-lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Foi perfeito?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Não.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Em retrospetiva, reconheço escolhas menos certas, certamente frutos da minha inexperiência, e imensos caminhos alternativos que teriam sido mais proveitosos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Relembro com pesar as ideias que faziam sentido mas não funcionaram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E imagino o potencial de tantas outras que não foram além de um rabisco esquecido numa folha qualquer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mas não estou só.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ainda não conheci um único aluno de doutoramento que tenha chegado ao fim do seu trajeto plenamente contente e verdadeiramente confiante no resultado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Resta a ténue calma na ideia de que esta tese de doutoramento não é a minha “obra-prima” nem aquilo que irá definir toda a minha carreira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mas sim, apenas, um livro de esboços, uma coleção de rascunhos de alguém que tropeçou, escorregou e errou inúmeras vezes durante quase cinco anos na tentativa de mapear um caminho inexplorado e, com sorte, tornar-se um investigador completo e autónomo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mudaria alguma coisa?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Também não.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Suponho que teria conseguido publicar mais artigos se não tivesse dividido a minha atenção entre tantas atividades paralelas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Talvez tivesse um trabalho com mais impacto se não tivesse orientado tantas teses de mestrado e estágios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Deveria talvez ter dado prioridade à importante experiência de dar aulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mas tudo isto é incerto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Certos são alguns momentos marcantes que deram um brilho especial a estes cinco anos de esforço e dedicação.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Relembro o convívio e a partilha de conhecimento no BTAS, no IbPRIA, no BIOSIG, nos vários RECPADs e em tantas outras conferências.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' O prazer de poder contribuir para a organização do IWBF 2020 e dos workshops xAI4Biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' O orgulho em ver o meu esforço reconhecido pela v vi comunidade europeia de biometria através do prémio Max Snijder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' O entusiasmo ao ver (e poder demostrar a tantas pessoas) o meu trabalho a funcionar, ao vivo, no fim do projeto Easy Ride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A felicidade ao assistir a cada uma das provas públicas dos meus alunos de mestrado, concluindo com sucesso as suas teses, depois do privilégio de os acompanhar ao longo de um ano de aprendizagem, dedicação, esforço e evolução.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A honra enorme de ser o escolhido da Inês, da Sofia e do Duarte para os cartolar na Imposição das Insígnias, um momento tão solene e significante a marcar o fim dos seus caminhos pela academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A satisfação dos últimos dias de cada VISUM, onde depois de tanto trabalho nos damos conta do imenso valor da nossa escola de verão (e ainda a minha estranha proficiência na apresentação de quizzes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E ainda, a amizade e entreajuda que encontrei todos os dias no grupo VCMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ao relembrar as pessoas encontradas e estes momentos vividos, chego à conclusão que não mudaria absolutamente nada, no receio de que perdesse sequer um deles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E afinal, talvez o meu doutoramento não esteja assim tão longe da perfeição.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Por tudo isto tenho a agradecer, em primeiro lugar, ao Professor Jaime Cardoso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' O melhor professor que tive a sorte de conhecer durante o meu percurso académico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ao aproximar-se o fim do meu mestrado, muitas foram as dúvidas relativamente à possibilidade de um doutoramento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Há muito que era algo que almejava fazer, mas a magnitude da tarefa impunha respeito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recebi muitas e variadas opiniões sobre o doutoramento e como fazer um com qualidade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' No meio de expectável discórdia, um ponto de consenso: “o orientador é, de longe, o mais importante”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mais vale o orientador certo numa universidade qualquer que o orientador errado na Ivy League, diziam em uníssono.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Estavam certos, todos eles, e sem dúvida estava certo eu também quando escolhi fazer o doutoramento consigo, Professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Um verdadeiro exemplo de integridade, profunda dedicação, impressionante inteligência e contagiante amor pelo que faz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Apesar das diferenças em experiência e currículo, tão comumente enfatizadas na academia, sempre me deixou à vontade para expressar livremente todas as ideias, dúvidas e problemas como se fosse um simples colega.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E tudo isto vale o mundo para um aluno de doutoramento a percorrer a sua maratona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recordo-me de certos momentos, como quando chegou a Professor Catedrático, em que me senti tão alegre como se tivesse sido eu próprio a conseguir essa conquista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E sei que não fui o único.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Deixar tal marca nos alunos é prova definitiva da sua capacidade, dedicação e entrega como Professor e orientador.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Desejo apenas que todos os seus objetivos se continuem a cumprir, com sucesso e felicidade, e que eu possa continuar a colaborar consigo e a tê-lo como meu mentor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ao Professor Miguel Velhote Correia, por ter aceitado o desafio de fazer parte deste projeto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Apesar dos nossos planos ambiciosos para este doutoramento, acabei por não ter muitas oportu- nidades para colaborar consigo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' No entanto, fico feliz por ter conseguido que deixasse a sua marca naquele que considero ser o meu melhor trabalho durante este doutoramento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Continuo a acreditar no valor da investigação em instrumentação para o futuro da biometria com electrocardiograma, e espero um dia voltar a poder contar com o seu conhecimento e a sua experiência neste e em outros tópicos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ao grupo VCMI e aos que dele fizeram parte e contribuíram para a sua história, plena de inúmeros sucessos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Tal como ninguém vive numa bolha, também ninguém faz um doutoramento sozinho (apesar de ser uma aventura bastante solitária).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recordo a afirmação atribuída a Isaac Newton - “if I have seen further, it is by standing on the shoulders of giants”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Os que por aqui passaram antes de mim abriram-me e mostraram-me o caminho com os seus sucessos, e aqueles com os quais tive o prazer de percorrer esta jornada ajudaram-me a encontrar novas oportunidades e a aprender com diversos desafios em biometria e tantos outros tópicos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Não teria certamente conseguido metade do que consegui neste doutoramento se não tivesse como alicerce este grupo de verdadeiros gigantes, repleto de genialidade, capacidade e dedicação.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Entre eles, um agradeci- mento especial ao Professor Jaime pela criação e constante dedicação ao grupo e à Filipa Sequeira pelos excelentes esforços recentes para aumentar a colaboração do subgrupo de biometria com vii a comunidade internacional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Obrigado por fazerem do grupo VCMI um símbolo de qualidade e excelência em biometria além-fronteiras, e por encherem o meu doutoramento de desafios, opor- tunidades, sucessos e ainda muitos momentos de felicidade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sem dúvida, não podia ter escolhido um melhor grupo para o meu doutoramento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' À VISUM, indubitavelmente a melhor escola de verão por esse mundo fora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Obrigado pela oportunidade de organizar estes momentos de confluência entre culturas e aprendizagem, com tal magnitude e visibilidade internacional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Agradeço a todos os que deram um pouco de si para tornar a VISUM possível (e excelente), em especial à Ana, à Sara, e ao Wilson, que sacrificaram bem mais que todos os restantes e puxaram a nossa escola de verão para uma 10ª edição que ficará para a história.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cada edição foi única, mas foram todas fantásticas, e fico feliz de ter tido a oportunidade de fazer parte de quatro delas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A todos os que confiaram em mim para co-orientar as suas tese de mestrado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ao Gabriel, à Carolina, ao Leonardo, ao Arthur, ao João, à Inês, à Sofia, à Telma, à Mariana, ao Duarte, ao Guilherme, ao Vítor e ao Erfan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cada um aceitou um tema único, com objetivos e dificuldades diferentes, e cada um o abordou com perspetivas e ideias diversas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mas todos estiveram em sin- tonia na vontade de aprender, na dedicação à procura do caminho certo, e no ânimo firme mesmo em momentos de maior dificuldade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sei que fui excecionalmente sortudo em ter tantos e tão ex- celentes alunos durante estes anos de doutoramento, e adorei trabalhar com cada um de vocês.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Espero que tenham também gostado de trabalhar comigo (ou que pelo menos não se arrependam da vossa escolha) e que eu tenha estado à altura das vossas expectativas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Espero ainda que tenham aprendido algo comigo, tal como eu aprendi com cada um de vocês.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Obrigado pelas vossas exce- lentes contribuições para este projeto, que na verdade também é um pouco vosso, e lembrem-se que continuarei aqui para vos ajudar, sempre que precisarem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A todos aqueles que colaboraram comigo durante este doutoramento, tanto em temas rela- cionados com biometria como naqueles que me permitiram alargar os meus horizontes e adquirir conhecimentos em tópicos diferentes, incluindo os projetos AUTOMOTIVE, Easy Ride e Au- rora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A todos na CardioID Technologies, em especial ao André, ao Carlos, ao Roberto, ao Pedro e ao Lourenço, pela ajuda em biometria com ECG e no projeto AUTOMOTIVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' À Bosch Car Multimedia, em especial ao Joaquim, ao Filipe, à Carolina, ao Ricardo, à Margarida, ao Niklas e ao Jochen pela excelente colaboração em in-vehicle monitoring no projeto Easy Ride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Aos da Fraunhofer IGD, em especial ao Fadi, ao Naser, ao Florian, ao Marco, ao Juan e à Meiling pela colaboração em masked face recognition e ainda pela calorosa receção em Darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Entre to- dos, um agradecimento especial ao André Lourenço pela colaboração, apoio e amizade já desde a minha tese de mestrado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Espero um dia poder voltar a colaborar com todos vós.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E por fim, mas certamente não em último lugar, à minha família e aos meus amigos, por tudo o resto que fez de mim o que sou hoje e que me ajudou a chegar aqui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Obrigado a todos, João Tiago viii Funding This work was financed by the Portuguese science and technology foundation, Fundação para a Ciência e a Tecnologia – FCT – and co-financed by the European Social Fund through the North Regional Operational Programme (NORTE 2020), under the grant “SFRH/BD/137720/2018”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Data The author wishes to thank the creators, contributors, and administrators of all databases and data collections used in the research work presented in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ECG signal databases and collections: the author acknowledges the creators of the PTB [49] and PTB-XL [443;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 444] databases at the Physikalisch-Technische Bundesanstalt, Germany, the cre- ators of the University of Toronto ECG Database (UofTDB) [445] at the University of Toronto, Canada, the creators of the Check Your Biosignals Here initiative [394] at Instituto de Teleco- municações, Portugal, the creators of the INCART database at the St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Petersburg Institute of Cardiological Technics, Russia, as well as the creators and administrators of the Physionet online data repository [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E-HOL-03-0202-003 (E-HOL) Data used for this research was provided by the Telemetric and Holter ECG Warehouse of the University of Rochester (THEW), NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Face biometric databases and collections: the author wishes to acknowledge the creators of the YouTube Faces [460] database at Tel Aviv University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Israel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the creators of the ROSE Youtu [249] database at the Nanyang Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the creators of the MFRC-21 [50] database at Fraunhofer IGD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Germany,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the creators of the VGGFace2 [58] at the University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the creators of the MS1MV2 [98] dataset at Imperial College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' and the creators of the Labelled Faces in the Wild (LFW) [178] dataset at the University of Massachusetts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Other databases: the author wishes to acknowledge the EmotiW 2020 Grand Challenge [101] organisers, and the authors of the Multi-Moments in Time (MMIT) [300] dataset and pretrained models at the Massachusetts Institute of Technology, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Credits All previously published copyrighted content reused, reprinted, or adapted in this thesis is appro- priately referenced and acknowledged and has been licensed as detailed below: Content in Chapter 2: Adapted from João Ribeiro Pinto, “Continuous Biometric Identification on the Steering Wheel”, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thesis, University of Porto, Portugal, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='12 was reprinted from Cognition, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 121, Rob Jenkins, David White, Xandra Van Montfort, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mike Burton, “Variability in photos of the same face”, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 313–323, Copyright 2011, with permission from Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 3 and Appendix A: © 2018 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reprinted, with permission, from João Ribeiro Pinto, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, André Lourenço, “Evolution, Current Challenges, and Future Possibilities in ECG Biometrics”, IEEE Access, June 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fundagao FCT REPUBLICA ORTUGAI UNIAO EUROPEIA NORTE2O2O **** CENSIN TEUNRIORIA para a Ciencia PORTUGUESA e a Tecnologia ***** ROGRAMA OPERACIONAL REGIONAL DO NORTix Content in Chapter 4 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Reproduced with permission of Taylor and Francis Group LLC (Books) US through PLSclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 5: © 2019 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reprinted, with permission, from João Ribeiro Pinto, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “An End-to-End Convolutional Neural Network for ECG-Based Biometric Authentica- tion”, 2019 IEEE 10th International Conference on Biometrics Theory, Applications and Systems (BTAS), September 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 6: Reprinted/adapted by permission from Springer Nature: Springer Na- ture “Don’t You Forget About Me: A Study on Long-Term Performance in ECG Biometrics” by Gabriel Lopes, João Ribeiro Pinto, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso © 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 9: Figures 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 were reprinted from Neurocomputing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 429, Mei Wang, Weihong Deng, “Deep face recognition: A survey”, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 30, Copyright 2021, with permis- sion from Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 10: © 2021 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reprinted, with permission, from Pedro C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neto, Fadi Boutros, João Ribeiro Pinto, Mohsen Saffari, Naser Damer, Ana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “My Eyes Are Up Here: Promoting Focus on Uncovered Regions in Masked Face Recognition”, 2021 International Conference of the Biometrics Special Interest Group (BIOSIG), September 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 12: © 2020 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reprinted, with permission, from João Ribeiro Pinto, Tiago Gonçalves, Carolina Pinto, Luís Sanhudo, Joaquim Fonseca, Filipe Gonçalves, Pedro Carvalho, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Audiovisual Classification of Group Emotion Valence Using Activity Recogni- tion Networks”, 2020 IEEE 4th International Conference on Image Processing, Applications and Systems (IPAS), December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 14: © 2021 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reprinted, with permission, from João Ribeiro Pinto, Miguel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Correia, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Secure Triplet Loss: Achieving Cancelability and Non- Linkability in End-to-End Deep Biometrics”, IEEE Transactions on Biometrics, Behavior, and Identity Science, April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Content in Chapter 15: © 2020 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reprinted, with permission, from João Ribeiro Pinto, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Self-Learning with Stochastic Triplet Loss”, 2020 International Joint Conference on Neural Networks (IJCNN), July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' “In theory, there is no difference between theory and practice, while in practice, there is.” Benjamin Brewster xi Contents I Prologue 1 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Signal denoising .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 55 xiii xiv CONTENTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Feature extraction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 97 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Handcrafted methodologies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 97 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Deep convolutional network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 107 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 127 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 127 III Face Biometrics 133 9 Prior Art in Face Biometrics 135 9.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 164 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Conducted Studies on PAD Interpretability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Representation of a model’s explanations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 165 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Comparison of explanations across different scenarios .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 166 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Interclass comparison in the unseen-attack scenario .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 168 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Intraclass comparison across different samples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 169 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 170 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Performance of the face PAD algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 170 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Comparison of explanations across different scenarios .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 170 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Interclass comparison in the unseen-attack scenario .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 175 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Intraclass comparison across different samples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 176 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 177 IV Wellbeing Monitoring 179 12 Emotion Valence Classification in the Wild 181 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 181 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Methodology .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 182 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 General overview .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 186 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Experiments .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 187 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results and Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 187 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Summary and Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 202 CONTENTS xvii 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Summary and Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 203 V Broader Topics on Biometrics and Pattern Recognition 205 14 Learning Template Security on End-to-End Biometric Models 207 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 212 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 ECG identity verification .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 212 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Face identity verification .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 214 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Evaluation frameworks and metrics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 214 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 216 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Verification performance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 217 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Cancelability evaluation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 221 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Unlinkability evaluation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 221 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Non-invertibility and secrecy leakage .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 235 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Stress experiments .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 238 VI Epilogue 239 16 Summary and Conclusions 241 17 Future Work Considerations 245 xviii CONTENTS VII Appendices 249 A ECG Biometrics Literature Methods 251 References 269 List of Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Schema of the topics covered during this doctoral project, their interconnec- tions, and their link to the central topic of personalised wellbeing monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 General structure of a biometric recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 The sequence of depolarisation and depolarisation events in the heart, and their relationship with the different heartbeat waveforms in an ECG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Medical acquisition settings: electrode placement and leads on the standard 12- lead configuration and Frank leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Acquisition settings with movement: example of a five-electrode Holter system for ambulatory recordings.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Examples of off-the-person ECG acquisition configurations, using thumb elec- trodes, index finger electrodes, metallic rods grabbed by the subjects, or elec- trodes mounted on a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Wearable and seamless acquisition: examples of surveyed configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9 Variability in off-the-person ECG heartbeats from the same subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Results of the proposed algorithm receiving raw five-second segments, with each technique of data augmentation, on the datasets of 25 and 100 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Results of the proposed algorithm, receiving raw five-second segments as input, with combinations of data augmentation techniques.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9 Direct benchmarking between the proposed architecture with the best baseline algorithm and the two implemented state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Schemes of the proposed identity verification model, including the weight trans- fer between networks for both proposed training methodologies.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 86 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Cross-database scenario: EER for the proposed methods IT-CNN and TL-CNN when trained with UofTDB data and directly applied to CYBHi or PTB, and comparison with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 86 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Fine-tuning scenario: EER results for the proposed methods IT-CNN and TL- CNN when directly trained with CYBHi or PTB data from 20 subjects, and comparison with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Fine-tuning scenario: EER results for the proposed methods when (DT) trained, from scratch, with data from CYBHi or PTB, or when (FT) trained with UofTDB data and fine-tuned to CYBHi/PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 87 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Dendrogram representing the taxonomy of template update techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Illustration of the search for the ideal threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Schema illustrating the use of each E-HOL record for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 97 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Identification performance over time corresponding to (a) the La- bati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' method, and (b) the implemented state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 98 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Comparison of the FIFO method applied with different thresholds to different identification methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 99 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Results using FIFO update with different thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 99 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Results using Fixation update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 101 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Illustration of the ECG waveforms on a sample PTB signal segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 104 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Architecture of the biometric identification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 106 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Explanations over an example five-second ECG segment from PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 109 LIST OF FIGURES xxi 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Explanations over an example five-second ECG segment from UofTDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 109 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Average explanations over heartbeat waveforms of subjects #1 and #2 on the subsets of the PTB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 122 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Example result of lead I to all conversion on the PTB test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 123 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Example cross-database result of lead II to all conversion on INCART.' metadata={'source': 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PTB-XL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 131 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Stages of a biometric recognition algorithm based on face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 138 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Examples of face detection in unconstrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 152 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Explanations obtained for each trained model with the Smooth Grad-CAM++ explainability tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} 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on the evalua- tion scenario x, and fixing Attack #i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 169 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Pairwise comparison of explanations produced by the models in unseen-attack scenarios, for the presentation attack sample Ik of type #i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 169 xxii LIST OF FIGURES 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Image Average mean and standard deviation (StD) results for bona fide (BF) and presentation attack (PA) samples in the comparison across the one-attack (OA) and unseen-attack (UA) scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 172 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Comparison of explanations in intraclass one-attack for an example PA sample of type 5 presenting high Iµ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 195 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Diagram of the visual submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' 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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 201 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Cascade results for the 21 selected classes from the MMIT database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 203 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Cascade results in the in-vehicle scenario with 42 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Detection Error Tradeoff (DET) curves for the face identity verification model when trained with the original triplet loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the proposed formulations of the Secure Triplet Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 238 List of Tables 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Main benefits and drawbacks of different biometrics traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Definition of the commonly used metrics for performance evaluation in identi- fication and identity verification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Summary of the technical specificities of the most relevant publicly available ECG collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Results of surveyed approaches evaluated with PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Results of surveyed approaches evaluated with ECG-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results of surveyed approaches evaluated with MIT-BIH NSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results of surveyed approaches evaluated with MIT-BIH Arrhythmia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Results of surveyed approaches evaluated with UofTDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Results of surveyed approaches evaluated with CYBHi.' 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state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 132 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Details on the main face recognition databases that are currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 136 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Results with the adapted triplet loss on synthetic masked face data (SMFD).' metadata={'source': 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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 154 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Ablation results with the multi-task contrastive learning approach on the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 171 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Accuracy (%) of the proposed method on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 188 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Accuracy (%) of the proposed method on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 188 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Confusion matrix of audio-based recognition on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 188 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Confusion matrix of video-based recognition on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Confusion matrix of multimodal recognition on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 188 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Accuracy (%) on the validation set for videos of small groups vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' large groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 189 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Summary of the accuracy (%) results obtained in the various experimental sce- narios.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 200 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Summary of the size, total number of parameters, and average run times per instance of the three pipeline submodules for the in-lab scenario.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 202 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Summary of the test results for ECG identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 216 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Summary of the test results for face identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 252 Abbreviations 1D Unidimensional 2D Bidimensional 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5D Bidimensional with Depth Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3D Tridimensional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='AC Autocorrelation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ADAS Advanced Driver Assistance Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='AE Autoencoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='AHA American Heart Association (dataset) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='AMAE Average Mean Absolute Error ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Var Variance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='VCMI Visual Computing and Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Intelligence (research group) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='VGAF Video-level Group Affect (dataset) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='VGG Visual Geometry Group (dataset) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='VISAPP International Conference on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Computer Vision Theory and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='VGA Video Graphics Array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='WACV Winter Conference on Applications of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Computer Vision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='WDIST Wavelet Distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='WWPRD Wavelet-Weighted Percent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Root-Mean-Square Difference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='xaFCM Extended-alphabet Finite-Context ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='xAI4Biometrics Explainable Artificial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Intelligence for Biometrics (workshop) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Part I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Prologue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Chapter 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Introduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation The interactions between humans and machines are increasingly mediated by intelligent sensing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nowadays, the default way to unlock a smartphone is using face or fingerprint bio- metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Highly-populated countries like India and China are building unprecedentedly massive national identity networks relying entirely on biometric characteristics (and dealing with the so- cietal intricacies of such colossal endeavours) [198;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 257;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 404].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sophisticated algorithms monitor our attention levels while we drive [119;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Meanwhile, the gaming and entertainment indus- tries are investing heavily in using affective computing to continuously personalise and enhance user experience [24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 97;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among all of these applications, few have been so revolutionised (and so quickly) as vehicles, and there are plenty of great reasons for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Whether in personal cars or public transport, people can typically spend up to multiple hours of their day inside vehicles, especially those with long daily commutes or those living/working in large urban centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, a considerable fraction of preventable deaths and serious injuries occur in accidents involving vehicles, caused mainly by driver fatigue or the influence of psychotropic substances [370;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 399].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While we wait for fully autonomous vehicles, advanced driver assistance systems (ADAS) based on artificial intelligence help drivers comply with speed limits, stay in their lanes, beware of their blind spots, and avoid accidents [81;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 82;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 410].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' One of the most interesting applications of this is drowsiness detection, where biometric data such as face video or physiological signals can be used to detect episodes of sleepiness and infer the fatigue levels of the driver [116;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 320;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 399].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When done continuously (or at least frequently) during vehicle usage, it can be used to warn the driver or even trigger the automatic safe interruption of the vehicle’s operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Drowsiness, just like most other wellbeing parameters, reveals itself on biometric data as in- trasubject variability: the way a person’s data varies over time and across diverse conditions [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This stands in stark opposition to biometric recognition applications, which rely on intersubject variability (the way a person’s data differs from another’s) to distinguish identities [213;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In 3 4 Introduction biometric recognition, intrasubject variability is typically regarded as a nuisance, as it blurs the boundaries between individuals’ data and hampers accurate and robust identity recognition [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In spite of their differences, biometric recognition and wellbeing monitoring can coexist in a vehicle scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Applications of biometric recognition for vehicles are typically focused on access control or the automatic personalisation of driving and environmental conditions (such as seat position, mirror adjustments, or infotainment settings) based on the recognised driver and occupants [91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, in this thesis, we defend that this coexistence could (and should) be intensified, and eventually evolve to a level of symbiotic integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite the efforts devoted to wellbeing monitoring technologies, one challenge remains un- vanquished: the fact that wellbeing patterns are deeply personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For example, no two individuals experience drowsiness in the exact same way, and similar levels of fatigue can have dramatically different effects on each person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is also true for the way wellbeing parameters reflect on bio- metric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A drowsiness monitoring system can present acceptable accuracy levels for a given set of subjects and, simultaneously, be inadequate at recognising the specific drowsiness patterns of other people [116;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 399].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Biometric recognition could be the key to unlocking the next generation of wellbeing mon- itoring technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Instead of using generalist models, identity predictions obtained through biometric recognition would enable the selection of specific models for each of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These models could also benefit from the influx of new data, continuously learning the subject-specific patterns of wellbeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, combining biometrics with wellbeing could enable the creation of more accurate and robust solutions for wellbeing monitoring [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the future, fully autonomous driving may put an end to the need for drivers, but not to the need for automatic intelligent sensing technologies inside vehicles [23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 302;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 348].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The same biometric algorithms and wellbeing monitoring systems that once were focused on the driver may easily be adapted to target the occupants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In fact, self-driving vehicles unveil a new scenario for automatic passenger monitoring: since there is no driver, shared autonomous vehicles (such as autonomous taxis) lack an authority figure, responsible for the integrity of the vehicle and the comfort and security of the passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The driver could be replaced by pattern recognition solutions to monitor the shared vehicle interior and its passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Overall, the advantages of introducing intelligent sensing technologies in vehicles are plenty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From a narrower subject-centric perspective, integrating the use of inter and intrasubject variability would enable the development of continuously personalisable models for more robust monitoring of wellbeing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From a broader perspective, taking advantage of seamless multimodal data acquisitions for automatic monitoring of the interior and passengers is of utmost importance to ensure security and comfort inside autonomous shared vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Objectives This doctoral work focused on advancing in-vehicle sensing technology through the conceptual- isation and development of novel computer vision and pattern recognition methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Objectives 5 goal has been to create automatic solutions for in-vehicle monitoring, robustly and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To achieve this, we targeted two specific scenarios, as follows: Personalised wellbeing monitoring systems using biometrics: Here, the objective was to advance biometric recognition technologies to be integrated with wellbeing monitoring methodologies, especially for driver assistance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' We build upon the ECG biometrics research conducted in [337], with additional work on face recognition and other impor- tant topics such as biometric security and learning from data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With this work, we aimed to pave the way towards a robust multimodal system to recognise the driver using ECG and face information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The automatic identity predictions enable the use of the drivers’ data to continuously learn their personal patterns of wellbeing for more accurate monitor- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Within wellbeing, this research work focused on driver drowsiness and emotions, as part of the AUTOMOTIVE project, but it stands to reason that biometrics could be used for personalised monitoring of any wellbeing parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Occupant monitoring for autonomous shared vehicles: Forecasting the advent of fully auto- nomous vehicles, this work aimed towards the use of data streams for monitoring occupants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Integrated within the Easy Ride project, we targeted the monitoring of emotions among pas- senger groups, as well as the recognition of activities and violence inside shared autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Just like the first scenario, this aims towards contributions to more intelligent and robust wellbeing monitoring systems using multimodal data sources, albeit in a less personal subject-centric way, focused instead on passenger groups as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although activity re- cognition is a relatively mature research topic, the in-vehicle environment offers very spe- cific challenges, mainly regarding perspective, lighting, and occlusions, which enabled the study of innovative solutions for robust passenger monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ultimately, the developed occupant wellbeing monitoring solutions could also be personalised, using the biometric recognition of individual passengers as additional information for improved accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite these two scenarios, used to contextualise and motivate the work conducted during this work, we aimed to build solutions that are applicable outside the target applications and beyond the fields of biometrics and wellbeing monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This work also aimed to result in significant advances to the target fields, which are ready for real-life applications and capable to withstand the test of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, throughout the entirety of this doctoral project, there was a constant concern to ensure the proposed methodologies were: Multimodal: The diversification of data sources is the key to truly robust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the topic of biometric recognition, considering the strengths and shortcomings of the ECG sig- nal as a biometric trait, the face is the best complementary trait for better performance and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beyond biometric recognition, the fusion of ECG and face results in a larger availability of anatomical and physiological measurements, which enable the more compre- hensive monitoring of wellbeing parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6 Introduction Seamless: Regardless of the possible consequences to accuracy, user comfort should be of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, subjects should be as unaware of the acquisition process as possible, to avoid attention or behaviour changes that could impact their comfort or the realism of the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, this work focused, as extensively as possible, on seamlessly acquired data, in nearly unconstrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When drivers are the target, and thus the subject is in physical contact with the system nearly continuously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', driving the car), ECG and other physiological signals can be acquired unobtrusively at the steering wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Face video can also be easily and inexpensively acquired with cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the other hand, for shared vehicle occupant monitoring, physiological signals have been avoided in favour of less contact-intensive alternatives, namely video and audio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Continuous: Continuous biometric methodologies offer unique advantages in usability and effectiveness, both for recognition and wellbeing monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the other hand, having a continuous stream of biometric data opens up new possibilities for improved accuracy, immersive systems, and error management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, beyond novelty and performance, the algorithms developed during this work aimed towards real-time operation, reflecting a constant preference for efficiency and simplicity whenever possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Realistic: Overall, performance results in ECG-based biometrics literature are unrealistic, mainly due to inadequate train-test splits and overly clean signal databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The same hap- pens in wellbeing monitoring when the problem of subject-independence is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This reveals the deeply flawed nature of typical evaluation frameworks, which should more realistically resemble actual application conditions and ensure reproducible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To achieve this, this doctoral work included the reformulation of testing procedures, especially for ECG-based biometrics, through the definition of adequate test protocols, the benchmark- ing of literature methods, and the development of more robust recognition and monitoring algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Contributions On the quest for personalised wellbeing monitoring, focused on the objectives detailed in the pre- vious section, this doctoral project comprised several research topics related to both biometric recognition and wellbeing monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 presents them and illustrates their interconnec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Most of these were the focus of research work during this doctoral project and resulted in innovative contributions towards the target of in-vehicle personalised wellbeing monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This thesis presents the innovative contributions of this work organised throughout four parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first two of these are focused on biometric recognition, aiming towards the development of personalised wellbeing monitoring solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The third part targets the scenario of passenger monitoring inside shared autonomous vehicles, specifically for emotion, activity, and violence re- cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The fourth presents broader contributions applicable to multiple scenarios in biometrics and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These contributions are concisely enumerated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Contributions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='PERSONALISED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='WELLBEING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MONITORING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='BIOMETRIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='RECOGNITION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='WELLBEING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MONITORING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='FACE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='BIOMETRICS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ECG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='BIOMETRICS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MASKED FACE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='RECOGNITION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='PRESENTATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ATTACK DETECTION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='OCCLUSIONS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='INTERPRETABILITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='INTERPRETABILITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='END-TO-END ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='DEEP MODELS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='LONG-TERM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='PERFORMANCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='LEARNED 2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='TRANSFORMS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='TEMPLATE UPDATE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='INTERLEAD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='CONVERSION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='TEMPLATE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='SECURITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='SELF-SUPERVISED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='LEARNING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ORDINAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='CLASSIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='EMOTION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='RECOGNITION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ACTIVITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='RECOGNITION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='DROWSINESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MONITORING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='FACE-BASED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ECG-BASED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MULTIMODAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='AUDIO + VIDEO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='GROUP EMOTION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='FACE-BASED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ECG-BASED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MULTIMODAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MULTIMODAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='AUDIO + VIDEO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='OPTIMIZATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='& EFFICIENCY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='IN-VEHICLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='SCENARIOS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='VIOLENCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='DETECTION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='DRIVERS’ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MONITORING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MULTIMODAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='FACE + ECG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='DATA AUGMENTATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='TRANSFER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='LEARNING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='OPEN WORLD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='FACE RECOGNITION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Schema of the topics covered during this doctoral project, their interconnections, and their link to the central topic of personalised wellbeing monitoring (black topics in bold correspond to the strongest direct contributions, which are presented in this thesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Part II, Electrocardiogram Biometrics: a comprehensive survey of one hundred and twenty-five state-of-the-art methodologies for ECG-based biometric recognition, describing the evolution of the topic from 1999 to 2022 and open challenges and opportunities for future research (Chapter 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the first end-to-end methodology for ECG biometrics, alongside tailored data augmentation strategies for ECG signals and a study on the advantages of integrating typically separate processes inside a single deep architecture (Chapter 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' an application of triplet loss and transfer learning for ECG-based identity verification, aiming towards higher robustness under realistic evaluation setups using off-the-person databases (Chapter 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' a study on long-term performance with multiple state-of-the-art ECG biometric methodolo- gies, including an assessment of the effect of diverse template and model update strategies (Chapter 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' a study on the relative importance of ECG waveforms for identification under diverse scenar- ios, less to more challenging, using explainability tools on our deep learning identification methodology (Chapter 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' a methodology for recovering missing ECG leads based on single-lead blindly-segmented input signals, paving the way towards more complete applications (even in clinical scenar- ios) with less obtrusive signal collection setups (Chapter 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Part III, Face Biometrics: 8 Introduction a custom training methodology tailored for face recognition with masks, aiming to promote the use of unmasked parts of the face and close the performance gap relative to typical face recognition (Chapter 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' a study using interpretability tools to understand the use of face image information in pre- sentation attack detection (PAD), alongside a discussion on the need for explainability and transparency in biometric recognition (Chapter 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Part IV, Wellbeing Monitoring: an approach for classifying emotion valence in groups of people, based on the late fusion of parallel visual and audio data streams using deep neural networks (Chapter 12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' a cascade strategy for improved efficiency in continuous audiovisual activity recognition and violence detection inside vehicles (Chapter 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Part V, Broader Topics on Biometrics and Pattern Recognition: the Secure Triplet Loss, a training methodology that promotes template cancelability and unlinkability, alongside identity discrimination, on end-to-end biometric algorithms without the need for separate encryption or hashing processes (Chapter 14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' a methodology for self-supervised learning based on the triplet loss, taking advantage of the nature of balanced multiclass datasets, especially those composed of sequential data, for more adequate learning of the target tasks (Chapter 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These research works may be categorised according to the specific contributions by the author of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The presented work in ECG-based biometric identification (Chapter 4), authenti- cation (Chapter 5), and explainability (Chapter 7), as well as emotion recognition (Chapter 12), activity recognition (Chapter 13), biometric template security (Chapter 14), and self-supervised learning (Chapter 15) comprise the main contributions, resulting completely or mostly from the work performed by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The research on long-term performance in ECG biometrics (Chap- ter 6), ECG interlead conversion (Chapter 8), masked face recognition (Chapter 10), and inter- pretability for face PAD (Chapter 11) are secondary contributions that resulted from collaborations within the scope of this thesis and benefitted partially from the work of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Further details on the specific contributions can be found at the start of each chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Other topics have been addressed which are not presented in this thesis, due to the relevance of the respective topics or the relative weight of the author’s contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' They are, however, also mapped in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 and listed in the sections below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Dissemination The contributions of the doctoral research to biometrics, wellbeing monitoring, and broader topics described in this thesis have been disseminated as part of twenty-four scientific publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are (clustered by type and in reverse chronological order): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Dissemination 9 Articles in journals: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Electrocardiogram Lead Conversion from Single-Lead Blindly-Segmented Signals,” BMC Medical Informatics and Decision Making, 22: 314, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [31] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Esteves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ferreira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Costa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Rodrigues, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Antunes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lopes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 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T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Gonçalves, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Silva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “An Exploratory Study of Interpretability for Face Presentation Attack Detection,” IET Biometrics, 10 (4): 441–455, 2021.' 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S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Stream- lining Action Recognition in Autonomous Shared Vehicles with an Audiovisual Cas- cade Strategy,” in 17th International Conference on Computer Vision Theory and Ap- plications (VISAPP), Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [348] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Focus- Face: Multi-task Contrastive Learning for Masked Face Recognition,” in Workshop on Face and Gesture Analysis for COVID-19 (FG4COVID19), Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [308] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Boutros, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Saffari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Damer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Car- doso, “My Eyes Are Up Here: Promoting Focus on Uncovered Regions in Masked Face Recognition,” in International Conference of the Biometrics Special Interest Group (BIOSIG 2021), Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [309] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lourenço, “Deep Neural Networks for Biomet- ric Identification Based on Non-Intrusive ECG Acquisitions,” in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Arya and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Bhadoria, Eds.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Dissemination 11 Abstracts in national conference proceedings: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “xECG: Using Interpretability to Understand Deep ECG Biometrics,” in 27th Portuguese Conference on Pattern Recognition (RECPAD 2021), Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Correia, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Achieving Cancellability in End-to-End Deep Biometrics with the Secure Triplet Loss,” in 26th Portuguese Conference on Pattern Recognition (RECPAD 2020), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Silva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Gonçalves, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, and Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Explain- able Artificial Intelligence for Face Presentation Attack Detection,” in 26th Portuguese Conference on Pattern Recognition (RECPAD 2020), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lopes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Rebelo, “Long-Term Performance of a Convolutional Neural Network for ECG-Based Biometrics,” in 25th Portuguese Con- ference on Pattern Recognition (RECPAD 2019), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beyond the aforementioned publications, the author has contributed to fourteen other scientific publications related to diverse pattern recognition and computer vision research topics not covered in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are listed below: Articles in journals: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Gonçalves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ross, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Explainable Biometrics in the Age of Deep Learning,” ACM Computing Surveys, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [311] (submitted) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Capozzi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Barbosa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pereira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Carvalho, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Toward Vehicle Occupant-Invariant Models for Activity Characterization,” IEEE Access, 10: 104215–104225, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [64] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Boutros, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Damer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Be- yond Masks: On the Generalization of Masked Face Recognition Models to Occluded Face Recognition,” IEEE Access, 10: 86222–86233, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [312] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Oliveira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': 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+page_content=' Oliveira, and Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “IHC Classification in Breast Cancer H&E Slides with a Weakly-Supervised Approach,” in 26th Portuguese Conference on Pattern Recognition (RECPAD 2020), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Costa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Silva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Rebelo, “Face Anti Spoofing: Handcrafted and Learned Features for Face Liveness Detection,” in 25th Portuguese Conference on Pattern Recognition (RECPAD 2019), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Fine Segmentation of Head and Torso Using Label Re- finement Networks,” in 25th Portuguese Conference on Pattern Recognition (RECPAD 2019), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The research conducted during these doctoral studies has also been partially presented to the scientific community at the Doctoral Consortium of the 2019 IEEE International Conference on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Collaborations 13 Biometrics: Theory, Applications and Systems (Tampa, FL, USA) and at the 2020 International Summer School for Advanced Studies on Biometrics for Secure Authentication (Alghero, Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Collaborations The doctoral work presented in this thesis included close collaborations with researchers from several institutions within the AUTOMOTIVE, Easy Ride, and Aurora projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The author also collaborated frequently with VCMI group colleagues within their research work and master disser- tations related to diverse biometrics, pattern recognition, and computer vision topics, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Research projects The AUTOMOTIVE project1 was focused on ushering in the next generation of driver drowsiness monitoring technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Led by the VCMI research group at INESC TEC, this project featured the participation of CardioID Technologies, Instituto Superior de Engenharia de Lisboa (ISEL), and Universidade Lusófona de Humanidades e Tecnologias (ULHT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The author of this thesis has participated in the AUTOMOTIVE project from June 2019 to its conclusion in November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' He mainly contributed to the development of novel algorithms for ECG-based biometric recogni- tion to enable personalised drowsiness models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of this project are nicely summed up in [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Easy Ride project2, under the motto “Experience is Everything”, was a large effort led by Bosch Car Multimedia and Universidade do Minho aiming to improve passenger comfort and safety in autonomous shared vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It featured INESC TEC’s participation in the SP5 sub- project, in close cooperation with Bosch Car Multimedia, which focused on occupant emotional monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The author of this thesis has participated in writing this project’s proposal, and then in the research work from its start in February 2020 to its conclusion in December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' He has mainly contributed to the design, implementation, and evaluation of efficient multimodal deep learning algorithms using audio and RGB video for in-vehicle emotion, activity, and violence recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The work encompassed all stages of research and development from scratch to in- vehicle deployment and is presented in [346;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 348].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Aurora project inherited Easy Ride’s goals of bringing forth the future of shared autono- mous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Specifically, it brought together Bosch Car Multimedia and INESC TEC’s Cen- tre for Telecommunications and Multimedia (CTM) and the High-Assurance Software Labora- tory (HASLab) to continue SP5’s mission of contributing towards accurate and efficient occupant emotional and activity monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The author of this thesis participated in this project since its 1AUTOMOTIVE (“POCI-01-0145-FEDER-030707”) was financed by the European Regional Development Fund (ERDF) through the Operational Programme for Competitiveness and Internationalisation (COMPETE 2020 Pro- gramme), and by national funds through the Portuguese funding agency, Fundação para a Ciência e a Tecnologia (FCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2Easy Ride (“POCI-01-0247-FEDER-039334”) was supported by European Structural and Investment Funds in the FEDER component, through the Operational Competitiveness and Internationalisation Programme (COMPETE 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 14 Introduction beginning in December 2021, having collaborated on the definition of the project’s objectives and task planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Synergies within the research group Within the VCMI research group, the author collaborated, in 2018, with Wilson Silva to develop a metric for ordinal classification which takes into account both accuracy and label ranking while re- taining robustness to class imbalance [396].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In 2020, the author collaborated with Sara P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Oliveira, Tiago Gonçalves, and Hélder Oliveira, alongside Rita C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Marques and Maria João Cardoso at Champalimaud Foundation (Lisboa, Portugal), on a weakly-supervised methodology based on multiple instance learning to classify HER2 expression in breast cancer histology slides [321].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Since 2020, the author has also been collaborating with Ana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, Wilson Silva, Tiago Gonçalves, and Pedro C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neto, alongside Arun Ross from Michigan State University (East Lansing, MI, USA) to study biometrics from the perspective of interpretability, rethink- ing how model accuracy should be measured, and calling for more transparent biometric algo- rithms [311;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 385;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 386].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In 2021, the author collaborated with Leonardo G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Capozzi and Ana Rebelo in their work related to person re-identification and scene geolocation for automatic missing person search- ing [60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Also in 2021 and 2022, the author has collaborated with Pedro C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neto, Mohsen Saffari, and Ana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira, alongside Fadi Boutros and Naser Damer at Fraunhofer IGD (Darm- stadt, Germany), on novel strategies for masked face recognition to uphold state-of-the-art bio- metric performance amidst a global pandemic [50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 308;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 309].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Organisation of scientific events This doctoral project and the aforementioned collaborations and synergies motivated the contribu- tion to the organisation of multiple scientific events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In regards to biometrics, the main example is that of the 2020 edition of the International Workshop on Biometrics and Forensics (IWBF), organised by INESC TEC and NTNU, where the author of this thesis collaborated as Demo Chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' He has also helped organise the Workshop on Explainable & Interpretable Artificial Intelligence for Biometrics (xAI4Biometrics), hosted yearly at the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), as Publicity Chair in 2021 and 2022 and Programme Committee member in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The collaboration with Pedro C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neto and Ana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira from INESC TEC and Fadi Boutros and Naser Damer from Fraunhofer IGD was enhanced by the co-organisation of the Advanced Occluded Face Recognition (OCFR) competition at the 2022 International Joint Conference on Biometrics (IJCB) [310].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Aligned with the topic of wellbeing monitoring, and as an extension to the Easy Ride and Au- rora projects, the author has also co-organised the In-Vehicle Sensing and Monitorization Work- shop (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This first edition of the workshop was hosted at the European Conference on Com- puter Vision (ECCV) in October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Collaborations 15 In a broader scope, the author of this thesis has also co-organised the special session on Ma- chine Learning in Healthcare Informatics and Medical Biology at the International Conference on Computational Intelligence Methods for Bioinformatics and Biostatistics (CIBB), in 2019 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' He was also a member of the Technical and Programme committee of this conference for its 2021 edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Since 2019, the author has also helped organise the VISUM Summer School on computer vision and machine intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the 2019 and 2020 editions, he was a member of the project team, while in 2021 and 2022 he was part of the main organisation team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Supervision of dissertations and internships The author of this thesis has collaborated, as co-supervisor or external supervisor, on the following master dissertations related to his doctoral studies (in reverse chronological order): 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Mariana Silva Xavier (2022), “Inside Out: Fusing ECG and Face Information to Recognise Emotions”, Master in Bioengineering, Universidade do Porto - as co-supervisor, alongside Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (supervisor) [464];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Guilherme Augusto Tiritan Romano Barbosa (2022), “Going 2D: Exploring Learnable Bidi- mensional Approaches for ECG Biometrics”, Master in Bioengineering, Universidade do Porto - as co-supervisor, alongside Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (supervisor) [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pedro Duarte da Cunha Nunes Lopes (2022), “Deep Neural Networks for Face-based Emotion Recognition”, Master in Bioengineering, Universidade do Porto - as external institution supervisor, alongside Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (supervisor) and Ana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira (co- supervisor) [271];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Erfan Omidvar (2022), “Single-Wrist Electrocardiogram Acquisition Application in Biometrics”, Master in Biomedical Engineering, Universidade do Porto - as second co-supervisor, alongside Miguel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Correia (supervisor) and Duarte Dias (first co- supervisor) [323];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Vítor Hugo Pereira Barbosa (2022), “Robust occupant action classification in shared au- tonomous vehicles”, Master in Informatics and Computing Engineering, Universidade do Porto - as external institution supervisor, alongside Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (supervisor) and Pe- dro Carvalho (co-supervisor) [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Telma Sofia Caldeira Esteves (2021), “Sleepy Drivers: Drowsiness Monitoring Using ECG and Face Video”, Master in Biomedical Engineering, Universidade Nova de Lisboa - as external institution supervisor, alongside Ricardo Vigário (supervisor), André Lourenço (co- supervisor), and Ana Rebelo (external institution supervisor) [115];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sofia Cardoso Beco (2021), “Make My Heartbeat: Generation and Interlead Conversion of ECG Signals”, Master in Bioengineering, Universidade do Porto - as co-supervisor, along- side Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (supervisor) [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 16 Introduction 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Inês Alexandra Teixeira Antunes de Magalhães (2021), “Feel My Heart: Emotion Recog- nition Using the Electrocardiogram”, Master in Bioengineering, Universidade do Porto - as co-supervisor, alongside Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (supervisor) [285];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Arthur Johas Matta (2020), “Open-World Face Recognition”, Master in Informatics and Computing Engineering, Universidade do Porto - as co-supervisor, alongside Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Car- doso (supervisor) [291];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Leonardo Gomes Capozzi (2020), “Face Recognition For Forensic Applications: Methods for Matching Facial Sketches to Mugshot Pictures”, Master in Informatics and Comput- ing Engineering, Universidade do Porto - as co-supervisor, alongside Ana Rebelo (supervi- sor) [62];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' João Manuel Guedes Ferreira (2020), “Head Pose Estimation for Facial Biometric Re- cognition Systems”, Master in Informatics and Computing Engineering, Universidade do Porto - as co-supervisor, alongside Ana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sequeira (supervisor) and Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (co- supervisor) [130];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Carolina Martins Barbosa Rodrigues Afonso (2020), “Changing Perspectives: Interlead Conversion in Electrocardiographic Signals”, Master in Network and Information Systems Engineering, Universidade do Porto - as co-supervisor, alongside Miguel Coimbra (super- visor) [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Gabriel Carneiro Lopes (2019), “Don’t You Forget About Me: Enhancing Long Term Per- formance in Electrocardiogram Biometrics”, Master in Bioengineering, Universidade do Porto - as co-supervisor, alongside Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso (supervisor) [270].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beyond these aforementioned dissertations, the author of this thesis also collaborated in the supervision of more than twenty students on curricular, extra-curricular, and summer internships related to biometrics, pattern recognition, and computer vision topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Awards and Distinctions Beyond the aforelisted peer-reviewed publications and presentations to the scientific community, the doctoral research work presented in this dissertation has also been the recipient of multiple awards and distinctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are listed below: The author of this thesis was granted the EAB Max Snijder Award at the European Biomet- rics Awards 2022 organised by the European Association for Biometrics (EAB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This award recognised the wider perspective and applicability of his work on ECG biometrics, which contributed towards a complete deep learning solution encompassing end-to-end models, learnable template security, and explainability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Document Structure 17 The initial work on the Secure Triplet Loss [345], focused on biometric template cancela- bility for end-to-end deep models, received the Computers Journal Best Paper Award at the 2020 International Workshop on Biometrics and Forensics (IWBF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The work on audiovisual group emotion valence recognition [346] received the Best Session Paper Award at the 2020 IEEE International Conference on Image Processing, Applications and Systems (IPAS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The extended work on the Secure Triplet Loss [347], presented at the 2021 NIS Workshop organised by INESC TEC’s Networked Intelligent Systems cluster, received the Best Pre- sentation Award by the official jury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Document Structure This thesis is composed of six parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part I is the prologue, which includes this introduction, Chap- ter 1, and offers an overview of the fundamental concepts related to biometric systems, biometric traits, and wellbeing monitoring in Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part II focuses on electrocardiogram biometrics, presenting the contributions to this topic pro- duced during the doctoral work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It begins with an overview of the existing data, related literature methodologies, and a discussion of open challenges and opportunities, in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 4 presents our study on end-to-end deep learning for ECG-based identification, including tailored unidimensional data augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 5 showcases the work on triplet loss and transfer learning for ECG-based identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A study on long-term performance and template update for identification is presented in Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 7 delves into the topic of explainability for ECG biometrics, aiming to better understand which parts of the signal are best for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Finally, Chapter 8 proposes a methodology for recovering missing ECG leads based on blindly-segmented single-lead segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part III deals with face biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 9 offers an overview of the data, existing method- ologies, and open challenges in this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Two methodologies to close the performance gap in masked face recognition are presented in Chapter 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 11 describes a study on inter- pretability to understand the decisions of deep learning models in face presentation attack detection and motivate a more widespread usage of interpretability for more transparent biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part IV is centred on wellbeing monitoring and covers the topics of emotion recognition, activ- ity recognition, and violence detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 12 describes an audiovisual approach developed to classify emotion valence in groups of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 13 presents an adaptation of the afore- mentioned approach for activity recognition and violence detection, alongside a cascade strategy for increased efficiency in in-vehicle scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part V covers broader topics related to biometrics and pattern recognition which have been addressed during the doctoral work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Specifically, Chapter 14 introduces the Secure Triplet Loss, a novel approach to ensure biometric template security on end-to-end deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lastly, 18 Introduction a methodology for self-supervised learning formulated for minimal performance gaps when using sequential data is presented in Chapter 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part VI is the epilogue, which concludes this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes an overview of the conducted work and the conclusions drawn from it, in Chapter 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 17 offers a discussion on future work opportunities related to the results of this doctoral thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 2 Fundamental Concepts Biometric systems are, in several ways, different from other pattern recognition applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The need for storage of personal data from users and the different modes on which the systems can operate are only some of the special characteristics of biometric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When developing one, one should be aware of these specificities to ensure the best performance and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, this chapter presents the fundamental concepts needed to build a biometric system, either for identity recognition or the monitoring of wellbeing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes an overview of the general structure and operation of biometric systems, their security vulnerabilities, the dif- ferent biometric traits (with a special focus on the electrocardiogram and face), and the metrics for thorough performance evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Biometric Systems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 General structure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Biometric recognition systems Biometric recognition systems are tools that use hardware and pattern recognition algorithms to compare the identity of a user with that of registered individuals based on their attributes (des- ignated as biometric characteristics or traits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Like traditional identification systems based on keys, cards, or codes, biometric systems are mostly used for access control to restricted places, confidential information, or personal data and belongings [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A biometric system is typically composed of an acquisition module, a storage module, and a biometric algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The algorithm can, in turn, be divided into three modules: quality as- sessment, feature extraction, and decision (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) [46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These modules are described below: Acquisition: The acquisition module is the interface between the system and the subject and is responsible for the measurement of the biometric characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The sensors used in this module should be carefully designed to fit the expected application settings and avoid, as much as possible, the noise and artefacts from environmental interference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 19 20 Fundamental Concepts ACQUISITION QUALITY ASSESSMENT FEATURE EXTRACTION Enrollment Recognition Stored Data STORAGE DECISION Extraction of meaningful attributes Quality check and enhancement of the collected trait Acquisition and quantifcation of biometric traits Identifcation or acceptance/rejection of identity claim Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: General structure of a biometric recognition system (from [343], based on [46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 201;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 351]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Quality Assessment: This module aims to evaluate the quality of the trait measurement and either accept it in its current form, enhance it to reduce noise and variability effects, or discard it if the quality is unacceptably low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Feature Extraction: The feature extraction module is focused on the processing of the ac- quired measurements, using pattern recognition tools, to extract the most meaningful at- tributes of the biometric trait and thus enable a robust decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The feature extraction pro- cess should be designed to provide attributes that present high intersubject discrimination power and low intrasubject variability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Decision: This module uses the output from the feature extraction module and the stored information from registered users to identify the user, or validate or reject their identity claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To achieve this, it compares the processed traits of the current user and the registered individuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Storage: The storage module is typically composed of a database that stores biometric tem- plates (processed biometric trait measurements) from all individuals registered on the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For security purposes, it can include template protection measures, such as hashing, to prevent leaks of sensitive personal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Wellbeing monitoring systems Wellbeing monitoring systems are very diverse due to the variety of parameters these can monitor, which include emotions, stress, fatigue, and health conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, most of these systems follow a general structure that is very similar to that of most biometric recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Just like biometric recognition systems, an acquisition module performs the recording of data, which are checked and processed by the quality assessment module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' After this, the feature extrac- tion module extracts meaningful attributes from the trait measurements to enable accurate labelling by the decision module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Most wellbeing monitoring systems do not require a storage module, as biometric templates from the specific set of enrolled subjects will not be required, unlike in recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is an advantage in terms of data security and privacy, which will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Biometric Systems 21 Enrollment True identity Biometric Trait(s) Database Identity Verifcation Claimed identity Biometric Trait(s) Database Feature Extraction Decision Claimed identity’s template Acceptance/Rejection Biometric Trait(s) Database All stored templates Identity/Rejection Quality Check Feature Extraction Quality Check Feature Extraction Acquisition Quality Check Acquisition Decision Acquisition Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Schematics of the operation of a biometric recognition system in identification and identity verification modes, and in the enrollment phase (adapted from [337], based on [133;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 201;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 351;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 413]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Operation modes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Biometric recognition systems Depending on the application context and its requirements, a biometric system can either operate in identification mode or identity verification mode [2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 201;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 351].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In identity verification mode, also commonly called authentication, the biometric system will receive an identity claim along with the biometric measurement (the current user will claim to be a specific enrolled individual).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, the decision module will only compare the current measurement with the stored data from the claimed identity, performing a one-vs-one comparison, and either accept or reject the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the identification mode, the biometric system will only receive the biometric measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, the decision module performs a one-vs-all comparison between the current biometric mea- surement and the data stored for each enrolled individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ultimately, the system will either assign one of the enrolled identities (corresponding to the strongest comparison) to the current user or reject to identify (if no comparison was strong enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, biometric systems include the enrollment phase, which comprises the acquisi- tion, processing, and storage of a biometric template of a subject for its registration on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 22 Fundamental Concepts Acquisition Processing Decision 1 2 3 4 5 7 Database 7 6 8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Attack points on a biometric system (based on [138;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 361]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' After this, the system will be able to correctly perform identification or identity verification when used by the subject [7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 351].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Wellbeing monitoring systems As aforementioned, wellbeing monitoring systems like emotion recognition devices rarely require the storage of personal information from the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, they dismiss enrollment phases and only operate in one mode: inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this mode, a trait acquisition is performed by the system, which will output a corresponding label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The output is composed of discrete categories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', sad, happy, or angry, in emotion recognition) or continuous scores (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', from very awake to very drowsy, in drowsiness recognition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Security and privacy concerns As key protectors of sensitive data, prized possessions, or restricted locations, biometric re- cognition systems are a prime target for attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The literature defines eight attack points (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3) that sum up the different ways to unlawfully gain access to a biometric sys- tem [202;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 297;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 317].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering a general structure (that groups the quality assessment and feature extraction into a single processing module) are described below: Type 1 – At the acquisition module: Such attacks are commonly called presentation attacks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' as they consist of physically forcing the biometric system to grant access to the attacker,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' either through the use of fake biometric traits (such as fake fingers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' voice recordings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' or prerecorded face videos) or even through the physical destruction of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Type 2 – Between the acquisition and the processing modules: In these attacks, called re- play attacks, the attackers will target the communication link between the sensor and the processing module, to steal the trait acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' They can then bypass the sensor module by injecting the stolen trait measurements directly into the processing module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Biometric Systems 23 Type 3 – At the processing module: The processing module can be attacked and overridden by another program, controlled by the attacker, that sends the desired features to the decision module upon request;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Type 4 – Between the processing and decision modules: These attacks are similar to replay attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The attacker will target the link between the processing module and the decision module and steal the features sent between them, to be later injected, bypassing the acquisi- tion and processing modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Type 5 – At the decision module: Here, the attackers can replace the decision algorithms so they can generate high matching scores as requested, thus granting them access whenever desired, or to always output negative decisions, amounting to a denial-of-service attack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Type 6 – At the storage module: This consists in exploiting database security flaws to add, modify, or delete templates, to ultimately grant access to unauthorised individuals or deny access to enrolled users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Type 7 – Between the storage and decision modules: Here, the attackers intercept the com- munications between the database and the decision module, to steal biometric templates and replay them later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Type 8 – Between the decision module and the application: These attacks consist in the manipulation of the data transmitted between the decision module and the application, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', to override a rejection decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In ECG-based biometrics, security vulnerabilities are still to be adequately addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite the pioneer studies of Eberz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [110] and Karimian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [211], no efforts have yet been devoted to better protect such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In general, biometric systems offer undeniable advantages when compared with traditional authentication systems, but this would be meaningless if the system introduced new vulnerabilities that paved the way to successful attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, it is very important to remember the attack points presented above and their specificities throughout all stages of the development and deployment of a biometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Of all modules, storage is one of the most sensitive, as it stores personal data that could be used to unlawfully access private information and belongings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These intimate data are not specific to a single application and can be used by their legitimate owner as a single credential on several biometric systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', a user could use fingerprint-based access on two different computers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is an obvious vulnerability as, just like using the same password for several online services, a single security failure can risk the privacy of the user on several applications [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Regardless of how sophisticated the database is, it can still be accessed or hacked by intruders who exert enough effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides working towards more secure databases, it is paramount to prepare for possible successful attacks and ensure biometric templates cannot be retrieved in those cases [186;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, regarding biometric template security, it is important to take into account the following factors: 24 Fundamental Concepts Non-Invertibility: The processing module should be designed in a way that eases the cre- ation of templates from biometric trait acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the retrieval of a close ap- proximation of the original trait measurement or feature set, from a stored template, should be difficult and sufficiently time-consuming to render the process unfeasible or unattractive for attackers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Revokability/Cancelability: Keys can be changed when moving to a new house, and users can easily change e-mail passwords after they have become compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, bio- metric systems rely on intrinsic personal characteristics, such as facial features or fingerprint minutiæ, which are very difficult (or even impossible) to be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, biometric sys- tems should include measures that allow the easy invalidation of templates when these have become compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, intruders will be denied access when using those credentials, but legitimate users will still be allowed to authenticate using their unchanged biometric trait;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Unlinkability: For improved performance, a single biometric system can store, separately, more than one template for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The data protection scheme should ensure that the comparison between stored templates does not enable an attacker to cluster them by iden- tity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This should also be difficult with templates from the same user in different biometric systems so that attackers are not able to attack multiple systems with a single stolen tem- plate [186;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 304].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides these factors, the biometric system should also be able to deal with the characteristic variability of biometric traits and their measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Methods like hashing are commonly used for passwords, but such traditional credentials do not present variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With biometric systems, small variations of the input should be considered normal and acceptable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', with the face, different haircuts or beard styles), and should not influence the final decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When designing a secure biometric system, it is necessary (although hard) to find an equilibrium between non- invertibility, unlinkability, and the controlled acceptance of natural intrasubject variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Biometric Traits 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 General overview Biometric traits are human attributes that include enough personal information to reliably serve as the basis for the recognition and discrimination of individuals [9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' According to the identity information they carry and the performance they can offer, biometric traits can either be considered hard traits, strong enough to be standalone traits in a reliable biometric system, or soft traits, which need further traits or information to offer acceptable recognition performance (see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Traits can be categorised according to their nature, as anatomical, physiological, or behaviour- al traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Anatomical traits result from measurements of parts of the human body and include the fingerprints, the face, and the iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Physiological traits are those that originate from physiological 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Biometric Traits 25 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Main benefits and drawbacks of different biometric traits (from [343], based on [2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 213]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Trait ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Benefits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Drawbacks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Electrocardiogram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='(ECG) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Universality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Hidden nature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Simple acquisition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Requires contact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Variability over time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Electroencephalogram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='(EEG) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Universality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Hidden nature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Expensive equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Vulnerability to noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Variability over time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Face ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Easily measurable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Affordable equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Easy circumvention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Depends on face visibility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='and lighting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Fingerprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='High performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Permanent over time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Requires contact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Gait ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Easy to measure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Affordable equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Low performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Variability over time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Iris ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='High performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Expensive equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Palmprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='High measurability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Permanent over time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Requires contact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Photoplethysmogram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='(PPG) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Easy to acquire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Hidden nature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Affordable equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Low performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Variability over time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Voice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Affordable equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Low performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='events in the body and include the heart rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' facial or hand thermography,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the electrocardiogram (ECG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' and the electroencephalogram (EEG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Behavioural traits originate from a person’s actions or behaviours, such as their gait (walking cadence), their signature, or their voice [2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The quality of a biometric trait can be defined, as proposed by Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [199], through seven different aspects: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Universality: the trait should be present in all subjects using the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Uniqueness: the trait should include enough personal information to present differences between all subjects, and thus allow their identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Permanence: despite the intersubject variability desired (uniqueness), the trait should be suf- ficiently stable over time (reduced intrasubject variability) to allow the identification through the comparison of measurements in different instances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Measurability: the trait should be easily and comfortably acquired and digitised, and its representation should allow easy processing and measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 26 Fundamental Concepts 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Performance: a system based on such a trait should meet or exceed the recognition accuracy requirements, set by the context in which it will be applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Acceptability: there should be no foreseeable reservations that could make the subjects un- willing to allow the trait acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Circumvention: the trait should be as hard as possible to mimic or counterfeit, in any way, to prevent spoofing of the biometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Abo-Zahhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [2] compared sixteen biometric traits according to their compliance with each of the seven defined qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In that comparison, it is possible to verify that the traits with the lowest overall quality are the behavioural ones (gait, keystroke, signature, and voice) and phonocardiogram (heart sounds), with low performance, permanence, and distinctiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Low circumvention and universality are also downsides of behavioural traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The traits with reported highest overall quality are the DNA, facial thermogram, fingerprint, iris, palm print, and ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Overall, the electrocardiogram excels in most factors, with just two ‘average’ scores (for collectability and acceptability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Traits like fingerprint, face, signature, iris, and voice have been the most studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, these have long seen a quick growth and evolution of spoofing methods (methods of counterfeit- ing a certain user’s trait to unlawfully gain access through the biometric system), which urges researchers to find more robust alternatives [13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Throughout this doctoral work, the focus will be on the electrocardiogram (ECG), an emerging biometric trait that offers unique advantages regarding inherent liveness, anti-spoofing abilities, and wellbeing insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Its performance and robustness drawbacks shall be mitigated through its fusion with face, a well-established and robust biometric trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, the next subsections consist of a presentation of both biometric traits, the ways to measure them, and the variability factors that provide them with identity and wellbeing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Electrocardiogram The electrocardiogram (ECG) is a physiological signal generated from the contraction and the recovery of the heart, that has been gaining traction as a biometric trait [343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The heart has three main functions: generate blood pressure to keep blood circulating, route venous and arterial blood to the respective parts of the body, and regulate blood supply according to the metabolic demands [428].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To do this, the heart needs to contract and relax its muscle, the myocardium, through the controlled generation and flow of depolarisation and repolarisation currents [377;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 428].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The measurement of such currents using electrodes placed on the body is designated as elec- trocardiography and results in the electrocardiographic (ECG) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In normal conditions, the ECG is a cyclic repetition of five easily recognisable deflections: the P, Q, R, S, and T waves (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A group of these deflections comprises a single heartbeat and each deflection can be traced back to the phase that originated it [286;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 377;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 428].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Biometric Traits 27 DEPOLARISATION DEPOLARISED REPOLARISATION Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: The sequence of depolarisation and depolarisation events in the heart, and their rela- tionship with the different heartbeat waveforms in an ECG signal (from [343], based on [286]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Acquisition The configurations used for the acquisition of ECG signals for biometric purposes have greatly evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From the first ECG-based biometric research works, considerable efforts have been devoted to more usable and comfortable acquisition technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This aims to place the ECG as a more attractive alternative to established biometric traits, mitigating the main disadvan- tage of the ECG, the obtrusive measurement techniques [343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In early ECG-based biometrics research, recordings from the standard 12-lead or Frank leads were commonly used for the development and evaluation of algorithms [144;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 349;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 463].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are two defined and established configurations of electrodes for standardised and comparable ECG measurement (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5), widely used for the diagnosis of cardiac disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Authors frequently selected certain leads for their biometric algorithms, especially Lead I [298;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 327;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 490] (because of its higher acceptability due to the electrode placement on the wrists), but also Lead II [233;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 234;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 303;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 332], or chest leads [122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 235;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 472].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some researchers opted for acquisitions without movement restrictions and with fewer elec- trodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Prominent choices in the literature include Holter systems (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6), which are pre- pared to acquire ECG signals for several hours while the subjects move and perform their daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These were first used for ECG biometrics by Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [389], using ambulatory record- ings from the MIT-BIH Normal Sinus Rhythm database, acquired for thirty minutes using Holter equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [236, 237] used 24-hour-long Holter acquisitions, from the E-HOL 24h signal collection, and seized the opportunity to study the effect of ECG variability over time on identification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Similarly, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [493] used a mini-Holter system to continuously record ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Medical and Holter systems are designated as on-the-person acquisition settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These present considerable drawbacks for biometric purposes, mainly concerning user comfort during 28 Fundamental Concepts MEDICAL ACQUISITION SETTINGS z x y Lead I Lead II Lead III 1 2 3 45 6 RA LA LL RL Standard 12-Lead Confguration Orthogonal/Frank Leads I E C A aVF aVR aVL F H Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Medical acquisition settings: electrode placement and leads on the standard 12-lead configuration and Frank leads (from [343], anterior electrodes depicted in blue, posterior elec- trodes depicted in lighter blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' HOLTER ACQUISITION EQUIPMENT Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Acquisition settings with movement: example of a five-electrode Holter system for ambulatory recordings (from [343], electrodes depicted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Biometric Traits 29 OFF-THE-PERSON CONFIGURATIONS Thumb Button Electrodes Index Finger Electrodes Metallic Rod electrodes Electrodes mounted on a table Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7: Examples of off-the-person ECG acquisition configurations, using thumb electrodes [69], index finger electrodes [289], metallic rods grabbed by the subjects [32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 261;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 390], or electrodes mounted on a table [394] (from [343], electrodes depicted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' acquisition due to the high number of electrodes and their placement on the chest and legs of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although allowing for longer acquisitions with movement and activity, Holter acquisi- tions still require the placement of electrodes on the torso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This significantly reduces acquisition acceptability and comfort and damages the ECG strength as a biometric trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To improve acceptability and acquisition comfort, and get closer to biometric systems deploy- able in real settings, wet electrodes are being replaced by dry metallic electrodes, their number has been reduced to two or three, and their placement has been confined to the upper limbs, especially the on wrists, hands, or fingers (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These acquisition configurations were designated as off-the-person settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first research works in ECG biometrics to use such signals were, to the best of our knowledge, Molina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [298], who used commercial metallic electrodes strapped to the wrists of the subjects, and Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [69], who acquired ECG signals using dry button electrodes held by the subjects in contact with their thumbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Since then, ECG signals have been recorded using metallic rod electrodes [32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 261;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 390], and dry metallic electrodes mounted on plaques [275] or attached to the users’ fingers [289], which offer increased comfort over on-the-person techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, off-the-person systems still require the user to hold the electrodes or deliberately place the fingers or palms over them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This prevents us from designating them as unconstrained systems, which puts the ECG at a disadvan- tage over other biometric traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides this, the use of dry electrodes in farther placements makes the acquisition more vulnerable to interference, thus affecting the quality of the signal [32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 394].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recently, some researchers have tried to improve off-the-person configurations and approach unconstrained settings in ECG biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' They aimed to close the gap to real, commercial appli- cations by developing wearable technologies for ECG acquisition or embedding the sensors into common objects (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In research, the first example of this highly acceptable acquisition 30 Fundamental Concepts WEARABLES AND SEAMLESS ACQUISITION Nymi Band CardioWheel miBEAT Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8: Wearable and seamless acquisition: examples of surveyed configurations (from [343], electrodes depicted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' was a sensor pad to be used alongside a computer keyboard, to acquire ECG signals continuously during computer use [86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 87;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 393].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' More recently, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [484] have shown it is possible to acquire ECG signals from a single arm and successfully used them for biometric recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As for commercial applications, the Nymi Band [11], a wearable wristband, acquires the ECG using two metallic electrodes on its inner and outer surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Identity verification is performed when the band is put on and the session remains open until the band is taken off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While a ses- sion is open, the Nymi Band broadcasts an identity signal to authenticate the user in other nearby systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The CardioWheel [279] is a steering wheel cover that uses conductive leather for seam- less and continuous biometric recognition and health monitoring of drivers, focused on automatic personalisation of driving settings and remote fleet supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The miBEAT [471] is a versatile platform for the simultaneous acquisition of ECG and photo- plethysmography (PPG) signals, which can be used for seamlessly integrated signal acquisition in smartphones or tablets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' AliveCor provides a set of commercial solutions for easy ECG acquisition in the Kardia 1 lineup, including the KardiaMobile, to be used with typical smartphones, and the KardiaMobile Card, a credit card-sized slim single-lead acquisition device with integrated metallic electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These recent efforts have brought ECG biometrics closer to viable and unconstrained appli- cability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, these newer technologies still require the users to wear certain products or perform specific actions and need contact with both limbs during acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides this, the quality of the acquired signals is typically very low, because of the loose contact with the subject’s skin, suffering from wide impedance variations, sensor saturation, and contact loss artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some researchers have already started to address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The single-arm acquisition settings studied by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [484] and the contactless electrodes developed by Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [75] 1Kardia by AliveCor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='kardia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Biometric Traits 31 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9: Variability in off-the-person ECG heartbeats from the same subjects (from [337], individual heartbeats superposed after denoising, amplitude normalisation, and outlier removal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' raise new and inspiring possibilities for wearable ECG devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For applications based solely on heart rate, techniques have been proposed to measure it at a distance, using microwave Doppler sensors [48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 316;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 376].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These efforts pave the way for better ECG acquisition technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Such systems could con- sist of seamlessly integrated biometric systems that can acquire ECG signals at short distances from one hand of the user, without requiring contact and thus suffering from signal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For wearables, the future could reside in products that can continuously monitor the users’ ECG while only contacting with one of their wrists, or when inside their pockets separated from the body by clothes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite all efforts devoted so far to ECG biometrics, much work is still needed to reach true applicability in the form of real, comfortable, and easy-to-use ECG-based biometric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Variability Although the ECG signals present, in normal conditions, the same deflections for all subjects at all times, these are characterised by a high degree of variability (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Variability in the ECG can be designated as intrasubject, the variations between cycles (heartbeats) in the electrocardio- gram of a single subject, or intersubject, the variations between heartbeats of different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Intrasubject variations on the ECG signal are mainly explored for health monitoring and medical 32 Fundamental Concepts diagnosis [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 164;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 370], while intersubject variations are especially useful to discriminate between subjects in biometric recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Both intrasubject and intersubject variability can originate from several factors, such as: Heart Geometry: Heart size, cardiac muscle thickness, and the overall shape of the heart influence the trajectories of electrical currents throughout the heart, the number of muscle cells that will conduct those currents, and the time to do it across the whole heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Athletes, because of intensive physical training, commonly have thicker myocardia, which affects the ECG with higher voltages in the QRS complex, and lower basal heart rates [171;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 172;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 441];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Individual Attributes: Age, weight, and pregnancy are some individual attributes that can cause shifts in the heart position and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These shifts will change the orientation of the electrical current conduction vectors across the heart, meaning the electrodes will detect the signal from a different perspective, thus altering the ECG waveforms [379];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Physical Exercise or Meditation: The duration of, and intervals between the different de- flections of the heartbeats in an ECG signal, vary with the heart rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These changes are especially visible on the interval between the QRS complex and the T wave in situations of tachycardia (higher heart rates) or bradycardia (lower heart rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Changes in the heart rate caused by physical exercise or meditation are reflected on the electrocardiogram [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardiac Conditions: Medical conditions of the heart can also interfere with the dynamics of electrical pulse conduction and generate variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In biometrics, one of the most studied conditions is Arrhythmia, which causes wide variations in the heart rate across time and, as reported by several researchers, can consistently shrink the performance of ECG-based biometric systems [9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 369;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 472].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Posture: Postures such as standing or laying down change the position and shape of internal organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The heart is also affected by this, changing its position in the thorax, and thus its position in reference to the electrode placement, which will cause variations in the collected ECG signal [379];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Emotions and Fatigue: The sympathetic and parasympathetic systems of the autonomous nervous system work to, respectively, increase or reduce the heart rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These systems are under the direct influence of psychological states and thus, under stress, fear and other strong emotions, fatigue or drowsiness, the heart rate and the ECG signal can be affected [10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 370];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Electrode characteristics and placement: The type, the size, and the number of electrodes, whether they are wet or dry, and the positioning on the chest or limbs, can influence the dominance of noise on the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The mispositioning of electrodes and reversal of leads are also sources of variability, as they change the perspective of detection of the electrocardio- graphic signal [171;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 379].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Biometric Traits 33 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='10: Comparison of face images acquired on the (a) visible light, (b) short-wave infrared, (c) mid-wave infrared, and (d) long-wave infrared spectra (from [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' All the previously presented factors change the morphology of the electrocardiographic sig- nals acquired from an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first two factors contribute more to intersubject variability and the biometric potential of the ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The remaining factors are the main origins of intrasubject variability, which may undermine the process of biometric recognition, but offer the ECG information on several health and wellbeing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When considering the acquisition of ECG for a specific application, whether for medical or biometric recognition, it is paramount to consider these factors, the way they can ease or difficult the task at hand, and how to mitigate their negative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Face The face may be considered the most intuitive of all biometric traits, since humans use facial features as the main clues for the identification of other people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite the challenges in its use for biometric recognition, the face offers the unique advantage of being the only trait that can be acquired using sensors at a considerable distance from the user, possibly without their knowledge [55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Below, the ways to measure the face trait and the factors that may affect the measurement are described and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Acquisition Among widespread and inexpensive cameras and sophisticated thermal sensors, the face trait can be acquired in the following settings [21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 123]: Visible Spectrum: This is the most common setting in face biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Visible light, with a wavelength in [400,750] nm, generally from natural and unconstrained sources in the ap- plication environment, is reflected by the face of the user and captured by the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using the visible spectrum of light has some disadvantages, discussed in subsubsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2, but it is commonly less expensive than the alternatives and, thus, the most fitting option for widespread applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (a) (b) (c) (d)34 Fundamental Concepts Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='11: Examples of tridimensional face models (from the Bosphorus 3D Face Data- base [375]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Infrared Spectrum: To overcome some limitations of the use of visible light, some sensors acquire the face trait using the infrared spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These use thermal cameras that can ob- tain reliable face images in a much wider range of lighting conditions, capturing the heat from blood vessels and tissues on the user’s face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Based on the wavelength range, these can be designated as Near-Infrared (NIR, [750,1400] nm wavelength), Short-Wave Infrared (SWIR, [1400,3000] nm), Mid-Wave Infrared (MWIR, [3000,8000] nm), or Long-Wave In- frared (LWIR, [8000,15000] nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are compared with visible light images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Depending on the frequency spectra used, the face trait can be categorised as either anatomical (visible light), or physiological (infrared spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is because the former mostly captures anatomical features of the face, while the latter is much more dependent on physiological factors that affect blood flow and face heat patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The face trait measurements can be acquired simultaneously with depth information, resulting in an upgrade of 2D images to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5D or 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While 2D data only includes colour or greyscale pixel information, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5D data combines that with depth information about the face of the subject, which is useful to increase recognition accuracy or avoid attack attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3D data is a further step, where both sources of information are fused to build tridimensional models of the subject’s face (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='11) [55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Variability The measurement of the face trait is affected by several intrinsic, environmental, or operational factors (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These can enhance intersubject or intrasubject variability, and thus make 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Biometric Traits 35 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='12: Variability in unconstrained face images of a subject (from [205]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the recognition task easier or more difficult [4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 102;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These factors are: Illumination and Heat Sources: This factor is most relevant for visible spectrum acquisition settings, as the illumination with visible light will directly affect the light received by the sensor and change the face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In infrared images, the influence of illumination can also be felt, due to the heat generated by light and heat sources, although generally in a lesser degree than in the visual spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pose Variations: Variations in pitch, roll, and yaw of the subjects head, relative to the sensor, will result in the capture of facial features in different perspectives, which may encumber pattern recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, the relative position of those facial features will also suffer from pose variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some authors argue infrared imaging is less susceptible to pose variations than visible spectrum sensors [135];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Facial Expressions: Like pose variations, facial expressions change the relative positions of the facial features, which may encumber the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Furthermore, it may create new features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', wrinkles) or hide facial features that would be needed for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As with the previous factor, facial expressions were found to have less impact in infrared imaging [135];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Stress, Emotions, and Illnesses: Stress, emotions like happiness and fear, or illnesses like headaches or tooth infections can influence facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' But more than that, they change the heat patterns on a subject’s head, which will influence the face images captured by infrared sensors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Age: Growing old is generally accompanied by expression wrinkles, grey hair, among other changes that may affect someone’s appearance and make it more difficult for them to be recognised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 36 Fundamental Concepts Cosmetics, Tattoos, Disguises, and Accessories: Makeup, face tattoos, disguises, and acces- sories like rings or piercings can change, hide, or create new facial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recognition algorithms may not be prepared to deal with this new information and be induced into errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Occlusions and Surrounding Objects: Like disguises and accessories, objects surrounding the person can sometimes occlude part of the face of the subject, and thus hide certain facial features that could be key for the recognition task at hand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sensor Quality and Stability: The sensor is another important variability source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using different cameras, with different characteristics, will result in different face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fur- thermore, the movement of the cameras (just like the movement of the subject) can cause blurred images that will disable the retrieval of finer facial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Like with the electrocardiogram, it is important to appropriately weigh all these factors when designing a biometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some will contribute more to intersubject variability, and will be desirable for biometric recognition, while others will enhance intrasubject variability, and em- power wellbeing monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering the task at hand, it is key to design the acquisition and processing stages to focus on the factors that fit the application needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Performance Evaluation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 System design considerations When designing, developing, and deploying a biometric system, it is important to be aware of how it will behave in realistic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, designers and developers should consider some central aspects, as described by Bolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [46]: System accuracy: This measures the frequency with which the biometric system makes correct or wrong decisions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Computation time: This corresponds to the time required by the system to output a deci- sion, starting from the moment the user initiates contact with the acquisition module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This time depends on the processes of trait acquisition, quality assurance, feature extraction, and decision, and should be as low as possible, to improve usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This aspect is especially important for continuous recognition systems, where decisions should be quickly output and frequently renewed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Exception handling: Because of universality issues and sensor flaws or unavailability, some users may find the system unable to acquire their biometric traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Also, software errors may occur and render the system incapable of adequately performing its function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These possibilities should be addressed during system design and development, to ensure the ap- plication works even without the biometric system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Performance Evaluation 37 System cost: The system cost includes all expenses related to acquisition and processing equipment needed, algorithm development and implementation, routine maintenance, and operational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It should be as low as possible, to make the biometric system more af- fordable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Security: Biometric systems decisions can serve as proof of the actions of a certain indi- vidual, and this, in some settings, can escalate to serious legal implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, it is important to minimise the possibility of decision flaws that allow impostors to act under the identity of an authorised person;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Privacy: Biometric systems require the storage of templates, consisting in discriminating information about each of the enrolled individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' That information, in the interest of anonymity and security, should be kept as safe as possible, resorting to encryption tech- niques that allow matching but minimise the possibility of reconstructing the original ac- quired trait data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some conflicts may exist between these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using more sophisticated approaches (such as deep learning) often leads to higher system accuracy, but at the expense of higher computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using passwords or credentials as a fallback in case of an exception weakens system security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Tighter security measures generally lead to more frequent rejection of legitimate users, which translates into reduced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ultimately, trying to fully cover all aspects will increase costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, all these aspects should be carefully analysed and weighted, considering the expected application to get an affordable, efficient, accurate, secure, and usable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the next subsection, the most relevant metrics are presented, regarding the specific aspect of system accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Recognition accuracy measurement 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Identity verification As described before, identity verification consists in accepting or rejecting an identity claim made by a user based on their biometric trait measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, there are four outcomes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The claim is true and the system correctly accepts it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The claim is false and the system correctly rejects it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The claim is true, but the system incorrectly rejects it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The claim is false, but the system incorrectly accepts it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In identity verification, the main goal is to minimise the frequency of outcomes three and four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Outcome three corresponds to a False Reject or False Non-Match error, and outcome four corresponds to a False Accept or False Match error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These errors are the foundation for most identity verification accuracy metrics (see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2), among which the two most commonly used are [7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 155]: 38 Fundamental Concepts Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Definition of the commonly used metrics for performance evaluation in identification and identity verification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Metric Definition Identity verification False Acceptance Rate FAR(T) = Number of impostor trials with prediction score above T Total number of impostor trials False Rejection Rate FRR(T) = Number of legitimate trials with prediction score below T Total number of legitimate trials Equal Error Rate EER = FAR(T), for T that gives FAR(T) = FRR(T) Area Under the Curve AUC = � 1 0 1−FRR(FAR(T)) dT Identification True Positive Identification Rate TPIR(R) = No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' of trials where one of the strongest R predictions is correct Total number of trials Identification Rate IDR = TPIR(1) = No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' of trials where the strongest prediction is correct Total number of trials Misidentification Rate MIDR = 1−IDR = No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' of trials where the strongest prediction is incorrect Total number of trials False Acceptance Rate (FAR or FMR, False Match Rate): measures the fraction of trials where the system accepted the identity claims, even though they were false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It represents how frequently the system erroneously grants access to impostors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The complement of FAR is called Convenience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' False Rejection Rate (FRR or FNMR, False Non-Match Rate): measures the fraction of trials where the system rejected true identity claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, it measures how frequently the system denies access to legitimate users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The complement of FRR is designated as Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For a given system, these two metrics commonly depend on the chosen operation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For example, in similarity or dissimilarity-based matching algorithms, the criterion for accepting or rejecting a claim is generally a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Different threshold values will correspond to different FAR and FRR results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, for a more complete performance evaluation, it is common to use performance characteristic curves such as the Receiver Operating Characteristic (ROC), which plots 1−FRR versus FAR for varying threshold values (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From the ROC curve, it is also common to extract two metrics that combine all results into a single performance value, easing performance comparison between algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Equal Error Rate (EER) is the error that corresponds to the operation point where FAR = FRR and represents an equilibrium between convenience and security (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Area Under the Curve (AUC) measures the area under the ROC curve and serves as a measure of overall quality of a biometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Identification In identification, the system does not receive an identity claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, it will compare the current biometric measurements with the stored templates to assign one of the enrolled identities to the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Performance Evaluation 39 AUC FAR R R F- 1 EER 0 1 1 Threshold 0 EER 1 1 FAR FRR Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='13: Example of a Receiver Operating Characteristic (ROC) curve for an identity veri- fication system (left) and the evolution of False Acceptance and False Rejection rates with the threshold value (right) (from [343], adapted from [419], example for a similarity-based matching method including the Equal Error Rate point and the Area Under the Curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, it can reject to identify when the strongest match is, still, not strong enough, which is generally asserted based on a threshold (as in identity verification mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, in the identification mode, there are five possible outcomes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The subject is enrolled and the system correctly identifies them, granting them access;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The subject is enrolled, but the system mistakes their identity for another enrolled subject, granting them access under a wrong identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The subject is enrolled, but the system fails to identify them with any enrolled subject, rejecting access;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The subject is not enrolled and the system correctly rejects access;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The subject is not enrolled, but the system erroneously identifies them as one of the enrolled subjects and grants them access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A good biometric system would maximise the frequency of outcomes 1 and 4 and minimise the frequency of the remaining outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Most metrics for identification mode (see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2) are based on these frequencies [7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Regarding the situations where the subject is enrolled (designated as legitimate trials), the most common metrics are: True-Positive Identification Rate (TPIR or Hit Rate): For a total number of legitimate trials, TPIR corresponds to the fraction of those where one of the system’s R strongest predictions corresponds to the true subject identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As most other metrics, TPIR depends on the de- fined threshold, the selected number of top ranks R, and the list of enrolled candidates (in identification, each enrolled subject is considered a candidate, and TPIR, as most metrics, varies not only with the size of L but also with the variety and individual characteristics of each subject in L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 40 Fundamental Concepts Identification Rate (IDR or Accuracy): IDR corresponds to TPIR when only the single high- est ranking prediction is considered (R = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It corresponds to the fraction of the legitimate trials where the true identity was the method’s strongest prediction above the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the literature, this is one of the most used metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reliability: Reliability corresponds to TPIR with R = N (where N is the number of enrolled subjects), we get the reliability metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This measures how frequently the true identity sat- isfies the minimum threshold constraint, regardless of its ranking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' False-Negative Identification Rate (FNIR or Miss Rate): Represents the fraction of trials where the true identity does not correspond to one of the R strongest predictions above the threshold T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Reject Rate (RR): This metric pertains to the very specific situations where all identity predictions stand below the defined threshold T, and the system has no choice but to reject to identify (RR = 1−TPIR−FNIR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Misidentification Rate (MIDR or, commonly, Misclassification Error): MIDR is the comple- ment of IDR (MIDR = 1−IDR), equivalent to FNIR with R = 1, and measures the fraction of legitimate trials where the true identity is not the system’s top ranking prediction above T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Regarding situations where the subject is not enrolled in the biometric system (designated as impostor trials), the most common metrics are: False Positive Identification Rate (FPIR): It is the fraction of impostor trials where at least one of the system’s predictions meets the threshold criterion, and the system thus grants access to the unenrolled subject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Selectivity: Similar to FPIR, selectivity counts the average number of predictions above the threshold T across all impostor trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As in identity verification mode, some characteristic curves can be drawn based on these met- rics and the defined thresholds, to help evaluate the algorithms more robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first is the Cu- mulative Match Characteristic (CMC), which plots TPIR with threshold T = 0 against R, varying R ∈ [1,N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The second is the Receiver Operating Characteristic (ROC) which, in the case of iden- tification, plots Reliability against FPIR, for various threshold values (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From these plots, we can also extract the AUC and EER metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Time-based performance measurement There are biometric systems that perform recognition upon request, for example in computers or smartphones, keeping the session open until the user ends it or it reaches an idle time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This creates security issues, specifically when the user leaves the device unattended and forgets to close the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Performance Evaluation 41 AUC Cumulative Match Characteristic Receiver Operating Characteristic R FPIR y tili b aile R 1 1 0 0 1 N Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14: Examples of a Cumulative Match Characteristic (CMC) curve, and a Receiver Operat- ing Characteristic (ROC) curve for an identification system (from [337], including a representation of the Area Under the Curve, AUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To solve this issue, there are continuous (or online) biometric systems, that aim to perform biometric recognition in real-time, acquire traits continuously, and renew decisions as frequently as possible based on the most recent acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With such systems, if the user leaves and is replaced by an attacker, the system would be able to detect this and close the session before any harm could have been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides accuracy, an important facet of performance evaluation for continuous biometric sys- tems is timeliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [400] have addressed this issue and proposed some time-based metrics for the evaluation of continuous identity verification systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These can be adapted for both identification and identity verification, as described below: Time to Correct Decision (TCD): TCD measures the time the system takes to detect an impostor, and take an appropriate decision, relative to the moment the impostor replaces a legitimate user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For an ideal system, this should be zero, but that is virtually impossible to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, Sim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [400] state that this window should at least be always lower than W, called Window of Vulnerability (the minimum access time required for the impostor or wrong individual to cause any kind of damage);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Probability of Time to Correct Decision (PTCD): PTCD measures the probability, for a given system, of TCD being lower than W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The higher this value, the lesser the probability of an impostor having time to cause damage before the system acts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Usability: In a normal continuous biometric system, we should expect some decisions to be incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Over t seconds of usage, Usability measures the fraction of t where the legitimate user is deprived of access due to wrong decisions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For any biometric system, this should be as close to zero as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 42 Fundamental Concepts AUC Usability-Security Curve Security (PTCD) y tili b a s U 1 1 0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15: Example of a Usability-Security characteristic curve (from [337], adapted from Sim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [400]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From these metrics, the authors also define the Usability-Security curve (USC), a new char- acteristic curve that plots Usability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' PTCD for a varying threshold T (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' USC is similar to a ROC curve and, thus, AUC can also be computed, being considered a good metric to evaluate the timeliness of a biometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Wellbeing monitoring performance measurement For wellbeing monitoring systems, several metrics have been used to measure their perfor- mance [299].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The choices depend on the nature of the ground-truths and system outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For example, emotion recognition or affective computing algorithms generally focus either on a limited set of discrete emotion categories or on continuous ranges of emotion qualifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the former case, they typically cluster all emotions onto six categories designated as the six basic emotions by Ekman [112]: happiness, sadness, anger, fear, surprise, and disgust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For systems based on categorical labels such as these, the most common metrics are: Accuracy: The accuracy is the fraction of test samples that have been correctly classified by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is widely used in wellbeing monitoring applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' F1-score: The F 1-score is the harmonic mean of precision p (the fraction of positive predic- tions that are truly positive) and recall r (the fraction of positive samples that are classified as such), through the expression F1 = 2(pr)/(p + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is commonly used to evaluate per- formance in binary tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' AUC: The Area Under the ROC Curve, which plots sensitivity (the fraction of positive sam- ples correctly classified) versus the complement of specificity (1−specificity, the fraction of negative samples incorrectly classified as positive) for several thresholds, is also commonly used for binary classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Performance Evaluation 43 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='16: Illustration of the bidimensional valence-arousal space with example emotion cate- gories (adapted from [392], with valence on the horizontal axis and arousal on the vertical axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the case of continuous labels, emotion recognition systems typically consider a bidimen- sional space with two main emotion qualifier variables: valence and arousal [240].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Valence measures the pleasure of the emotion being felt, ranging from negative (unpleasant) to positive (pleasant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Arousal measures the level of activation or intensity of the emotion, ranging from low (passive) to high (active).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='16 illustrates these concepts by offering examples of categorical emotions in the bidimensional valence-arousal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For systems focused on regression tasks such as these, outputting continuous scores, the most common metrics are: Root Mean Squared Error (RMSE): RMSE is the root square of the mean of all squared differences between corresponding predictions and ground-truths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pearson’s Correlation Coefficient (CC): To correct the limitations of RMSE, the Pearson’s correlation coefficient uses the covariance between predictions ˆθ and ground-truths θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It follows the expression CC = COV{ ˆθ,θ}/(σ ˆθσθ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Concordance Correlation Coefficient (CCC): This metric is used for time-series predictions, based on the Pearson’s correlation coefficient and the mean value of each time series, fol- lowing the expression ρc = {2CCσ ˆθσθ}/{σ2 ˆθ + σ2 θ(µ ˆθ − µθ)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is a very common metric for performance evaluation of emotion recognition over time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sign Agreement Metric (SAGR): The sign agreement metric combines the magnitude of the prediction error with a penalisation for sign errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It can be computed with SAGR = 1 n ∑n i=1 δ(sign( ˆθi),sign(θi)), where δ is the Kronecker delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This metric has been mainly used for valence and arousal prediction, where the concordance of signs between the predictions and the ground-truths can be more important than the magnitude of the scoring errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ACTIVE ALARMED EXCITED AFRAID AMUSED ANGRY GLAD PLEASED NEGATIVE SAD SATISFIED POSITIVE GLOOMY RELAXED BORED TIRED SLEEPY PASSIVE44 Fundamental Concepts Among these alternatives, it is important to consider the task at hand when selecting metrics for an adequate performance evaluation that also enables a simple and thorough comparison with the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part II Electrocardiogram Biometrics 45 Chapter 3 Prior Art in Electrocardiogram Biometrics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Data Numerous researchers, when working with ECG signals, for biometric recognition purposes or automatic diagnosis of medical cardiac conditions, opt for private acquisitions of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, as the needs grow for more complete datasets, with more subjects, including medical conditions, on more sessions, spread across wider time frames, and under different posture and activity con- ditions, researchers became more aware of the importance of public signal collections [394].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, public ECG databases are needed to enable the comparison and benchmarking of al- gorithms in challenging conditions, across different publications, without requiring authors to im- plement algorithms and evaluate them again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Below, we delve into the important aspects behind a well-structured ECG signal collection, we present the most relevant publicly available collections, and we discuss the current needs and future possibilities regarding data in ECG biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Building a complete ECG data collection A well-structured ECG signal collection is key to appropriately guiding the development towards the exploitation of the best possibilities for the system, and accurately predicting its performance upon real-life application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To achieve such a complete collection, a few aspects have to be consid- ered: Number of electrodes: Fewer electrodes and leads have been shown to provide more chal- lenging settings for biometrics [122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 350];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Electrode placement: As shown by [490], the use of chest leads is less challenging than limb leads, and the distance of the electrodes to the heart has a significant negative impact on the system’s performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 47 48 Prior Art in Electrocardiogram Biometrics Sampling frequency: Sampling causes the loss of fine details that influence the recognition process [350].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The lower the sampling frequency, the larger the amount of detail that can be lost, and the higher the risk of aliasing of high-frequency noise (such as electromyogram interference);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Subject posture, activity, and fatigue: Several studies have shown that fatigue, exercise, or different postures have a negative effect on recognition performance if the systems have not been trained accordingly [332;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 350;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 445];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Subject health: Some health issues, mainly arrhythmia, can generate intrasubject signal variability that encumbers the recognition process [95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 96;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 369].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, systems should be made robust against this, by including subjects with heart conditions in the datasets used during the development and validation of the methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Number of subjects: The diversity of individuals and their own characteristics may ease or difficult the job of the biometric systems [104;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 468], and successful state-of-the-art algo- rithms have been shown to be significantly worse when evaluated on larger datasets [319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The use of a collection with a large number of subjects ensures the presence of subject di- versity, increasing the thoroughness of the performance assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As discussed in [343], the vast majority of literature in ECG biometrics reports the use of data from less than 100 subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Acquisition sessions: The ECG signal varies enough to cause recognition errors in most biometric systems, even over a short 24-hour period [236;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Systems should be prepared with data from several sessions, weeks or months apart, to ensure their robustness [350;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 393].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' All these factors can have an impact on the performance of an ECG-based biometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In order to correctly assess the capabilities of such systems, it is of the highest relevance to not only build a database that fits the system’s expected application context, but also one that reflects all possibilities mentioned above, in order to study the use of the same biometric system in a wider set of contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Publicly available data Currently, there are several collections, publicly available for ECG biometrics research1, which try to cover the aforementioned factors to create a challenging environment for the development of robust biometric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Many are stored by Physionet2, while others are ceded by their owners upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Below, we present and characterise the most relevant of the currently available 1Some of these databases may require prospective users to contact the respective administrators to request access to the data and/or sign agreements beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, all presented databases are made available by the creators for research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2Physionet ECG databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='org/physiobank/database/#ecg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Data 49 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Summary of the technical specificities of the most relevant publicly available ECG collections (from [343], OP – off-the-person;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' NS – number of subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fs – sampling frequency (Hz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' L / E – number of leads/electrodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Collection OP NS Fs Electrode Placement L / E Health Conditions Activity/Posture Sessions AHA No 154 250 Chest 2 / - Various 3 h CEBSDB [139] No 20 5000 Chest 2 / - None At rest, listening to music 60 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' CYBHi [394] Yes 128 1000 Palms + Fingers 2 / 4 None Reactions triggered by sound and video Up to two 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' sessions, 3 months apart DriveDB [168] No 9 456 Chest 1 / - Rest, highway, and city driving 50 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 h ECG-ID [281;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 307] No 90 500 Wrists 1 / - Sitting, unrestrained movement Various 20 s rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' per subject over 6 months E-HOL 24h No 203 200 Chest 3 / 4 None Ambulatory recordings 24 h European ST-T [421] No 79 250 Chest 2 / - Various Ambulatory recordings 2 h sessions FANTASIA [195] No 40 250 1 / - None Supine, at rest, watching a movie 120 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' LTST [196] No 80 250 Chest 2-3 / - Arrhythmia and ischaemia Ambulatory recordings 21-24 h MIT-BIH Arrhythmia [287;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 301] No 47 360 Chest 2 / - None Ambulatory recordings 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' MIT-BIH NSR [287;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 301] No 18 360 Chest 2 / - None Ambulatory recordings 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Physionet 2017 CinC Challenge Yes 8528 300 Fingers 1 / 2 Various At rest 10-60s PTB [49] No 290 1000 Chest + Limbs 15 / - Various At rest only 1-5 per subject, 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4-104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 s PTB-XL [443;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 444] No 18 885 500 Chest + Limbs 12 / 10 Various At rest only 1-2 per subject, 10 s QT [239] No 105 250 Chest / - Various Rest and exercise 15 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UofTDB [445] Yes 1019 200 Fingers 1 / 2 None Sit, stand, supine, exercise, and tripod Up to six 2-5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' recordings over 6 months 50 Prior Art in Electrocardiogram Biometrics ECG collections (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 for the number of publications that have used them), and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 summarises the characteristics of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' AHA: The AHA ECG database3 was created by the American Heart Association to guide the training of health professionals on the diagnosis of arrhythmias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes 154 ECG recordings from real patients, donated by various institutions, each three hours long and composed of two lead signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The last 30 minutes of each recording are annotated for seven types of arrhythmia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' CEBSDB: The Combined measurement of ECG, Breathing and Seismocardiograms (CEB- SDB) database [139] is a multimodal database available on Physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes two chan- nels of ECG (standard leads I and II), thoracic respiratory signals, and seismocardiograms (SCG) from twenty healthy subjects at rest and in the supine position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recordings include 50 minutes of classical music listening preceded and followed by five minutes at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ECG signals are sampled at 5 kHz and were acquired with foam tape and gel electrodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' CYBHi: The Check Your Biosignals Here initiative4 [394] is a collection of off-the-person ECG signals acquired with two dry electrodes at the palms, and two electrolycras at the middle and index fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It consists of a short-term dataset, with single-session recordings of 65 volunteers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' and a long-term dataset, where 63 subjects were recorded in two sessions, three months apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In each session, for 5 minutes, the subjects were exposed to videos designed to cause emotional reactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' DriveDB: Resulting from the Stress Recognition in Automobile Drivers initiative, this data- base was created with the purpose of monitoring stress in drivers [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Various physiolog- ical parameters (electrocardiogram, electromyogram, and skin conductivity) were recorded from 9 subjects over a total of 18 driving sessions, including periods of rest (lower stress levels), highway driving, and city driving (higher stress levels);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ECG-ID: The ECG-ID is a database entirely focused on biometrics [281;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 307].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 20-second ECG recordings were collected from 90 subjects, and are currently available on Physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For each subject, the database has between 2 and 20 recordings (a total of 310) collected over six months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The signals were acquired from Lead I using limb-clamp electrodes at the wrists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' E-HOL 24h Holter: This is an ECG database, focused on biometrics, from the University of Rochester5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A total of 203 healthy subjects were recorded using a Holter monitor for 24 hours, with four electrodes placed on the chest, from 3 leads following a pseudo-orthogonal configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3American Heart Association ECG database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ecri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='org/components/Pages/AHA_ECG_ USB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='aspx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4CYBHi dataset for off-the-person ECG biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='org/record/2381823# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='YwDLmHbMKMp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5University of Rochester Medical Center, Telemetric and Holter ECG Warehouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Database E-HOL-03-0202-003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: http://thew-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='org/Database/E-HOL-03-0202-003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Data 51 European ST-T: The European ST-T database [421] was originally intended for the analysis of ST and T-wave changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The database is composed of 90 two-hour excerpts of record- ings from 79 subjects, from 2 leads, and includes abnormalities with origin in myocardial ischaemia, hypertension, ventricular dyskinesia, and effects of medication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' FANTASIA: The FANTASIA database [195], available on Physionet6, is composed of 120 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' recordings of ECG, respiration, and blood pressure signals from forty people (twenty young and twenty old).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' All signals were acquired at 250 Hz as the subjects remained at rest, supine, watching Disney’s Fantasia movie;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Long-Term ST: The LTST database [196], available on Physionet, includes a variety of ST segment changes for the development of algorithms for the diagnosis of myocardial ischaemia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This database includes 86 records from 80 subjects, from ambulatory recordings between 21 and 24 hours, from two and three leads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' MIT-BIH Arrhythmia: The MIT-BIH Arrhythmia database [287;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 301], one of the most used in ECG-based biometrics research, is available at the Physionet repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The database is composed of a total of 48 signals, 30 minutes long excerpts from ambulatory two-lead recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The 47 subjects were selected to obtain a representation of a wide variety of arrhythmias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' MIT-BIH Normal Sinus Rhythm: This database is composed of excerpts from 18 subjects, from the MIT-BIH Arrhythmia database [287;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 301], presented above, deemed to be free from arrhythmias or other abnormalities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Physionet 2017 CinC Challenge: This ECG database is available on the Physionet repos- itory, and was used for the 2017 Computers in Cardiology (CinC) Challenge, consisting of arrhythmia detection in short single-lead ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes individual 10-60s recordings from a total of 8528 subjects, acquired at 300 Hz sampling frequency using the AliveCor KardiaMobile off-the-person acquisition device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' PTB: The PTB Diagnostic ECG database [49;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 230] includes 549 recordings from 290 heal- thy subjects and individuals with various cardiac conditions (such as myocardial infarction, dysrhythmia, hypertrophy, or heart failure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It has 1 to 5 recordings per subject, ranging between 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 and 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 seconds, from all 12 standard and 3 Frank leads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' PTB-XL: From the creators of the PTB Diagnostic ECG database, described above, the PTB- XL database [443;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 444] gathers a very large number of ten-second ECG signals (21 837) from over eighteen thousand subjects in clinical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The signals are sampled at 500 Hz (but also available at 100 Hz) and include the twelve standard leads, annotated by up to two cardiologists considering diagnostic, form, and rhythm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6FANTASIA database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: https://physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='org/content/fantasia/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 52 Prior Art in Electrocardiogram Biometrics AHA CYBHi DriveDB ECG-ID E-HOL 24h Euro ST-T LTST MIT-BIH Arrh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' MIT-BIH NSR PTB Physionet CinC 2017 QT UofTDB 0 2 4 6 10 8 12 14 16 22 5 Number of publications NUMBER OF PUBLICATIONS THAT USED EACH ECG COLLECTION 0 3 28 26 36 0 2 2 0 12 PTB-XL CEBSDB 6 0 18 22 24 26 30 28 32 34 36 20 12 FANTASIA 4 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Currently available ECG collections and the number of surveyed publications that have used them (adapted from [343]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' QT: The QT database aims to aid the development of automatic methods of measurement of QT waveforms [239].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This collection is a compilation of 105 15-minute relevant recording extracts from other public databases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UofTDB: The University of Toronto ECG Database [445] was specifically created for bio- metrics and addresses several important criteria for a thorough evaluation of biometric per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The off-the-person ECG signals were captured using dry electrodes at the thumbs of a total of 1019 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For each subject, the database includes up to six recordings over a period of six months, in various postures: supine, tripod, exercise, sitting, and standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While many researchers opt to use private acquisitions of data for their studies on ECG bio- metrics, public datasets have been crucial in allowing the appropriate comparison of results across publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, if our goal is to increase competitiveness between ECG-based biomet- rics and more developed traits, we should address some concerns regarding public collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Currently, countries like India, China, and the United States, are starting to invest in nationwide identification systems for their large populations [198], which awakens the need for biometric systems that can robustly discriminate between several million enrolled subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To keep up with this trend, we need to work towards the creation of public ECG collections with a larger number of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' PTB-XL is impressive in this regard, as it includes signals from over eighteen thousand subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, it is still quite limited considering the extent and diversity of data per subject, which are very important aspects for biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, researchers can currently choose from small on-the-person datasets that include 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 53 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: General structure of an ECG-based recognition system (from [344]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' health conditions and longer acquisition times (such as the AHA, European ST-T, and the MIT- BIH Arrhythmia databases), or the off-the-person UofTDB collection with short recordings from several healthy subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This calls for the creation of a public database with a number of sub- jects similar or superior to UofTDB, with several longer off-the-person recordings (ideally over one hour), taken over long periods (months to years), during different activities and postures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ECG-BG, used by Ingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [187], is composed of off-the-person data from 1119 healthy and unhealthy subjects, but it is still not publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Finally, it would also be very beneficial to have publicly available collections of signals ac- quired using recent wearable and seamless technologies, such as the CardioWheel and the Nymi Band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The highly acceptable acquisition settings offered by such products places, undoubtedly, new challenges on signal noise and variability, that would be very useful for the development of robust biometric algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work Despite being more recent and less developed than the face or fingerprint, the electrocardiogram (ECG) is quickly growing as a biometric trait, especially due to its inherent liveness and anti- spoofing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A comprehensive review of prior art in ECG-based biometric recognition is available in [343], published in the scope of this doctoral work, and succinctly summarised in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this section, we present a summary of the survey, organised in the four common stages of an ECG- based recognition algorithm: signal denoising, signal preparation, feature extraction, and decision (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' We also offer a discussion on the approaches based on deep learning and the current challenges and possibilities in the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Signal denoising ECG signals are highly susceptible to interference during the acquisition stage [366].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The ampli- tude of their waveforms may vary depending on the electrode characteristics and placement but, 54 Prior Art in Electrocardiogram Biometrics under ideal conditions (using chest leads in medical settings), the QRS complex only reaches 2−3 mV, the largest amplitude of the whole cyclic beat [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This means that when the electrodes are placed far from the heart, the signal is weaker and the noise is more dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This can result in different interference types, such as powerline inter- ference (PLI) from alternating current energy lines, baseline wander from breathing movements, electrode movement from motion, lead reversal due to electrode mispositioning, or pacemaker interference [124;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 403].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The stage of signal denoising is, thus, of utmost importance for an ECG biometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the first initiatives in ECG biometrics, using on-the-person acquisitions, the signal-to- noise ratio was higher, and noise sources were mainly limited to powerline interference and base- line wander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, filters, such as bandpass (BPF), lowpass (LPF), highpass (HPF), or notch (NF), were the first and have been the most frequent option, due to their simplicity and lower computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Bandpass filters have been most common, with bands between 1 − 40 Hz [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 272], 2 − 40 Hz [193;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 366;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 369], or 2 − 30 Hz [86–88], aiming to keep most useful individual information of the ECG while attenuating low and high-frequency noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recently, Choudhary and Manikandan [78] proposed the use of the Discrete Cosine Trans- form (DCT) for simultaneous removal of baseline wander and powerline interference, which proved more successful than bandpass filters, when compared on simulated scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Dis- crete Wavelet Transform (DWT) has also been proposed for denoising of on-the-person signals [80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 124;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 373], as it allows to decompose the signal into several levels, which may be separately processed to eliminate noise in certain frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When considering off-the-person approaches, wearables, or seamlessly integrated acquisition settings, it is reasonable to expect a considerable increase in the noise influence, with a lower signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The ability to capture the ECG signal weakens, so the amplitude of the ECG components is smaller when compared with chest leads, and movement artefacts are much more frequent and dominant [278;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 289;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 394].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For these, filters have also been widely applied [272;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 289], as well as DWT [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the enhanced noise content motivated the proposal of new approaches based on line fitting algo- rithms, such as fitting of polynomial curves and the Savitzky-Golay algorithm [374].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Their use or combination with moving average or median filters has been shown more successful than filters or transform denoising [298;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342], likely because noise is widely present across the ECG frequency range, and such methods avoid restricting their operation to narrow frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering all this, it is possible to conclude that the trend in signal denoising has been the evolution towards methods that can adapt to increasingly unexpected and dominant noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering the efforts devoted to more acceptable and comfortable acquisition settings, with an increasing focus on wearables and seamless settings, it is unreasonable to expect this trend would be reversed in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While filters appear to be a wise option if the noise is confined to known frequency ranges outside the ECG frequency range, for on-the-person signals, transforms (especially DCT) have shown to be good alternatives for denoising without causing distortions [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, when 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 55 the noise is widespread and/or its frequency range is unpredictable (such as with off-the-person signals), line fitting algorithms such as the Savitzky-Golay filter may be a better option, as they smooth the signal without making strong assumptions about its noise content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, research must continue to work towards increasingly robust and adaptable de- noising methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Researchers have recently started to use deep learning methodologies (as dis- cussed further on in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5), that have shown remarkable robustness to noise and vari- ability in several pattern recognition applications [163;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 484].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These, along with a deep study of data augmentation, may result in better alternatives to current and future methods devoted to signal denoising, and should certainly be explored in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Signal preparation ECG biometric algorithms frequently resort to the application of several processing operations over the acquired ECG signal, between denoising and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These have the main goal to prepare the signal for the feature extraction phase, maximise the performance of the system (by reducing persistent noise and variability), segmenting specific useful parts of the acquired signal, and/or discarding undesirable or prejudicial parts [191;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 278;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The noise and variability that may remain after the signal denoising stage, which this stage will aim to attenuate, are generally segment length and alignment inconsistencies, amplitude variations, heart rate variability, movement artefacts, and contact loss or impedance artefacts [189;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 191;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To fulfil its objective, this stage generally consists of reference point detection, signal segmen- tation, amplitude normalisation, time normalisation, and/or outlier detection processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The most common in the literature methods are fiducial point detection, signal segmentation, and amplitude normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Below, an overview of these processes is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fiducial Point Detection: To aid posterior processes, such as signal segmentation, the prepa- ration of the signal for recognition can include a step of detection of heartbeat reference points, designated as fiducials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The majority of the surveyed research works have used this technique, varying in the methods used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the other hand, some researchers have opted to make their algorithms completely non-fiducial, discarding the processes included in this stage [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 170;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 349].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Pan-Tompkins algorithm [328], based on moving-window integration and adap- tive thresholding, has been the most frequent choice for fiducial detection [272;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 303;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 327;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 390;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 446].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Alternatives include the Discrete Wavelet Transform (DWT) (used in [124;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 125;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 326;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 373]), the Trahanias algorithm [436] (based on morphological operations and adaptive thresholding, used in [298;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342]), and the Engelse-Zeelenberg algorithm [113;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 277] (based on differentiation, negative lobe detection, and adaptive thresholding, used in [272;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 276;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [342] have applied these methods and found Pan-Tompkins and Engelse-Zeelenberg gave better results for on-the-person signals, while Trahanias per- formed better in off-the-person settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 56 Prior Art in Electrocardiogram Biometrics Signal Segmentation: Signal segmentation is used to limit the signal span for feature extrac- tion, or to set a fixed size to ease template matching when the feature is the signal itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In some cases, the segmentation uses the fiducial point locations and is used to crop QRS complexes and/or other waveforms [414;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 429;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 446].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It can also be used to crop the whole heartbeat (or a majority of it), thus being performed at fixed distances before and after de- tected R-peaks or QRS complexes [66;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 272;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 493].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some research works included signal segmentation using sliding windows, with or without overlap, regardless of the complete- ness of the heartbeat cycles inside it [114;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 292;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 318].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The alignment and averaging of various signal segments are closely related to the signal segmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The alignment is generally performed using the R-peaks as a refer- ence after these are located, or it is performed using cross-correlation [32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 78;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 88;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 278].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It usually serves as a way to ensure the template and the collected signal are not affected by variability, which distorts the personal information the signal contains, and could threaten the recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Amplitude and Time Normalisation: As previously discussed, the electrocardiogram varies over time with several factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These include differences in acquisition equipment or in the interaction of the subject that may cause differences in signal amplitude and DC offset [189], or heart rate variability that causes significant changes in the duration of the heartbeats and their waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To mitigate this, some methods include amplitude and time normalisation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Amplitude normalisation techniques include the min-max technique [189] (used in[122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 250;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 369]), which normalises the signal to the range [0,1], the z − score method [318], which subtracts the signal and divides it by its standard deviation, or the max-div method [274;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 429], which divides the signal by the maximum amplitude value (generally, the R-peak amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Time normalisation techniques aim to reduce the impact of heart rate variability on the electrocardiogram’s heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some methods perform normalisation by simply shrinking the segmented signal to a predefined length through resampling [250;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 274;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 368].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As the heartbeat does not expand uniformly with lower heart rates, Tawfik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [429] normalised only the QT waveform, more prone to heart rate variations, using the Framingham study formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This formula computes the linearly corrected QT duration (QTLC), based on the time between the nearest R-peaks (RR) and the original duration of the waveform (QT), us- ing QTLC = QT +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='154(1−RR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fatemian and Hatzinakos [124] went further, segmenting each ECG heartbeat into its key waveforms (P, QRS, and T), and individually resampling them, before joining them back together, with regulated intervals between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' By reduc- ing the effects of heart rate variability and avoiding the typical distortion of the individual waveforms, this is likely the best technique for time normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, it requires the detection of several waveforms’ onset and offset fiducial points, making it potentially unreliable for off-the-person or seamlessly acquired signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 57 Outlier Detection: Outlier detection is generally applied to discard false or deflected heartbeats, segmented from unacceptably noisy signal portions affected by movement or impedance artefacts or contact loss [342].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Such methods should be able to discriminate between normal deflections, noise interference, and health-related deflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' DMEAN was proposed by Lourenço et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [273] specifically to reject heartbeat outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It verifies the compliance of candidate heartbeats with four rules, regarding the distance to the aver- age template, the minimum and maximum amplitudes, and the position of the maximum heartbeat amplitude (which must correspond to the R-peak location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Louis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [272] opted to use Gaussian Mixture Models (GMM) as a supervised method for outlier detection, after being trained on a set of known clean and desirable heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [342] proposed a clustering algorithm, NCCC, based on normalised cross-correlation between candidate heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While GMM, due to the supervised data, can easily be bi- ased towards certain patterns or subjects seen during training, clustering-based approaches like NCCC are not susceptible to this issue but can become unreliable for small sets of can- didate heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, with the rise of off-the-person, wearable, and seamlessly integrated acquisition settings, it is expected that robust outlier detection methods will be increasingly necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Feature extraction The stage of feature extraction aims to translate the acquired signal into a representation that reduces the effects of remaining noise and intrasubject variability and emphasises differences be- tween subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Several feature extraction methods have been proposed for ECG biometrics, which are generally grouped into three types – fiducial, non-fiducial, or hybrid approaches [288;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 292].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fiducial Approaches: Fiducial approaches are those that exclusively use as features mea- surements of fiducial landmarks of the ECG signal in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These measurements vary widely throughout the state-of-the-art, including time intervals, amplitude, widths, and angles based on the heartbeat waveforms P, Q, R, S, and T, their onset and offset points [193;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 366;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 390;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 446;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 490].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, these approaches present the significant drawback of requiring the previous localisation of several fiducial points in the ECG heartbeats (see subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2), which proves difficult to satisfy when using off-the-person or seamless signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, fiducial feature extraction approaches were considerably more frequent in early research works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Non-Fiducial Approaches: Non-fiducial approaches are those that use the entirety of the sig- nal (or segments of it), holistically, to extract features related to the waveform morphology [274;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 288;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 292].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These approaches include the use of Fourier, Wavelet, or cosine transforms 58 Prior Art in Electrocardiogram Biometrics [32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 288;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 289;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 318;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 368;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 429;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 472], autocorrelation coefficients [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 170;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 349], car- dioid graphs [188;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 414], generated tridimensional vectorcardiograms (TVCG) [105], mul- tiresolution local binary patterns [272], and information-theoretical approaches based on Lloyd-Max quantisation [87;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 88] and Kolmogorov complexity [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some methods do not perform feature extraction, alternatively using segmented heartbeats, average ensemble heartbeats, or segments between consecutive R peaks as features [66;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 237;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 276;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 298;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 493].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While more applicable to noisier signals, non-fiducial have still to reach the near-perfect performance reported by earlier works using fiducial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hybrid Approaches: Hybrid approaches are those that use features from both fiducial and non-fiducial origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are rare among the surveyed literature works and include the approach from Palaniappan and Krishnan [327], which combined common amplitude, in- terval and width fiducial features with a non-fiducial QRS complex form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ergin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [114] proposed the fusion of QRS fiducials, with several time-domain, Wavelet transform, and Power Spectral Density (PSD) features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Also, Dar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [96] opted for the extraction of a total of 46 features from Haar transform and heart-rate-variable R-R intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Through the analysis of the surveyed research works, it is possible to conclude that fiducial approaches generally contribute more towards a high-performance biometric system, as the use of specific measurements reduces useless information to a minimum, and allows for feature sets with fewer dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, as noise increases, the relevance of robustness overcomes that of accuracy, and the former can only be offered by non-fiducial methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The ideal feature extrac- tion method would be one that combines the conditions for high performance offered by fiducial approaches with the robustness to noise and variability offered by non-fiducial approaches, per- haps using deep learning networks and their characteristic robustness to noise and versatile feature extraction capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Extracted features may, additionally, suffer dimensionality reduction to improve performance [445].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although frequently overlooked, dimensionality reduction has a very important goal in biometric systems, as the number of features extracted by biometric algorithms can easily become too high for a time-efficient and reliable recognition process [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, dimensionality reduction aims to select or transform the extracted features, to reduce its number to a more computationally viable number, while keeping the maximum discriminant power to ensure the system’s recognition performance [445].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Dimensionality reduction methods in the surveyed literature range from common methods such as Linear Discriminant Analysis (LDA) [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 47;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 170;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 292] and its Fisher (FLDA) [332] and Heteroscedastic variants (HLDA) [250], Principal Component Analysis (PCA) [170] and Ker- nel PCA (KPCA) [170], or Greedy Best-First Search (GBFS) [95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 96], to rarer methods such as Discrete Cosine Transform (DCT) [349], Wilkes’ lambda stepwise correlation [193], correlation matrices [42;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 43], or bin selection based on symmetric Kullback-Leibler divergence [288;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 289].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The work performed by Plataniotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [349], Agrafioti and Hatzinakos [8], and Hejazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [170] provides an adequate platform for the comparison of dimensionality reduction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 59 According to their findings, LDA offers better performance than unsupervised techniques such as PCA and DCT coefficients, despite its supervised nature that requires knowledge of the subjects prior to the deployment of the biometric system [292].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' More recently, other supervised techniques such as the non-linear KPCA method [170;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 326] rendered even better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, research should probably focus on more sophisticated dimensionality reduction methods and deep learning methodologies, which are tunable to provide optimised non-linear dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Decision Based on the representation of the ECG acquisition, obtained through processes of feature ex- traction and dimensionality reduction, the decision stage aims to accurately attribute one of the enrolled identities to the user, in the case of identification tasks, or to accept or reject an identity claim, for authentication tasks [9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the case of identification, the decision stage usually consists of a classification process while, for authentication, the acceptance or rejection of the identity claim is generally based on a reference threshold T that is applied to the prediction score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The decision stage can be based on: Classifiers: A classifier can be trained on enrollment templates from a set of subjects, and then be used to discriminate them, to output an accurate decision when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Classi- fiers are more commonly used for identification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The most common classifiers in ECG-based biometrics are Support Vector Machines (SVM) [170;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 250;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 261;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 276;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 278;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 366;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 393;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 472], mostly using Radial Basis Function (RBF) and Polynomial functions as kernels, Nearest Neighbour (kNN) classifiers [10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 54;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 66;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 144;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 276;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 350;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 449;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 463], or Multilayer Perceptrons [144;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 188;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 327;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 447].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Metric-based Matching: Some methods are based on the comparison between the currently acquired trait and the previously acquired templates, stored in the system database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The comparison is performed based on similarity or dissimilarity metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In ECG-Based bio- metrics, most metric-based matching methods have been based on distance metrics, among which the most popular was the Euclidean distance [80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 274;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 292;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 332;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 349;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 369;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 393;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 405].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, since the Euclidean distance is regarded by some as unreliable in high dimensional spaces, some researchers have opted to use the cosine [393] or the Mahalanobis distances [156;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 233–235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among similarity metrics, literature methods include the correlation coef- ficient [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 69;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 124;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 125;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 373;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 389;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 390], normalised cross-correlation (NCC) [78], Gaussian log-likelihood [288;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 289;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 318], and Dynamic Time Warping (DTW) [298;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 303;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 493].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the literature, it is hard to perform a thorough and fair comparison between the algorithms based on the results reported by the respective authors, as the data used to evaluate such algorithms is commonly not the same, or is used differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, it is important to compare algorithms to find the advantages and disadvantages of each and find opportunities for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, to help the comparison of state-of-the-art methods in terms of reported performance, the results of the surveyed publications that have used the six most common data collections (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) 60 Prior Art in Electrocardiogram Biometrics – PTB, ECG-ID, MIT-BIH NSR, MIT-BIH Arrhythmia, UofTDB, and CYBHi – are presented in Tables 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' SVM and kNN have shown superior performance among traditional classifiers, even in situ- ations with increased noise and variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, it is safe to assume that these would be wise options for new ECG biometric algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, there is the need for an equally accurate alternative that would not require re-training with every subject enrollment or update (as SVM does) or the memory-heavy storage of all subjects’ templates (as kNN does).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recent studies in- dicate that Deep Learning models could solve these issues, but researchers will need to dedicate efforts to overcome the challenge of scarce supervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Deep learning Deep learning methodologies are quickly revolutionising several fields in pattern recognition, gal- vanising the machine learning community with outstanding results and unforeseen robustness to input noise and variability in diverse tasks [163;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 164;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 242;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 440].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It achieved these milestones mainly due to the flexibility and robustness of convolutional layers for feature learning, the se- lective memory of recurrent layers connected to their previous instances, and the versatility of fully-connected layers [242;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 483].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Their adaptability to scarce data through techniques such as data augmentation, fine-tuning, transfer learning, and weakly supervised learning, just add to their power for pattern recognition applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the topic of ECG biometrics, the study of deep learning is still a pioneering affair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It has, however, been gathering steam throughout the past four years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite a few works which contin- ued focusing on traditional feature extraction and decision models [37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 252;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 358;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 433;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 450], the majority of literature methods proposed since 2020 already include some deep learning architec- ture responsible for the processes of feature extraction, decision, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Initially, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [483] proposed a multiscale CNN that receives, in parallel, selected autocorrelation coefficients of approximation and detail Wavelet transform coefficient sets of two- second ECG segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [111] replaced the feature extraction stage using an Autoen- coder to learn lower-dimensional representations of segmented heartbeats, which were ultimately fed to a kNN classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, most researchers aim to integrate several stages into the deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sal- loum and Kuo [371], after signal preprocessing and segmentation, replaced the stages of feature extraction and decision with an RNN with Long Short-Term Memory (LSTM) and Gated Recur- rent Unit (GRU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [484] replaced the stages of feature extraction and classification, by feeding 2D representations of single-arm ECG signals to a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Luz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [284] also integrated the feature extraction and decision stages, proposing the combined use of two separate CNN, one receiving segmented heartbeats as input and the other receiving the respective heartbeats’ spectro- grams, fused at score level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [238] detected, segmented, and selected QRS complexes from ECG signals, and concatenated them into a QRS vector that served as input to a unidimensional CNN that fulfilled the purposes of feature extraction and decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With a softmax output, the method attained 100% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 61 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Results of surveyed approaches evaluated with PTB (adapted from [343], ordered by the number of subjects, NS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' works that joined PTB with other databases are not included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Author Year NS Results Ingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [187] 2020 290 IDR EER 100% 2% Ibtehaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [185] 2021 290 IDR EER 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='66% Srivastva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [412] 2021 290 IDR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% Thentu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [431] 2021 290 IDR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [79] 2019 290 IDR EER 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='59% Byeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [56] 2020 290 IDR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [451] 2021 290 IDR 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% Hammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [161] 2019 290 IDR 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% Pinto and Cardoso [339] 2020 290 IDR 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% Jyotishi and Dandapat [209] 2020 290 IDR 97.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='55% Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [180] 2022 248 IDR EER 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='26% Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [487] 2019 234 IDR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% Zhang et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [463] 2007 74 IDR EER 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [251] 2022 71 IDR 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [238] 2019 52 IDR 100% Brás and Pinho [54] 2015 52 IDR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% Coutinho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [88] 2013 51 IDR EER 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='01% Plataniotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [349] 2006 14 IDR FAR 100% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='02% Waili et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [472] 2010 18 IDR FPIR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% Tan and Perkowski [426] 2017 18 IDR 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% Li and Narayanan [250] 2010 18 IDR EER 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% Ergin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [114] 2014 18 F-score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='97% Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [487] 2019 18 IDR 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [483] 2017 18 IDR 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% Zhang et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% 64 Prior Art in Electrocardiogram Biometrics Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Results of surveyed approaches evaluated with MIT-BIH Arrhythmia (adapted from [343], ordered by the number of subjects, NS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' works that joined MIT-BIH Arrhythmia with other databases are not included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Author Year NS Results Salloum and Kuo [371] 2017 47 IDR EER 100% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4% Ingale et al.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='02% Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [472] 2010 47 IDR FPIR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% Ibtehaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [185] 2021 47 IDR EER 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='36% Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [451] 2021 47 IDR 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% Jyotishi and Dandapat [209] 2020 47 IDR 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% Lee and Kwak [245] 2022 47 IDR 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0% Dar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [96] 2015 47 IDR FAR FRR 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [180] 2022 47 IDR EER 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='36% Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [79] 2019 47 IDR EER 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='74% Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [450] 2020 47 IDR EER 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='73% Dar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [95] 2015 47 IDR 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [483] 2017 47 IDR 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% Jahiruzzaman and Hossain [197] 2015 11 IDR 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% Sasikala and Wahidabanu [373] 2010 10 IDR 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% Sufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [413] 2010 MIDR EER 1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 65 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Results of surveyed approaches evaluated with UofTDB (ordered by the number of subjects, NS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' works that joined UofTDB with other databases are not included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Author Year NS Results Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [344] 2019 1019 IDR 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% Luz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [284] 2018 1019 EER (raw) EER (spect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') EER (fusion) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% Pinto and Cardoso [339] 2020 1018 IDR 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% Pinto and Cardoso [338] 2019 1018 EER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86% Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [345] 2020 1018 EER 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6% Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [347] 2021 1018 EER 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6% Louis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [272] 2016 1012 EER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='89% Komeili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [225] 2017 82 EER (sess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') EER (post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% Ciocoiu and Cleju [83, 84] 2019/20 52 IDR EER 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='48% Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [450] 2020 46 IDR EER 100% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='17% Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [180] 2022 46 IDR EER 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='36% Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7: Results of surveyed approaches evaluated with CYBHi (ordered by the number of subjects, NS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' works that joined CYBHi with other databases are not included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Author Year NS Results Pinto and Cardoso [338] 2019 128 EER 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% Ingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [187] 2020 125 IDR EER 100% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% Hammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [161] 2019 65 IDR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% Ciocoiu and Cleju [83, 84] 2019/20 65 IDR EER 95% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6% Belo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [36] 2020 63 IDR EER 100% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0% Srivastva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [412] 2021 63 IDR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [451] 2021 63 IDR 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [180] 2022 63 IDR EER 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='48% Jyotishi and Dandapat [209] 2020 63 IDR 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4% Ibtehaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [185] 2021 63 IDR 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% Luz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [284] 2018 61 EER (raw) EER (spect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') EER (fusion) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8% 66 Prior Art in Electrocardiogram Biometrics IDR on the PTB database and, with Hamming distance matching, achieved 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='75% EER with long- term signals from the E-HOL 24h collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Several researchers have also tried to explore two-dimensional representations of ECG signals in order to use 2D deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ciocoiu and Cleju [83, 84] have studied S-Transforms, Gramian Angular Fields, Phase-Space Trajectories, and Recurrence Plots of segmented heartbeats on a custom two-dimensional convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' They found that S-Transforms offered the best performance, achieving around 95% identification rates on the UofTDB and CYBHi off- the-person databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Bento et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [38] explored spectrograms as inputs to a custom 2D CNN architecture and a DenseNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The latter achieved the best performance, with almost 97% IDR on ECG-ID and 100% on FANTASIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Byeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [56] also studied DenseNets, alongside XCeption and ResNet ar- chitectures to build ensembles of models receiving spectrograms, log-spectrograms, melspectro- grams, scalograms, and MFCCs, achieving approximately 99% IDR on the PTB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' More recently, Srivastva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [412] used images of ECG heartbeat plots as inputs to an ensem- ble of ImageNet-pretrained DenseNet and ResNet models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With this methodology, they achieved 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% IDR on PTB and CYBHi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thentu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [431] explored continuous wavelet transform-based multi-scale representations of ECG heartbeats on various ImageNet pretrained two-dimensional architectures achieving over 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% accuracy on both CEBSDB and PTB databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, beyond the approaches proposed within this doctoral work, no truly end-to-end methodologies are present in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Two-dimensional representations are promising, especially considering the possibility of using pretrained deep models, but the transformations themselves are still separately optimised processes that may lose important information and limit achievable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Another promising category of approaches is temporal networks, such as LSTMs, which have attained interesting results and are a natural match with ECG signals (time series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the aforementioned problem is still valid: if separate processes of denoising, preparation, and/or feature extraction are added to the pipeline, the model may be limited in the information received and thus in the performance it can achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Open Challenges and Opportunities Much of the great potential of deep learning for ECG biometrics is still to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering the information presented and discussed throughout this chapter, one can align the main challenges in ECG biometrics with the corresponding trends in ECG acquisition, thus painting a panorama of the history and near future of ECG biometrics (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As illustrated in the aforementioned figure, the use of deep learning is a major research op- portunity as we move into the future of ECG biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although deep learning typically brings significantly increased computational costs to biometric systems, these should be compensated by considerable boosts in performance and robustness due to the flexibility offered by deep models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Open Challenges and Opportunities 67 ELECTROCARDIOGRAM BIOMETRICS Lead I Lead II Lead III 1 2 3 45 6 RA LA LL RL aVF aVR aVL MEDICAL ACQUISITION HOLTER SYSTEMS OFF-THE-PERSON ACQUISITION WEARABLES & SEAMLESS ACQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='FUTURE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='SYSTEMS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='EVOLUTION OF ACQUISITION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='MAIN TOPICS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Fiducial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Decision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Long-Term ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Variability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Intersubject ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Variability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Health Conditions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Activity & Posture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Non-Fiducial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Higher Comfort ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='& Usability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Efcient Noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Suppression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Non-Fiducial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Integration in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Everyday Objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Continuous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Biometrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Contact Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Artifacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Deep Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Improved Multimodal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Biometrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Contactless/Distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='ECG Acquisition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Biometric Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Larger Databases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Learning with few data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: A panorama of ECG biometrics across time: the past, present, and future trends in ECG acquisition for biometrics and the corresponding research challenges (from [343]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The integration of all processing stages in a single end-to-end deep model, alongside new techniques of data augmentation and regularisation, could enable the coordinated optimisation for individual recognition and lead us to new levels of robustness against noise and variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, as we move towards “black-box” deep models, it is important to address the problem of trustworthiness and transparency through the study of model interpretability and explainability in ECG biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The growth of deep learning should further unveil a serious problem in ECG biometrics: the scarcity of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This places significant hurdles on the development of accurate and robust biomet- ric models, which should only worsen with data-hungry deep learning methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As we delve deeper into deep learning methodologies for ECG biometrics, it is important to build larger and more complete off-the-person ECG collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, researchers should devote further efforts to data augmentation, siamese architectures, triplet learning methodologies, unsupervised and self-supervised learning, and other strategies towards the mitigation of the effects of data scarcity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Partially linked to data scarcity, another large problem currently plaguing ECG biometrics is the prevalence of unrealistic and mismatching evaluation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The diversity of databases and data subsets is noticeable throughout this literature review and makes it impossible to adequately compare different methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The high frequency of random train/test subset splits makes results unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, the rarity of long multi-session data acquisitions makes long-term 68 Prior Art in Electrocardiogram Biometrics performance a hidden problem waiting to be truly unveiled and solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, this thesis part focuses on five contributions to these open challenges and opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Specifically: In Chapter 4, we propose the first true end-to-end methodology for ECG-based identifica- tion, complete with a study on the progressive integration of pipeline stages within the deep model and various tailored data augmentation strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Chapter 5, we adapt the previous model for identity verification and explore identification vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' triplet loss training for template similarity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed methodology is also benchmarked against state-of-the-art approaches on a restructured evaluation setup for more realistic results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Chapter 6, we study the effect of long-term variability on the performance of state-of-the- art approaches using a database of day-long Holter acquisitions, along with the application of template/model update strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Chapter 7, we study the relative importance of ECG waveforms, with a special focus on the QRS, throughout experimental setups with varying database size and noise/variability using interpretability tools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Chapter 8, we propose an end-to-end methodology for the recovery of the entire set of twelve standard leads requiring as input just one single-lead blindly-segmented ECG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 4 End-to-End Models and Augmentation Strategies for Identification Foreword on Author Contributions The research work described in this chapter was conducted entirely by the author of this thesis, under the super- vision of Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso and André Lourenço.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of this work have been disseminated in the form of a chapter in a book and an abstract in national conference proceedings: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lourenço, “Deep Neural Networks for Biometric Identification Based on Non- Intrusive ECG Acquisitions,” in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Arya and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Bhadoria, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', The Biometric Computing: Recognition and Registration, CRC Press, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [344] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lourenço, “Improving ECG-Based Biometric Identification Using End-to-End Convolutional Networks,” in 24th Portuguese Conference on Pattern Recognition (RECPAD 2018), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation The state-of-the-art in ECG-based recognition mostly consists of pipeline algorithms, composed of separate stages of denoising, signal preparation, feature extraction, and decision, as discussed in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Even the most recent methods using deep learning techniques still rely on some of these separate processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, Convolutional Neural Networks (CNNs) possess the tools to integrate all phases of processing, from acquisition to decision, into a single model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This integration replaces separate, step-by-step tuning with a holistic optimisation process, synergically adapting the model to attain the best performance possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Furthermore, the flexibility of convolutional and fully-connected layers makes deep networks able to autonomously learn the most fitted features for the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Meanwhile, these keep the ability to generalise and be robust against high variability and noise dominance over the signals [242;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 483].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, it could be the key to improving the inferior performance results verified in off-the-person ECG biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 69 70 End-to-End Models and Augmentation Strategies for Identification Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Architecture of the proposed CNN model for ECG-based identification (the number of neurons on the fully connected layer refers to the entire dataset with 1019 possible identities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This work aimed to study the full extent of the capabilities of CNNs for biometric identi- fication using non-intrusive ECG signal acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A CNN architecture is proposed for the complete integration of traditional pipeline stages in a single model, for higher accuracy and ro- bustness in off-the-person settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To obtain further improved performance, unidimensional data augmentation strategies are designed specifically for ECG-based biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Methodology 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Model The proposed convolutional neural network (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) integrates all common pipeline stages into a single, end-to-end model receiving raw five-second ECG segments and delivering the corre- sponding identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It follows the typical structure of a convolutional neural network: the first part includes convolutional and max-pooling layers, and the second part includes one fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Convolutional layers hold filter banks to learn the most advantageous representation of the input signal segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pooling quickly reduces the number of parameters, controlling the compu- tational cost and training time, and improves robustness to small input variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed architecture uses Rectified Linear Unit (ReLU) activations, filters’ size 5, stride 1, pooling size 5, and pooling stride 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The feature maps output by the last convolutional layer are concatenated into a single uni- dimensional vector of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This serves as input to the fully-connected layer, which weighs and combines the received features at each neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The fully-connected layer is composed of N neurons (where N is the number of enrolled subjects), with softmax activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The neuron that outputs the highest value will correspond to the predicted identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Based on a batch of train samples fed to the network, a measure of loss is computed by com- paring the output of the network with the true labels of the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The weights/parameters that compose the neural network are adjusted to reduce that loss, using an optimiser function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Methodology 71 this work, the optimiser Adam [219] was used, with empirically adjusted initial learning rate in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='001], and Sparse Categorical Cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To avoid overfitting, the network used dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Dropout will avoid learning overly specific patterns in the training data [231;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 411].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' They are placed between two layers and act upon the con- nections between them, setting the corresponding input to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the proposed method, dropouts are used on the connections between the flattened vector of features and the fully-connected layer, effectively blocking the access of the classifier to a part of the features, and requiring it to become less specific to the training set, and more robust to unexpected variability and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Data augmentation strategies Data augmentation is used to obtain a more robust classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It consists of the application of small transformations or changes to the train samples while protecting the integrity of the underlying label of each sample, to simulate larger datasets and ensure the network is robust to such variabil- ity [71;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Like deep learning in general, data augmentation techniques are significantly more frequent in 2D networks (for images) than in 1D (signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Based on the recent work of [440], and taking into account the unique characteristics of the ECG signals, seven different types of data augmentation are proposed and explored for 1D convo- lutional neural networks (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These types are: Baseline Wander: simulates a periodic undulation on the signal, by adding a sinusoidal wave with a frequency near 1 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cropping: takes a contiguous subsegment and resamples it to match the original length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the case of ECG signals, this technique simulates slower cardiac frequencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Flip: inverts the signal along the time axis, which causes the inversion of the heartbeat waveforms and their relative locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Gaussian Noise: introduces Gaussian noise (with mean zero and standard deviation about ten times lower than the signal amplitude) to cause high-frequency distortions on the signal, similar to movement artefacts and powerline interference noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Magnitude Scaling: rescales the original train sample by multiplying it by a factor inferior or superior (but close) to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Magnitude Warping: similar to the previous technique, it rescales the signal in a non- uniform fashion, using a sinusoidal wave instead of a fixed factor, so that different parts of the signal will have their amplitude reduced or increased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Random Permutations: divides the signal into N contiguous subsegments with similar lengths, and their order is randomly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This may cause discontinuities in the heart- beats and their waveforms, simulating sensor faults or abrupt segment terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 72 End-to-End Models and Augmentation Strategies for Identification Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Illustration of the effects of the different data augmentation techniques on an exam- ple five-second ECG segment (for easier visualisation, the original segment was denoised with a bandpass filter 1–30 Hz and had its amplitude z-score normalised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Experimental Setup 73 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Illustration of the progressive phases of integration of the traditional pipeline stages into the CNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Experimental Setup The performance of the proposed convolutional neural network architecture, as previously de- scribed, was evaluated on off-the-person ECG recordings of the University of Toronto ECG Data- base (UofTDB) [445].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides the entire database of 1019 subjects, two subsets were also used, with 25 and 100 subjects, to evaluate the performance in smaller datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The datasets were divided with 70% of the data for training and 30% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed method suffered slight adaptations to allow the study of the progressive inte- gration of the traditional pipeline stages into the CNN model (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, besides the proposed end-to-end version that receives raw five-second ECG segments, three other variants were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first receives five-second ECG segments denoised using a 1 − 30 Hz band- pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The second receives the average of heartbeats detected using the Engelse-Zeelenberg algorithm and normalised to zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The third variant receives DCT fea- tures extracted from the average heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The pool size of max-pooling, which was set at 5 for five-second segments as input, was changed to 3 for ensemble heartbeats, or 2 for DCT features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed method was compared with a baseline algorithm, adapted from the method proposed by Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [342] using SVM and kNN for decision, and the state-of-the-art algorithm based on autoencoders proposed by Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [111], and the algorithm based on AC/LDA features by Matta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [292], evaluated in the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results and Discussion With DCT features as input (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4), the performance of the proposed method is similar to that of the baseline algorithm for 25 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, with the increase of subjects on the dataset (with 100 and 1019 subjects), the proposed algorithm falls behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This may be caused by the very concise information that the input carries, fitted for typical pipeline algorithms as the baseline but not for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of the evaluation using ensemble heartbeats as input support 74 End-to-End Models and Augmentation Strategies for Identification CNN (Proposed) Baseline SVM Baseline kNN 85 90 95 100 Identification Rate (%) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Results with DCT Features as Input 25 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1019 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: Results of the proposed and baseline algorithms, when using DCT features as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' this hypothesis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5), as the performance increases and approaches that of the baseline methods, and even surpasses that of the kNN classifier on the two smaller datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Integrating additional stages into the deep learning model allows us to simplify its structure, and use longer signal segments as inputs (in this case, five seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This means an increase in complexity of the input, which can harm the performance of the network, but also an increase in available information and variability, which can allow for a more robust model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of the use of denoised and raw five-second segments (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6) illustrate the trade-off between signal complexity and the increase of robustness due to extra information and variability, as the results were similar to those of the CNN receiving ensemble heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' More- over, in general, the results of the CNN with raw segments surpassed those of the CNN with denoised segments, which likely result from the benefit of increased variability during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Increased variability is, in turn, the goal of data augmentation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The aforementioned techniques were separately tested on the subsets of 25 and 100 subjects (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Most of the techniques of data augmentation bring improvements to the algorithm’s identification rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The exceptions were magnitude scaling in the smallest dataset, cropping and magnitude warping in the 100 subject dataset, and Gaussian noise in both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Likely, this performance decay is the result of a corruption of the underlying labels with these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The most promising data augmentation techniques were random permutations (that excelled in both datasets), baseline wander, and flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These were evaluated in groups to assess if the combina- CNN (Proposed) Baseline SVM Baseline kNN 85 90 95 100 Identification Rate (%) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Results with Ensemble Heartbeats as Input 25 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1019 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Results of the proposed and baseline algorithms, when using ensemble heartbeats as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results and Discussion 75 CNN (Denoised) CNN + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (Denoised) CNN (Raw) CNN + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (Raw) 85 90 95 100 Identification Rate (%) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Results with Denoised or Raw Five-Second Segments as Input 25 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1019 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Results of the proposed and baseline algorithms, when using five-second ECG seg- ments as input, raw or denoised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' tion of two or three techniques would offer performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8) show that the sole use of random permutations is the best option, although the combinations also caused an improvement in identification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' We compared the proposed and baseline algorithms with state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As they were implemented and tested in the same conditions, the algorithms of Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [111] and Matta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [292] can be used for a direct benchmarking (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed method presents better results than the alternatives and a slightly slower decay with the increase of the number of subjects, denoting better scalability to larger populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The state-of-the-art algo- rithms likely suffer from using nearest neighbour classifiers, prone to overfit, as the results of the baseline algorithm with kNN were also consistently worse than with SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The method of Ed- uardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [111], despite showing remarkably good results in the denoising of signals during our experiments (using the entire encoder-decoder), falls short in these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Finally, the results of the proposed and baseline algorithm can be compared with the results reported by the most recent prior artworks (see Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The IDR of the proposed and baseline algorithms may pale in comparison with some results reported in some of the considered prior works, but it is important to consider the evaluation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Only Wieclaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [457] used an off-the-person database, as opposed to the much cleaner signals of on-the-person databases still used by most researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Also, the UofTDB collection enabled the evaluation of the proposed algorithm with a much larger set of subjects than any other identification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' None Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Perm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Scaling M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Warping B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Wander Flip Gauss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Noise Cropping 95 96 97 98 99 100 Identification Rate (%) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Results with Data Augmentation Techniques 25 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7: Results of the proposed algorithm receiving raw five-second segments, with each technique of data augmentation, on the datasets of 25 and 100 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 76 End-to-End Models and Augmentation Strategies for Identification None Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Perm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Perm+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Wander R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Perm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Wander+Flip 95 96 97 98 99 100 Identification Rate (%) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Results with Combinations of Data Augmentation 25 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8: Results of the proposed algorithm, receiving raw five-second segments as input, with combinations of data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Proposed Method Baseline DCT+SVM Eduardo (2017) Matta (2011) 80 85 90 95 100 Identification Rate (%) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Benchmarking of CNN with Data Augmentation on Raw 5s Segments 25 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1019 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9: Direct benchmarking between the proposed architecture with the best baseline algo- rithm and the two implemented state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Comparison of the proposed and baseline algorithms with recent state-of-the-art meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Authors Brief Description Dataset IDR Proposed Method Raw segments + CNN with data augment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UofTDB (off-the-person) - 1019 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1% Baseline DCT features + SVM UofTDB (off-the-person) - 1019 sub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3% Salloum and Kuo [371] LSTM-GRU RNN ECG-ID (on-the-person) - 90 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100% Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [483] Multiscale CNN Several (on-the-person) - 18-47 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% Wieclaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [457] Heartbeats + MLP Private (off-the-person) - 18 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0% Tan and Perkowski [426] Fiducials + RF fused with DWT + WDIST kNN Several (on-the-person) - 184 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% Carreiras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [66] Heartbeats + kNN Private (on-the-person) - 618 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4% Brás and Pinho [54] Kolmogorov-based compression PTB (on-the-person) - 52 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [449] Max-pooling of sparse coding coefficients PTB (on-the-person) - 100 subj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Summary and Conclusions 77 However, it is important to recall that deep learning both requires and benefits greatly from large datasets where each class is represented by a large number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While, as visible in the results presented here, data augmentation attenuates the prejudicial effects of scarce data, it is difficult to acquire sufficient ECG signals from each subject to compensate for the increased noise and variability in off-the-person settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the datasets used, each subject was represented, on average, by just 170 five-second ECG segments, which is arguably too few to train a convolutional neural network to robustly discrimi- nate between 1019 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering this, with future efforts devoted to adequately dealing with scarce data, deep learning methodologies could see their potential for ECG biometrics be bet- ter harnessed and place themselves as clearly better alternatives to traditional pipeline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Summary and Conclusions This work proposed a convolutional neural network for biometric identification based on non- intrusive electrocardiogram acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed method was evaluated for incremental integration of traditional ECG biometric pipeline stages, including a complete substitution by the CNN architecture, that received raw five-second ECG segments and output a decision on the corresponding identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides this study, seven data augmentation techniques for unidimensional signals were ex- plored and their individual and collective impact on the algorithm’s performance was assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results on the UofTDB database were compared with those of a baseline algorithm and two promising state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results show that the total integration of traditional pipeline processes in the CNN ar- chitecture was successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed CNN with data augmentation and receiving raw five- second segments surpassed, in all settings, the baseline and state-of-the-art algorithms in direct benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among other recent state-of-the-art methods, considering the diverse dataset char- acteristics, the proposed method has also shown promise as an accurate and robust biometric identification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 5 Triplet Loss and Transfer Learning for Identity Verification Foreword on Author Contributions The research work described in this chapter was conducted entirely by the author of this thesis, under the supervi- sion of Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of this work have been disseminated in the form of an article in international conference proceedings: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “An End-to-End Convolutional Neural Network for ECG-Based Biometric Au- thentication,” in 10th IEEE International Conference on Biometrics: Theory, Applications and Systems (BTAS 2019), Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [338] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation The field of ECG biometrics has been steadily evolving from on-the-person signals to off-the- person acquisition setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite the enhanced usability and comfort, the increased dominance of noise and variability in off-the-person signals places serious hurdles to the real application of ECG biometric systems (more detailed information in Chapter 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some researchers have resorted to deep learning in order to fight off noise and variability and achieve better performance and robustness [111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 284;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 344;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 483;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 484].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, these still rely on separate predefined feature transforms and/or noise removal techniques, which are not optimised for the task at hand and therefore limit the achievable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In fact, the work presented in Chapter 4 shows that end-to-end deep models offer considerable performance benefits in off-the- person ECG biometric identification, especially when using tailored augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Building upon the work in Chapter 4, this work studied the use of end-to-end convolutional neural networks (CNN) for ECG identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The main goal was to discover if dismissing all separate processes of denoising or preparation in favour of a single integrated model (granted complete control over the robustness to signal noise and variability) would also improve per- formance and robustness in the task of identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Besides the use of metric learning through the triplet loss, this work introduces the technique of weight transfer from a similar model 79 80 Triplet Loss and Transfer Learning for Identity Verification Conv1 24@1x5 ReLu MaxPool 1x5 Conv2 24@1x5 ReLu MaxPool 1x5 Conv3 36@1x5 ReLu MaxPool 1x5 Conv4 36@1x5 ReLu FC1 100 ReLu Reference Sample 1x1000 1st Stored Template Kth Stored Template Conv1 MaxPool Conv2 MaxPool Conv3 MaxPool Conv4 FC1 Current Segment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Model trained with triplet loss Trained weights transfer (TL-CNN) Euclidean distance 1 Euclidean distance K Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (score) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Positive Sample Negative Sample Shared weights Positive Euclidean distance Negative Euclidean distance Triplet Loss Conv1 24@1x5 ReLu MaxPool 1x5 Conv2 24@1x5 ReLu MaxPool 1x5 Conv3 36@1x5 ReLu MaxPool 1x5 Conv4 36@1x5 ReLu FC1 100 ReLu FC2 N Softmax Input ECG Segment 1x1000 Output 1xN (Identity scores) Trained weights transfer (IT-CNN) Identification model Identity verification model Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Schemes of the proposed identity verification model, including the weight transfer between networks for both proposed training methodologies (the input shape 1×1000 refers to the five-second length of the segments used in this work, 1000 samples at 200 Hz sampling frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' trained for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This aimed to assess whether parameters optimised for identification tasks would offer performance benefits in identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed network and both training methodologies were extensively evaluated on three ECG collections, which include on-the-person and off-the-person signals with varying signal qual- ity, multi-session recordings from several subjects, and the influence of emotions, posture, and ex- ercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This evaluation included the assessment of the trained model’s applicability to other signal collections, through cross-database tests using transfer learning and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Methodology 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Model architecture The proposed method for ECG biometric identity verification is based on a CNN (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1, darker grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' All enrolled users have one or more fixed-length ECG segments (templates) stored in the system, that have been blindly segmented (without requiring any process of reference point detection) from a recording obtained upon enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When a user claims to be an enrolled individual, the model receives and processes, simulta- neously, the K stored templates of the claimed identity and 1 current segment of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Methodology 81 comparison between the processed current segment and each of the K stored templates allows the model to output a dissimilarity score, which can be used to accept or reject the identity claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' After sample-wise normalisation to zero mean and unit variance, the processing of each input segment or template starts with a succession of convolutional and pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1, four unidimensional convolutional layers are alternated with three max-pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' All have 1×5 filters, and the convolution is performed with unit stride and no padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first two convolutional layers hold 24 feature maps, while the last two hold 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The second part of the network is composed of a fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The outputs of this fully-connected layer for each stored template (a) and for the current segment (b) are compared using normalised Euclidean distance [461] (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1)), using their variance (Var) so the output lies in [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among the K distances computed, the minimum is output as the final dissimilarity score for identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' d(a,b) = Var(a−b) 2(Var(a)+Var(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Model training The weights for the identity verification model layers are transferred either from a model trained for identification or from a model trained using triplet loss (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The training methodol- ogy of transferring weights from an identification model aimed to take advantage of the training process of identification deep neural networks and assess how it could benefit a neural network for identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the other hand, triplet loss has been recently and successfully used in biometrics, for identity verification and other similar tasks [73;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 74;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The training process requires specific structural changes to the model, which are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 and described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In all cases, during training, the optimiser used was Adam [219] with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='001, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='999, and no decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Dropout [411] and data augmentation (random permutations, as in [344]) were used to prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' After training, the weights are transferred to the respective layers on the identity verification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Transfer from identification network (IT-CNN) In the case of identification training (IT-CNN), the model is structured to receive 1 input segment and contain one additional fully-connected layer (FC2), using softmax activation, that will out- put N scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is trained for identification with data from N identities (following the work of Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [344]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' After receiving a training segment, considering its true label and the network’s output, the sparse categorical cross-entropy loss [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 76] is computed and used during training to ultimately prepare the model to adequately discriminate the subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 82 Triplet Loss and Transfer Learning for Identity Verification 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Triplet loss training (TL-CNN) To be trained using triplet loss (TL-CNN), the identity verification model, which has K + 1 in- puts and 1 output, is restructured to receive 3 inputs and offer 2 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The three inputs are the reference template, a positive template (whose identity is the same as the reference), and a negative template (of a different identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The network processes each input and computes the dissimilarities between the reference and the positive template (p) and between the reference and the negative template (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using adequate triplets of signal segments, the goal is to minimize p and maximize n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, the model is trained using triplet loss [73], which can be computed for each triplet of inputs through the function: l(p,n) = max(0,α + p−n), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2) where α controls the margin to be enforced between the scores of positive and negative pairs (in this work, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This margin eases the choice of an effective threshold for the purpose of identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Experimental Setup In this work, one of the main concerns was ensuring the performance results were as realistic as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To achieve this, all databases were split between training subjects and testing subjects, to ensure the model can be trained and applied to data from two entirely disjoint sets of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Furthermore, cross-database tests were performed to ensure the model can generalise to other population samples and acquisition settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Subject enrollment was limited to realistic durations (5, 10, 15, or, at most, 30 seconds of the earliest data from each subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Data and reference methods The three selected databases were UofTDB [445], CYBHi [394], and PTB [49;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UofTDB (off-the-person, 1019 subjects) was used for most experiments due to its intermediate but realistic signal quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The PTB (on-the-person, 290 subjects) and CYBHi (off-the-person, 128 subjects) databases were used to assess performance in better and worse signal quality settings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To match UofTDB, CYBHi and PTB signals were resampled to 200 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For PTB, only Lead I signals were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Three literature methods were used as reference: the AC/LDA method, proposed by Agrafi- oti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the Autoencoder method, proposed by Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [111];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' and the DCT method, proposed by Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [342;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 344] (adapted for identity verification, using cosine distance nor- malised to [0,1] for matching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results and Discussion 83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Evaluation scenarios The proposed and implemented methods were evaluated across four scenarios, as detailed below, using the Equal Error Rate (EER, see [343] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Here, each signal segment used as input for the proposed model was five seconds long (1000 samples at 200 Hz sampling frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the single-database scenario, the proposed model was evaluated on UofTDB data, and compared with the aforementioned reference state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The last 100 subjects were reserved for training, while the data from the remaining 919 subjects were used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The number of enrollment templates varied between 1, 2, 3, or 6 five-second segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The varying identity set size scenario aimed to study how the performance is affected by the number of subjects used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Instead of the original 100 subjects, training was performed using the 20, 50, or 150 last subjects of UofTDB, and the remaining 999, 969, or 869 subjects, respectively, were used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The cross-database scenario was designed to assess the proposed model’s applicability to sig- nals from other databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed model, previously trained on 100 subjects from UofTDB, was directly tested on data from CYBHi and PTB, without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' At last, in the fine-tuning scenario, the goal was to assess the performance benefits brought by fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As in the cross-database scenario, the proposed model trained on UofTDB data (from 100 subjects) was fine-tuned to CYBHi/PTB data (from 20 subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This was compared to the model directly trained, from scratch, on data from CYBHi or PTB (from 20 subjects, following the single-database scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With 20 subjects reserved for training, the tests on this scenario were performed for 108 (CYBHi) or 270 (PTB) subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results and Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Single-database scenario The results obtained in the single-database scenario are presented in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In all cases, the IT- CNN model, which used weights trained for identification, attained better results than TL-CNN, which was trained using triplet loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With 30 seconds of user enrollment, IT-CNN achieved 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86% EER, while TL-CNN offered 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='94% EER in the same circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When considering shorter enrollment recordings (5 s, 10 s, and 15 s), the performance of both proposed methods worsens, but always remained below 14% EER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is noteworthy that IT-CNN presented a wider advantage over TL-CNN with more enrollment data, which may denote it takes better advantage of the availability of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among the reference methods, AC/LDA presented the best results in most settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When compared with these results, both proposed methods offered consistently lower EER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering the best reference method for each enrollment duration, IT-CNN attained an EER reduction of around 7−8%, which can be regarded as a significant improvement over the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 84 Triplet Loss and Transfer Learning for Identity Verification Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Single-database scenario: EER results (%) when trained with data from 100 UofTDB subjects and tested with 919 UofTDB subjects (in italics: proposed methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' in bold: best results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Enrolment duration Method 5 s 10 s 15 s 30 s IT-CNN 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='70 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='92 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='52 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86 TL-CNN 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='93 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='89 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='90 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='94 AC/LDA [10] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='27 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='90 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='55 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82 Autoencoder [111] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='68 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='84 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='09 DCT [342;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 344] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='05 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='41 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='55 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 TL-CNN Output 0 20 40 60 IT-CNN Output 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 20 40 60 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Network outputs for all training samples of five example subjects from the UofTDB collection (one subject for each row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the average output feature vector is presented as a black line, and the standard deviation as a grey area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among other state-of-the-art works, Luz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [284], under similar settings, reported 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='27% EER with UofTDB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' All IT-CNN and TL-CNN performance results are better, even when considering only 5 seconds of enrollment (much less than what was used by Luz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, Louis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [272] reported 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='89% EER, but only using single session data from 1012 UofTDB subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using only data from subjects with more than one session (82 subjects), Louis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' reported 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='10% EER, while Komeili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [225] reported 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% EER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although the evaluation settings are different, the proposed method’s results are aligned with these (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86% for IT-CNN with 30 s enrollment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The statistical significance of the results was assessed, repeating the evaluation on one-hundred random subject data divisions between enrollment and testing (Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Overall, the results were better, as this test is arguably less realistic than the remaining tests performed in this study (a real 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results and Discussion 85 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Single-database scenario: Mean and standard deviation of the EER results (%) obtained on 100 random data divisions (in italics: proposed methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' in bold: best results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Enrolment duration Method 5 s 10 s 15 s 30 s IT-CNN 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14 TL-CNN 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='11 AC/LDA [10] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='18 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='17 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='17 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='31 Autoencoder [111] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='16 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='12 DCT [342;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 344] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='16 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='14 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='21 biometric system will always use the very first data of a subject for enrollment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Applying a paired two-sided t-test to the EER estimates, the results of the proposed methods IT-CNN and TL-CNN were significantly different in all cases (the differences are statistically significant at the 1% level), not only from each of the implemented state-of-the-art methods but also between themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, the outputs of the network for five-second training segments from different sub- jects were visualised (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are, effectively, the feature vectors used for the identity verification decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is possible to observe that, despite the blind segmentation and the noise and variability carried by each five-second segment, the trained network was able to represent each input segment in a way that maximises similarity with other segments from the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although some variability is still present, it is reduced to a manageable level for the biometric identity verification task, and the differences between the subjects’ output patterns are noticeable even through a simple visualisation of the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Varying identity set size scenario In the varying identity set size scenario, multiple numbers of UofTDB subjects reserved for train- ing were explored (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In all cases, an increase in the number of training subjects resulted in performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The best results were obtained with 150 training subjects and 30 seconds enrollment, with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='46% EER and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='71% for IT-CNN and TL-CNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Never- theless, even with just 20 training subjects, IT-CNN offered performance under 10% EER (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='92%, with 30 s enrollment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As in the single-database scenario, it was noticeable that the performance advantage of IT- CNN over TL-CNN was greater when more data was available, either for model training or user enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For example, the EER difference between IT-CNN and TL-CNN grew from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='25% when increasing the number of training subjects from 20 to 150 and the enrollment duration from 5 to 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite this, one could expect the IT-CNN method to perform better than the state-of-the-art, even under scarce data conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Based on these results, when pretrained with only 20 subjects with 10 s enrollments, IT-CNN should offer an EER lower than 13% on a population of nearly one thousand individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 86 Triplet Loss and Transfer Learning for Identity Verification 20 50 100 150 Number of Training Subjects 6 8 10 12 14 16 EER (%) IT-CNN 5 s 10 s 15 s 30 s 20 50 100 150 Number of Training Subjects 10 12 14 16 EER (%) TL-CNN 5 s 10 s 15 s 30 s Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Varying identity set size scenario: EER evolution with number of subjects reserved for training, for diverse enrollment durations, for the proposed methods IT-CNN and TL-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5 10 15 30 Enrollment duration (s) 20 30 40 EER (%) CYBHi Direct Application IT-CNN TL-CNN DCT AC/LDA Autoen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5 10 15 30 Enrollment duration (s) 10 12 14 16 18 EER (%) PTB Direct Application IT-CNN TL-CNN DCT AC/LDA Autoen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: Cross-database scenario: EER for the proposed methods IT-CNN and TL-CNN when trained with UofTDB data and directly applied to CYBHi or PTB, and comparison with state-of- the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Results and Discussion 87 5 10 15 30 Enrollment duration (s) 20 30 40 EER (%) CYBHi Direct Training vs State-of-the-Art IT-CNN TL-CNN DCT AC/LDA Autoenc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5 10 15 30 Enrollment duration (s) 10 12 14 EER (%) PTB Direct Training vs State-of-the-Art IT-CNN TL-CNN DCT AC/LDA Autoenc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Fine-tuning scenario: EER results for the proposed methods IT-CNN and TL-CNN when directly trained with CYBHi or PTB data from 20 subjects, and comparison with state-of- the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5 10 15 30 Enrollment duration (s) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 EER (%) CYBHi Direct Training vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fine-Tuning IT-CNN DT IT-CNN FT TL-CNN DT TL-CNN FT 5 10 15 30 Enrollment duration (s) 10 12 14 EER (%) PTB Direct Training vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fine-Tuning IT-CNN DT IT-CNN FT TL-CNN DT TL-CNN FT Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Fine-tuning scenario: EER results for the proposed methods when (DT) trained, from scratch, with data from CYBHi or PTB, or when (FT) trained with UofTDB data and fine-tuned to CYBHi/PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 88 Triplet Loss and Transfer Learning for Identity Verification 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Cross-database scenario In the cross-database scenario, the proposed methodologies were directly applied to CYBHi and PTB data, after training on data from 100 UofTDB subjects (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With CYBHi, IT-CNN offered better performance than TL-CNN when using 30 s enrollment (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='30% against 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='56% EER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, with reduced enrollment duration (5 s), TL-CNN per- formed better (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='66% against 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='89% EER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This reinforces the idea that TL-CNN is better in scarce data situations, while IT-CNN takes better advantage of the greater availability of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With PTB, IT-CNN was, in all cases, the most successful proposed method (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='83% EER with 5 s enrollment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among the state-of-the-art methods, AC/LDA behaved as in the single-database scenario (see Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1), offering the worst results when using 5 s enrollment, but sharply improving with more enrollment data, offering the best result with PTB (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='03% EER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' DCT presented the best result with CYBHi (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='40% EER), while IT-CNN offered the second-best result (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='30% EER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Both proposed methods were, in general, worse than the state-of-the-art with the PTB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Fine-tuning scenario In the fine-tuning scenario, the model was trained with CYBHi/PTB data and compared with the state-of-the-art (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5) and when trained with UofTDB data and fine-tuned to CYBHi/PTB (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Directly trained on CYBHi data, TL-CNN attained 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='04% EER, but it offered 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='56% EER if trained with UofTDB data, and further improving to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='37% EER if fine-tuning is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' TL-CNN was able to attain better performance than IT-CNN in more difficult settings, once again indicating that this method may be better fitted for scarcer data or noisier signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On PTB, TL-CNN did not offer competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For IT-CNN, fine-tuning (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='06% EER with 30 s enrollment) improved the results over the direct application, but it was not enough to significantly improve the results of direct training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Apparently, training with UofTDB data over- prepared the network for a degree of noise and variability that is not verified on PTB signals, which ultimately harmed its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A hybrid method where, before regular training, the neural network would be encouraged to mimic the behaviour of traditional methods, could be beneficial in cross-database settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Overall, the proposed methodologies presented more competitive results on CYBHi than on PTB, likely due to PTB signals’ lesser noise and variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, while the proposed model has shown robustness to noise and variability in off-the-person settings, the state-of-the-art methods are more fitted to cleaner on-the-person signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Summary and Conclusions In this work, an end-to-end model, based on a CNN, was proposed for biometric identity verifica- tion using ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It was designed to use a set of stored templates of a claimed identity and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Summary and Conclusions 89 an ECG segment of the current user, and output a dissimilarity score used to accept or reject the identity claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The model was trained using triplet loss or by transferring weights from a similar model trained for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed model was successful in improving the performance of state-of-the-art methods, especially in off-the-person signals, increasingly used in ECG-based biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using identifica- tion training has offered better performance than triplet loss when more training and enrollment data are available and could bring benefits for other tasks or biometric traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Both methods have shown the ability to overcome increased noise and variability of off-the-person signals, focusing on subject-specific signal patterns for accurate identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, further efforts should be devoted to improving performance and turning the ECG into a reliable biometric trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 6 Long-Term Performance and Template Update Foreword on Author Contributions The research work described in this chapter was conducted in collaboration with Gabriel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lopes, under the supervision of Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The author of this thesis contributed to this work on the formulation and im- plementation of the biometric recognition models, the conceptualisation of the template update methodologies, the preparation and conduction of experiments, the discussion of the results, and the writing of the scientific publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of this work have been disseminated as an article in international conference proceedings and an abstract in national conference proceedings: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lopes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Don’t You Forget About Me: A Study on Long-Term Performance in ECG Biometrics,” in IbPRIA 2019: 9th Iberian Conference on Pattern Recognition and Image Analysis, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [269] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lopes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Rebelo, “Long-Term Performance of a Convolutional Neural Net- work for ECG-Based Biometrics,” in 25th Portuguese Conference on Pattern Recognition (RECPAD 2019), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation Modern ECG biometric techniques generally report relatively high identification rates and low verification error, while current off-the-person ECG acquisition techniques contribute towards in- creased simplicity, usability, and comfort [342;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, as with most alternatives, the performance decays over time, especially when considering long-term usage [224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The natural variability of the input biometric data, the effects of ageing, and variations caused by the subject’s interaction with the sensor contribute to intrasubject variability [201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This causes stored individual templates to quickly lose representativity, resulting in poor recognition perfor- mance and placing serious challenges on long-term recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, long-term biometrics ben- efits from the frequent update of stored templates to keep up with the variability and ageing of the users, thus maintaining acceptable performance over time [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 91 92 Long-Term Performance and Template Update Specifically for ECG biometrics, long-term performance and template update remain open challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the most thorough work yet on this topic, Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [236] have studied the performance decay over time on their proposed algorithm for ECG-based authentication, finding that performance decays significantly even over relatively short periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the focus of this study was limited to authentication and the algorithm proposed by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, we build upon the study of Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' We aim to more thoroughly explore the problem of long-term performance decay in ECG biometrics and how to correctly address it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Specifically, we extend it to (a) focus on the task of ECG biometric identification, (b) study diverse state-of-the-art biometric methods, and (c) evaluate how different update techniques may be able to improve long-term performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work In the literature, it is difficult to find a strong and widely accepted rule for template update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Most methods are based on heuristics and empirically determined thresholds, which are highly depen- dent on the data and the application scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For example, Komeili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [224], for authentication, have set the acceptance threshold equal to the point of zero false acceptance rate, thus ensuring updates with only genuine samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, it is possible to identify some common mechanisms that may vary depending on different factors: these include the choice of the update criterion (based on thresholds or graphs), the update periodicity (online or offline), the selection mechanism, and the template update work- ing mode system (supervised or semi-supervised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The taxonomy of template update (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) divides the existing techniques into two categories: supervised and semi-supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Supervised methods are offline methods in which label attribution is given by a supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These contain the Clustering subcategory, which includes the MDIST that aims to search for the templates that minimise the distance among all the samples in the database (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', the most similar) and DEND that aims to search for the templates that exhibit large intraclass variations resort to the dendrogram (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', the most different) [282].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The second subcategory comprises Editing-based methods, which are independent of the num- ber of templates and give focus on the whole collected training set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A subset E ∈ T is generated, maintaining the classification performance offered by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The best subsets were obtained by re- viewing the structure of the data (which needs to be done for each subject) [134;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' All the algorithms (based on k-Nearest Neighbours) must be representative of T and can be roughly de- scribed as incremental when the E starts empty and grows, or decremental when E starts equal to T and in each iteration some instances are deleted until some criterion is reached [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Semi-supervised methods merge labelled (in biometrics, these correspond to the initial train- ing samples) and unlabelled (corresponding to the samples available during system operation) data to improve the system’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This category comprises the Single Modality (for unimodal biometric systems) and Multiple Modality (for systems using more than one biometric trait) sub- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Single Modality subcategory includes the Self-Training approaches such as FIFO 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 93 Template-Based Adaptive Biometric Systems Supervised Semi-supervised Clustering Editing-based MDIST DEND Single Modality Multimodal Graph Approach Min-Cut Co-update Self Training Nearest Neighbour Selective Condensing Edited Reduced Penalised Fixation MDIST DEND Super Template Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Dendrogram representing the taxonomy of template update techniques (based on [364]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (first-in-first-out), Fixation, Super Template (X composed by N templates x) where new genuine date is always fused to a common single template [237] updated online during the execution of continuous verification, Penalised template update method based on the mean of the past ECG’s and the actual ECG [80] and clock method where the current template is tested against all the others stored in the database [378].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Generally, a new unknown trait measurement is used for template update if its score (returned by the biometric recognition system) is above a set threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, the future performance of the system relies heavily on the chosen threshold value [364].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The update threshold is commonly estimated using enrollment templates or training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When training data are scarce or when using short enrollments, this may lead to some problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' First, important intrasubject variability information may be missed since only the patterns similar to the stored templates are used (and all others are discarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Second, the effectiveness of the online methods depends on the order of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Third, the methods are vulnerable to large intraclass variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' At last, since the algorithms normally look for the minimal cost (high scores), they may get stuck in local maxima and always only use high-confidence data for updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Semi-supervised methods also include Graph approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These commonly define a graph where the nodes are either labelled (the identity is known) or unlabelled (unknown identity) data, and the edges (which can have different weights) are the similarity between those samples [364;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 494].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To be considered a graph-based semi-supervised method, it must estimate a function f, approximate the known Y on the labelled nodes, and include two terms to smooth the graph: a loss function and a regulariser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These two terms are what define each approach (as can be seen in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) [494], among which the most common in biometrics is min-cut graphs [364].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 94 Long-Term Performance and Template Update Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Graph-based template update methods and their respective loss and regulariser func- tions (based on [494]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Method Source Loss Regulariser Min Cut [44] ∑ i∈L (yi −yi|L)2 1 2 ∑ i j wi j(yi −yj)2 Gaussian Random Fields and Harmonic Function [495] ∑ i∈L (fi −yi)2 f T∆ f Local and Global Consistency [479] n ∑ i=1 (fi −yi)2 D− 1 2 ∆D 1 2 Tikhonov Regularisation [35] 1 K ∑ i (fi −yi)2 γ f TS f Manifold Regularisation [402] 1 l l ∑ i=1 V(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' f) γA∥ f∥2 k +γI∥ f∥2 I Graph Kernel from the Spectrum of Laplacian [70] min 1 2wTW exp(−σ 2 λ) Spectral Graph Transducer [495] minc(f −γ)TC(f −γ) f TL f Local Learning Regularisation [222] min 1 k k ∑ i=1 (yi − fk(xi))2 γ k∥ fk∥2 Considering the topic of template update is still to be adequately addressed in ECG biomet- rics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' this work studies the effect of ECG permanence and variability in long-term identification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Furthermore, it aimed to evaluate the effect of template update techniques, on the performance of several state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Methodology 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Biometric identification methods To fully and objectively evaluate the effects of ECG variability on the performance of biometric algorithms, a study was conducted on four literature methods: Plataniotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [349] proposed an ECG biometric recognition method using a non-fiducial approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Signals are preprocessed using a bandpass filter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 − 40 Hz), followed by fea- ture extraction with autocorrelation (AC) and dimensionality reduction using discrete cosine transform (DCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The fifteen most relevant features were selected, and Euclidean distance was used for classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Tawfik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [429] used a bandpass filter (1 − 40 Hz) in the preprocessing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' QRS complexes (the most stable part of ECG) were cut from the signal using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='35 second window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The average ensemble QRS was computed and features were extracted using the DCT technique (the thirty most relevant features were selected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A multilayer perceptron (MLP) is used for classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Methodology 95 Belgacem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [32] also preprocessed signals with a bandpass filter (1 − 40 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The QRS complexes were located and cut from the signal, and the average QRS was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The feature extraction resorted to Discrete Wavelet Transform (DWT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From all DWT de- composition levels, only the most relevant were selected, and a Random Forest is used for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This method was originally proposed for authentication and adapted here for identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [111] used a Finite Impulse Response (5 − 20 Hz) filter for preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Heartbeats were cut with a fixed length of [−200,400] ms around each R peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Outliers were detected and removed using DMEAN (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5, with Euclidean distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For decision, the k-nearest neighbours (kNN) classifier was used with k = 3 and cosine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beyond these literature methods, this work also explored the deep learning-based method- ology presented in Chapter 4 (proposed in Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [344]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This method uses an end-to-end unidimensional convolutional neural network, that receives five-second blindly-segmented z-score normalised ECG segments to perform biometric identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The feature extraction part of the model is composed of four convolutional layers, interleaved with three max-pooling layers, with filter/pooling size 1 × 5 and ReLU activation units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For classification, the network uses a single fully-connected layer and softmax activation units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Template update methods FIFO: First-In-First-Out is the most common strategy and, computationally, is very lightweight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Here, the database is updated using new samples whose score is above or below a threshold (whether the score represents similarity or dissimilarity, respectively), or between two threshold values (discarding previously stored sample) [87;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The score of a new sample can either be output by a classifier or be a measure of distance or similarity between that sample and the stored templates [274].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, the training data were used to search for threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Among all training samples, 75% were used to train a model, which was used to obtain scores for the remaining data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Comparing the scores with several thresholds, the error at each threshold was analysed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2) to find one that simultaneously maximises true positives and minimises false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fixation: This method consists of fixing certain templates, allowing only the remaining stored samples to be updated [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, 25, 50, or 75% of the enrollment templates of the individual are fixed, while the rest of the samples are free to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This ensures some initial, labelled information of the subjects remains on the system over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' An adaptation of this technique was explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Here, n + j × n samples were fixed, where n ∈ [1,2,3] is the number of fixed initial templates, and j increases over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, j ∈ [0,6] increased by one at each testing moment (j ∈ [0,6]), which allowed the system to fix more and more samples over time, thus storing information on the subject’s variability over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 96 Long-Term Performance and Template Update Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Illustration of the search for the ideal threshold (the values were chosen near the intersection, inside the yellow zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In a real system with potentially endless use, the parameters n and j should be carefully chosen to avoid the eventual fixation of the entire template gallery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fine-Tuning: In this technique, the model is briefly optimised with the samples accepted for update, using the predicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The model retains knowledge of the users’ supervised training samples, as it was trained using their enrollment samples, but is slightly adapted to the new per- sonal patterns carried by the new signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is explored exclusively for the CNN method [344].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Experimental Setup 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Dataset For evaluation, the ECG signals used were from the E-HOL-03-0202-003 database1 (most com- monly designated as E-HOL 24h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This database consists of a study of 202 healthy subjects (only 201 were provided), recorded using three leads at 200 Hz sampling frequency, after an initial rest- ing supine period of 20 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From the available data of 201 subjects, thirteen were discarded due to saturation or unacceptable noise (subjects 1043, 9003, 9005, 9020, 9021, 9022, 9025, 9046, 9061, 9064, 9071, 9082 and 9105), similar to what was done by Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From each of the remaining 188 subjects, only the lead most closely resembling Lead I ECG was selected, to approximate off-the-person settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Experiments Standard sample wise normalisation was performed following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) for all methods except that of Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [111], which required [−1,1] min-max normalisation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2), where x 1THEW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: http://thew-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='org/Database/E-HOL-03-0202-003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Search for the optimal update threshold 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 False Positives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 False Negatives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 5 0 1 2 3 4 6 Threshold6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 97 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Schema illustrating the use of each E-HOL record for training and testing (in orange - training segment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' in blue - each test segment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' represents the input signal and ˜x the normalised signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ˜x[n] = x[n]−x σ(x) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) ˜x[n] = 2 � x[n]−min(x) max(x)−min(x) � −1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2) In order to fit the used data, some changes were introduced to the original methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the method from Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', the cutoff frequencies of the bandpass filter were changed to 1 and 40 Hz, to retrieve important information on higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Outlier removal was reparametrized with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The autoencoder had the topology [120,60,40,20,40,60,120] and was trained using the Adam optimiser with a learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Classification used k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For the method of Belgacem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', DWT feature extraction was performed in four decomposition levels, due to lower data sampling rate, and cd4, cd3, cd2, and cd1 coefficients were used as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Data were divided into train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The training phase used the last 30 seconds (mim- icking short enrollments on real-life applications) of the first 60 minutes (avoiding unrealistic calm after the initial resting period) of each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Five-second overlap was used to obtain 26 samples from every 30 seconds of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To study performance over time, testing was performed over seven time points (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3): one immediately after enrollment, another after one hour, and regularly until the end of the records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From each point, from 15 minutes of data, thirty 30 s samples are extracted, and batches are built with one sample from each of the 188 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Handcrafted methodologies After implementing, for identification, the method proposed by Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [236] (replicating their evaluation conditions), it was possible to conclude that the ECG signal is not fully permanent over 24 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, similarly to what was stated by Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', the results are relatively good over the first two hours (see Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4a), although permanence was not verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The performance results at each test hour, obtained through the weighted average of the corre- sponding batches, for the state-of-the-art methods can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It was found that the → 30 seconds fortraining 15 minutes fortesting 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 98 Long-Term Performance and Template Update (a) (b) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: Identification performance over time corresponding to (a) the Labati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' method, and (b) the implemented state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' performance is mostly acceptable in the first test point, but performance decays significantly over time and variability changes considerably over the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, a minimum around the 15th hour occurs independently of the chosen method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Con- sidering that most of the records start between 8-12 am, after 15 hours the subjects must be sleep- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this perspective, it appears that the ECG is most different from normal when the subject is asleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering the previous results, template update was applied to the methods, in an effort to avoid performance decay over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 presents the results using the FIFO technique, with diverse thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For the methods of Plataniotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' and Eduardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', the best results were obtained using two thresholds, respectively, {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% accuracy improvement) and {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3} (+ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7% accuracy), improving all performance results until the 15th hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, for the method of Belgacem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', the performance worsens with template update after the first two hours (best results were obtained when the difference between the highest and second highest scores ∆score ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The same was verified for the method of Tawfik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' which, in the first two hours, offered the best results with ∆score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In general, using two thresholds instead of one offered the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering this, it appeared that the Random Forest and MLP classifiers are not suitable for these kinds of template/model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This was confirmed after a repetition of the evaluation of these methods, with kNN replacing the classifiers (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With kNN, the template update was able to reduce the performance decay over time, improving accuracy, on average, by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2%, respectively, for the methods of Belgacem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' and Tawfik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As for the Fixation technique, the obtained results were more promising (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This template update technique brought performance improvements for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The fixation tech- nique that offered the best results was j×3+3, improving the baseline identification accuracy, on average, by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Variability study in Labati conditions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Labati method Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 20 40 60 80 100 Time (min)Baseline 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Plataniotis Tawfik Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Belgacem Eduardo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 99 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Comparison of the FIFO method applied with different thresholds to different identi- fication methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Results using FIFO update with different thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Plataniotis FiFO technigue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 < score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 < score < 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 < score < 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Eduardo FiFO technigue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 < score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 < score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 < score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Tawfik FiFO technique 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 < Ascore < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='65 < △score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9 △score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 △score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Belgacem FiFO technigue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15 < △score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 < △score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 △score> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Ascore > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Tawfik with kNN FiFO technique 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline 5 < Ascore < 7 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 4 < △score < 8 △score > 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 △score < 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Belgacem with kNN FiFO technique 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 < Ascore < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 < △score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Ascore > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Ascore < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)100 Long-Term Performance and Template Update Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7: Results using Fixation update (the corresponding value represents the number of sam- ples that were fixated per subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Deep convolutional network The results for the implemented end-to-end convolutional neural network are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Model update offered a small improvement in performance in the first test point (91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='48% versus 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15 without update).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the model experiences sharp performance decay and, after the fifth hour, the model update is unable to improve identification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In fact, model update caused a decrease in identification rate, which is coherent with the findings regarding update with multilayer perceptron classifiers reported by Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [269].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Different results were obtained when the fully-connected layer of the network was replaced by a kNN classifier (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although the results with kNN are slightly worse than those of the end-to-end network (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='89% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15% for the first test point), the template update technique is more successful and is able to offer performance improvements for almost all test points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When compared with the results reported by Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', the CNN (either end-to-end or with kNN classification) offers the best performance in the first test points after enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, it gradually loses that advantage as time passes, even with template update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Likely, the network will require more data with more variability during the first training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Increasing the training data to thirty minutes or even a few hours per subject would enable the Plataniotis Fixation technigue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline 1/4 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 1/2 3/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 j+1 j*2 + 2 j*3 + 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Eduardo Fixation technigue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline 1/4 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 1/2 3/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 j+1 j*2 + 2 j*3 + 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Tawfik with kNN Fixation technique 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Baseline 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 1/2 3/4 j+1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 j*2 ＋ 2 j* 3 + 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)Belgacem with kNN Fixation technique 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Baseline 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 1/2 3/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 j+1 j* 2 + 2 j*3+3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions 101 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8: Performance results over time for the CNN model with fine-tuning-based model up- date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9: Performance results over time for the CNN model adapted with kNN decision and FIFO template update, for several threshold criteria (samples with scores between the presented values are accepted for update).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' network to better learn the common variability patterns of the ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This should not only increase the initial performance, immediately after enrolment but also reduce the performance decay over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions This work studied how ECG variability affects the performance of state-of-the-art biometric algo- rithms, and how template update could mitigate performance decay over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results have shown long-term identification performance in ECG biometrics is generally weak, despite the promising results often presented in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Template update techniques proved successful in enhancing the long-term performance of handcrafted state-of-the-art methods, especially when using template fixation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Addi- tionally, with a deep learning algorithm, results are better than traditional methods immediately after enrollment, although it offers slightly worse performance as time progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Generally, one can conclude that further efforts are needed for the study and development of more advanced techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The obtained results in these more realistic settings show that the CNN Results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 Baseline CNN with fine-tuning update Identification Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)CNN+kNN Results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Baseline 13 < score < 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 12 < score < 18 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 < score < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 14 < score < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 10 < score < 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 0 5 10 15 20 25 Time (h)102 Long-Term Performance and Template Update performance levels commonly reported in the literature would likely not be verified upon real application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Special focus should be devoted to supervised update techniques, so that ECG-based biometric systems can offer reliable performances over long periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 7 Leveraging Explainability to Understand ECG Biometrics Foreword on Author Contributions The research work described in this chapter was conducted entirely by the author of this thesis, under the supervi- sion of Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of this work have been disseminated in the form of an article in international conference proceedings and an abstract in national conference proceedings: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Explaining ECG Biometrics: Is It All In The QRS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=',” in International Conference of the Biometrics Special Interest Group (BIOSIG 2020), Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [339] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “xECG: Using Interpretability to Understand Deep ECG Biometrics,” in 27th Portuguese Conference on Pattern Recognition (RECPAD 2021), Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation Throughout the past twenty years, research on biometrics based on the electrocardiogram (ECG) has largely been a success story [343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' After successful proofs-of-concept in cleaner medical signals (on-the-person), the focus is quickly shifting to acquisitions in more realistic scenarios (off- the-person).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Deep learning approaches [162;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 238;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 284;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 338;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 344] have been essential in dealing with the increased noise and variability in off-the-person settings, despite the performance and robustness issues that still hinder application in real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, deep learning decisions are obscure: unlike traditional methods based on fiducial features, we don’t know what information the model uses to distinguish people [107;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 372].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' One can assume that the models look mainly to the QRS since it is the most stable part of the ECG in the face of noise and variability [172;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 379].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Several methods have thus focused on QRS complexes for ECG biometrics [238;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 446], but this practice has become uncommon in recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This indicates the true role of this waveform complex in identity discrimination is still to be adequately recognised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 103 104 Leveraging Explainability to Understand ECG Biometrics P Q R S T Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Illustration of the ECG waveforms on a sample PTB signal segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Currently, pattern recognition researchers understand the importance of knowing what specific information is relevant for their models to reach decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Retreating to easily explainable tradi- tional models (such as decision trees) is often unacceptable due to their performance limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, various interpretability tools are being developed to peek into the inner workings of deep networks applied to diverse tasks [67;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 385;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 397].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This work uses, for the first time in the literature, such interpretability tools on a deep ECG biometric model, to understand what parts of the ECG are most useful for automatic human iden- tification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The model is a competitive state-of-the-art method [338;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 344] applied for ECG-based identification in data subsets with diverse signal quality and number of identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With this, we aim to assert the importance of the QRS and other waveforms for ECG biometrics and discuss future possibilities as this topic evolves towards more challenging and realistic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Addi- tionally, we propose an intuitive way to visualise interpretations for unidimensional signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The code and additional results are available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 The Electrocardiogram as a Biometric Trait As presented in Chapter 2, subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2, the ECG is approximately a cyclical repetition of a set of waveforms (P, Q, R, S, and T) that corresponds to a heartbeat (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) [286;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Each of these waves corresponds to specific phenomena involved in the heart’s contraction and relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As a measurement of the electrical currents spread across the heart, the ECG signals will reflect the geometry of this organ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For example, larger hearts, with more cells to depolarise and repolarise, will result in ECG waveforms with larger amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Higher or lower basal heart rates will also result in different signal morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Since heart geometry and basal heart rates vary across individuals, this intersubject variability is what makes the ECG sufficiently unique to be used in biometric recognition [172;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 441].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the ECG signals are also susceptible to intrasubject variability factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Noise sources during acquisition, the short-term and long-term effects of exercise, emotional states, stress, drowsiness, and fatigue are some of the factors that reflect mainly in the heart rate variability, changing the morphology of the P-R and S-T segments [10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 379].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are the sources of un- certainty that hinder the use of the ECG as a biometric trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While these are largely controlled 1xECG Github Repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='com/jtrpinto/xECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Methodology 105 in medical or on-the-person settings (where the subject is at rest, laying down, and signals are acquired using several high-quality gel electrodes), their effects are dominant for realistic off-the- person signals (acquired using fewer dry electrodes on the hands, during common daily activi- ties) [338;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 342;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When compared with the P and T waves, the QRS corresponds to a larger polarisation event over a shorter period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In practice, this makes the QRS more dominant over noise and intrasubject variability than the other ECG waveforms [342;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, the QRS is considered more stable over time and across variable conditions, which makes it better suited for biometric recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite this, it is still unclear how much identity information is carried by the QRS complex compared to the other waveforms, and whether it is enough for an accurate and robust biometric recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Studies on ECG-based biometric identification have shown it is possible to distinguish small sets of individuals in on-the-person settings using only the QRS complex or QRS fiducial amplitude and time measurements [238;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 446].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, this practice is becoming uncommon as research evolves towards realistic off-the-person signals and larger databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This denotes that the sole use of the QRS may not be adequate for off-the-person settings, or the individual information carried by the QRS may not be enough to distinguish individuals in large populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This work aimed to address these doubts through a study on the role and rele- vance of the QRS and the other waveforms in ECG-based biometric identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Interpretability tools are used to assess which parts of the ECG are more relevant to the decisions of an end-to-end identification model [344], with on-the-person and off-the-person signals and data subsets with a varying number of identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Methodology 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Biometric identification model The biometric model for identification followed the architecture proposed by Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [344], which has attained state-of-the-art results in off-the-person settings for both identification and, later, identity verification [338].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The model (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2) receives five-second blindly segmented ECG signals and outputs probabilities for each of the N identities considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Finding the highest probability score allows us to assign the respective identity to the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The model consists of an end-to-end 1D convolutional neural network (CNN) with four con- volutional layers (with 1 × 5 filters, two layers with 24 followed by two with 36), followed by ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Neighbouring convolutional layers are separated by 1 × 5 max-pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The last convolutional layer is followed by two fully-connected layers (100 neurons with ReLU and N neurons with softmax activation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Interpretability tools To capture the dynamics behind the decisions of the biometric model, four interpretability meth- ods are applied to the trained model: Occlusion, Saliency, Gradient SHAP, and DeepLIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Oc- 106 Leveraging Explainability to Understand ECG Biometrics ID Conv1D 24@1x5 ReLU MaxPool 1x5 Conv1D 24@1x5 ReLU MaxPool 1x5 Conv1D 36@1x5 ReLU MaxPool 1x5 Conv1D 36@1x5 ReLU FullyConn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 ReLU FullyConn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' N Softmax Signal 1x1000 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Architecture of the biometric identification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' clusion and Saliency are two of the simplest interpretability methods, while Gradient SHAP and DeepLIFT are more sophisticated and powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These are implemented in the Captum tool- box [221] for PyTorch and are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Occlusion The Occlusion method [478] consists in measuring the influence of hiding a portion of the input on the output of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' When hidden, the more relevant input parts will cause larger changes in the output, and will thus be assigned greater relevance in the explanations offered by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is the simplest method to interpret a model, although the size of occluded regions should be carefully defined to obtain meaningful explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Saliency The Saliency method [401] is based on the gradients of a model given a certain in- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Through backpropagation, the gradient of target class scores w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' the input is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A saliency map is then generated by rearranging the class score derivatives, generating saliency maps that assign higher relevance to input regions that correspond to higher gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Requiring a single backpropagation pass, this method is a simple and fast way to obtain explanations of model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Gradient SHAP Gradient SHAP [283] is an approach based on game theory which considers the explanations of a model’s predictions as models themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For sophisticated deep learning models, the explanation models are simplified and interpretable approximations of the respective models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' SHapley Additive exPlanation (SHAP) values, inspired by game theory’s Shapley values, are computed through the gradient of a random point between a baseline and the input with added random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The SHAP values denote how much a given part of the input raises the probability for the considered class, and are reportedly better aligned with human intuition and effective in discriminating among output classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' DeepLIFT DeepLIFT (Deep Learning Important FeaTures) [391] performs backpropagation to track the contributions to the output to the responsible parts of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Throughout this process, it compares the difference in inputs and outputs considering a reference (or baseline) input, as- signing contribution scores to each neuron of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It also allows for the study of negative contributions: how much a specific part of the input contributes to lower the probability for the considered class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Experimental Setup 107 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Visualisation Decision explanations obtained using interpretability tools are visualised using the multicoloured line plot feature of Matplotlib [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' ECG signals are plotted so that the colour of each signal component represents its relative relevance to the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this case, lighter yellow colours represent less relevant time samples, whereas more relevant samples assume darker purple colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This way, both the ECG morphology and the relevance of each of its components are easily and intuitively presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Experimental Setup The data used for model training and evaluation have been drawn from the Physikalisch- Technische Bundesanstalt ECG Database (PTB) [49;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 146] and the University of Toronto ECG Database (UofTDB) [445].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The PTB database includes on-the-person (high-quality) 12-lead ECG signals acquired at 1 kHz from 290 subjects at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The UofTDB includes single-lead off-the- person (more noisy and realistic) data acquired from 1019 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To match the UofTDB, PTB signals were downsampled to 200 Hz and only Lead I was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Five-second segments were blindly extracted (without fiducial detection) from the recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Fifty per cent of those segments (per identity) were used during training and the remaining were reserved for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This provided more challenging test settings than those commonly found in the literature, but also deliberately avoided the most realistic settings (see [338]), for the sake of obtaining meaningful interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To simulate gradually increasing identification difficulty within each database, subsets of N identities are considered, with N ∈ {2,5,10,20,50,100,200,500,1019}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The identities in each subset are the first N in lexicographical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Each subset includes all identities that compose smaller subsets, so subjects #1 and #2 are the main focus of analysis since these are present in all subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Throughout this paper, TN denotes the subset of UofTDB data from N subjects and PN denotes the subset of PTB data from N identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As stated in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1, P290 was used instead of P200 to take advantage of the entire PTB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Model training details can be found online at this project’s repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Performance evaluation is based on the True Positive Identification Rate (or accuracy): the fraction of test samples that are correctly assigned to their true identity by the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Interpretations are examined through the proposed visualisation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion The results of the performance evaluation are presented in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These results roughly follow the expected patterns considering the use of on-the-person versus off-the-person ECG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The model is able to attain high true positive identification rates in both databases when the population 108 Leveraging Explainability to Understand ECG Biometrics Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: True positive identification rate results (%) on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Database Number of Identities 2 5 10 20 50 100 2001 500 1019 PTB 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='63 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='50 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='92 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='76 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='73 UofTDB 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='26 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='30 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='46 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='16 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='70 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='20 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='45 1For PTB, this column corresponds to the entire set of 290 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' is small, but as the set of subjects grows, performance decreases and a wide gap distinguishes the more challenging off-the-person settings from the more controlled on-the-person settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, one can find some unusual patterns in the performance results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering M > N, one would expect identification performance with subset TN to be higher than with subset TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With UofTDB off-the-person data this is not always verified: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', from T5 to T10, performance increases from 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='26% to 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In these cases, we need to consider that datasets with fewer identities have fewer data and, thus, more unstable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Alternatively, the identities added to TN to create TM may be easier to discriminate (“sheep”, according to the concept of biometric menagerie [104;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 468]) and thus contribute to improving accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, one should also regard the substantial regularisation needed to avoid overfitting and the instability during training as possible causes for these discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is a very important insight into the increased difficulties of using off-the-person data and the need for improved and more robust biometric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Analysing the explanations obtained using the four interpretability tools (examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4), a trend is verified from smaller to larger identity subsets, consisting on the deviation from focusing mainly on the QRS complex to the increasing relevance of other parts of the heart- beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is also confirmed when combining the explanations of all heartbeats of each person into a single average heartbeat (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With the cleaner medical signals from PTB, the focus is mostly on the QRS complex, but information from other waveforms starts to become more and more relevant as more identities are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is noteworthy how, when discriminating PTB subjects #1 and #2 in a two-subject scenario (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5), the model still focuses mainly on the QRS, even though subject #2 has a very specific characteristic, the inverted T-wave, that is arguably their most distinctive feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This denotes how, in these cleaner signals, the QRS complex is so stable that the remaining waveforms, more susceptible to heart rate variability, are largely ignored by the model regardless of any visually obvious intersubject differences they may present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With the more realistic off-the-person signals from UofTDB, the QRS retains high importance but the relevance is more evenly spread among the signal waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the specific case of subject #2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6), it is evident that the QRS retains the highest importance for the decision, even in T1019 (the largest subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This may denote that, even in these more challenging settings, the identification models will still give preference to the QRS over other waveforms if it is sufficiently unique among the considered identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, in such large sets of identities, the expected behaviour is that of subject #1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6), since the limited identity information carried by the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 109 2 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Occlusion Saliency Gradient SHAP DeepLIFT 5 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 10 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 20 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 50 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 290 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Explanations over an example five-second ECG segment from PTB (in each subplot, the yellow to dark purple colours correspond to increasing time sample relevance and vertical grey lines denote R-peak locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' signals were filtered for easier visualisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Occlusion Saliency Gradient SHAP DeepLIFT 5 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 10 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 20 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 50 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 100 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 200 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 500 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1019 id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: Explanations over an example five-second ECG segment from UofTDB (in each sub- plot, the yellow to dark purple colours correspond to increasing time sample relevance and vertical grey lines denote R-peak locations, signals were filtered for easier visualisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 110 Leveraging Explainability to Understand ECG Biometrics Subject #2 Subject #1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Average explanations over heartbeat waveforms of subjects #1 and #2 on the subsets of the PTB database (in each subplot, the yellow to dark purple colours correspond to increasing time sample relevance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' signals were filtered for easier visualisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Subject #2 Subject #1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Average explanations over heartbeat waveforms of subjects #1 and #2 on the subsets of the UofTDB database (in each subplot, the yellow to dark purple colours correspond to increasing time sample relevance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' signals were filtered for easier visualisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions 111 QRS will lead the model to also look to other parts of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' One interesting aspect is the difference between the results with Occlusion versus the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Occlusion generally grants the QRS complex much more relevance, regardless of the settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the state-of-the-art approaches, the QRS complex is not only a source for identity features but also frequently used as an easily detectable reference landmark for the location of other ECG waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This may also be the case in this end-to-end deep model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although there are challenging contexts where the QRS may not be the main contributor to the decision, it may be essential to the deep model as a reference landmark to locate other waveforms in the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, when occluded, it will be the signal component that most impacts the decision, causing the occlusion method to generally consider it the most relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions This work aimed to explain how deep models use ECG signals to distinguish people, using inter- pretability tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Overall, the obtained results partially confirm the claim that the QRS is the key to ECG-based biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With small populations in on-the-person settings, it can alone be used for reliable recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, as we evolve towards larger populations and off-the-person settings, other components become relevant in discriminating people, as the models require more identity information to overcome the hurdles placed by enhanced intrasubject variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, even though relevance is more evenly shared in off-the-person identification in large sets of identities, the QRS is shown as essential by the occlusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It appears that, just like several literature methods, the implemented end-to-end model learnt to use the QRS as a landmark for the location of other ECG components in the signal, resulting in large output changes when the QRS is occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, despite the literature claims, one should avoid relying too heavily on any single part of the ECG, including the QRS complex, since all waveforms carry identity information that proves increasingly useful in more realistic settings and larger populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beyond these insights, further efforts should be devoted to extending this study and offering a deeper, more thorough, and more objective analysis of the contribution of each ECG waveform to the model’s decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Obtaining more systematic and complete explanations could create new opportunities for the use of interpretability tools during model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using explanations to regularise models and promote the focus on the most relevant signal components or the distributed use of the whole signal (instead of just the QRS) could lead to improved recognition accuracy and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Chapter 8 Interlead Conversion of Electrocardiographic Signals Foreword on Author Contributions The research work described in this chapter was conducted in collaboration with Sofia C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beco, under the super- vision of Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The author of this thesis contributed to this work on the formulation, implementation, and improvement of the interlead conversion methodology, the preparation and conduction of the extended exper- iments, the discussion of the results, and the writing of the scientific publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The results of this work have been disseminated in the form of an extended journal article and a short paper presented at an international conference: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Electrocardiogram Lead Conversion from Single-Lead Blindly- Segmented Signals,” BMC Medical Informatics and Decision Making, 22: 314, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Pinto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Cardoso, “Interlead Conversion of Single-Lead Blindly-Segmented Electrocar- diogram Signals,” in 17th International Conference on Computational Intelligence Methods for Bioinformatics and Biostatistics (CIBB 2021), Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [30] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Context and Motivation The electrocardiogram (ECG) is the measurement of electrical potentials that make the heart con- tract and relax as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The morphology of the ECG signal depends on the location of the electrodes used for acquisition: different electrode placement results in different perspectives over the heart [343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For medical purposes, the standard configuration acquires the ECG over twelve leads for more information, but it requires ten electrodes placed on the patient’s arms, legs, and chest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using fewer electrodes allows for more comfortable and inexpensive acquisitions, at the expense of certain leads that could be ideal for a more accurate diagnosis of certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To get the best of both worlds, researchers have proposed methods for the automatic interlead conversion of ECG signals [244;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 293;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 395;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 407;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 408].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These transform short ECG segments to mimic other perspectives, using acquired leads to reconstruct any leads that were not recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, these methods still present limited applicability, since they typically require multiple leads as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Even the most advanced methods [244;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 293], that only use one input lead, still 113 114 Interlead Conversion of Electrocardiographic Signals require the inputs to be single heartbeat segments aligned in time, which makes them dependent on separate processes and, overall, less flexible and robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This chapter presents a study on the feasibility of ECG interlead conversion using short seg- ments from just one limb lead without any kind of temporal alignment (blindly-segmented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With such input, the proposed methodology is trained to reconstruct other leads as faithfully as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This aims to open up new possibilities for more comfortable ECG acquisition in clinical scenarios or wearable devices without giving up the benefits of multi-lead recordings for medical diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed methodology, based on deep learning encoder-decoder structures, is explored for interlead conversion using either lead II or lead I (limb leads) signals as reference, and using a single shared encoder or an individual encoder for each target lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Beyond the training and testing on the widely used PTB database, the conversion models are evaluated on cross-database scenarios with the INCART and PTB-XL databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, the clinical annotations of the PTB-XL database are also used for a differential performance evaluation in the presence of medical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The code is available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work At the onset of research on interlead conversion, methodologies commonly required several leads as reference for robust lead reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [496] performed a preliminary study on the conversion of ambulatory ECG recordings into standard 12-lead ECG signals using lead-field theory and the least-squares method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nelwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [306] learned generic and patient-specific linear regression coefficient templates to reconstruct up to four missing leads with high correlation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Later, Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [474] used 12 lead acquisitions to synthesise additional leads (right ven- tricular leads V3R, V4R, and V5R and posterior chest leads V7, V8, and V9) which provide important information for the diagnosis of acute myocardial infarction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Their algorithm was based on the transfer coefficient estimated from the learning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [395] developed three methods for obtaining the Frank leads using the 12 standard leads as reference: the Kors Quasi-Orthogonal method, the Kors Linear Regression method, and the Dower Inverse Matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The conversion was successful for signals from healthy subjects but presented limitations on signals from subjects with pathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The recent work by Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [407] was one of the first to use machine learning techniques for interlead conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' They used a focused time-delay neural network (FTDNN), which is well suited for time series prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, their methodology required seven input leads (all limb leads and V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Atoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [22] used ensembles of fully-connected neural networks to learn to synthesise V1, V3, V4, V5, and V6 heartbeats from three-lead inputs (I, II, and V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Schreck and Fishberg [380] performed the first study on the synthesis of the entire set of 12 standard leads and scalar 3-lead derived vectorcardiogram from just three measured leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Their proposed methodology used nonlinear optimisation to construct a universal patient transformation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1Interlead ECG Conversion Github Repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Available on: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='com/jtrpinto/ecg-conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Methodology 115 [165] applied linear generic and subject-specific transforms to convert recordings from adhesive patch-type ECG monitors to the standard 12-lead ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In [435;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 438], researchers also explored personalised statistically determined linear transforms and went on to achieve improved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [243] proposed methods based on linear regression and artificial neural networks to reconstruct the 12 standard leads from subsets of 35 channels acquired using one single large patch covering the subject’s chest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although accurate, the method is arguably incompatible with scenarios focused on ease of use and patient/user comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Similarly, Grande-Fidalgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [154] used linear regression and fully-connected networks to reconstruct the entire set of twelve stan- dard leads from a subset of just three input leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [408] used long short-term memory (LSTM) networks to reconstruct the twelve ECG standard leads from a three-lead patch-type de- vice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Their results show their method was able to correctly retain pathological abnormalities from medical conditions on the reconstructed signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The work of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [244] was one of the few that studied the synthesis of standard leads using only one reference lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In their study, chest leads (V1 to V6) were synthesised from lead II using a generative adversarial network (GAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, input segments had to be single heart- beats, aligned according to the R-peaks, which decreases the difficulty of the proposed method but also its applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Matyschik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [293] developed patient-specific models to more accurately reconstruct eleven missing ECG signals from a single available lead of the standard 12-lead sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the reference lead was either V1, V2, or V3 which, being chest leads, do not enable the usage in less obtrusive setups which would preferentially use limb leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, we explore the more challenging scenario of reconstructing the entire set of twelve standard leads using only one reference lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, the reference signals are blindly- segmented (without any kind of temporal alignment) and pertain to one of the limb leads to allow for applications on the least obtrusive setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Our main goal is to assess whether it is possible to reconstruct the electrocardiogram signal in such challenging scenarios and discuss the next steps towards the use of interlead conversion in less obtrusive clinical setups and wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Methodology 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 General overview The proposed methodology for interlead ECG conversion follows the encoder-decoder structure typically used for deep image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The encoder receives an input signal and processes it to create a compressed representation that retains relevant information for the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The decoder receives this representation and processes it so that the output matches the ground-truth as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Here, the input to the encoder is a short ECG segment of one lead (X) and the ground-truth is the corresponding segment in a different lead (Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, the encoder is in charge of selecting the information from X that is needed for Y, and the decoder will use that information to reconstruct the corresponding lead Y signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 116 Interlead Conversion of Electrocardiographic Signals 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Model architectures The general encoder-decoder structure allows for diverse specific model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This work focuses on the U-Net model, a fully convolutional architecture that has found many applications related to semantic segmentation and can also be adapted for the task of ECG lead conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 U-Net The U-Net was initially proposed by Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [367] as a tool for biomedical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, the implemented architecture (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) receives an input segment of lead X, which initially goes through a chain of three sequential blocks, each with half the signal resolution of the previous block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Each block includes two convolutional layers (each followed by batch normalisation and ReLU activation) and ends with a max-pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Between the encoder and the decoder, two convolutional layers compose the latent space or bottleneck block, which corresponds to the maximum point of information compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The decoder mirrors the encoder in its structure, with three similar blocks composed of an upsampling layer and two transposed convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The last transposed convolutional layer outputs a single-channel signal whose size corresponds to the input segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The activation function of this last layer is the hyperbolic tangent for an output signal with amplitudes in [−1,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' One aspect of the U-Net which is often cited as the key to its widespread success is the skip- connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' U-Nets typically include skip-connections between corresponding blocks on the en- coder and the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This means the feature maps from the encoder blocks are directly routed to the corresponding decoder blocks, allowing the model to propagate context information from multiple resolutions between the encoder and the decoder for higher flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Convolutional autoencoder (AE) Beyond the aforementioned U-Net architecture, adapted for unidimensional signal inputs, we also explore a convolutional autoencoder (AE, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Its architecture is very similar to the U- Net, albeit without skip-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As a result, the structure is simplified, when compared to the U-Net, and the latent representation sent from the encoder to the decoder is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Experiments with the AE architecture aim to assess if the skip-connections are essential for the task at hand or if the simplified structure could avoid overfitting and bring performance benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Label refinement network (LRN) The third architecture explored in this work was based on Label Refinement Network (LRN, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3) was originally proposed by Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [190] for semantic image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Its archi- tecture is identical to the aforementioned U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The singularity of the LRN lies in the supervision strategy: while the U-Net only uses the output of the last decoder block in the reconstruction loss, the LRN computes the loss at the outputs of every decoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This results in supervision at several resolution levels, leading the decoder to offer a coarse reconstruction right after the first 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Methodology 117 1 32 32 32 32 1 64 64 64 128 128 256 256 512 512 256 128 256 128 64 Input lead X segment Output lead Y segment conv 5x5, BatchNorm, ReLU copy and crop max pool 5x5 up-conv 5x5 conv 1x1, Tanh (1, 5000) (1, 5000) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Schema of the U-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 1 32 32 32 32 1 64 64 64 128 128 256 256 128 256 128 64 Input lead X segment Output lead Y segment conv 5x5, BatchNorm, ReLU max pool 5x5 up-conv 5x5 conv 1x1, Tanh (1, 5000) (1, 5000) 32 64 128 128 64 Encoder Latent Space Decoder Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Schema of the convolutional autoencoder (AE) architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' block, which should be gradually refined by the subsequent blocks for improved results at higher resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Experiments with the LRN architecture aim to assess if the multi-level resolution could bring improved performance to the task of signal lead conversion as they have for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Shared vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders The conversion of one lead into multiple missing leads requires multiple decoders - each one will fulfil the task of reconstructing their respective lead based on the compressed latent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the case of the encoder, however, it is possible to have a single one whose output will be shared by all decoders or have multiple encoders, each one dedicated to one individual decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 118 Interlead Conversion of Electrocardiographic Signals 1 32 32 32 32 1 64 64 64 128 128 256 256 512 512 256 128 256 128 64 output 4 output 3 output 2 output 1 (1, 1000) (1, 200) (1, 40) (1, 8) conv 5x5, BatchNorm, ReLU copy and crop max pool 5x5 up-conv 5x5 conv 1x1, Tanh Input lead X segment (1, 5000) Output lead Y segment (1, 5000) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Schema of the architecture based on label refinement networks (LRN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, we explore both possibilities for 12-lead reconstruction - using one shared en- coder connected to all 11 decoders, for all 11 output leads except the one corresponding to the input, or using one individual encoder for each of the 11 decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using individual encoders grants more flexibility to each lead conversion process, as each encoder will be able to learn a unique way to obtain compressed representations and each encoder-decoder pair will work inde- pendently from all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the other hand, using one shared encoder results in a much lighter and faster algorithm and the added simplicity may contribute to avoiding overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Experimental Setup 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Data The experiments conducted in this work used mainly the data provided in the PTB Diagnostic ECG Database [49], available on Physionet [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The PTB database includes data from 16 channels, including all 12 standard leads, sampled at 1 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It contains a total of 549 records from 290 indi- viduals, with one to five records per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recordings were cropped into segments of 5 s (5000 samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' A second-order Butterworth bandpass filter with cut-off frequencies fc = [1,40] Hz was applied to each segment to remove noise while retaining the most useful ECG information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The amplitudes of the n values of each signal x were then min-max normalised to the interval [−1,1] following the equation: xn = 2× xn −xmin xmax −xmin −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Experimental Setup 119 The data from PTB was divided into train and test sets, with approximately 63%, 7% and 30% of the segments, respectively, for a total of 7086, 787, and 3509 ECG segments for each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For a more thorough and challenging evaluation, subjects are divided between the train/validation and test sets: the latter had recordings from subjects 1 to 50 while the former had recordings from subjects 51 to 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The INCART database (officially the St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Petersburg INCART 12-lead Arrhythmia Database), also available on Physionet, was used to test the performance of trained models on cross-database scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This database contains 75 Holter recordings from 32 subjects undergoing tests for coronary artery diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Each record is 30 minutes long and contains twelve standard leads sampled at 257 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Recordings from this database were resampled to 1 kHz and processed as described above for PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The PTB-XL database [443;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 444], created by the same team as the PTB, includes 21837 clini- cal ECG recordings from a total of 18885 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Each recording is 10 seconds long, includes all twelve standard ECG leads, and is originally sampled at 500 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The waveforms were annotated by up to two cardiologists, who assigned annotations to each record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The 71 possible annota- tion statements have been clustered into five superclasses: NORM (normal ECG), MI (myocardial infarction), STTC (ST/T change), CD (conduction disturbance), and HYP (hypertrophy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This dataset was originally created for the training and evaluation of automatic ECG interpretation al- gorithms but also shows great promise for the development of lead conversion algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In this work, we take advantage of expert clinical annotations to study the effect of medical conditions on the quality of the lead conversion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From the total of 21837 recordings, we selected the 16272 that did not have conflicting superclass annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From each recording, the first 5 seconds were cropped, resampled to 1 kHz, and processed as described above for PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Model training and evaluation The models were trained using the l1-loss between the model outputs and the corresponding ground-truth signals as the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The l1 was chosen empirically as it allowed the model to learn most adequately both the overall morphology of the signals and their finer details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Adam optimiser was used with an initial learning rate of 1 × 10−3, over a maximum of 500 epochs with batch size 32 (shared encoder) or 16 (individual encoder) and early stopping patience of 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To compare lead conversions with the corresponding measured ground-truth signals, this work used the following metrics: the average and median Pearson correlation coefficient (r, used in the majority of the related literature), the average root mean square error (RMSE), and the average Structural Similarity Index Measure (SSIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 120 Interlead Conversion of Electrocardiographic Signals Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Comparison of encoder-decoder architectures on one-to-one lead conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Model r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') U-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='78 Autoencoder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='78 LRN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='75 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Average correlation between lead II signals and the remaining leads on the PTB, IN- CART, and PTB-XL databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Average Correlation to Lead II I III aVR aVL aVF V1 V2 V3 V4 V5 V6 PTB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='81 INCART 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='77 PTB-XL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='84 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Test results of the U-Net used for multi-lead conversion from lead II, with shared or individual encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Shared Encoder Individual Encoders Lead r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='67 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='49 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Architecture comparison To compare the selected architectures, the first experiment entailed the one-to-one lead conversion from II to I, two of the most used ECG leads for medical purposes (see Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' According to the results, the U-Net performs better than both alternatives AE and LRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although the AE achieves the same median r as the U-Net, the average r is lower, meaning that the least successful results are generally worse with the AE than with the U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The skip-connections give it the capability to send more information (and at more resolution levels) from the encoder to the decoders, granting it more flexibility and ultimately better perfor- mance than the AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The multi-resolution supervision of the LRN, expected to improve overall performance, appears to excessively draw the model’s attention away from the details, which re- sults in worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Following the results of this comparison, subsequent experiments 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 121 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: Average correlation between lead I signals and the remaining leads on the PTB, IN- CART, and PTB-XL databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Average Correlation to Lead I II III aVR aVL aVF V1 V2 V3 V4 V5 V6 PTB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='68 INCART 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='44 PTB-XL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='83 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Test results of the U-Net used for multi-lead conversion from lead I, with shared or individual encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Shared Encoder Individual Encoders Lead r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM II 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='80 V3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='39 focus solely on the U-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 One-to-all leads conversion Not all leads can be converted equally: the correlation between leads depends on their perspectives of the heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 presents an overview of the average correlation between lead II and the remaining eleven standard leads, computed using the PTB, INCART, and PTB-XL test segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Specifically for the PTB data, one can observe that some leads such as aVF or aVR are highly (positively or negatively) correlated with lead II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the other hand, aVL is almost orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, one should expect aVL to be much harder to accurately convert from lead II than aVF or aVR, since the former shares much less information with lead II than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is verified in the results for multi-lead conversion on the PTB database (see Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Conversion from lead II to aVF, aVR, and V6 consistently offer good results, while the conversions to aVL, lead I, or V4 were overall the least successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This behaviour is also visible in the example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='42 where the model is unable to capture the finer details of the signals in lead aVL and leads V1-V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The opposite happens in lead III, aVF, V6, and especially aVR, where the model was consistently able to capture the morphological details of the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 2Examples were selected among all test samples to correspond to the median overall r result for each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, they represent a median result and the methodology should offer better results in half of the occasions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 122 Interlead Conversion of Electrocardiographic Signals PTB - Lead II to I (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='613) Measured Converted PTB - Lead II to I (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='599) Measured Converted PTB - Lead II to III (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='915) Measured Converted PTB - Lead II to III (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='93) Measured Converted PTB - Lead II to aVR (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='965) Measured Converted PTB - Lead II to aVR (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='98) Measured Converted PTB - Lead II to aVL (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='754) Measured Converted PTB - Lead II to aVL (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='791) Measured Converted PTB - Lead II to aVF (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='976) Measured Converted PTB - Lead II to aVF (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='988) Measured Converted PTB - Lead II to V1 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='794) Measured Converted PTB - Lead II to V1 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='788) Measured Converted PTB - Lead II to V2 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='583) Measured Converted PTB - Lead II to V2 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='635) Measured Converted PTB - Lead II to V3 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2) Measured Converted PTB - Lead II to V3 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='357) Measured Converted PTB - Lead II to V4 (shared, r=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='088) Measured Converted PTB - Lead II to V4 (individual, r=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='02) Measured Converted PTB - Lead II to V5 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='853) Measured Converted PTB - Lead II to V5 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='881) Measured Converted PTB - Lead II to V6 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='97) Measured Converted PTB - Lead II to V6 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='972) Measured Converted 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: Example result of lead II to all conversion on the PTB test dataset (each row depicts one converted lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' shared encoder on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 123 PTB - Lead I to II (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='53) Measured Converted PTB - Lead I to II (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='099) Measured Converted PTB - Lead I to III (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='866) Measured Converted PTB - Lead I to III (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='859) Measured Converted PTB - Lead I to aVR (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='833) Measured Converted PTB - Lead I to aVR (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='776) Measured Converted PTB - Lead I to aVL (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='956) Measured Converted PTB - Lead I to aVL (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='96) Measured Converted PTB - Lead I to aVF (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='767) Measured Converted PTB - Lead I to aVF (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='309) Measured Converted PTB - Lead I to V1 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='794) Measured Converted PTB - Lead I to V1 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='764) Measured Converted PTB - Lead I to V2 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='617) Measured Converted PTB - Lead I to V2 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='572) Measured Converted PTB - Lead I to V3 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='41) Measured Converted PTB - Lead I to V3 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='152) Measured Converted PTB - Lead I to V4 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7) Measured Converted PTB - Lead I to V4 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='653) Measured Converted PTB - Lead I to V5 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='748) Measured Converted PTB - Lead I to V5 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='88) Measured Converted PTB - Lead I to V6 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='643) Measured Converted PTB - Lead I to V6 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='626) Measured Converted 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: Example result of lead I to all conversion on the PTB test dataset (each row depicts one converted lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' shared encoder on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 124 Interlead Conversion of Electrocardiographic Signals While lead II ECG signals are generally better for medical diagnosis in clinical scenarios, lead I is becoming increasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The widespread implementation of ECG acquisition equipment in smartwatches, fitness bands, and other gadgets for daily use allows for the collection of lead I signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Combining these growing applications with robust conversion algorithms would enable the recovery of missing leads on wearables and empower the next generation of robust continuous health monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 presents the average correlation between lead I and the remaining eleven standard leads on the PTB, INCART, and PTB-XL test segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Like lead II, lead I is more correlated (positively or negatively) with certain leads, such as aVR, aVL, or V6, while it is almost orthogonal with aVF or V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, one can observe, in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5, that the proposed methodology obtains better performance with aVR and aVL while struggling to convert from lead I to lead aVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The same can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5: for aVR and aVL, the model is able to correctly capture the target morphology, while the reconstructions of aVF and V3-V6 are largely unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' From the example result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5, one can also identify a shortcoming of the proposed methodology: the occasional offsets between the baseline of the measured and converted signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' We suspect this is due to the min-max normalisation of the signals, drawing them into the [−1,1] amplitude range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Alternatives to this normalisation, such as standard normalisation, should be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering the overall results, no lead is perfect for converting all twelve standard leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Hence, lead II should be chosen as reference input when aVF or V5-V6 are the most important leads for the application at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Lead I serves better as a reference when aVR, aVL, or V1- V2 are more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Otherwise, other leads (such as lead III) should probably be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, the results show it is possible to nicely reconstruct several leads using only one input lead without temporal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using either lead as a reference, there is apparently no considerable or consistent difference between using one single shared encoder or using an individual encoder for each target lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It appears as if the additional flexibility of having multiple encoders is only beneficial up to a point, and the higher complexity ends up opening the door to overfitting and loss of robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, for this application, one should expect a shared encoder to be the best option, considering its higher simplicity and similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3 Comparison with the state-of-the-art For a comparison with the state-of-the-art, we implemented the method recently proposed by Grande-Fidalgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [154] as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This method is based on a simple fully-connected model that receives each signal point’s amplitude in three reference leads as inputs and returns the same point’s amplitude in all twelve leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Here, we adapt the methodology so it receives signal point amplitudes from one single lead (leads I or II), to exactly match the evaluation conditions of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Unlike what has been reported in [154], the baseline was not successful in learning to retrieve the entire set of leads from just one reference lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In fact, across all leads, the average test r of this 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Results and Discussion 125 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Cross-database test results for INCART conversion from lead II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Shared Encoder Individual Encoders Lead r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM I 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='39 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7: Cross-database test results for INCART conversion from lead I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Shared Encoder Individual Encoders Lead r (avg.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='21 method ranged from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='005 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='002, considerably worse than the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' One can assume that, although such a simplistic model presents advantages in terms of lightweight operation and robustness to overfitting, single-lead information is not enough for it to achieve reliable interlead conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The fact the baseline method reconstructs signals point-by-point, unable to analyse broader local context information, makes it hard to reconstruct the signal without already having data from more than one channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' On the other hand, using convolutional layers allows the proposed method to use broader local information as context to adequately learn to reconstruct signals using only one lead as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4 Cross-database evaluation The cross-database tests aimed to assess the behaviour of the proposed methodology in more diverse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Here, the models used were the same as in the previous experiments (trained 126 Interlead Conversion of Electrocardiographic Signals Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8: Cross-database test results for PTB-XL conversion from lead II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Shared Encoder Individual Encoders Lead r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='58 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9: Cross-database test results for PTB-XL conversion from lead I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Shared Encoder Individual Encoders Lead r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM r (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') r (med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=') RMSE SSIM II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='60 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='51 with PTB data), and the evaluation was conducted using data from the INCART and PTB-XL databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For both INCART and PTB-XL, some differences in interlead correlations can be observed when compared to PTB (see Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 and Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This can be explained due to the different acquisition setups, especially the positioning of the electrodes, which potentially causes each lead to offer a different perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For INCART (see Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 and Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7), the overall quality of the results is inferior to those with PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is as expected since PTB data was seen by the models during training and the INCART database is arguably more challenging regarding noise and variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite these metrics, it is noticeable in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 and Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 that both reference leads can offer good conversion results in some leads, especially with lead II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Using this lead as reference, the proposed methodology is relatively good at converting most leads except I, V2, and V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For PTB-XL (see Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 and Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9), results are, overall, the worst, although some leads (namely V4, V5, and V6), due to higher correlation with the reference leads, are better 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions 127 reconstructed than with the PTB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9, it is possible to observe that, despite occasional baseline offset and prevalent noise, both reference leads enable the approximate reconstruction of most of the set of twelve standard leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' For either database, differences in acquisition settings and electrode placement result in in- ferior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The ideal solution is to always make sure the acquisition details of training and inference data match, to ensure optimal performance upon deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, the robustness in cross-database scenarios is a relevant issue that merits further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='5 Influence of medical conditions As aforementioned, medical conditions may affect differently the various leads of an ECG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' While this is the main motivation behind the quest to reconstruct missing leads it may also be one of the main hurdles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' If the medical condition is somehow not evident in the input lead, the algo- rithm could be led to reconstruct the remaining leads incorrectly without the proper information on the respective medical condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, we conducted a differential performance evaluation according to the existence and type of diagnosed medical conditions on the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To do this, we use the expert clinical an- notations on the PTB-XL database and separate the results by the superclass labelling of each test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The average r results for each converted lead and each superclass are presented in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='10 (using lead II as reference) and Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='11 (using lead I as reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Overall, no dominant difference could be observed between the results with normal signals and the results with signals with medical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Similarly, no specific medical condition superclass presents considerably different performance results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is likely due to the presence of medical conditions on the PTB signals originally used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, although the behaviour of the proposed methodology should be expected to vary slightly in the presence of medical conditions, it should not have a considerable impact on its baseline performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions This work implemented and compared the performance of three deep learning architectures for interlead conversion of ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Unlike the literature, this work focused on the more chal- lenging scenario of single-lead blindly-segmented inputs from limb leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The proposed model was explored on 12-lead acquisitions from three different databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Ablation studies were con- ducted on the architectures used for conversion and on the use of a shared encoder vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, the model was evaluated in both single-database and cross-database scenar- ios, including an experiment on the effect of medical conditions on signal reconstruction and the study of diagnosis performance with original vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' converted signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Despite the considerably more challenging scenario, the proposed methodology based on a U- Net was capable of obtaining relatively good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Each reference lead enabled the high-quality reconstruction of several of the twelve standard ECG leads, in some cases reaching state-of-the-art 128 Interlead Conversion of Electrocardiographic Signals INCART - Lead II to I (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='733) Measured Converted INCART - Lead II to I (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='569) Measured Converted INCART - Lead II to III (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='902) Measured Converted INCART - Lead II to III (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86) Measured Converted INCART - Lead II to aVR (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='982) Measured Converted INCART - Lead II to aVR (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='989) Measured Converted INCART - Lead II to aVL (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='779) Measured Converted INCART - Lead II to aVL (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='711) Measured Converted INCART - Lead II to aVF (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='985) Measured Converted INCART - Lead II to aVF (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='986) Measured Converted INCART - Lead II to V1 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='895) Measured Converted INCART - Lead II to V1 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='916) Measured Converted INCART - Lead II to V2 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='643) Measured Converted INCART - Lead II to V2 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='705) Measured Converted INCART - Lead II to V3 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='166) Measured Converted INCART - Lead II to V3 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='206) Measured Converted INCART - Lead II to V4 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='609) Measured Converted INCART - Lead II to V4 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='156) Measured Converted INCART - Lead II to V5 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='904) Measured Converted INCART - Lead II to V5 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='916) Measured Converted INCART - Lead II to V6 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='935) Measured Converted INCART - Lead II to V6 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='916) Measured Converted 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6: Example cross-database result of lead II to all conversion on INCART (each row depicts one converted lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' shared encoder on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions 129 INCART - Lead I to II (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='52) Measured Converted INCART - Lead I to II (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='461) Measured Converted INCART - Lead I to III (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='582) Measured Converted INCART - Lead I to III (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='755) Measured Converted INCART - Lead I to aVR (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='934) Measured Converted INCART - Lead I to aVR (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='918) Measured Converted INCART - Lead I to aVL (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='915) Measured Converted INCART - Lead I to aVL (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='912) Measured Converted INCART - Lead I to aVF (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='464) Measured Converted INCART - Lead I to aVF (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='614) Measured Converted INCART - Lead I to V1 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='862) Measured Converted INCART - Lead I to V1 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='85) Measured Converted INCART - Lead I to V2 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='79) Measured Converted INCART - Lead I to V2 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='753) Measured Converted INCART - Lead I to V3 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='574) Measured Converted INCART - Lead I to V3 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='412) Measured Converted INCART - Lead I to V4 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='078) Measured Converted INCART - Lead I to V4 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='093) Measured Converted INCART - Lead I to V5 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='366) Measured Converted INCART - Lead I to V5 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='432) Measured Converted INCART - Lead I to V6 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82) Measured Converted INCART - Lead I to V6 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='75) Measured Converted 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7: Example cross-database result of lead I to all conversion on INCART (each row depicts one converted lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' shared encoder on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 130 Interlead Conversion of Electrocardiographic Signals PTB-XL - Lead II to I (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='18) Measured Converted PTB-XL - Lead II to I (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='153) Measured Converted PTB-XL - Lead II to III (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='484) Measured Converted PTB-XL - Lead II to III (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='772) Measured Converted PTB-XL - Lead II to aVR (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='578) Measured Converted PTB-XL - Lead II to aVR (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='536) Measured Converted PTB-XL - Lead II to aVL (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='147) Measured Converted PTB-XL - Lead II to aVL (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='225) Measured Converted PTB-XL - Lead II to aVF (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='896) Measured Converted PTB-XL - Lead II to aVF (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='957) Measured Converted PTB-XL - Lead II to V1 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='843) Measured Converted PTB-XL - Lead II to V1 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='766) Measured Converted PTB-XL - Lead II to V2 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='763) Measured Converted PTB-XL - Lead II to V2 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='871) Measured Converted PTB-XL - Lead II to V3 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='741) Measured Converted PTB-XL - Lead II to V3 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='816) Measured Converted PTB-XL - Lead II to V4 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='71) Measured Converted PTB-XL - Lead II to V4 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='574) Measured Converted PTB-XL - Lead II to V5 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='874) Measured Converted PTB-XL - Lead II to V5 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='919) Measured Converted PTB-XL - Lead II to V6 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='896) Measured Converted PTB-XL - Lead II to V6 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='903) Measured Converted 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8: Example cross-database result of lead II to all conversion on PTB-XL (each row depicts one converted lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' shared encoder on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='6 Summary and Conclusions 131 PTB-XL - Lead I to II (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='879) Measured Converted PTB-XL - Lead I to II (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='885) Measured Converted PTB-XL - Lead I to III (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='708) Measured Converted PTB-XL - Lead I to III (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='677) Measured Converted PTB-XL - Lead I to aVR (shared, r=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='491) Measured Converted PTB-XL - Lead I to aVR (individual, r=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='446) Measured Converted PTB-XL - Lead I to aVL (shared, r=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='378) Measured Converted PTB-XL - Lead I to aVL (individual, r=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='477) Measured Converted PTB-XL - Lead I to aVF (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='856) Measured Converted PTB-XL - Lead I to aVF (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='765) Measured Converted PTB-XL - Lead I to V1 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='57) Measured Converted PTB-XL - Lead I to V1 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='707) Measured Converted PTB-XL - Lead I to V2 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='578) Measured Converted PTB-XL - Lead I to V2 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='507) Measured Converted PTB-XL - Lead I to V3 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='785) Measured Converted PTB-XL - Lead I to V3 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='866) Measured Converted PTB-XL - Lead I to V4 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='768) Measured Converted PTB-XL - Lead I to V4 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='743) Measured Converted PTB-XL - Lead I to V5 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='976) Measured Converted PTB-XL - Lead I to V5 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='893) Measured Converted PTB-XL - Lead I to V6 (shared, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='969) Measured Converted PTB-XL - Lead I to V6 (individual, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='98) Measured Converted 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='9: Example cross-database result of lead I to all conversion on PTB-XL (each row depicts one converted lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' shared encoder on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' individual encoders on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 132 Interlead Conversion of Electrocardiographic Signals Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='10: Average correlation results for PTB-XL conversion from lead II according to medical condition class (using the U-Net with a shared encoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Converted leads Class I III aVR aVL aVF V1 V2 V3 V4 V5 V6 NORM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='90 MI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='77 STTC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='79 0.' metadata={'source': 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conversion from lead I according to medical condition class (using the U-Net with a shared encoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Converted leads Class II III aVR aVL aVF V1 V2 V3 V4 V5 V6 NORM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='23 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='82 level performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Both lead I and II appear to be especially suitable for certain sets of leads, and could be used on specific target applications that focus on those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' In the cross-database scenario, despite the acquisition setup differences, results were promising especially with the INCART database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Finally, the analysis of the influence of medical conditions has shown no considerable effect of pathologies on the performance of the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, a state-of-the-art methodology for automatic diagnosis revealed lower accuracy when using reconstructed signals, a problem that should be addressed in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although the results are promising, further efforts should be devoted towards the improvement of the methodologies for interlead conversion using single-lead blindly-segmented inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Namely, the pre-processing and normalisation of the signals, as well as the robustness to diverse acquisition setups, should be the target of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, task-oriented objective functions should be explored to ensure useful signal information is kept and avoid performance losses in subsequent diagnoses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' With some consolidation, the proposed methodology could be the key to better cardiac health monitoring in wearable devices and less obtrusive clinical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Taking the example of emer- gency rooms, if we can retrieve all twelve leads (or the most important among these) from Lead I signals, then patients will only need two electrodes placed on the wrists to have their ECG col- lected, instead of the full set of 10 electrodes on wrists, ankles, and chest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This is a meaningful step towards higher comfort and usability for patients in clinical settings or users in other scenarios involving the monitoring of ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Additionally, albeit outside the scope of this work, this methodology could also be applicable to other multi-channel signals where the different channels correspond to different perspectives over the same physiological phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Part III Face Biometrics 133 Chapter 9 Prior Art in Face Biometrics 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Data There are several publicly available databases for research purposes, to develop and benchmark face biometric recognition algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Considering the face is one of the most developed biometric traits, the databases available are some of the largest and most complete, thoughtfully structured for deep and adequate evaluation of recognition algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 compiles some relevant in- formation about the most important databases currently available, which are also described below: CASIA NIR-VIS: Also known as CBSR NIR, the CASIA NIR-VIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='0 database was cre- ated by the Institute of Automation of the Chinese Academy of Sciences (CASIA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It in- cludes pairs of mugshots and corresponding NIR images from 725 people, acquired over four recording sessions, from 2007 to 2010 [253];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' CASIA WebFace: This database was created and made available by the Institute of Au- tomation of the Chinese Academy of Sciences (CASIA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes almost five hundred thousand images from more than ten thousand identities collected to support the research in unconstrained face recognition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' CelebA: This database results from a previous one, CelebFaces+, which has been enriched with fiducial and attribute annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes over two hundred thousand pictures from over ten thousand celebrities, with five fiducial locations, and forty binary attributes per image [267];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' COX Face: The COX Face database was designed to study recognition across still images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Thus, it includes still images from one thousand subjects in a controlled envi- ronment with high quality, and surveillance videos from the subjects in unconstrained and low-quality settings [181];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 135 136 Prior Art in Face Biometrics Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Details on the main face recognition databases that are currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Database Spectrum Subjects Images Videos Resolution Unconstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' CASIA NIR-VIS [253] Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' + NIR 725 17 580 None 640x480 \x15 CASIA WebFace Visible 10 575 494 414 None 250x250 CelebA [267] Visible 10 177 202 599 None COX Face [181] Visible 1000 1000 3000 \x15 CSIST Lab1 [466] Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' + NIR 50 1000 None 100x80 \x15 CSIST Lab2 [466] Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' + NIR 50 2000 None 200x200 \x15 FERET Visible 1199 14 126 None 512x768 IJB-C Visible 3531 138 836 11 779 IMDb-Face [448] Visible 59 000 1 700 000 None LFW [177;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 178] ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='768x576 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='UMD Faces [25] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Visible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='8277 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='367 888 ' 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+page_content='3 310 000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Diverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Yale Face ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Visible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='165 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='320x243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='YouTube Faces [460] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='Visible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1595 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3495 CSIST Lab1: The CSIST database was developed by the Chung-Shan Institute of Science ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' with images from volunteers at the Harbin Institute of Technology of Shen- zhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Lab1 dataset contains ten visible light and ten NIR images from each of fifty subjects [466];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' CSIST Lab2: The CSIST Lab2 dataset is part of the CSIST database, and includes twenty visible light and twenty NIR images from fifty volunteers, with natural lighting and artificial lighting [466];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' FERET: The Facial Recognition Technology (FERET) Program was sponsored by the US Department of Defence, and the FERET database is distributed by the National Institute of Standards and Technology (NIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The database, with more than fourteen thousand face images collected between 1993 and 1996, has the goal to support the development of new techniques for automatic recognition of faces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Data 137 IJB-C: The IARPA Janus Benchmark-C (IJB-C) is a database resulting from a group of challenges created by NIST addressing verification, identification, detection, clustering, and processing of videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes over one hundred and thirty-eight thousand images and eleven thousand face videos from over three thousand identities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' IMDb-Face: This dataset includes approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='7 million images with faces from fifty- nine thousand celebrities on the IMDb movie database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' According to the authors, efforts have been devoted to ensuring this database is cleaner than most other large available databases, making it better for training robust algorithms [448];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' LFW: The Labelled Faces in the Wild dataset was one of the first databases of unconstrained face images, and includes over thirteen thousand face images of almost six thousand identi- ties [177;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 178];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' MegaFace: The MegaFace collection includes an average of 7 unconstrained face photos (between 2 and 2469) for each of almost seven hundred thousand identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is the sur- veyed database with the most identities, and the second with most total images [305];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' MS-Celeb: The Microsoft Celeb is a dataset of over eight million images obtained online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes faces from nearly one hundred thousand celebrities in unconstrained settings, aiming to accelerate research in face recognition with large target sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' PaSC: The Point and Shoot Face Recognition Challenge (PaSC) dataset was created by NIST to encourage the development of face recognition algorithms that are more robust to very unconstrained settings and inexpensive camera technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes over nine thousand images and almost three thousand videos with faces from almost three hundred identities with different distances to the camera, perspective, and locations [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' PolyU: This database was created by the Biometric Research Centre (UGC/CRC) at The Hong Kong Polytechnic University, using their own real-time NIR face capture device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It includes approximately thirty-four thousand NIR face images from over three hundred sub- jects [480];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UMD Faces: This dataset was created at the University of Maryland (UMD) and contains almost four hundred thousand face images from over eight thousand subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Each image includes human-curated face bounding boxes and annotations on pose, gender, and twenty- one keypoints [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' UMD Videos: This dataset is similar to UMD Faces, only it includes over three million video frames from twenty-two thousand videos with over three thousand identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The frames are also annotated with pose, keypoints, and gender information [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' VGGFace: This database was created by the Visual Geometry Group (VGG) of the Univer- sity of Oxford, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The group developed a method for minimally-supervised online image 138 Prior Art in Face Biometrics Face Detection Feature Extraction Recognition Image Identity Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1: Stages of a biometric recognition algorithm based on face images (based on [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' collection, which enabled the creation of this large database, with over two million faces in unconstrained conditions [331];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' VGGFace2: The VGGFace database was later extended to create the VGGFace2 database, which includes more than three million unconstrained face images from almost ten thousand identities [58];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Yale Face: The Yale Face database includes eleven images from each of fifteen subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Although not an unconstrained database, it includes annotations on certain expressions and configurations simulated by the subjects, which can be useful in training models for other tasks such as emotion recognition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' YouTube Faces: This dataset is composed exclusively of faces on YouTube videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It in- cludes over three thousand videos with over one thousand identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The videos range from 48 to 6070 frames (average of 181 frames per video) [460].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These databases already cover most bases and offer a good starting point for the study and development of strong biometric algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Nevertheless, it is important to acknowledge the growth of heterogeneous approaches in facial recognition, and the subsequent need for databases of face images acquired in different modalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', visual spectrum vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' NIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These databases are still too small and too controlled for the development of robust algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Moreover, it would be useful to have larger databases focused on more specific applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=', face images and videos of car drivers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work According to Barnouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [28], face recognition can be decomposed into three processes: face detection, feature extraction, and face recognition (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Below, we delve into the state- of-the-art in each of these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Due to the currently common practice of joining feature extraction and face recognition into a single model using deep learning, these two processes are jointly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='1 Face detection Given an image or video stream, the process of face detection has the goal of locating and extract- ing all human faces visible in the received input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' It is an extremely important process not only 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 139 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2: Examples of face detection in unconstrained settings (images from the FDDB data- base [203], ground-truths in green and predictions in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' for face-based biometric recognition, but also for face tracking and person re-identification across surveillance cameras, recognition of expressions and emotions, and analysis of soft-biometrics such as gender or age [232;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 476].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some earlier, simpler methods relied on skin colour for face detection on images [89;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These presented the advantage of being orientation-invariant, as it would serve to detect a face even if it did not present a frontal pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, they fail to consider the great variety of skin colours, both due to natural differences between individuals, and due to diverse illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' More sophisticated algorithms have been proposed, including the method by Sirovich and Kirby [406] based on eigenvectors from large face image datasets, the method by Viola and Jones [442] which uses cascades of Haar transform filters selected using AdaBoost, or the method by Dalal and Triggs [92] which uses histograms of intensity gradients from image regions and their orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These commonly offer very fast detection, adequate for real-time systems, but often fail on non-frontal face detection and faces of very diverse scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Like most pattern recognition tasks, traditional methods from earlier literature have been re- cently replaced with deep learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These offer more robust and accurate face detec- tions, especially for non-frontal face detection, making better use of very large datasets currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Some of these datasets currently offer a public benchmark for fair and direct comparison with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These include the WIDER face dataset [469] and the Face Detection Database (FDDB) [203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These benchmarks are currently largely dominated by deep learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The FDDB benchmark is currently dominated by the S3FD, the DeepIR, and the RSA algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' S3FD [486] is based on a single deep network specifically fitted to better detect small faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The DeepIR method [416] uses a Faster Region-based CNN (RCNN) adapted with fea- ture concatenation, multiscale training, and hard negative mining for more robust detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The RSA algorithm [264] is based on a convolutional network with a recurrent strategy for detection at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The best scores on the WIDER Face benchmark belong to the AInnoface and the RetinaFace algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The RetinaFace algorithm [99] is a single-stage pixelwise face detector that takes advantage of extra-supervised and self-supervised multitask learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' AInnoface [481] is based 140 Prior Art in Face Biometrics on RetinaFace with two-step classification and regression, an IoU loss function, improved data augmentation, and several other structural network changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These sophisticated algorithms perform accurate and robust detection for faces in different poses and at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' But they are still computationally heavy and require GPUs for real-time operation in images with VGA resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Having overcome most challenges in uncon- strained face detection, research should now focus on making algorithms faster and more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Feature extraction and recognition Having extracted the detected faces from the input, face-based recognition systems need to extract appropriate features from those faces to accurately decide on their identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Wang and Deng [452], in their survey of deep learning face recognition, have pointed out how the field of face bio- metrics has moved from traditional machine learning approaches to deep learning (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, the best results were only obtained when the development of sophisticated tailored ob- jective functions began (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' As such, Wang and Deng [452] divide approaches into four categories: holistic learning, local handcrafted, shallow learning, and deep learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Below, the most relevant examples of each category are presented, along with their advantages and shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Holistic learning approaches are those that use the whole face image to obtain representa- tions that ease the process of face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The Eigenface method [406], described for face detection, is one of these methods, along with the Fisherface method [34], which is similar to Eigenface, but uses the Fisher Linear Discriminant Analysis (FLDA, instead of PCA) for dimen- sionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Such methods are simple and fast, but lack robustness to several variability factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Eventually, researchers started to explore methods that extracted features from regions of the face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' These methods mostly used Gabor filters and Local Binary Patterns for feature extrac- tion based on intensity gradients and image edges [12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' 262].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Methods like these and the Elastic Bunch Graph Matching [459] were able to improve recognition accuracy, but not to make it high enough for real use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' To improve accuracy and robustness to pose variations, researchers proposed learning-based methods, that used available data to learn the best features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first method to use shallow learning was proposed by Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' [59], using gradient filtering after facial landmark alignment and clustering methods to learn encodings for better recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' But truly high accuracy and robustness in face recognition were only attained with the rise of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' The first models were convolutional neural networks with conventional archi- tectures, such as DeepFace [423], VGG-Face [331], or VGG-Face2 [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Over time, researchers started to focus on adapting the networks for specific details of face recognition, such as custom loss functions that force increased intersubject separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' This resulted in improved methods such as DeepID [417], L2-Softmax [360], and ArcFace [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Both Facenet and DeepID are among the top five non-commercial methods in the LFW bench- mark, with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='63% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='45% accuracy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' However, as discussed by Wang and Deng 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='2 Related Work 141 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='3: Evolution of face recognition approaches, from holistic to deep learning (from [452]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content='4: Recent history of face recognition, from deep learning to tailored objective functions (from [452]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf'} +page_content=' Representation Deepface %L62.0.CO;2. +Kingma, D. 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HEX: Hyperbolic Event eXtractor, a Seismic +Phase Associator for Highly Active Seismic Regions. +Seismological Research Letters, 91(5): +2769–2778, July 2020. doi: 10.1785/0220200037. +Yeck, W. L., Patton, J. M., Johnson, C. E., Kragness, D., Benz, H. M., Earle, P. S., Guy, M. R., +and Ambruz, N. B. GLASS3: A Standalone Multiscale Seismic Detection Associator. Bulletin of +the Seismological Society of America, 109(4):1469–1478, May 2019. doi: 10.1785/0120180308. +14 + +Zhang, M., Ellsworth, W. L., and Beroza, G. C. Rapid Earthquake Association and Location. +Seismological Research Letters, 90(6):2276–2284, Sept. 2019. doi: 10.1785/0220190052. +Zhu, W. and Beroza, G. C. PhaseNet: a deep-neural-network-based seismic arrival-time picking +method. Geophysical Journal International, 216(1):261–273, Jan. 2019. doi: 10.1093/gji/ggy423. +Zhu, W., McBrearty, I. W., Mousavi, S. M., Ellsworth, W. L., and Beroza, G. C. +Earth- +quake Phase Association Using a Bayesian Gaussian Mixture Model. Journal of Geophysical +Research: Solid Earth, 127(5):e2021JB023249, 2022. +doi: 10.1029/2021JB023249. +_eprint: +https://onlinelibrary.wiley.com/doi/pdf/10.1029/2021JB023249. +15 + diff --git a/z9E0T4oBgHgl3EQftwG-/content/tmp_files/load_file.txt b/z9E0T4oBgHgl3EQftwG-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18fb4d3536c07b365481869cf580bdeb3af3a529 --- /dev/null +++ b/z9E0T4oBgHgl3EQftwG-/content/tmp_files/load_file.txt @@ -0,0 +1,673 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf,len=672 +page_content='Neural mixture model association of seismic phases Zachary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Ross Seismological Laboratory California Institute of Technology zross@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='edu Weiqiang Zhu Seismological Laboratory California Institute of Technology wqzhu@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='edu Kamyar Azizzadenesheli Nvidia Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' kamyara@nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='com Abstract Seismic phase association is the task of grouping phase arrival picks across a seismic network into subsets with common origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Building on recent successes in this area with machine learning tools, we introduce a neural mixture model association algorithm (Neuma), which incorporates physics-informed neural networks and mixture models to address this challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Our formulation assumes explicitly that a dataset contains real phase picks from earthquakes and noise picks resulting from phase picking mistakes and fake picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The problem statement is then to assign each observation to either an earthquake or noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We iteratively update a set of hypocenters and magnitudes while determining the optimal class assignment for each pick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We show that by using a physics-informed Eikonal solver as the forward model, we can impose stringent quality control on surviving picks while maintaining high recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We evaluate the performance of Neuma against several baseline algorithms on a series of challenging synthetic datasets and the 2019 Ridgecrest, California sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Neuma outperforms the baselines in precision and recall for each of the synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Furthermore, it detects an additional 3285 more earthquakes than the best baseline on the Ridgecrest dataset (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='5%), while substantially improving the hypocenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Keywords: 1 Introduction Seismicity catalogs are the foundation for a wide array of seismological analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' They are generally produced in automated fashion with detection and location algorithms, which include phase picking algorithms and array processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The phase picking approach is the one adopted most commonly by real-time seismic networks, and consists of two main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' First, the raw continuous waveform data is processed by phase detection/picking algorithms, one station at a time, to identify candidate earthquake signals and measure their onset times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Then, a second algorithm is applied to consider combinations of these detections across the network of sensors and decide whether any disjoint subsets could originate from coherent sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' This step is referred to as seismic phase association and is responsible for determining which of the candidate phase picks should be used to 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='02597v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='geo-ph] 6 Jan 2023 locate the event and determine its magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' For large magnitude events with high signal-to-noise ratios, phase association is fairly straightforward, as the number of phase picks across the network is generally large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Detecting small earthquakes, particularly during aftershock sequences and swarms, is quite challenging however due to relatively fewer stations having picks available, the presence of lots of noise picks, and overprinting of seismic wave moveouts due to nearly concurrent events [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' For many years, phase association algorithms were largely based around grid search algorithms, in which phase arrival times were back-projected to see if any subset focused at a coherent origin time [Le Bras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 1994, Draelos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2015, Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2016, Yeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' These methods are strictly based on wave propagation physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' More recently, there have been several phase association algorithms that incorporate machine learning or more modern elements from applied mathematics, [Reynen and Audet, 2017, McBrearty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2019, Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2019b, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2019, Woollam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2020, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' These methods all incorporate aspects of the wave physics (either explicitly or implicitly) in solving the association problem, but are wide-ranging in the types of algorithms employed to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Of these, the Gaussian Mixture Model Association algorithm [GaMMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2022] is particularly appealing because it is able to not only use travel time information, but also amplitude information, to better constrain the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' GaMMA formulates the association problem as one of unsupervised (probabilistic) clustering, where the centroids of the clusters are hypocenter-magnitude pairs and the error in the picks and forward model is captured by a Gaussian distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' the problem statement then is to optimize the hypocenters while simultaneously grouping together phase picks that match them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' GaMMA was shown to perform especially well on the very active 2019 Ridgecrest, California earthquake sequence [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' However, the method has a number of shortcomings that hinder its potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' One clear limitation is that the technique uses a forward model for the travel times that is based on a homogeneous velocity structure, resulting in travel times that can be substantially inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022] overcome this in part by using a very large (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 s) uncertainty on the travel times, but this comes with the cost of potentially including noise picks too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Another significant limitation of GaMMA is that it assumes that all picks were produced by earthquakes when fitting the model, even if in reality a pick was a false detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' As we have strong a priori knowledge that a considerable fraction of the input picks will be noise picks output from the phase pickers [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='g Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2018b], it would be ideal to include this knowledge explicitly into a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' In this study, we introduce a Neural mixture model association algorithm, Neuma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Our contributions are three-fold: (i) We modify the Gaussian mixture model association algorithm to use a neural network Eikonal solver, eliminating the need for a homogeneous velocity model and leading to considerably improved phase associations, more detections, and more precise hypocenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' (ii) We introduce a new probability formulation that includes latent noise variables in addition to real picks, allowing for explicit noise labels to be assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' (iii) Finally, we introduce a separate warmup scheme for initializing the hypocenters that helps to achieve better global convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We demonstrate the performance of our method on several synthetic datasets and the 2019 Ridgecrest, California sequence and compare the results to several established baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' For the Ridgecrest dataset, we detect an additional 3283 earthquakes over the best baseline, an improvement of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='4%, while reducing the number of associated picks by 2 per event on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' In addition, we reducethe bias in hypocentral 2 depth by nearly 5 km per event on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 2 Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1 Preliminaries Seismic phase association is the problem of organizing a set of tentative phase detections from different sensors into subsets that have common hypocenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Let y = {(ti, ai)}N i=1 denote a set of observed phase arrival picks and corresponding peak amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' In most real-world cases, some of these picks will correspond to earthquakes, whereas the remainder will be noise that needs to be filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The problem statement is therefore to assign each yi to one of K + 1 classes, where K is the number of earthquakes and the (K + 1)th class represents noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' An earthquake is defined by a hypocenter xi ∈ R4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' location and origin time) and magnitude Mi ∈ R, and the observations y occur over some time interval ∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Furthermore, let M denote the set of all magnitudes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', M := {Mi}N i and x denote the set of all hypocenters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', x := {xi}N i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' If the probability of an observation belonging to class k is φk, and φ denotes the set of all event probabilities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', φ := {φk}K+1 k , then the joint likelihood of the set of observations can be written as, p(t, a|φ, x, M) = N � i p(ti, ai|φ, x, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' (1) For a mixture model with K + 1 classes, this (log)-likelihood can be written as, log p(t, a|φ, x, M) = N � i=1 log � K � k=1 φkp(ti, ai|xk, Mk) + φK+1pG(ti, ai) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' (2) where pG(ti, ai) = pG T (ti)pG A(ai) is a joint density for noise picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Neuma is schematically illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We are given an initially unclustered set of phase picks and amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We are tasked with identifying which of the picks should be labeled as noise, and which of the picks should be assigned to disjoint subsets that correspond to individual earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The subset of picks associated to each earthquake is described by a forward model and a probability model for the observation uncertainty (pick and amplitude error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Expectation-Maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Optimizing the likelihood in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 2 is complicated by the summation inside the logarithm, which results from the class assignments for each observation being unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' One of the most common approaches to maximizing likelihoods with latent variables is expectation- maximization [E-M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Dempster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 1977], in which the class assignments are first computed under the fixed probability model (the "E" step) and then the cluster parameters are optimized using these newly determined class assignments (the "M" step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' By iterating between the E-step and M-step, the E-M algorithm has provable non-decreasing convergence, making it appealing for solving likelihood problems with latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 3 P/S arrival times predicted by EikoNet Associated earthquakes Laplace distributions Figure 1: A cartoon depicting neural mixture model association (Neuma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' A set of initially disordered phase picks and amplitudes are iteratively grouped into disjoint subsets that have common hypocenters and magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' E-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Let γik denote the probability of the ith phase pick belonging to class k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Given a set of hypocenters and magnitudes, we can compute γ to update φ as follows, γi,k = φkpE T (ti|xk, θT )pE A(ai|xk, Mk, θA) �K k φkpE T (ti|xk, θT )pE A(ai|xk, Mk, θA) + φK+1pG T (ti)pG A(ai) (3) for all k ∈ [K], and γi,K+1 = φK+1pG T (ti)pG A(ai) �K k φkpE T (ti|xk, θT )pE A(ai|xk, Mk, θA) + φK+1pG T (ti)pG A(ai) (4) Therefore, ∀k ∈ [K + 1] we have, φk = �N i=1 γi,k N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' M-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The maximization step of the E-M algorithm uses the newly determined class assignments from the E-step to compute the expectation of the likelihood in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 2 and then maximize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Mathematically this corresponds to, x, M = arg max x,M log p(y|φ, x, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 4 The γik from the E-step are thus used to update x and M, xk = arg min xk N � i=1 γik|ti − t(xi, xk)| (5) Mk = �N i=1 γikM(xk, xi, ai) �N i=1 γik (6) where M(xk, xi, ai) is an empirical magnitude definition e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' local magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Here, we use the same equation as Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022], taken from Picozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2018], log10 PGV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='08 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='93(M − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='5) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='68 log10 R where PGV is peak ground velocity in units of cm/s, M is magnitude, and R is hypocentral distance in km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 Implementation Choice of density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The densities pE T , pE A, pG T , and pG A define the arrival time and amplitude likelihoods for the earthquake and noise classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The earthquake densities were taken to be Gaussian by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022], with pE T (ti|xk, θT ) = N(t(xk, xi), ΛT k ) pA T (ai|xk, Mk, θA) = N(a(xk, xi, Mk), ΛA k ), where t(xk, xi) and a(xk, xi, Mk) are forward models for arrival times and amplitudes for a source with hypocenter xk at receiver xi with magnitude Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' For this study, we instead use Laplace distributions, pE T (ti|xk, θT ) = Laplace(µ = t(xk, xi), bT ) pA T (ai|xk, Mk, θA) = Laplace(µ = a(xk, xi, Mk), bA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The scale parameters bT ∈ R and bA ∈ R in these distributions are taken to represent the combined uncertain from our measurement error and forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The use of a Laplace distribution over a Gaussian distribution is motivated by the observations that arrival time residuals made with deep learning algorithms are closer to a Laplace distribution than a Normal distribution [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2018a, Zhu and Beroza, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Additionally, the ℓ1 norm in the Laplace distribution is more robust against outliers, which are common in phase picking datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We also need the densities pG T , and pG A for the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' For the arrival time noise distribution, we assume that pG T (t) is a uniform distribution on ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The noise amplitude distribution, pG A, does not have any strong theoretical motivation, particularly at the high frequencies of interest in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' As a result we model this distribution empirically using the distribution of amplitudes for picks rejected as earthquakes by GaMMA in the Ridgecrest dataset produced in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' This distribution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 2 and is approximately log-normal, with best-fitting parameters 5 µ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='46 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='72 in log10 m/s units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We use these parameters throughout this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' While there is the possibility that some of these amplitudes actually correspond to earthquakes that were missed by the associator, the majority are likely to be noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Forward Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The forward model used to compute µk for Neuma is an EikoNet [Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2020], a physics-informed deep neural network trained to solve the Eikonal equation for ray tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We use a simplified network architecture to maximize computational efficiency, with 5 dense layers of 128 neurons followed by exponential linear unit activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We perform the M-step update of the K hypocenters using the Adam optimizer [Kingma and Ba, 2014] in a non-stochastic (all data points used) fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Warmup Iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Our implementation of the E-M equations includes a few changes from that described by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We found that if the uncertainties on the likelihood functions, bT and bA, were set from iteration 1 onward to values representative of the true velocity model and picking errors, that the E-M would have poor convergence and assign most picks automatically to the noise class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The reason for this is because if the initial hypocenters assigned to the K clusters are not sufficiently close to the true position, that the E-step assignments for γ would always start out being effectively zero for all picks, which is lower than the probability of the noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' To circumvent this problem, we designed an implementation that instead has a warmup period in which the noise variables are turned off (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' γi,K+1 = 0) to allow for the K hypocenters to first be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' During this period, we also decay the numerical values of Λk for both the travel times and the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' At iteration 1, Λk is set to 5 times the actual (assumed) uncertainty of the travel times and amplitudes, and then is decayed linearly to the actual uncertainty over 10 warmup iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The initial value of the uncertainty was chosen as a compromise being much larger than the true (final) uncertainty, but not so large that all clusters have the same likelihood of pick assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' After the warmup period is over, γi,K+1 is computed at each iteration rather than being forced to zero, and Λk is set to the constant values taken at the final iteration of the warmup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We found that the E-M converged typically within about 5 iterations after the warmup period was finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 10 7 10 6 10 5 10 4 10 3 10 2 PGV (m/s) 0 2500 5000 7500 10000 12500 15000 17500 20000 Frequency Figure 2: noise pick amplitude distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 6 Selection of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022] showed that the parameter K has a major influence on the performance of the method, particularly in terms of computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' This is because the computational complexity scales non-linearly with K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' If too low, however, then some earthquakes may be missed entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The computational performance also scales with the number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' These issues generally require that a dataset be broken up into smaller chunks, which has the additional advantage of being fully parallelizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022] defined the oversample factor to help select K, in which K is set to some integer multiple of the ratio between the total number of picks in an observation window to the total number of stations available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' It was shown that choosing a value of K that is larger than the number of earthquakes usually results in the presence of some clusters with few or no picks associated, making the main tradeoff in K being poor computational performance if too large, and having false negative detections if too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' In this regard, Neuma is essentially unchanged from GaMMA and we use in this paper an oversample factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Breaking up a set of picks into subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Limiting the value of K to something computationally manageable also requires breaking up a larger dataset into smaller chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022] achieved this using the density-based clustering algorithm DBSCAN [Ester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 1996] to cluster the picks in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' This requires some hyperparameter tuning to achieve clusters that are long enough in time that they capture entire earthquakes, but not so long that the clusters run on for more than a few minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We also use DBSCAN for this purpose and set the parameter ϵ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 s with a minimum of 1 pick to merge clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We discard clusters with fewer picks than the minimum number required for detection (details to come in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' These values will however vary strongly based on the dataset attributes including the number of stations and their geometry, the number of false picks, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Thus for any new dataset these hyperparameters will require some tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 3 Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1 Baselines We use several different baselines in this section to evaluate the performance of Neuma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' In addition to Neuma, we test three other association algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' PhaseLink [Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2019b] is an association algorithm that uses deep recurrent neural networks trained in supervised fashion and incorporates an unsupervised cluster analysis to group picks into events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' GaMMA [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2022] is an approach based on Gaussian mixture models in which the centroids of the clusters are earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' For the real dataset, we use a catalog produced by the SCSN that was manually reviewed [SCEDC, 2013], which was produced with the Binder phase associator as part of the Earthworm package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 Synthetic datasets For the first experiment, we generate a suite of four synthetic datasets of earthquakes (picks) that consist of both "true" earthquakes and garbage picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The advantage of these datasets is that we have exact ground truth for every pick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We construct the datasets such that each one is increasingly more challenging than the last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We specify for each dataset a fixed number of earthquakes that are given random origin times and random uniform hypocenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We then generate synthetic arrival times and amplitudes for the Southern California Seismic Network stations located within the study area [SCEDC, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We design each dataset such that it spans exactly 24 hours (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The 7 synthetic amplitudes are drawn from the empirical amplitude distribution defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The easiest dataset (D1) contains 1080 earthquakes in it, making the average time between earthquakes 80 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The remaining datasets, D2, D3, and D4, have an average interevent time of 60, 40, and 20 seconds, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We add 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 sec Gaussian noise to each arrival and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 log10 unit of noise to the amplitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' we set bA and bT to these values for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' In addition, we create 57600 garbage picks with uniform times, randomly chosen stations and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Magnitudes for each event are set to a fixed value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 for these tests for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' To Figure 3: Samples of the synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Only the first 10 minutes of each dataset is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Colored points indicate real phase picks and are colored according to the associated event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Black points indicate noise picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' evaluate the performance of Neuma on these ground truth datasets, we use two metrics as introduced by Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Since earthquake detections are really defined by a collection of phase picks, we use analogs of precision and recall that are more appropriate for collections of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' To determine whether the kth cluster of picks, Ak, corresponds to a successful event detection, we define the precision between it and the best matching cluster of picks in the ground truth, Bi, Pk = max i N(Ak ∩ Bi), ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='., C} , 8 Dataset 1 Dataset 2 Dataset 3 Dataset 4 600 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 400 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 8 C 8 Time (sec) 300 : 8 200- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' C 100 : : 8 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 0 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9 Longitude Longitude Longitude LongitudeTable 1: Precision scores for the synthetic datasets Algorithm D1 D2 D3 D4 PhaseLink 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='928 GaMMA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='901 Neuma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='952 Table 2: Recall scores for the synthetic datasets Algorithm D1 D2 D3 D4 PhaseLink 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='228 GaMMA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='743 Neuma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='947 where, C is the total number of events (clusters) in the ground truth and N(·) denotes the cardinality of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We also define the recall as, Ri = max k N(Ak ∩ Bi), ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='., D} , where D is the number of detected events (clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' From these definitions, we can define the set-averaged precision, P, and recall, R, as, P = �D k Pk �D k N(Ak) R = �C i Ri �C i N(Bi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' As with the classical definitions of precision and recall, P describes how likely a single phase association is to be correct, whereas R describes how likely each phase association in the ground truth is correctly recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' These metrics naturally account for the potential for erroneously split and merged detections as both of these scenarios will impact the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The precision and recall scores for each dataset are shown in Tables 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We can see that Neuma outperforms both baselines for all datasets in both precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Even for the hardest dataset, D4, Neuma achieves P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='952, which is substantially better than the P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='901 and P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='928 for GaMMA and PhaseLink, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The recall differences are even more stark, as Neuma achieves R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='947 on D4, whereas GaMMA gets R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='743 and PhaseLink only obtains R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3 Ridgecrest Dataset To test Neuma on an established real dataset, we apply it to the 2019 Ridgecrest earthquake sequence [Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', 2019a] for the time period 2019-07-04 to 2019-07-09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' This dataset was previously used as a benchmark by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022] in tuning GaMMA, so it provides a clear comparison with a state of the art automated algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' This dataset has a manually reviewed seismicity catalog 9 produced by the Southern California Seismic Network (SCSN), including phases [SCEDC, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Since this dataset does not have ground truth available, we use a variety of measures to evaluate the overall quality of the results, including magnitude distributions, temporal analysis, phase association counts, and location quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Since PhaseLink does not compute magnitudes, we do not use it as a baseline for this dataset, and instead rely on the SCSN catalog which does have them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' To facilitate analysis strictly at the association level, rather than the phase detection level, we use the same input set of phase detections between GaMMA and Neuma that were produced by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' [2022];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' these were made by the PhaseNet algorithm [Zhu and Beroza, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' It should be noted however that the SCSN results use moving average phase detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Figure 4: Ridgecrest frequency-magnitude statistics for the three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Figure 5: Summary of detection results over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Left and right panels show magnitude and counts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Note the considerably larger drop in magnitude completeness after the M7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1 for the SCSN catalog than the Neuma and GaMMA catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' For this experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' we set Neuma to output detections with at least 10 phase picks and set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='GaMMA: 24465 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='Neuma: 27748 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='SCSN: 11389 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='MagnitudeGaMMA: 24465 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='Neuma: 27748 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='SCSN: 11389 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='Magnitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-05:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-07:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-08:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-09:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-10:001400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='GaMMA: 24465 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='Neuma: 27748 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='SCSN: 11389 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-05:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-07:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-08:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-09:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='07-10:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='DateFigure 6: Ridgecrest dataset phase association results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' bT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='35 s and bA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We use a 1D velocity model for southern California that is the same as the one employed by the SCSN for their real-time catalog [Hadley and Kanamori, 1977].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The frequency-magnitude statistics for the three catalogs are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Both GaMMA and Neuma detect more than twice as many earthquakes as the SCSN, which is almost a full magnitude unit decrease in the completeness magnitude due to the low b-value for this sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Neuma detects an additional 3283 events over GaMMA on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Additional gains of the algorithm are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 5, where the results are parsed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Neuma detects more earthquakes than the other catalogs consistently over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' In comparison with the SCSN catalog, Neuma and GaMMA do not see such a large drop in the magnitude of completeness over the two days following the M 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1 mainshock;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' these high-resolution catalogs have a much more gradual change in completeness over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We also examine the performance of Neuma at a phase association level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Figure 6 shows the phase association results for the three catalogs in two forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Out of the 1,338,285 total picks made by PhaseNet, which represents an upper bound for the number of possible picks to be associated, Neuma associates 1,090,701 picks, while GaMMA only associates 1,005,348 – a difference of about 85,000 picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' These numbers are much larger than the 409,115 associated by the SCSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' However, at an event level, Neuma associates almost 2 picks fewer per event than GaMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' This decrease is expected because Neuma uses a much more appropriate model for the velocity and picking errors, which would result in GaMMA being more likely to include noise picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The more efficient association of Neuma results in locations that are substantially better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Figure 7 shows the hypocenters for the three catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The SCSN results are the cleanest, which is expected because the picks have all been manually reviewed, and the detected events have the highest completeness magnitude, which results in a subset of events with the highest signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Neuma locations in general closely track those of the SCSN, although with some minor differences expected to arise from differences in the velocity model used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' GaMMA locations in contrast are significantly worse, with depths systematically more than 5 km deeper that reflects 11 1e6 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 35 Total associated picks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='8 30 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='4 15 Avg 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 0 SCSN GaMMA Neuma SCSN GaMMA NeumaFigure 7: Comparison of hypocenters for the Ridgecrest dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' the large errors produced by the homogeneous velocity model and inclusion of more false positive associated picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 4 Conclusion We formulate a new mixture model-based phase association algorithm called Neuma, which in- corporates a physics-informed deep neural network as the forward model to rapidly and reliable compute travel times in 3D media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Our formulation also explicitly includes a class for noise picks, which are widespread in automated picking datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Our formulation is easily implemented with the expectation-maximization algorithm by iterating between optimizing a set of hypocenters and magnitudes, and determining the optimal event to associate each pick to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' We demonstrate significant improvements over several baseline methods on challenging synthetic and real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' The developed approach is expected to help streamline the automation of seismicity catalog construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 12 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 (b) GaMMA 5 Neuma (km) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='SCSN Depth ( 10 Stations 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 D 15 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9 20 Longitude 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} 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Longitude 0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='7 5 (km) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6 10 Depth 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='5 15 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='4 20 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='8-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='7-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='5-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='4-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='3-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='1 36.' metadata={'source': 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+page_content='6980 [cs], Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' arXiv: 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='6980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Le Bras, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Swanger, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Sereno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Beall, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Jenkins, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Nagy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', and Henson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Global association;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' final report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Science Applications International Corporation Technical Report, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' McBrearty, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Gomberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Delorey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', and Johnson, P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Patton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Guy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Benz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Buland, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Erickson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', and Kragness, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Hydra—The National Earthquake Information Center’s 24/7 seismic monitoring, analysis, catalog 13 GaMMA: 24465 8000 Neuma: 27748 SCSN: 11389 7000 6000 5000 4000 3000 2000 1000 0 0 5 10 15 20 25 Depth (km)production, quality analysis, and special studies tool suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' USGS Numbered Series 2016-1128, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Geological Survey, Reston, VA, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Code Number: 2016-1128 Code: Hydra—The National Earthquake Information Center’s 24/7 seismic monitoring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' catalog production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' quality analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' and special studies tool suite Publication Title: Hydra—The National Earthquake Information Center’s 24/7 seismic monitoring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' catalog production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' quality analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' and special studies tool suite Reporter: Hydra—The National Earthquake Information Center’s 24/7 seismic monitoring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' catalog production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' quality analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' and special studies tool suite Series: Open-File Report IP-074998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' Picozzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Bindi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Spallarossa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', Di Giacomo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=', and Zollo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQftwG-/content/2301.02597v1.pdf'} +page_content=' A rapid response magnitude scale for timely assessment of the high frequency seismic radiation.' metadata={'source': 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b/zNFLT4oBgHgl3EQfni-P/content/tmp_files/2301.12128v1.pdf.txt @@ -0,0 +1,5109 @@ +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES IN +GENERIC CONFORMALLY FLAT HYPERSURFACES +NOZOMU MATSUURA AND YOSHIHIKO SUYAMA +Abstract. We study generic conformally flat (analytic-)hypersurfaces in the Euclidean 4- +space R4. +Such a local-hypersurface is obtained as an evolution of surfaces issuing from a +certain surface in R4, and then, in consequence, the original surface is a (principal-)curvature +surface of the hypersurface. The hyperbolic 2-metric ˇgH of the upper half plane leads to a +6-dimensional set of singular (analytic-)Riemannian 2-metrics g0 of R2: on a simply connected +open set in the regular domain of g0, a curvature surface with the metric g0 is determined. +In this paper, we choose a suitable singular metric g0 for ˇgH and clarify the structure of the +curvature surfaces: there would be an analytic extension of the surfaces beyond the regular set +of g0; for the surface, the singularities and the points of infinity of g0 could be caught explicitly. +In this case, all principal curvature lines in the extended surface are expressed by a frame field +of R4 induced on the surface from hypersurfaces and they lie on some standard 2-spheres S2, +respectively. We also provide a general method constructing an approximation of such frame +fields, and obtain the figures of those lines including the singular points of g0. +Key words: conformally flat hypersurface, principal curvature surface, integrability condi- +tion, cuspidal edge, envelope, point of infinity, numerical solution for orthonormal frame field. +2020 Mathematics Subject Classification: 53A07, 53C40, 68W25 +1. Introduction +We study (principal-)curvature surfaces of generic conformally flat (analytic-)hypersurfaces +in the Euclidean 4-space R4, arising from the hyperbolic 2-metric ˇgH of the upper half plane. +The 2-metric ˇgH gives rise to many generic conformally flat local-hypersurfaces, but there is +no known explicit representation of such hypersurfaces or their curvature surfaces. +In this +paper, we clarify the property and the structure of the analytically extended curvature surfaces +by using a frame field of R4 induced on the surface from hypersurfaces. Here, we say that a +hypersurface is generic if it has distinct three principal curvatures at each points, and that a +surface is curvature if it is woven of the curvature lines for two principal curvatures of some +generic conformally flat hypersurface, because such lines always make a surface. The principal +curvature line of a curvature surface is also that of the hypersurface including the surface. For +n-dimentional hypersurfaces with n > 3, there are no generic conformally flat hypersurfaces by +the result due to Cartan[4]. +Any generic conformally flat local-hypersurface is regarded as a one-parameter family of +curvature surfaces, in other words, it is obtained by an evolution of surfaces issuing from a +certain surface in R4, and then, in consequence, the original surface is a curvature surface of +the hypersurface. Thus, for generic conformally flat hypersurfaces, it would be important to +study the structure of the surfaces to be curvature. A certain analytic (local-)surface φ in the +standard 3-sphere S3 leads to a curvature surface: φ gives rise to an orthonormal frame field +of R4, and the frame field induces a curvature surface. In particular, any metric ˇg of a certain +family Met0, consisting of orthogonal analytic Riemannian 2-metrics on simply connected open +sets V ⊂ R2 with constant Gauss curvature −1, leads to a 6-dimensional set of Riemannian +This work was partially supported by JSPS KAKENHI Grant Number JP19K03507. +1 +arXiv:2301.12128v1 [math.DG] 28 Jan 2023 + +2 +N. MATSUURA AND Y. SUYAMA +2-metrics g0 on V such that, for each metric g0, a surface φ in S3 mentioned above is determined +and the curvature surface obtained has the metric g0 (cf. [1], [18]). +Now, we regard the hyperbolic 2-metric ˇgH := ((dx)2 + (dy)2)/y2 on the upper half plane +as a (singular) metric on R2. +For the 2-metric ˇgH, a pair Φ = (ϕ(x, y), ϕz(x, y)) of func- +tions on R2 is determined (see (1.4) below). For a simply connected open set V satisfying +(ϕz(ϕz)x(ϕz)y)(x, y) ̸= 0, the metric ˇgH on V belongs to the family Met0. Moreover, for the +metric ˇgH on R2 and Φ, a 5-dimensional set of singular (analytic-)Riemannian 2-metrics g0 on +R2 is determined [18, Example 3.2]. We choose a suitable singular metric g0 on R2 among them +to get nice curvature surfaces. Our first aim in this paper is to study an analytic extension of +curvature surfaces beyond the regular set of the metric g0 and further to clarify the structure +of the extended surface including the points at infinity. Then, all extended principal curvature +lines in the surface are expressed by the frame field determining curvature surfaces and they lie +on some standard 2-spheres, respectively. The second aim is to construct generally an approx- +imation of such frame fields. Then, the approximation of each principal curvature line also lies +on a standard 2-sphere. By the approximation, we give several figures for the extended surface: +principal curvature lines, the image of the singular curves for g0 and so on. +Now, let f be a generic conformally flat hypersurface in R4 defined on a domain U of R3, +and κi (i = 1, 2, 3) be the principal curvatures1 of f. Then, for f a principal curvature line +coordinate system (x, y, z) and a function ϕ = ϕ(x, y, z) on U are determined such that the +(non-degenerate) 3-metric +g = cos2 ϕ(dx)2 + sin2 ϕ(dy)2 + (dz)2 +(1.1) +is conformally flat and the first fundamental form If of f is given by +If := P 2g = P 2(cos2 ϕ(dx)2 + sin2 ϕ(dy)2 + (dz)2) +(1.2) +with a function P = P(x, y, z) ̸= 0 on U. Let the principal curvatures κi (i = 1, 2, 3) correspond +to x, y and z-lines in order. Then, κ1, κ2 are determined from (ϕ, P, κ3) as +κ1 = P −1 tan ϕ + κ3, +κ2 = −P −1 cot ϕ + κ3. +(1.3) +The conformally flat metric 3-metric g of (1.1) is called the (principal) Guichard net2 of f. +Note that the above coordinate system (x, y, z) and the metric g in (1.1) are defined only +on the domain where f is generic. Conversely, for any conformally flat 3-metric g of (1.1), +there is a generic conformally flat hypersurface with the Guichard net g uniquely up to a +conformal transformation, if U is simply connected (cf. [5]–[7], [19]). Then, in order to realize +the hypersurface in R4, it is necessary to find out a function P in (1.2) from ϕ such that the +Gauss and the Codazzi equations are satisfied (cf. [9]). +For the examples of Guichard nets and generic conformally flat hypersurfaces, see a series of +papers ([7], [8], [10], [15]–[17]). The other construction of such hypersurfaces, which do not use +the Guichard net, is given by the papers ([2], [3], [14]), where [14] is another proof of [7]. +1.1. Our problems. For the function P in (1.2), we study the curvature surfaces including the +case P ≤ 0 not only the case P > 0. Here, we state the setting in this paper, and briefly review +the results for generic conformally flat local-hypersurfaces in the papers [1] and [18], which +make our problems clearer. Let ˇgH be the hyperbolic 2-metric on R2. From now on, we assume +that the domain U, where hypersurfaces f(x, y, z) are defined, is given by U = V × I ⊂ R2 × R +or U = V ′ × I ⊂ R2 × R for a simply connected open set V or V ′ in R2 = R2 +(x,y) and a suitable +open interval I ⊂ R = R(z) with 0 ∈ I. +1In (1.1)–(1.3), we have assumed that κ3 is the middle principal curvature: κ1 > κ3 > κ2 or κ1 < κ3 < κ2, +for the sake of simplicity for the description later. +2We call the canonical principal Guichard net of f only the Guichard net (see [2]). + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +3 +(R1) For ˇgH, a pair Φ = (ϕ(x, y), ϕz(x, y)) of functions is determined as +cos ϕ(x, y) = x2 − y2 +x2 + y2, +sin ϕ(x, y) = +2xy +x2 + y2, +ϕz(x, y) = +y +x2 + y2, +(1.4) +which are analytic functions on R2 with the pole at the origin. Actually, a one-parameter family +Φc(x, y) = (ϕ(x, y), cϕz(x, y)) with parameter c ̸= 0 is determined for ˇgH, where we have chosen +Φ = Φ1 in our study. +(R2) Let S1 be the set defined by +S1 := +� +(x, y) ∈ R2 | (ϕz(ϕz)x(ϕz)y)(x, y) = 0 +� += +� +(x, y) ∈ R2 | xy(x2 − y2) = 0 +� +. +(1.5) +For a domain V such that V ⊂ R2 \ S1, the pair Φ leads to an analytic function ϕ(x, y, z) on +U = V ×I uniquely as an evolution in z-direction under the initial conditions ϕ(x, y, 0) = ϕ(x, y) +and ϕz(x, y, 0) = ϕz(x, y)3, and then ϕ(x, y, z) defines a conformally flat metric g on V × I in +(1.1) (by replacing V × I with a sub-domain V ′ × I such that ϕzϕzxϕzy ̸= 0 holds on V ′ × I, +if necessary)4. Thus, a generic conformally flat hypersurface fV (x, y, z) on V × I with the +Guichard net g is determined from Φ|V . Note that the Guichard net g is not defined on S1. +(R3) As the solutions to a certain system of differential equations defined from Φ, a class +Ψ = ( ¯P(x, y), ¯Pz(x, y), ¯κ3(x, y)) consisting of three functions on R2 are also determined. In order +to describe the class Ψ explicitly, we prepare a generalized hypergeometric function X0 on R of +type 1F2: for a sequence {ak}∞ +k=1 given by a1 = 1 and 2(k + 1)(4k2 + 5/4)ak+1 + (2k − 1)ak = +0 (k ≥ 1), X0(x) is defined by +X0 = X0(x) := 5/2 + �∞ +k=1 akx2k = 5/2 + x2 − (1/21)x4 + · · · . +(1.6) +The class Ψ is determined from X0 as follows: +¯P −1(x, y) = +x +x2 + y2h(x, y), +¯κ3(x, y) = − +y +x2 + y2 +� +2(X0 − xX′ +0) + +√ +5 +� +, +( ¯P −1)z(x, y) := − +¯Pz +¯P 2(x, y) = +1 +(x2 + y2)2 +� +(x2 − y2) +� +X0 + +√ +5 +2 +� ++ 2xy2X′ +0 +� ++ +√ +5, +where h(x, y) = 2X0 − xX′ +0 + y2X′ +0/x + +√ +5 (> +√ +5) on R2 (Corollary 2.3 in §2) and that the +functions ¯P −1(x, y), ( ¯P −1)z(x, y) and ¯κ3(x, y) are analytic on the whole R2 with the pole at +the origin. For the class Ψ a singular Riemannian 2-metric g0 on R2, which is our object, is +determined by +g0 := ¯P 2 � +cos2 ϕ(dx)2 + sin2 ϕ(dy)2� += +1 +h2(x, y) +�(x2 − y2)2 +x2 +(dx)2 + 4y2(dy)2 +� +. +(1.7) +Then, S1 in (1.5) coincides with the singular set of g0, and we have g0(x, y) = g0(−x, y) = +g0(x, −y) on R2 \ S1 by the property of h (Corollary 2.3). Actually, a 5-dimensional set of +classes {( ¯P, ¯Pz, ¯κ3)} is determined for Φ, and we have chosen one class Ψ from that set. +Next, we review the relation between the class Ψ and the generic conformally flat hyper- +surfaces fV (x, y, z) mentioned in (R2). For the class Ψ, we define two functions ¯κ1 and ¯κ2 on +R2 \ S1 by +¯κ1(x, y) := +2y +x2 − y2 +� +X0 + +√ +5 +2 +� +, +¯κ2(x, y) := 1 +y +� +x2 + y2 +2 +X′ +0 +x − +� +X0 + +√ +5 +2 +�� +. +Then, ¯κ1¯κ2 ̸= 0 holds on R2 \ S, where S is the set defined by S := S1 ∪ S2 and +S2 := +� +(x, y) +�� (x2 + y2)X′ +0 = (2X0 + +√ +5)x +� +. +For a domain V such that V ⊂ R2 \ S, we have the following facts (R4) and (R5): +3We applied the Cauchy-Kovalevskaya theorem for analytic evolution equations (more precisely, see [1], [18]). +4We shall omit this remark from now on. + +4 +N. MATSUURA AND Y. SUYAMA +(R4) For the pair Φ and the class Ψ, an analytic surface φ(x, y) in S3 is determined on V such +that (x, y) is a principal curvature line coordinates of φ. Let us denote the surface φ(x, y) on +V by the frame field F 0 +V (x, y) := +� +φ, X0 +α, X0 +β, ξ +� +(x, y), where X0 +α and X0 +β are the unit principal +directions corresponding to the coordinates (x, y) and ξ is a unit normal vector field of φ. Then, +an analytic surface f 0 +V (x, y) in R4 with the metric g0 is determined on V as a certain integral +surface of (X0 +α, X0 +β) (see Theorem 1 below). +(R5) There is a generic conformally flat hypersurface fV (x, y, z) in (R2) defined on V × I +such that fV (x, y, z) satisfies the following conditions (1) and (2): +(1) fV (x, y, 0) = f 0 +V (x, y) holds on V . +(2) For fV (x, y, z), the conformal element P 2 of IfV in (1.2) and the principal curvatures κi +satisfy the equations +P(x, y, 0) = ¯P(x, y), +Pz(x, y, 0) = ¯Pz(x, y), +κi(x, y, 0) = ¯κi(x, y). +Let FV (x, y, z) := [N, Xα, Xβ, Xγ] (x, y, z) be the orthonormal frame field determined by fV (x, y, z), +where N(x, y, z) is normal and (Xα, Xβ, Xγ)(x, y, z) are the principal directions corresponding +to the coordinates (x, y, z). Actually, fV (x, y, z) is determined as an evolution of surfaces in z +issuing from the surface f 0 +V (x, y) on z = 0 under the condition FV (x, y, 0) = F 0 +V (x, y) on V , and +then the condition ¯κ1¯κ2 ̸= 0 is necessary. Here, note that the other classes {( ¯P, ¯Pz, ¯κ3)} for Φ +determine the conformal transformations of fV (x, y, z). +Now, by (R4) and (R5), f 0 +V (x, y) is an analytic curvature surface with the metric g0|V , and +f 0 +V (x, y) and F 0 +V (x, y) are determined only by Φ and Ψ. Furthermore, each coordinate line of +f 0 +V (x, y) is a principal curvature line of fV (x, y, z). We emphasize that the curvature surfaces +f 0 +V (x, y) are defined only on each domain V ⊂ R2 \S (or at most on V ⊂ R2 \S1). Our first aim +is to study the existence and the structure of an extended curvature surface f 0(x, y) including +these singularities of g0: for example, let us take an interval (0, 1] × {y} ⊂ R2 for each y, then +the length by the metric g0 of the interval converges to a finite value if y = 0, but it diverges +to ∞ if y ̸= 0; hence, it would be interesting to study the curvature surface as x → ±0. +1.2. The results. For the hyperbolic 2-metric ˇgH, the singular Riemannian metric g0 on R2 +in (1.7) and the curvature (local-)surfaces are determined by Φ and Ψ. Let D := {(x, y) ∈ +R2 | x > 0}. In Sections 3 and 4, we study the singular Riemannian space (D, g0) and an +extended curvature surface only on D. In Section 5, we shall study them on the whole R2. +Now, the function X0(x) in (1.6) determines the properties of the space (D, g0) and the +curvature surfaces. In Section 2 we study the property of X0(x) for later use. Then, we also +review how the class Ψ is determined from Φ. In Section 3, we define a regular coordinates +(ˆx, ˆy) of the space (D, g0). Let ˆx and ˆy be the functions on D defined by ˆx := x3/3 − xy2, +ˆy := y2 respectively, and ι be an analytic map given by ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ R2. The +determinant of the Jacobi matrix for ι is a prime polynomial defining the singular set S1 ∩D of +g0, where S1∩D = {(x, y) | y(x2−y2) = 0}. From the property ι(x, y) = ι(x, −y), we define the +sub-domains Di (i = 1, 2) of D by D1 := {(x, y) ∈ D | y ≥ 0} and D2 := {(x, y) ∈ D | y ≤ 0}. +Then, there is a Riemannian space ( ˆD, ˆg0) without singularity and an analytic map ˆι : D → ˆD +is induced from ι such that ( ˆD, ˆg0) and ˆι satisfy the following conditions (C1), (C2) and (C3): +(C1) For i = 1, 2, the restricted maps ˆι : (Di \ S1, g0) → ( ˆD \ ˆι(S1), ˆg0) are isometric. +(C2) At each point ˆι(x, ±x) = ι(x, ±x) or ˆι(x, 0) = ι(x, 0), the tangent space of ˆD is defined +only as a half plane: any x-curve ˆι(x, y) with fixed y ̸= 0 has a cusp at x = |y| and any y-curve +ˆι(x, y) reflects at y = 0; the curve ˆι(|y|, y) of y is the envelope of the family of y-curves ˆι(x, y) +with x. Let T + +ˆι(x,±x) ˆD and T + +ˆι(x,0) ˆD be those tangent half spaces. +(C3) The metric ˆg0 is well defined even on the tangent spaces T + +ˆι(x,±x) ˆD and T + +ˆι(x,0) ˆD. + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +5 +By the above property, it is natural to consider both sides of the space ( ˆD, ˆg0): we denote by +( ˆD+, ˆg+ +0 ) and ( ˆD−, ˆg− +0 ) the front and the back of ( ˆD, ˆg0), respectively, and by ( ˆD±, ˆg± +0 ) the space +equipped with both sides of ( ˆD, ˆg0). That is, ( ˆD+, ˆg+ +0 ) and ( ˆD−, ˆg− +0 ) are isometric with ( ˆD, ˆg0) +and have only the common points ˆι(x, 0) and the common tangent spaces T + +ˆι(x,0) ˆD. Thus, we +regard the map ˆι : (D1, g0) → ( ˆD, ˆg0) (resp. ˆι : (D2, g0) → ( ˆD, ˆg0)) as ˆι : (D1, g0) → ( ˆD+, ˆg+ +0 ) +(resp. ˆι : (D2, g0) → ( ˆD−, ˆg− +0 )). +Then, we could call ( ˆD±, ˆg± +0 ) (resp. (ˆx, ˆy)) a regular Riemannian space associated to (D, g0) +(resp. a regular coordinate system for (D, g0)). +In Section 4, we firstly verify that Φ and Ψ give rise to an analytic curvature surface f 0(x, y) +on D with the metric g0. Here, we say that f 0(x, y) is a curvature surface on D if, for a domain +V such that V ⊂ D \ S, there is a generic conformally flat hypersurface fV (x, y, z) on V × I +such that fV (x, y, 0) = f 0(x, y) holds on V . +Theorem 1. +(1) The functions Φ and Ψ lead to an analytic orthonormal frame field F 0(x, y) := +� +φ, X0 +α, X0 +β, ξ +� +(x, y) of R4 defined on D such that the structure equation of F 0(x, y) sat- +isfies the integrability condition. +(2) An analytic curvature surface f 0(x, y) on D is determined as an integral surface of +(X0 +α, X0 +β) by df 0 = (xh(x, y))−1((x2 − y2)X0 +α dx + 2xyX0 +β dy). +In Theorem 1, the field F 0(x, y) is uniquely determined up to a transformation AF 0(x, y) by +a constant orthogonal matrix A. The coordinate lines of f 0(x, y) satisfy the following condition. +Theorem 2. The surface f 0(x, y) on D in Theorem 1 satisfies the following conditions (1), +(2), (3) and (4): +(1) The unit analytic vector u(y) := (1 + y2)−1/2 [yX0 +β − φ](x, y) depends only on y. +(2) Any x-curve f 0(x, y) with fixed y lies on a standard 2-sphere S2 +y of radius +(2 +√ +5)−1� +(5 + 4y2)/(1 + y2) in an affine hyperplane perpendicular to u(y). +(3) The unit analytic vector ˜u(x) := (B2 +2 +C2 +2)−1/2(x)(−B2X0 +α +C2ξ)(x, y) depends only on +x, where B2(x) = −(1/2)(X′ +0(x)/x) − +√ +5 and C2(x) = X′′ +0 (x) − X′ +0(x)/x. +(4) Any y-curve f 0(x, y) with fixed x lies on a standard 2-sphere S2 +x of radius (B2 +2+C2 +2)−1/2(x) +in an affine hyperplane perpendicular to ˜u(x). +In Theorem 2, when we regard the surface f 0(x, y) as a one-parameter family of x-curves, +the surface f 0(x, y) is expressed as +f 0(x, y) = (2 +√ +5)−1� +(5 + 4y2)(1 + y2)−1 f(x, y) + A(y) +(1.8) +with the unit analytic vector f(x, y) := ((1 + y2)(5 + 4y2))−1/2(X0 +β − 2(1 + y2)ξ + yφ)(x, y), +where A(y) is a R4-valued analytic function of y determined uniquely up to a parallel translation. +Similarly, when we regard the surface f 0(x, y) as a one-parameter family of y-curves, the surface +f 0(x, y) is expressed as +f 0(x, y) = (B2 +2 + C2 +2)−1/2(x)˜f(x, y) + ˜A(x) +(1.9) +with the unit analytic vector ˜f(x, y) := (B2 +2 + C2 +2)−1/2(x) (B2ξ + C2X0 +α) (x, y), where ˜A(x) is a +R4-valued analytic function of x determined uniquely up to a parallel translation. +Next, in the singular set of g0, the surface f 0(x, y) has the following features: +(F1) Any x-curve f 0(x, y) with fixed y ̸= 0 has a cusp of type (2, 3, 4) at x = |y|. +(F2) Any y-curve f 0(x, y) with fixed x has a cusp of type (2, 3, 4) at y = 0. +(F3) The curves f 0(|y|, y) of y are the envelopes of the family of y-curves f 0(x, y) with x. +(F4) We translate the surface f 0(x, y) such that f 0(p0) = 0 holds at some point p0 := (x, 0), +where 0 is the origin of R4. Then, there is a reflection B of R4 such that (B ◦ f 0)(x, y) = +f 0(x, −y), (B ◦ φ)(x, y) = −φ(x, −y) and (B ◦ ξ)(x, y) = ξ(x, −y) hold. + +6 +N. MATSUURA AND Y. SUYAMA +From these facts, we may recognize that the map ˆι : (D, g0) → ( ˆD±, ˆg± +0 ) is a plane model of +the curvature surface f 0(x, y) in R4, that is, there is an isometric map ¯f 0 : ( ˆD±, ˆg± +0 ) ∋ (ˆx, ˆy) �→ +¯f 0(ˆx, ˆy) ∈ R4 such that f 0(x, y) = ( ¯f 0 ◦ ˆι)(x, y) holds on D. +In Section 5, we study the limits of x-curves f 0(x, y) as x → 0 and y-curves f 0(x, y) as +y → ∞. Then, we can clarify the structure of the surface f 0(x, y) (and also the space (D, g0)). +Here, in the case of x → 0, we replace x with u given by x = e−u and consider the case of +u → ∞. Then, the vectors u(y) and ˜u(x) in Theorem 2 have the following properties (see +Definition 5.2 for the uniform convergence): +Theorem 3. There is an orthonormal frame [b∞, c∞, ˜b∞, ˜c∞] of R4 (consisting of constant +vectors) such that it satisfies the following conditions: +(1) The vector u(y) moves on the unit circle in a plane spanned by ˜b∞ and ˜c∞. Let v1(y) +be a unit vector of y determined by (∇′ +d/dyu)(y) = (y2/(1 + y2))v1(y). +Then, u(y) +(resp. v1(y)) uniformly converges to the circle ˜b(y) := cos y ˜b∞ + sin y ˜c∞ (resp. ˜c(y) := +− sin y ˜b∞ + cos y ˜c∞) as y tends to ∞. +(2) The vector ˜u(x) moves on the unit circle in a plane spanned by b∞ and c∞. Let ˜v1(x) be +a unit vector of x determined by (∇′ +d/du˜u)(e−u) = −T(u)˜v1(e−u), where T(u) > 0. Then, +˜u(e−u) (resp. ˜v1(e−u)) uniformly converges to the circle b(u) := cos +√ +5u +2 b∞ − sin +√ +5u +2 c∞ +(resp. −v2(u) := sin +√ +5u +2 b∞ + cos +√ +5u +2 c∞) as u tends to ∞. +Next, let v0 +2(y) and v0 +3(y) be the functions of y defined by +v0 +2(y) := +1 +h(0, y) +� +5 + 4y2 +�5 +2(5 + +√ +5) + (10 + +√ +5)y2 +� +, +v0 +3(y) := −2 +√ +5y2 +h(0, y) +� +1 + y2 +5 + 4y2. +Then, we have (v0 +2(y))2 + (v0 +3(y))2 = 5/4, v0 +2(y) > 0 and v0 +3(y) < 0 if y ̸= 0, v0 +3(0) = 0. For the +vectors v1(y) and v2(y) in Theorem 3, we define the vector Γ(u, y) by +Γ(u, y) := +1 +2 +√ +5 +� +5 + 4y2 +1 + y2 a(u, y) + A(y), +a(u, y) := 2 +√ +5 +� +−v0 +2(y)v1(y) + v0 +3(y)v2(u) +� +. +Then, for each y ̸= 0, a(u, y) is a circle with parameter u and a(u, 0) degenerates to one point. +Note that −(2/ +√ +5)v0 +2(y)v1(y) is the center of the circle a(u, y) and that v1(y) and v2(u) are +always perpendicular. +Theorem 4. We have the following facts (1)–(4): +(1) Any u-curve f 0(e−u, y) with y uniformly converges to the circle Γ(u, y) as u tends to ∞: +only the u-curve f 0(e−u, 0) converges to the point Γ(u, 0) = −(1/2)v1(0) + A(0). +(2) All u-curve X0 +α(e−u, y) with y uniformly converge to the circle b(u) as u tends to ∞. +(3) Any y-curve f 0(x, y) with x converges to the point (B2 +2 + C2 +2)−1/2(x)˜v1(x) + ˜A(x) as y +tends to ∞. +(4) All y-curve X0 +β(x, y) with x uniformly converge to the circle ˜b(y) as y tends to ∞. +In (1) and (2), the convergences for those u-curves are also uniform with respect to y ∈ R. In +(3) and (4), the convergences for those y-curves f 0(x, y) are also uniform in the wider sense +with respect to x ∈ (0, ∞). +Furthermore, we have some other results: (1) As x → 0 and x → ∞, each x-curve f 0(x, y) +with y ̸= 0 converges uniformly to parallel small circles in S2 +y. (2) limy→∞ f 0(x, y) = limy→−∞ f 0(x, y) +holds for any x ∈ (0, ∞). (3) The curve A(y) in (1.8) (resp. ˜A(x) in (1.9)) lies on a plane H +spanned by ˜b +∞ and ˜c∞ (resp. on a plane ˜H spanned by b∞ and c∞). In consequence, a curvature +surface on D has the following structure. The curves −(v0 +2(y)/5) +� +(5 + 4y2)(1 + y2)−1v1(y) + + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +7 +A(y) and (B2 +2 +C2 +2)−1/2(x)˜v1(x)+ ˜A(x) in Theorem 4 lie on those planes H and ˜H, respectively. +For the spheres S2 +y in Theorem 2, two points ±(2 +√ +5)−1� +(5 + 4y2)(1 + y2)−1v1(y) + A(y) are +the antipodal points of S2 +y to each other: the tangent spaces of S2 +y at these points are spanned +by b∞ and c∞. Similarly, for the sphere S2 +x, two points ±(B2 +2 + C2 +2)−1/2(x)˜v1(x) + ˜A(x) are the +antipodal points of S2 +x to each other: the tangent spaces of S2 +x at these points are spanned by +˜b +∞ and ˜c∞. +We also give the figures of several curves in the curvature surface f 0(x, y) on D explicitly: +some coordinate lines, the cusps and the enveloping curves and so on, by using the approxima- +tion F δn(x, y) for some integer n of the frame field F 0(x, y), which will be defined in Section +6. +Next, a curvature surface ˆf 0(x, y) on D(−) = {(x, y) | x < 0} is also defined from Φ and +Ψ, and we can determine it by ˆf 0(−x, y) := f 0(x, y) for f 0(x, y) on D. We regard the surfaces +ˆf 0(x, y) on D(−) as the back side for f(x, y) on D. Then, two surfaces on D and D(−) connect +at (0, 0) continuously in a sense. Hence, we could recognize that the curvature surface formed +by both sides of f 0(x, y) on D is a natural realization in R4 of the space (D∪{(0, 0)}∪D(−), g0). +In Section 6, we define an approximation F δn(x, y) := [φδn, Xδn +α , Xδn +β , ξδn](x, y) of the frame +field F 0(x, y) = [φ, X0 +α, X0 +β, ξ](x, y) for each positive integer n. We construct F δn(x, y) on a +compact square E ⊂ D, by regarding φ (or F 0(x, y)) as a (singular) surface in the standard 3- +sphere S3. Then, the Gauss and the Codazzi equations for φ are important in the construction. +Let E := [x0, x0+a]×[y0, y0+a], and x0 < x1 < · · · < xn = x0+a and y0 < y1 < · · · < yn = y0+a +be the divisions of [x0, x0 + a] and [y0, y0 + a] of equal length δn = a/n. Then, for an initial +orthogonal matrix at (x0, y0), an orthonormal frame field F δn(x, y), not to depend on the width +δn, is determined on the lattice in E made from the divisions. Furthermore, the approximation +of every coordinate line in f 0(x, y) also lies on a 2-sphere S2. +The approximation will be +constructed by a kind of polygonal line method: on each edge [xi, xi+1]×{yj} or {xi}×[yj, yj+1], +we approximate F 0(x, y) by a rational curve (not by a line). We further define F δn(x, y) at +all points (x, y) ∈ E by a little change of the divisions. Then, F δn(x, y) converges to F 0(x, y) +uniformly on E as n → ∞. By using F δn(x, y), we can draw the curves given in Section 5, +since each coordinate curve is expressed by the frame field F 0(x, y) as in (1.8) and (1.9). +2. Choice of a singular metric g0 determining curvature surfaces +As mentioned in the introduction, for the hyperbolic metric ˇgH = (dx2 + dy2)/y2 on R2, we +have chosen the following pair Φ = (ϕ, ϕz) of functions on R2, +cos ϕ(x, y) = x2 − y2 +x2 + y2, +sin ϕ(x, y) = +2xy +x2 + y2, +ϕz(x, y) = +y +x2 + y2, +(2.1) +and the class Ψ = ( ¯P, ¯Pz, ¯κ3) of three functions on R2, +¯P −1 = +x +x2 + y2h(x, y), +¯κ3 = − +y +x2 + y2 +� +2(X0 − xX′ +0) + +√ +5 +� +, +( ¯P −1)z := − +¯Pz +¯P 2 = +1 +(x2 + y2)2 +� +(x2 − y2) +� +X0 + +√ +5 +2 +� ++ 2xy2X′ +0 +� ++ +√ +5. +(2.2) +Here X0 = X0(x) is the hypergeometric function on R2 given in (1.6) and h(x, y) is the function +defined by +h(x, y) = 2X0 − xX′ +0 + y2(X′ +0/x) + +√ +5. +(2.3) +The class Ψ is selected from the 5-dimensional set of classes {( ¯P, ¯Pz, ¯κ3)} in [18, Example 3.2] +determined by Φ and it leads to a singular metric g0 determining curvature surfaces. In this +section, we firstly explain that the choice of the class Ψ is suitable, and next study the property + +8 +N. MATSUURA AND Y. SUYAMA +of X0(x) for the argument later. +Choice of the class Ψ and the singular metric g0: We define two 1-variable functions X1 = +X1(x) of x and Y = Y (y) of y by the equations +xX′′′ +1 − X′′ +1 + (x + 9/(4x)) X′ +1 − X1 = cx2, +Y ′′ + Y = cy2, +(2.4) +respectively, where c is a constant. For the solutions X1 and Y to the equations (2.4), we define +the functions GX1(x) and HY (y) by +GX1(x) := (X′′ +1 )2 + 4cX1 + +� +1 + 9/(4x2) +� +(X′ +1)2 + ((2/x)X2 − 4cx) X′ +1, +HY (y) := (Y ′ − 2cy)2 + (Y − cy2 + 2c)2 − 4c2, +where X2 := −X′′ +1 − X1 + cx2. Furthermore, for a pair (X1, Y ) of solutions, let ¯X = ¯X(x), +¯Y = ¯Y (y) and A = A(x, y) be the functions defined by +¯X := xX′ +1 − X1, +¯Y := yY ′ − Y, +A := ¯X + ¯Y . +The following proposition are verified in [18, Example 3.2] except for the equations for GX1(x) +and HY (y) in (3) and (4). Hence, we only prove these equations. In the proposition, we say +that a triplet ( ¯P, ¯Pz, ¯κ3) is a class for Φ, if curvature surfaces f 0 +V (x, y) and generic conformally +flat hypersurfaces fV (x, y, z) in R4 are determined by Φ and the class ( ¯P, ¯Pz, ¯κ3). +Proposition 2.1. Let (X1, Y ) be a pair of solutions to the equations (2.4). Then, we have the +following facts (1) and (2): +(1) The functions GX1(x) and HY (y), respectively, are constant. +(2) For any pair (X1, Y ) such that GX1(x) + HY (y) = 0, a class ( ¯P, ¯Pz, ¯κ3) for Φ is deter- +mined as follows: +¯P −1 = X′ +1 − +2x +x2 + y2A, +¯κ3 = −Y ′ + +2y +x2 + y2A, +( ¯P −1)z = − +¯Pz +¯P 2 = +1 +x2 + y2 +� +xX′ +1 − A + +2y2 +x2 + y2A +� +. +Conversely, all classes ( ¯P, ¯Pz, ¯κ3) for Φ are determined by the above forms from the pairs +(X1, Y ) such that GX1(x) + HY (y) = 0. +Furthermore, we have the following facts (3) and (4): +(3) X0(x) in (1.6) is a solution to the first equation in (2.4) with c = 0, and GX0(x) = −5 +holds. +(4) Let us take c = +√ +5 in two equations of (2.4). Then X1(x) = X0(x)+ +√ +5 (x2 + 5/2) and +Y (y) = +√ +5 (y2 − 2), respectively, are solutions to the equations, and the pair satisfies +GX1(x) + HY (y) = 0. In particular, we have GX1(x) = 20. +Proof. Firstly, note that X1(x) = X0(x) + c(x2 + 5/2) and Y (x) = c(y2 − 2) are the solutions of +the first and the second equations of (2.4), respectively. Now, we verify that GX1(x) = −5+5c2 +holds. From +(X′′ +1 (0))2 = 4(1 + c)2, +4cX1(0) = 10c(1 + c), +� +(1 + 9/(4x2))(X′ +1)2� +(0) = 9(1 + c)2, +((2X2/x − 4cx)X′ +1) (0) = +� +(2X2 − 4cx2)(X′ +1/x) +� +(0) = −18(1 + c)2 +by (1.6), we have GX1(0) = −5 + 5c2. Then, GX1(x) = −5 + 5c2 holds for any x, since the +function GX1(x) is constant. In the same way, we have HY (y) = −4c2 for Y (x) = c(y2 − 2). +In consequence, we also have verified that the pair X1(x) = X0(x) + +√ +5(x2 + 5/2) and +Y = +√ +5(y2 − 2) is a solution to (2.4) and satisfies GX1(x) + HY (y) = 0. +□ + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +9 +Let X1(x) and Y (y) be a pair of solutions to (2.4) given in Proposition 2.1-(4). From the +pair, our class Ψ = ( ¯P(x, y), ¯Pz(x, y), ¯κ3(x, y)) of (2.2) is determined by Proposition 2.1-(2), +and then for the class Ψ, the singular metric g0 on R2 and the functions ¯κi (i = 1, 2) mentioned +in §1 are also determined as follows: +g0 := ¯P 2 � +cos2 ϕ(dx)2 + sin2 ϕ(dy)2� += h−2(x, y) +� +x−2(x2 − y2)2(dx)2 + 4y2(dy)2� +, +(2.5) +¯κ1 := ¯P −1 tan ϕ + ¯κ3 = 2y(x2 − y2)−1� +X0 + +√ +5/2 +� +, +¯κ2 := − ¯P −1 cot ϕ + ¯κ3 = y−1� +((x2 + y2)/2)(X′ +0/x) − +� +X0 + +√ +5/2 +�� +. +(2.6) +Property of the function X0(x): Since X0(x) determines the properties of g0 and ¯κi, we study +the property of X0(x). +Proposition 2.2. The function X0(x) on R in (1.6) satisfies the following conditions: +(1) X0(−x) = X0(x) ≥ 5/2. +(2) X′ +0(x)/x > 0 for x ∈ R and (X′ +0(x)/x)(0) = 2. +(3) (2X0 − xX′ +0)(x) > 0 for x ∈ R. +Proof. X0(−x) = X0(x) follows from the definition (1.6) of X0. For X0(x) ≥ 5/2 and (2), +suppose that there is a point x0 > 0 such that X′ +0(x0) = 0. +Then, we have GX0(x0) = +(X′′ +0 )2(x0) = −5, which is a contradiction. Moreover, we have (X′ +0/x)(0) = 2 by (1.6). Hence, +we have X′ +0(x) > 0 for x > 0 and (X′ +0/x)(x) > 0 for x ∈ R. Furthermore, since X′ +0(x) > 0 for +x > 0, X0(x) is an increasing function on [0, ∞) and X0(0) = 5/2. Hence, we have X0(x) ≥ 5/2 +for x ∈ R. For (3), we have +GX0(x) = (X′′ +0 − X′ +0/x)2 + (1 + 5/(4x2))(X′ +0)2 − 2X0(X′ +0/x) = −5. +Hence, we have (X′ +0)2 − 2X0(X′ +0/x) = (X′ +0/x)(xX′ +0 − 2X0) < −5. By X′ +0/x > 0 for x ∈ R, we +have 2X0 − xX′ +0 > 0 for x ∈ R. +□ +Corollary 2.3. For the function h(x, y), we have the following facts. +(1) h(x, y) > +√ +5 on R2. +(2) There is a number t1 (0 < t1 < 1) such that 5 + +√ +5 + y2 < h(x, y) < 6 + +√ +5 + 2y2 and +|h(x, y) − h(0, y)| ≤ x2(1 + 2y2)/7 hold for 0 ≤ x < t1 and any y ∈ R. +Proof. The fact (1) follows from the definition (2.3) of h(x, y) and Proposition 2.2. The fact +(2) follows from the definition (1.6) of X0(x). In fact, as x tends to 0, for any y ∈ R we have +2X0 − xX′ +0 = 5 + (2/21)x4 + O(x6), +X′ +0/x = 2 − (4/21)x2 + O(x4), +|h(x, y) − h(0, y)| ≤ |2X0 − xX′ +0 − 5| + y2|X′ +0/x − 2| < (x2/7)(1 + 2y2), +where l(x) = O(xk) for a function l(x) implies that c1 < limx→0(l(x)/xk) < c2 holds for some +constants ci. +□ +The function h(x, y) also satisfies the following equations: +h(x, y) = h(−x, y) = h(x, −y), +h(0, y) = 5 + +√ +5 + 2y2, +h(x, 0) = (2X0 − xX′ +0) + +√ +5, +h(0, 0) = 5 + +√ +5. +Now, X′ +0(x) is an oscillating function, since X0(x) is a generalized hypergeometric function. +We study whether X′ +0(x) oscillates even at x = ∞. Here, we say that X′ +0(x) oscillates at x = ∞, +if there is a bounded interval J (not one point) satisfying the following condition: for any point +p ∈ J, there is a sequence xn (xn → ∞) such that X′ +0(xn) converges to p. +Proposition 2.4. The function X0(x) on R satisfies the following conditions (1)–(4): + +10 +N. MATSUURA AND Y. SUYAMA +(1) The function τ(x) := (X0 + X′′ +0 )(x)/x is a decreasing function on (0, ∞) and τ(x) +converges to a non-negative constant τ(∞) := limx→∞ τ(x) as x tends to ∞. Then, we +have τ(∞) > +√ +5, which implies that X′ +0(x) oscillates even at x = ∞.5 +(2) We have +τ(∞) − +� +τ 2(∞) − 5 ≤ lim +x→∞ X′ +0(x) ≤ τ(∞) + +� +τ 2(∞) − 5, +where limx→∞ X′ +0(x) implies the oscillation of X′ +0(x) at x = ∞. +(3) For x ≥ 0, we have (X0(x) − τ(x)x)2 = (X′′ +0 (x))2 ≤ τ 2(x) − 5. +(4) We have +τ(∞) − +� +τ 2(∞) − 5 ≤ lim +x→∞((2X0 − xX′ +0)/x) ≤ τ(∞) + +� +τ 2(∞) − 5, +where limx→∞ ((2X0 − xX′ +0)/x) implies the oscillation at x = ∞. +Proof. The function X0(x) is a solution to the first equation in (2.4) with c = 0. Hence, X0(x) +satisfies the following equations: +xX′′′ +0 + (x + 9/(4x)) X′ +0 − (X′′ +0 + X0) = 0, +(2.7) +GX0(x) = (X′′ +0 )2 + (1 + 9/(4x2))(X′ +0)2 − 2τ(x)X′ +0 = −5. +(2.8) +Now, for (1) and (2), we have τ ′(x) = −9/(4x3)X′ +0 by (2.7). Hence, we have τ ′(x) < 0 for +x > 0 by X′(x) > 0 in Proposition 2.2. Next, we have τ(x)X′ +0 > 5/2 by (2.8). Hence, there is +the limit τ(∞) := limx→∞ τ(x) by τ(x) > 0 for x ≥ 0: τ(∞) is non-negative. Next, we have +(X′ +0)2 − 2τX′ +0 + 5 < 0 by (2.8). Hence, we have +τ(x) ≥ +√ +5, +τ(x) − +� +τ 2(x) − 5 < X′ +0(x) < τ(x) + +� +τ 2(x) − 5, +(2.9) +since X′ +0(x) is a real-valued function and τ(x) > 0. Furthermore, τ(x) − +� +τ 2(x) − 5 (resp. +τ(x) + +� +τ 2(x) − 5) is an increasing function (resp. a decreasing function). +Thus, we have +obtained (1) τ(∞) ≥ +√ +5 and (2). We shall verify τ(∞) ̸= +√ +5 after the proofs of (3) and (4). +Here, note that τ(∞) = +√ +5 is equivalent to limx→∞ X′′ +0 (x) = 0 and limx→∞ X′ +0(x) = +√ +5 by the +argument above. Hence, τ(∞) > +√ +5 implies that both functions X′′ +0 (x) and X′ +0(x) oscillate +even at x = ∞. +For (3), by X′′ +0 = τ(x)x − X0 and (2.8), we have +(X0 − τ(x)x)2 = (X′′ +0 )2 = −(1 + 9/(4x2))(X′ +0)2 + 2τ(x)X′ +0 − 5 +< −(X′ +0)2 + 2τ(x)X′ +0 − 5 = −(X′ +0 − τ(x))2 + τ 2(x) − 5 ≤ τ 2(x) − 5. +For (4), we have limx→∞ ((2X0 − xX′ +0)/x) = limx→∞(2τ(x)−X′ +0(x)) by (3). Hence, we obtain +(4) by the existence of τ(∞) and (2). +Finally, we show τ(∞) ̸= +√ +5. Firstly, we have +� � +1 + +9 +4x2 +�−1/2 τ(x) +�′ = +9 +4x3 +� +1 + +9 +4x2 +�−3/2 � +τ(x) − +� +1 + +9 +4x2 +� +X′ +0 +� += +9 +4x3 +� +1 + +9 +4x2 +�−3/2 X′′′ +0 +by (2.7) and τ ′(x) = −9/(4x3)X′ +0. The equation shows that (1 + 9/(4x2))−1/2 τ(x) is almost +equal to τ(∞) for x ≥ 10, because X′′′ +0 (x) is an oscillating function taking small values around +0 and limx→∞ +� +(1 + 9/(4x2))−1/2τ(x) +� += τ(∞) holds. We shall precisely verify it below. +We integrate the equation on the interval [x, C], where C > x > 0. As C → ∞, we have +���� +� ∞ +x +�� +1 + +9 +4x2 +�−1/2τ +�′dx +���� = +���τ(∞) − +� +1 + +9 +4x2 +�−1/2τ(x) +��� +(2.10) +5Actually, we can show τ(∞) = +√ +5 coth( +√ +5π/4) ≈ +√ +5 × 1.0614 (> +√ +5), by using the asymptotic expansion +formula for large x of generalized hypergeometric functions of type 1F2 (cf. [12], [13]). + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +11 +for any x ∈ (0, ∞), since τ(C) converges to τ(∞). And we have +� C +x +9 +4t3 +� +1 + +9 +4t2 +�−3/2 X′′′ +0 (t)dt = +� +9 +4t3 +� +1 + +9 +4t2 +�−3/2 X′′ +0 (t) +�C +x − +� C +x +� 9 +4t3 +� +1 + +9 +4t2 +�−3/2 �′X′′ +0 (t)dt. +In the equation, we have +� 9 +4t3 +� +1 + +9 +4t2 +�−3/2�′ = − 27 +4t4 +� +1 + +9 +4t2 +�−5/2, +(X′′ +0 (t))2 ≤ 4τ 2(t), +where the second inequality follows from (2.8), (2.9) and the fact that τ(x) is a decreasing +function. Hence, by +����− +� C +x +� 9 +4t3 +� +1 + +9 +4t2 +�−3/2�′X′′ +0 (t)dt +���� ≤ 27 +2 +���� +� C +x +τ(t) +t4 dt +���� ≤ +9 +2x3τ(x), +we have +���� +� ∞ +x +9 +4t3 +� +1 + +9 +4t2 +�−3/2X′′′ +0 (t)dt +���� ≤ +9 +2x3τ(x) + 9 +2x3τ(x) = 9 +x3τ(x). +(2.11) +By (2.10) and (2.11), we obtain the inequality +��τ(∞) − (1 + 9/(4x2))−1/2τ(x) +�� ≤ (9/x3)τ(x) +for any x ∈ (0, ∞). That is, for any x ∈ (0, ∞), the following inequalities are satisfied +� +(1 + 9/(4x2))−1/2 − 9/x3� +τ(x) ≤ τ(∞) +� +(1 + 9/(4x2))−1/2 + 9/x3� +τ(x). +Now, if there is an x0 ∈ (0, ∞) such that +� +(1 + 9/(4x2 +0))−1/2 − (9/x3 +0) +� +τ(x0) > +√ +5, then we +have τ(∞) > +√ +5. Actually, for x0 = 10, the desired inequality holds. We can make sure the +fact as follows. The function τ(x) is expressed as the following alternating power series: +τ(x) = (9/2)x−1 + (9/4) �∞ +k=1 bkx2k−1, +bk = ak/(4k2 + 5/4), +where ak is the coefficients of X0(x) given in (1.6). At x = 10, the sequence |bk|x2k−1 is strongly +decreasing for k ≥ 4, and |(9/4)bk102k−1| ≤ (3/2) · 10−12 holds for k = 20. Hence, we have +|τ(x) − +� +(9/2)x−1 + (9/4) �20 +k=1 bkx2k−1� +| ≤ (3/2) · 10−12 at x = 10. Furthermore, at x = 10 we +have the inequality +� +(1 + 9/(4x2))−1/2 − 9/x3� � +(9/2)x−1 + (9/4) �20 +k=1 bkx2k−1� +≥ 2.35 (> +√ +5). +In consequence, the proof of the proposition has been completed. +□ +Proposition 2.4-(4) implies that the first term of h(x, y) = 2X0 − xX′ +0 + y2(X′ +0/x) + +√ +5 +satisfies 2X0 −xX′ +0 = O(x) as x tends to ∞, and that (2X0 −xX′ +0)/x oscillates even at x = ∞. +In the next section, we study the property of the singular metric g0 in (2.5). The singular set +of g0 is given by S1 = {(x, y) ∈ R2 | xy(x2 − y2) = 0}: g0 diverges along the line x = 0 except +for the origin (0, 0) and degenerates along the lines y = 0 and x2 − y2 = 0; limx→±0 g0(x, 0) +totally degenerates. +3. Regular coordinates of the singular metric g0 on D +Let D := {(x, y) ∈ R2 | x > 0}. In this section, we define a regular coordinate system of the +singular Riemannian metric space (D, g0) and study the property of the metric g0|D by using +the coordinate system. In the next section, we define a curvature surface on D from Φ and Ψ, +and then the curvature surface will be recognized as a regularization in R4 of the space (D, g0), +by the results in this section. For the metric g0 on D(−) := {(x, y) ∈ R2 | x < 0}, we shall +study it in Section 5. + +12 +N. MATSUURA AND Y. SUYAMA +Now, let g0 be the metric on D given by +g0 = +1 +h2(x, y) +�(x2 − y2)2 +x2 +(dx)2 + 4y2(dy)2 +� +and D = {(x, y) | x > 0}, +(3.1) +where g0 (resp. h(x, y)) has been defined in (2.5) from Φ(x, y) and ¯P(x, y) (resp. defined in +(2.3) from X0(x)): h(x, y) > +√ +5 holds on R2 by Corollary 2.3. The metric g0 degenerates on +the set S1 ∩ D = {(x, y) | y(x2 − y2) = 0}, where S1 = {(x, y) | xy(x2 − y2) = 0}. The aim in +this section is to verify that g0 leads to a positive-definite metric on some domain ˆD. +We firstly divide D into several domains according to the singularity of g0: +D1 := {(x, y) ∈ D | y ≥ 0}, +D2 := {(x, y) ∈ D | y ≤ 0}, +Di1 := {(x, y) ∈ Di | x2 ≤ y2}, +Di2 := {(x, y) ∈ Di | x2 ≥ y2}. +(3.2) +Then, the Riemannian spaces (D1, g0|D1) and (D2, g0|D2) are isometric by g0(x, y) = g0(x, −y). +Let e1(x, y) and e2(x, y) be the orthonormal vector fields on D \ S1 defined by +e1(x, y) := +1 +¯P cos ϕ +∂ +∂x = xh(x, y) +x2 − y2 +∂ +∂x, +e2(x, y) := +1 +¯P sin ϕ +∂ +∂y = h(x, y) +2y +∂ +∂y, +(3.3) +respectively: e1(x, 0) and e2(|y|, y) are also determined by (3.3). Here, we adopt x2 − y2 not +|x2 − y2| (resp. y not |y|) in the definition of e1(x, y) (resp. e2(x, y)): the functions ¯P cos ϕ +and ¯P sin ϕ are analytic. We extend the orthonormal frame field (e1, e2)(x, y) on D \ S1 to the +singular set of g0 as follows: +e1(|y|−, y) := +lim +Ker(Di1)∋(x,˜y)→(|y|,y) e1(x, ˜y), +e1(|y|+, y) := +lim +Ker(Di2)∋(x,˜y)→(|y|,y) e1(x, ˜y), +(3.4) +e2(x, +0) := +lim +Ker(D12)∋(˜x,y)→(x,0) e2(˜x, y), +e2(x, −0) := +lim +Ker(D22)∋(˜x,y)→(x,0) e2(˜x, y), +(3.5) +where, for Dij ⊂ R2, Ker(Dij) is the open kernel of Dij in R2. These vectors e1(|y|±, y) and +e2(x, ±0) are well defined: for example, e1(|y|+, y) is a unit vector at (|y|, y) with the same +direction as ∂/∂x and g0(e1(|y|+, y), e2(|y|, y)) = 0 holds, because e1(|y|+, y) is the limit of +unit vector e1(x, y) on Di2 \ S1. Then, we have e1(|y|−, y) = −e1(|y|+, y) on x = |y| and +e2(x, +0) = −e1(x, −0) on y = 0. In Lemma 3.3 below, we shall obtain an explicit expression +of these vectors. Now, on each curve (|y|, y) ⊂ Dij (i, j = 1, 2), we determine e1(|y|, y) as +follows: +e1(|y|, y) := e1(|y|−, y) if (|y|, y) ∈ Di1 and e1(|y|, y) := e1(|y|+, y) if (|y|, y) ∈ Di2. +Similarly, on each curve {y = 0} ⊂ Di2 (i = 1, 2), we determine e2(x, 0) as follows: +e2(x, 0) := e2(x, +0) if (x, 0) ∈ D12 and e2(x, 0) := e2(x, −0) if (x, 0) ∈ D22. +Then, (e1, e2)(x, y) is an orthonormal frame field defined on each domain Dij. +The following lemma follows directly from the sign of (x2 − y2) or y. +Lemma 3.1. The frame field (e1, e2)(x, y) on each Dij satisfies the following facts (1) and (2): +(1) If (x, y) ∈ Di1 (i = 1, 2), then the two vectors e1(x, y) and (∂/∂x)(x, y) have the inverse +directions to each other. If (x, y) ∈ Di2 (i = 1, 2), then the two vectors e1(x, y) and +(∂/∂x)(x, y) have the same direction. +(2) If (x, y) ∈ D1j (j = 1, 2), then the two vectors e2(x, y) and (∂/∂y)(x, y) have the same +direction. If (x, y) ∈ D2j (j = 1, 2), then the two vectors e2(x, y) and (∂/∂y)(x, y) have +the inverse directions to each other. + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +13 +Next, for the functions ˆx := (1/3)x3 − xy2 and ˆy := y2, we define an analytic map ι : D ∋ +(x, y) �→ (ˆx, ˆy) ∈ R2. The Jacobi matrix of the map ι is given by +J(x, y) = +� +ˆxx +ˆxy +ˆyx +ˆyy +� += +� +x2 − y2 +−2xy +0 +2y +� +, +det J(x, y) = 2y(x2 − y2). +(3.6) +The singular set S1 ∩ D of the metric g0 coincides with the vanishing points of det J(x, y), and +hence det J(x, y) is an irreducible polynomial determining the singularities of g0|D. +Now, the following lemma is obtained by direct calculation. +Lemma 3.2. The function ˆx = (1/3)x3 − xy2 (x > 0) satisfies the following facts (1) and (2): +(1) For a fixed y ̸= 0, ˆx has only one critical point x0 = |y|, and then ˆx decreases (resp. +increases) on the interval (0, x0] (resp. [x0, ∞)). In particular, we have limx↘0 ˆx = 0 +and limx→∞ ˆx = ∞. +(2) For y = 0, ˆx is an increasing function on (0, ∞). In particular, we have limx↘0 ˜x = 0 +and limx→∞ ˜x = ∞. +We have ι(|y|, y) = (−(2/3)(ˆy)3/2, ˆy), ˆy(ι(x, 0)) = 0, ι(x, y) = ι(x, −y) and g0(x, y) = +g0(x, −y). Let η(ˆy) := (−(2/3)(ˆy)3/2, ˆy) = ι(|y|, y), and ˆDi (i = 1, 2) be the domains defined by +ˆD1 := {(ˆx, ˆy) ∈ R2 | ˆy ≥ 0, −(2/3)(ˆy)3/2 ≤ ˆx < 0}, +ˆD2 := {(ˆx, ˆy) ∈ R2 \ {(0, 0)} | ˆy ≥ 0, −(2/3)(ˆy)3/2 ≤ ˆx < ∞}. +x +y +y = x +D1 +x +y +y = x +D11 +ι11 +ˆx +ˆy +η +�D1 +x +y +y = x +D12 +ι12 +ˆx +ˆy +η +�D2 +1 +Figure 1 +We have the following facts (1) and (2) by the definitions of ˆDi. +(1) At the points η(ˆy) ∈ ˆDi (i = 1, 2), only the tangent half planes T + +η(ˆy) ˆDi are deter- +mined, and similarly, at the points (ˆx, 0) ∈ ˆD2, only the tangent half planes T + +(ˆx,0)) ˆD2 are +determined: precisely, let v = c1∂/∂ˆx + c2∂/∂ˆy be a tangent vector at η(ˆy) in R2 +(ˆx,ˆy) with +∂/∂ˆx := (∂/∂ˆx)(η(ˆy)) and ∂/∂ˆy := (∂/∂ˆy)(η(ˆy)) (or a tangent vector at (ˆx, 0) in R2 +(ˆx,ˆy) with +∂/∂ˆx := (∂/∂ˆx)(ˆx, 0) and ∂/∂ˆy := (∂/∂ˆy)(ˆx, 0)), then we have +T + +η(ˆy) ˆDi := {v | ⟨v, n⟩ ≥ 0}, +T + +(ˆx,0) ˆD2 := {v | c2 ≥ 0}, +where n and ⟨v, n⟩ are the inward unit normal vector at η(ˆy) for the domain ˆDi and the +Euclidean inner product, respectively. +(2) For the maps ιi2 := ι|Di2 : Di2 → ˆD2 (i = 1, 2), we have dι12(e2(x, +0)) = dι22(e2(x, −0)) +by ι(x, y) = ι(x, −y). + +14 +N. MATSUURA AND Y. SUYAMA +Now, although ˆD1 ⊂ ˆD2 as the subsets in R2, we have to recognize that ˆD1 and ˆD2 are +the distinct domains without intersection. In fact, for i = 1, 2, the map ιi1 := ι|Di1 : Di1 ∋ +(x, y) �→ (ˆx, ˆy) ∈ ˆD1 (resp. ιi2 : Di2 ∋ (x, y) �→ (ˆx, ˆy) ∈ ˆD2) gives a new coordinate system +for the Riemannian space (Di1, g0) (resp. (Di2, g0)). For the metrics ˆg1 on ˆD1 and ˆg2 on ˆD2 +determined by g0|Dij = (ιij)∗ˆgj (j = 1, 2), the spaces ( ˆD1, ˆg1) and ( ˆD2, ˆg2) are not isometric +even on the common domain ˆD1, since (Di1, g0|Di1) and (Di2, g0|Di2) are not isometric. Then, +with ˆx = (1/3)x3 − xˆy, the metrics ˆgj (j = 1, 2) are explicitly given by +ˆgj := +� +xh(x, (ˆy)1/2) +�−2 � +(dˆx)2 + 2x(dˆx)(dˆy) + 2x2(dˆy)2� +(3.7) +on ˆDj. The metrics ˆgj (j = 1, 2) are positive-definite on ˆDj by xh(x, (ˆy)1/2) > 0: they are +defined even on the tangent half spaces T + +η(ˆy) ˆDj (j = 1, 2). The domain ˆD2 does not extend to +ˆD2 ∪ {(0, 0)}, since the metric ˆg2(ˆι(x, y)) diverges as (x, y) → (0, 0). +By considering the Riemannian spaces ( ˆDi, ˆgi), we have verified that ˆD1 and ˆD2 are the +distinct domains without intersection. However, in the following Lemmata 3.3 and 3.4, we +shall return to the original definition of the map ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ ˆD1 ∪ ˆD2 ⊂ R2. +Lemma 3.3. Let ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ R2 be the analytic map given as above. Then, we +have the following facts (1), (2) and (3): +(1) The Riemannian metrics ˆg1 and ˆg2 in (3.7) are positive-definite on ˆD1 and ˆD2, respec- +tively. In particular, at each point η(ˆy), the metrics ˆg1 and ˆg2 are also well defined on +the tangent space T + +η(ˆy) := T + +η(ˆy) ˆD1 = T + +η(ˆy) ˆD2 ⊂ Tη(ˆy)R2 and ˆg1|T + +η(ˆy) = ˆg2|T + +η(ˆy) holds; +similarly, at each points (ˆx, 0), ˆg2 is also well defined on T + +(ˆx,0) := T + +(ˆx,0) ˆD2 ⊂ T(ˆx,0)R2. +(2) We have |y|h(|y|, y) · (∂/∂ˆx)(η(ˆy)) = dι(e1(|y|−, y)) = dι(e1(|y|+, y)) at each point η(ˆy). +That is, e1(|y|−, y) and e1(|y|+, y) are identified with the vector of T + +η(ˆy) through the map +ι. +(3) We have h(x, 0) · (∂/∂ˆy)(ˆx, 0) = dι (e1(x, 0) + e2(x, +0)) = dι (e1(x, 0) + e2(x, −0)) at +each point (ˆx, 0). That is, e1(x, 0) + e2(x, +0) and e1(x, 0) + e2(x, −0) are identified +with the vector of T + +(ˆx,0) through the map ι. +Proof. The fact (1) follows from (3.7) directly. For (2), by ˆx = (1/3)x3 − xy2 and ˆy = y2, we +have dˆx = (x2 − y2)dx − 2xydy, dˆy = 2ydy on D, and hence we have +∂ +∂ˆx = +1 +x2 − y2 +∂ +∂x, +∂ +∂ˆy = +x +x2 − y2 +∂ +∂x + 1 +2y +∂ +∂y +(3.8) +on D \ S1. Then, since the map ι|Dij = ιij is given by ιij : Dij ∋ (x, y) �→ (ˆx, ˆy) ∈ ˆDj and (ˆx, ˆy) +is a coordinate system of ˆDj, the equations in (3.8) are equivalent to the equations +∂ +∂ˆx = dι +� +1 +x2 − y2 +∂ +∂x +� +, +∂ +∂ˆy = dι +� +x +x2 − y2 +∂ +∂x + 1 +2y +∂ +∂y +� +(3.9) +on ι(D \ S1), and then the first equation also holds at ι(x, 0). Now, the equations in (3.9) are +also satisfied even at each point η(ˆy): these right hand sides at η(ˆy) are determined as the dual +base of (dˆx, dˆy)(η(ˆy)) as follows, +dι +� +1 +x2−y2 +∂ +∂x +� +(ι(|y|, y)) = +lim +Ker(Dij)∋(x,˜y)→(|y|,y) dι +� +1 +x2−y2 +∂ +∂x +� +(ι(x, ˜y)), +dι +� +x +x2−y2 +∂ +∂x + 1 +2y +∂ +∂y +� +(ι(|y|, y)) = +lim +Ker(Dij)∋(x,˜y)→(|y|,y) dι +� +x +x2−y2 +∂ +∂x + 1 +2y +∂ +∂y +� +(ι(x, ˜y)), +(3.10) + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +15 +respectively. For example, we have +lim +Ker(Dij)∋(x,˜y)→(|y|,y) +� +(ι∗dˆx) +� +(x2 − y2)−1∂/∂x +�� +(x, ˜y) += +lim +Ker(Dij)∋(x,˜y)→(|y|,y) +�� +(x2 − y2)dx − 2xydy +� � +(x2 − y2)−1∂/∂x +�� +(x, ˜y) = 1, +lim +Ker(Dij)∋(x,˜y)→(|y|,y) +� +(ι∗dˆx) +� +x(x2 − y2)−1∂/∂x + (2y)−1∂/∂y +�� +(ι(x, ˜y)) += +lim +Ker(Dij)∋(x,˜y)→(|y|,y) +�� +(x2 − y2)dx − 2xydy +� � +x(x2 − y2)−1∂/∂x + (2y)−1∂/∂y +�� +(x, ˜y) = 0. +The other equations are also verified in the same way. +Next, by (3.3), (3.4) and (3.9), the equations (3.10) are equivalent to the following equations, +|y|h(|y|, y) · (∂/∂ˆx)(ι(|y|, y)) = dι(e1(|y|−, y)) = dι(e1(|y|+, y)), +h(|y|, y) · (∂/∂ˆy)(ι(|y|, y)) = dι (e1(|y|−, y) + e2(|y|, y)) = dι (e1(|y|+, y) + e2(|y|, y)) . +(3.11) +In particular, we have obtained the fact (2) by (3.11). +The fact (3) is obtained in the same way as in (2), by (3.3), (3.5) and (3.9). In consequence, +the proof has been completed. +□ +The spaces ( ˆD1, ˆg1) and ( ˆD2, ˆg2) are not isomeric. However, in the two spaces ( ˆDi, ˆgi) (i = +1, 2), the curves η(ˆy) ⊂ ˆDi, the tangent half spaces T + +η(ˆy) ˆDi and the metrics ˆgi on T + +η(ˆy) ˆDi +have been identified. Furthermore, the orthonormal frame fields (dι(e1(x, y)), dι(e2(x, y))) on +ˆDi (i = 1, 2) are uniquely determined from (e1, e2) on each Dij in Lemma 3.1 as +xh(x, y) · ∂ +∂ˆx(ι(x, y)) = dι(e1(x, y)), +h(x, y) · ∂ +∂ˆy(ι(x, y)) = dι(e1(x, y) + e2(x, y)) +(3.12) +by (3.9) and Lemma 3.3: in particular, these equations also hold on ι(S1 ∩ D). +Lemma 3.4. Let ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ R2 be the map given as above. Then, the curve η(ˆy) +is the envelope of the family of y-curves ι(x, y) with x. +Proof. In this proof, we can assume y ≥ 0 by ι(x, y) = ι(x, −y). Now, any y-curve ι(x, y) is +a half line ˆx + xˆy = (1/3)x3 in the domain ˆy ≥ 0 of R2 +(ˆx,ˆy). The tangent vector of the curve +µ(ˆy) = (−(2/3)(ˆy)3/2, ˆy) is given by +dµ/dˆy = −(ˆy)1/2∂/∂ˆx + ∂/∂ˆy = −x ∂/∂ˆx + ∂/∂ˆy +from (ˆy)1/2 = x on η(ˆy) = ι(y, y) = ι(x, x). Hence, we have +h(y, y)dµ/dˆy = −dι(e1) + dι(e1 + e2) = dι(e2) = (h(y, y)/(2y))dι (∂/∂y) +by (3.3) and (3.12). The equation shows that any y-curve ι(x, y) is tangent to the curve η(ˆy) +at ι(y, y). +□ +In Lemma 3.4, each y-curve ι(x, y) with x is included in ˆD1 if |y| ≥ x and included in ˆD2 if +−x ≤ y ≤ x. Now, let ˆD1 and ˆD2 be the distinct domains in R2 without intersection. On the +direct sum ˆD1 ⊕ ˆD2, we define the Riemannian metric ˆg0 := ˆg1 ⊕ ˆg2 by +ˆg0(X, Y ) := ˆg1(X, Y ) for X, Y ∈ Tp ˆD1, +ˆg0(X, Y ) := ˆg2(X, Y ) for X, Y ∈ Tp ˆD2. +Then, by Lemma 3.3, we can define the domain ˆD := ˆD1 ⊕ ˆD2/ ∼, where ∼ implies two +identifications: the first one is the identification of two curves η(ˆy) in both domains ˆD1 and ˆD2, +and the second one is the identification with T + +η(ˆy) ˆD1 and T + +η(ˆy) ˆD2 at each point of η(ˆy). Thus, +we have obtained the Riemannian space ( ˆD, ˆg0) without singularity, that is, the space ( ˆD, ˆg0) + +16 +N. MATSUURA AND Y. SUYAMA +is regular. Furthermore, an analytic map ˆι : D → ˆD = ˆD1 ⊕ ˆD2/ ∼ is defined from ι : D → R2 +by +ˆι : Di1 ∋ (x, y) → ι(x, y) ∈ ˆD1, +ˆι : Di2 ∋ (x, y) → ι(x, y) ∈ ˆD2, +which satisfies g0 = ˆι∗ˆg0. +We summarize the above results as the following theorem, where D = D1 ∪ D2 in (3.2): +Theorem 3.5. Let (D, g0) and ( ˆD, ˆg0) be the Riemannian spaces and ˆι : D → ˆD be the analytic +map, given as above. Let e1(x, y) and e2(x, y) be the orthonormal frame field on each Dij in +Lemma 3.1. Then, they are satisfied the following facts (1), (2), (3) and (4). +(1) The space ( ˆD, ˆg0) is an isometric regularization of the two singular spaces (Di, g0|Di) (i = +1, 2). That is, ( ˆD, ˆg0) is a regular Riemannian space and g0 = (ˆι|Di)∗ˆg0 holds. +(2) The vector field dˆι(e1(x, y)) is determined on ˆD as xh(x, y) · ∂/∂ˆx(ˆι(x, y)), and in par- +ticular, dˆι(e1(|y|+, y)) = dˆι(e1(|y|−, y)) holds. Any x-curve ˆι(x, y) with y ̸= 0 has a +cusp at the point ˆι(|y|, y). +(3) The vector field dˆι(e2(x, y)) is determined on ˆD as h(x, y) · (∂/∂ˆy − x∂/∂ˆx)(ˆι(x, y)), +and in particular, dˆι(e2(x, +0)) = dˆι(e2(x, −0)) holds. Any y-curve ˆι(x, y) reflects at +the point ˆι(x, 0). +(4) The curve ˆι(|y|, y) = η(ˆy) is the envelope of the family of y-curves ˆι(x, y) with x. +Proof. Almost all facts have been verified in the argument above. Here, we only prove the fact +in (2) that any x-curve ˆι(x, y) with y ̸= 0 has a cusp at ˆι(|y|, y). Then, we assume y > 0 by +ˆι(x, y) = ˆι(x, −y). Now, the vector field dˆι(e1) has the same direction as ∂/∂ˆx at any point in +ˆD by h(x, y) > 0, and in particular, the fact holds even on η(ˆy). Hence, dˆι(e1) is a standard +vector on ˆD to know the direction of the other vector. Next, for each x-curve ˆι(x, y) with y, +two tangent vector fields dˆι(e1) and dˆι(∂/∂x) of the curve has the inverse directions on the +interval (0, y) to each other but the same direction on the interval (y, ∞), by Lemma 3.1. The +fact means that the x-curve reverses the direction at ˆι(y, y) as the curve is passing though the +point. Furthermore, we have ˆι(x, y) ⊂ ˆD1 for x ∈ (0, y] and ˆι(x, y) ⊂ ˆD2 for x ∈ [y, ∞), and +( ˆD1, ˆg0| ˆD1) and ( ˆD2, ˆg0| ˆD2) are not isometric. By these facts, we can recognize the point ˆι(y, y) +as a cusp (not a reflection) of the x-curve. +□ +By (1) and (3) of Theorem 3.5, it is natural to consider both sides of the space ( ˆD, ˆg0), which +are the front ( ˆD+, ˆg+ +0 ) and the back ( ˆD−, ˆg− +0 ): the spaces ( ˆD+, ˆg+ +0 ) and ( ˆD−, ˆg− +0 ) are isometric +with ( ˆD, ˆg0), respectively, and ( ˆD+, ˆg+ +0 ) and ( ˆD−, ˆg− +0 ) have only the common points (ˆx, 0) and +tangent spaces T(ˆx,0) ˆD. Let ( ˆD±, ˆg± +0 ) be the space equipped with both sides of ( ˆD, ˆg0). Then, +we can regard the analytic map ˆι as a bijection ˆι : (D, g0) → ( ˆD±, ˆg± +0 ) defined by +ˆι : D1 ∋ (x, y) → ˆι(x, y) ∈ ˆD+, +ˆι : D2 ∋ (x, y) → ˆι(x, y) ∈ ˆD−. +(3.13) +From Theorem 3.5 and the above argument, we could say that ( ˆD±, ˆg± +0 ) is a regular Riemannian +space associated with the singular Riemannian space (D, g0) and the coordinate system (ˆx, ˆy) +of ( ˆD, ˆg0) is a regular coordinate system of (D, g0). +In the next section, we construct a curvature surface f 0(x, y) defined on D as a realization in +R4 of the space (D, g0), by replacing ˆι(x, y) with f 0(x, y). Then, the vector fields df 0(e1) and +df 0(e2) give an orthonormal frame field on the surface f 0(x, y) in place of dˆι(e1) and dˆι(e2) +and the map f 0 satisfies a similar property to the map ˆι. +4. Extended frame field and analytic curvature surface defined on D +For the hyperbolic metric ˇgH on R2, a pair Φ = (ϕ, ϕz) in (2.1) and a class Ψ = ( ¯P, ¯Pz, ¯κ3) +in (2.2) are determined, and these functions give rise to a singular metric g0 in (2.5) on D = + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +17 +{(x, y) | x > 0}. In this section, we verify that an orthonormal frame field of R4 is determined +on the whole space D from Φ and Ψ and that the frame field leads to an analytic curvature +surface in R4 defined on D, which is a realization in R4 of the space (D, g0). +Now, as mentioned in (R4) and (R5) of the introduction, for a simply connected open set V +such that V ⊂ D \ S, a curvature surface f 0 +V (x, y) on V and a generic conformally flat hyper- +surface fV (x, y, z) on V ×I are determined from Φ and Ψ such that f 0 +V (x, y) = fV (x, y, 0) holds +on V . Then, the coordinates (x, y, z) of fV (x, y, z) and (x, y) of f 0 +V (x, y) are principal curvature +line coordinate systems. Let FV (x, y, z) = [N, Xα, Xβ, Xγ] (x, y, z) be the orthonormal frame +field on fV (x, y, z), where N is a normal vector field of fV (x, y, z) and (Xα, Xβ, Xγ)(x, y, z) +are the principal curvature directions corresponding to x, y, and z-lines: let ϕ(x, y, z) be the +function determined from Φ by ϕ(x, y, 0) = ϕ(x, y) and ϕz(x, y, 0) = ϕz(x, y), and P(x, y, z) +be the function determined from Ψ by P(x, y, 0) = ¯P(x, y) and Pz(x, y, 0) = ¯Pz(x, y), then the +frame field (Xα, Xβ, Xγ)(x, y, z) is given from (1.2) by +Xα := (P cos ϕ)−1∂fV /∂x, +Xβ := (P sin ϕ)−1∂fV /∂y, +Xγ := P −1∂fV /∂z. +Hence, the differential dfV (x, y, z) of fV (x, y, z) is expressed as +dfV (x, y, z) = P(x, y, z) ((cos ϕ dx)Xα + (sin ϕ dy)Xβ + dzXγ) (x, y, z), +For the frame FV (x, y, z), we put φ(x, y) := N(x, y, 0), X0 +α(x, y) := Xα(x, y, 0), X0 +β(x, y) := +Xβ(x, y, 0) and ξ(x, y) := Xγ(x, y, 0). Then, we have +X0 +α := ( ¯P cos ϕ)−1∂f 0 +V /∂x = e1(f 0 +V ), +X0 +β := ( ¯P sin ϕ)−1∂f 0 +V /∂y = e2(f 0 +V ) +(4.1) +with the vector fields e1(x, y) and e2(x, y) in (3.3), and then F 0 +V (x, y) := +� +φ, X0 +α, X0 +β, ξ +� +(x, y) +is an orthonormal frame field on f 0 +V (x, y). The vector fields X0 +α(x, y) and X0 +β(x, y) are the +curvature directions on f 0 +V (x, y): if we regard φ(x, y) as a surface in S3, then X0 +α and X0 +β are the +curvature directions and ξ is a normal vector field of φ(x, y). Thus, the differential df 0 +V (x, y) of +f 0 +V (x, y) is determined as +df 0 +V (x, y) = ¯P(x, y) +� +cos ϕ dxX0 +α + sin ϕ dyX0 +β +� +(x, y) += h−1(x, y) +� +((x2 − y2)/x)dxX0 +α + 2ydyX0 +β +� +(x, y), +(4.2) +and in particular f 0 +V (x, y) has the metric g0|V . For the functions ¯κi(x, y) (i = 1, 2) in (2.6), the +principal curvatures κi(x, y, z) (i = 1, 2, 3) of fV (x, y, z) satisfy κi(x, y, 0) = ¯κi(x, y) (i = 1, 2, 3) +on V . The structure equation of fV (x, y, z) (i.e., the equation for dF(x, y, z)) is determined +from (ϕ(x, y, z), P(x, y, z), κi(x, y, z)), and then the structure equation of f 0(x, y) (i.e, the equa- +tion for dF 0 +V (x, y)) is determined from Φ and Ψ, as the restriction to FV (x, y, 0) of those for +FV (x, y, z) (see the proof of Lemma 4.1 below). +Let Dij (i, j = 1, 2) be the sub-domains in D defined in (3.2). For each Dij, we arbitrarily +fix a simply connected open set Vij such that Vij ⊂ Dij \ S. For the canonical connection ∇′ of +R4, the structure equation of f 0 +Vij(x, y) is determined on each domain Vij as the following form: +∇′ +X0αX0 +α = ¯κ1φ − B1ξ − C1X0 +β, +∇′ +X0αφ = −¯κ1X0 +α, +∇′ +X0αξ = B1X0 +α, +∇′ +X0αX0 +β = C1X0 +α, +(4.3) +and +∇′ +X0 +βX0 +β = ¯κ2φ − B2ξ − C2X0 +α, +∇′ +X0 +βφ = −¯κ2X0 +β, +∇′ +X0 +βξ = B2X0 +β, +∇′ +X0 +βX0 +α = C2X0 +β, +(4.4) +where Bk and Ck are functions on Vij and ¯κi (i = 1, 2) are the functions given in (2.6): +¯κ1 = (2y/(x2 − y2)) +� +X0 + +√ +5 +2 +� +, +¯κ2 = y−1 � +((x2 + y2)/2)(X′ +0/x) − +� +X0 + +√ +5 +2 +�� +. + +18 +N. MATSUURA AND Y. SUYAMA +The equations (4.3) and (4.4) imply that X0 +α and X0 +β are the principal directions of f 0 +Vij(x, y). +Lemma 4.1. On each Vij given above, the functions Bk, Ck in (4.3) and (4.4) are determined +as follows: +B1(x, y) = −(x2 − y2)−1� +X0 + +√ +5 +2 +� +− +√ +5, +C1(x, y) = −2(x2 − y2)−1� +X0 + +√ +5 +2 +� +, +B2(x) = −(1/2) (X′ +0/x) − +√ +5, +C2(x) = X′′ +0 − X′ +0/x. +Then, for these functions Bk and Ck, the equations in (4.3) and (4.4) extend to D \ S1. +Proof. The structure equation of the hypersurface fVij(x, y, z) is given at [18, Equations (2.2.3) +and (2.2.4) in §2.2]. Then, by F 0 +Vij(x, y) = FVij(x, y, 0), the derivative of F 0 +Vij = [φ, X0 +α, X0 +β, ξ] +is obtained from the equation by taking as ϕ(x, y, 0) = ϕ(x, y), ϕz(x, y, 0) = ϕz(x, y) and +P(x, y, z) = ¯P(x, y), Pz(x, y, 0) = ¯Pz(x, y). For B1: We have +B1 = +¯P −2 +cos ϕ( ¯P cos ϕ)z = − +1 +cos ϕ +� +( ¯P −1)z cos ϕ + ¯P −1ϕz sin ϕ +� +. +Then, we obtain B1 by +ϕz = y/(x2 + y2), +( ¯P −1)z = +√ +5 + +� +(x2 − y2) +� +X0 + +√ +5 +2 +� ++ 2xy2X′ +0 +� +/(x2 + y2)2. +For C1: We have +C1 = +¯P −2 +sin ϕ cos ϕ( ¯P cos ϕ)y = +−1 +sin ϕ cos ϕ +� +( ¯P −1)y cos ϕ + ¯P −1ϕy sin ϕ +� +. +Then, we obtain C1 by +ϕy = 2x/(x2 + y2), +( ¯P −1)y = 2xy(x2 + y2)−2� +− 2X0 + 2xX′ +0 − +√ +5 +� +. +For C2: We have +C2 = +¯P −2 +sin ϕ cos ϕ( ¯P sin ϕ)x = +−1 +sin ϕ cos ϕ +� +( ¯P −1 sin ϕ)x − 2 ¯P −1ϕx cos ϕ +� +. +Then, by ϕx = −2y/(x2 + y2), we have +( ¯P −1 sin ϕ)x = 2 ¯P −1ϕx cos ϕ + 2y(x2 − y2)(x2 + y2)−2 (X′ +0 − xX′′ +0 ) . +Hence, we obtain C2 = X′′ +0 − X′ +0/x. For B2: We have +B2 = +¯P −2 +sin ϕ( ¯P sin ϕ)z = − +1 +sin ϕ +� +( ¯P −1)z sin ϕ − ¯P −1ϕz cos ϕ +� +. +Then, in the same way as above, we obtain B2 from the equation. +The last statement follows from the fact that the functions Bk and Ck above are independent +of choice of the domains Vij. +□ +Next, we obtain the following lemma directly from (4.1) and Lemma 4.1: +Lemma 4.2. On each Vij, we have the following equations: +∇′ +∂/∂xφ = −a1X0 +α := +−2y +xh(x, y) +� +X0 + +√ +5 +2 +� +X0 +α, +∇′ +∂/∂xX0 +β = c1X0 +α := +−2 +xh(x, y) +� +X0 + +√ +5 +2 +� +X0 +α, +∇′ +∂/∂xξ = b1X0 +α := +−1 +xh(x, y) +� +X0 + +√ +5 +2 + +√ +5(x2 − y2) +� +X0 +α, +∇′ +∂/∂yφ = −a2X0 +β := +1 +h(x, y) +� +2X0 + +√ +5 − (x2 + y2)X′ +0 +x +� +X0 +β, +∇′ +∂/∂yX0 +α = c2X0 +β := +2y +h(x, y) +� +X′′ +0 − X′ +0 +x +� +X0 +β, +∇′ +∂/∂yξ = b2X0 +β, := +−2y +h(x, y) +�X′ +0 +2x + +√ +5 +� +X0 +β, + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +19 +Then, all equations above are independent of choice of the domains Vij and they are analytic +equations defined on D. +We also have ∇′ +∂/∂xX0 +α = a1φ − b1ξ − c1X0 +β and ∇′ +∂/∂yX0 +β = a2φ − b2ξ − c2X0 +α by (4.3) and +(4.4). +For the analytic functions (ai, bi, ci) on D in Lemma 4.2, we define the matrix-valued functions +Ω1 and Ω2 by +Ω1(x, y) = +� +��� +0 +a1 +0 +0 +−a1 +0 +c1 +b1 +0 +−c1 +0 +0 +0 +−b1 +0 +0 +� +��� (x, y), +Ω2(x, y) = +� +��� +0 +0 +a2 +0 +0 +0 +−c2 +0 +−a2 +c2 +0 +b2 +0 +0 +−b2 +0 +� +��� (x, y). +(4.5) +For a frame field F 0(x, y) := [φ, X0 +α, X0 +β, ξ](x, y) and the matrix-valued differential 1-form Ω = +Ω1dx + Ω2dy on D, the equations of Lemma 4.2 are summarized as +dF 0 = F 0Ω, +(4.6) +which is an analytic equation on the whole domain D, and in particular, it is independent of +choice of Vij. Now, in Theorem 4.3 below, we shall verify that the equation (4.6) has a solution +F 0(x, y) on D. Then, the solution F 0(x, y) is uniquely determined up to a transformation +AF 0(x, y) by a constant orthogonal matrix A. For the sake of simplicity for the argument, we +determine an initial condition of (4.6) by +F 0(x0, y0) = Id +(4.7) +for a while, where (x0, y0) is a point of V12 and Id is the unit matrix. It is possible to take +such an initial condition. In fact, for the frame field FV12(x, y, z) determining the hypersur- +face fV12(x, y, z), suppose FV12(x0, y0, 0) ̸= Id. We take a constant orthogonal matrix A such +that AFV12(x0, y0, 0) = Id holds. Then, the frame AFV12(x, y, z) determines the hypersurface +AfV12(x, y, z) by AdfV12 = d(AfV12). +Theorem 4.3. An analytic orthonormal frame field F 0(x, y) is uniquely determined on D such +that it is a solution to the structure equation (4.6) under the initial condition (4.7). Then, +the surface f 0 +V12(x, y) on V12 determined above extends to an analytic surface f 0(x, y) with the +metric g0 defined on the whole domain D. Furthermore, for any simply connected domain V ′ +ij +satisfying V ′ +ij ⊂ Dij \ S, there is a generic conformally flat hypersurface fV ′ +ij(x, y, z) on V ′ +ij × I +such that f 0(x, y) = fV ′ +ij(x, y, 0) holds on V ′ +ij. +Proof. The frame field FV12(x, y, z) on V12 × I is determined from the hypersurface fV12(x, y, z). +Since F 0 +V12(x, y) = FV12(x, y, 0) and f 0 +V12(x, y) = fV12(x, y, 0), the 1-form Ω(x, y) satisfies the +Maurer-Cartan equation dΩ + Ω ∧ Ω = 0 on the open domain V12. Then, since Ω is analytic +on D, the Maurer-Cartan equation is satisfied on the whole domain D. Hence, a frame field +F 0(x, y) on D is uniquely determined under the condition (4.7), which is the extension of +F 0 +V12(x, y) on V12. +Furthermore, for the vector fields X0 +α and X0 +β of F 0(x, y) on D, an analytic surface f 0(x, y) +on D is determined by +df 0 = θ1X0 +α + θ2X0 +β, +θ1 := (x2 − y2)/(xh(x, y))dx, +θ2 := 2y/h(x, y)dy, +(4.8) +since f 0 +V12(x, y) satisfies (4.2) on V12. +In fact, df 0(x, y) in (4.8) satisfies d(df 0) = 0 on D +by Lemma 4.2. Hence, f 0(x, y) is an extension of f 0 +V12(x, y) to D. Then, φ and ξ are the +normal vector fields of f 0(x, y), which are distinguished by the condition for the hypersurface +fV12(x, y, z). Furthermore, the surface f 0(x, y) on D has the metric g0 by (4.8). +Finally, the existence of the generic conformally flat hypersurface fV ′ +ij(x, y, z) in the last +statement follows from the fact that F 0(x, y) is determined by Φ and Ψ. +□ + +20 +N. MATSUURA AND Y. SUYAMA +Definition 4.4 (Curvature surface defined on D). By Theorem 4.3, we may recognize that the +analytic surface f 0(x, y) on D is an extended curvature surface. Our aim is to study the property +and the structure on f 0(x, y). We call f 0(x, y) and F 0(x, y) in Theorem 4.3, respectively, a +curvature surface defined on D and a frame field determining the curvature surface f 0(x, y) +on D. Then, we can arbitrarily give the initial condition of F 0(x, y) by F 0(x0, y0) = A not +only (4.7), where (x0, y0) and A are a point D and an orthogonal matrix, respectively. For a +fixed frame field F 0(x, y), the curvature surface f 0(x, y) is determined uniquely up to a parallel +translation. +Here, we remark on a curvature surface f 0(x, y) on D. Since we have fixed F 0(x, y) by (4.7), +det F 0(x, y) = 1 holds on D. Then, with W(x, y) := det[φ, ∂f 0/∂x, ∂f 0/∂y, ξ](x, y), we have +W ≤ 0 on D11, +W ≥ 0 on D12, +W ≥ 0 on D21, +W ≤ 0 on D22. +(4.9) +Next, we study the coordinate lines of a curvature surface f 0(x, y) on D. Let ∥v∥ be the +Euclidean norm for a vector v ∈ R4. +Theorem 4.5. Let f 0(x, y) be a curvature surface on D. Then, for an x-curve f 0(x, y) with +fixed y, we have the following facts (1), (2) and (3): +(1) Along any x-curve f 0(x, y), the vector (yX0 +β − φ)(x, y) is constant. That is, an analytic +unit vector u(y) of y is determined by u(y) = (1 + y2)−1/2(yX0 +β − φ)(x, y). +(2) Let f(x, y) be an analytic unit vector defined by +f(x, y) := ((1 + y2)(5 + 4y2))−1/2 � +X0 +β − 2(1 + y2)ξ + yφ +� +(x, y). +When we regard the surface f 0(x, y) as a one-parameter family of x-curves, it is ex- +pressed as +f 0(x, y) = (2 +√ +5)−1� +(5 + 4y2)(1 + y2)−1f(x, y) + A(y), +where A(y) is a R4-valued analytic function of y determined uniquely up to a parallel +translation. +(3) Any x-curve f 0(x, y) lies on a 2-sphere S2 +y of radius (2 +√ +5)−1� +(5 + 4y2)(1 + y2)−1 in +an affine hyperplane R3 +y perpendicular to u(y). +Proof. We firstly verify (1)–(3) on the domain U := {(x, y) ∈ D12 | x > y ≥ 0}, and then we +use the equations in Lemma 4.1. Now, we have ∇′ +X0α(yX0 +β − φ) = 0. Hence, we have (1). Next, +we have ∇′ +X0α(X0 +β − 2ξ) = 2 +√ +5X0 +α. The component ˆf(x, y) of (X0 +β − 2ξ)(x, y) perpendicular to +u(y) is given by +ˆf(x, y) := X0 +β − 2ξ − y(1 + y2)−1/2u = (1 + y2)−1 � +X0 +β − 2(1 + y2)ξ + yφ +� +. +Hence, we have ∇′ +X0α ˆf = 2 +√ +5X0 +α. Then, since f(x, y) is the normalization of ˆf(x, y), we have +(2) by X0 +αf 0 = X0 +α. Finally, since X0 +α(x, y) ⊥ u(y) (or f(x, y) ⊥ u(y)), we obtain (3) by the +norm ∥ˆf(x, y)∥. By the above argument, we have verified the theorem for x-curves on U. +Next, all x-curves on D are also expressed as the form (2), since all our objects: the frame +field F 0(x, y), the surface f 0(x, y) and the vector f(x, y), are analytic on D. +Actually, by +Lemma 4.2, we have the following equation, +∇′ +∂/∂x +� +(2 +√ +5)−1� +(5 + 4y2)(1 + y2)−1 f 2 + A +� += [(x2 − y2)/(xh(x, y))] X0 +α +on D, which coincides with ∂f 0/∂x on D. We can verify directly by Lemma 4.2 that all x-curves +on D also satisfy (1) and (3). In consequence, the proof has been completed. +□ +Theorem 4.6. Let f 0(x, y) be a curvature surface on D. Then, for a y-curve f 0(x, y) with +fixed x, we have the following facts (1), (2) and (3): + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +21 +(1) Along any y-curve f 0(x, y), the vector (−B2X0 +α + C2ξ)(x, y) is constant, where B2(x) = +−(1/2)(X′ +0/x) − +√ +5 and C2(x) = X′′ +0 − (X′ +0/x). That is, an analytic unit vector ˜u(x) +of x is determined by ˜u(x) = (B2 +2 + C2 +2)−1/2(−B2X0 +α + C2ξ)(x, y) and (B2 +2 + C2 +2)(x) = +(X′ +0/x)(2X0 − xX′ +0 − X′ +0/x + +√ +5) > 0 holds for x ∈ R. +(2) Let ˜f(x, y) be an analytic unit vector defined by +˜f(x, y) := (B2 +2 + C2 +2)−1/2(x)(B2ξ + C2X0 +α)(x, y). +When we regard the surface f 0(x, y) as a one-parameter family of y-curves, it is ex- +pressed as +f 0(x, y) = (B2 +2 + C2 +2)−1/2(x)˜f(x, y) + ˜A(x), +where ˜A(x) is a R4-valued analytic function of x determined uniquely up to a parallel +translation. +(3) Any y-curve f 0(x, y) lies on a standard 2-sphere S2 +x of radius (B2 +2 + C2 +2)−1/2(x) in an +affine hyperplane R3 +x perpendicular to ˜u(x). +Proof. We firstly prove the theorem on the domain ˜U := {(x, y) ∈ D1| y > 0}, and then we use +the equations in Lemma 4.1. Now, we have ∇′ +X0 +β(−B2X0 +α + C2ξ) = 0 and (B2 +2 + C2 +2)(x) > 0 on +R by B2(x) < 0. Furthermore (B2 +2 + C2 +2)(x) is expressed as the form in (1) by (2.8). Hence, we +have obtained (1). Next, we have ∇′ +X0 +β(B2ξ+C2X0 +α) = (B2 +2 +C2 +2)X0 +β. Then, by (B2 +2 +C2 +2)(x) ̸= 0 +and X0 +βf 0 = X0 +β, we have (2). Finally, since X0 +β ⊥ ˜u(x) (or ˜f(x, y) ⊥ ˜u(x)), we have (3) by +∥(B2 +2 + C2 +2)−1/2(x)˜f(x, y)∥ = (B2 +2 + C2 +2)−1/2. Thus, we have verified the theorem for y-curves +on ˜U. These results also hold for all y-curves on D by Lemma 4.2 similarly to the proof of +Theorem 4.5. In consequence, the proof has been completed. +□ +Remark 4.7. (1) In Theorem 4.5 and Theorem 4.6, the vectors u(y) and ˜u(x) are perpendic- +ular for all (x, y) ∈ D. In fact, by the definitions of u(y) and ˜u(x), we have +⟨u(y), ˜u(x)⟩ = C(x, y) ⟨yX0 +β − φ, −B2X0 +α + C2ξ⟩(x, y) = 0, +where C(x, y) := ((1 + y2)(B2 +2 + C2 +2)(x))−1/2. +(2) The vector u(y) of y (resp. ˜u(x) of x) moves on a circle S1 in a plane. We shall give a +simple proof of these facts in the next section (see Theorem 5.3). Certainly, we can also make +sure these facts by direct calculation, but it is very hard. Here, we only give the norms of the +first derivatives of u(y) and ˜u(x): +∥∇′ +∂/∂yu(y)∥ = +y2 +1 + y2, +∥∇′ +∂/∂x˜u(x)∥ = +� +X′ +0 + 2 +√ +5x +� � +2X0 + +√ +5 +� +4xX′ +0 +� +2X0 − xX′ +0 − X′ +0/x + +√ +5 +�. +These norms show that the length of the curve u(y) (resp. ˜u(x)) diverges to ∞ as y tends to ±∞ +(resp. as x tends to ∞): in the right hand side of the second equation, we have (2X0−xX′ +0)(x) = +O(x) and X′ +0(x) is a positive bounded function, by Proposition 2.4. Furthermore, the length +of ˜u(x) on (ε, 1] also diverges to ∞ as ε(> 0) → 0, since we have ∥∇′ +∂/∂x˜u(x)∥ ≈ +√ +5/(2x) by +X0(0) = 5/2 and X′ +0(x) ≈ 2x. +By Theorems 4.5 and 4.6, any x-curve f 0(x, y) with fixed y (resp. any y-curve f 0(x, y) with +fixed x) belongs to an affine hyperplane perpendicular to u(y) (resp. ˜u(x)). We can determine +the following orthonormal frame fields along the curves: let f(x, y) and ˜f(x, y) be vectors in +Theorems 4.5 and 4.6, respectively; along each x-curve, the frame field is given by +� +u(y), X0 +α(x, y), f(x, y), u2(x, y) +� +(4.10) + +22 +N. MATSUURA AND Y. SUYAMA +where u2(x, y) is defined by u2(x, y) := (5 + 4y2)−1/2(2X0 +β + ξ + 2yφ); along each y-curve, the +frame field is given by +�˜u(x), X0 +β(x, y), ˜f(x, y), φ(x, y) +� +. +(4.11) +Now, in the following theorem, we verify that each x-curve with y ̸= 0 (resp. each y-curve +with x) of a curvature surface f 0(x, y) on D has a cusp at the point x = |y| (resp. at the point +y = 0). Then, we say that a curve p(t) in R3 has a cusp of type (2, 3, 4) at t = 0, if p(t) is +expressed as p(t) = (at2, bt3, ct4) around t = 0 with constants a, b and c (abc ̸= 0). +Theorem 4.8. For any coordinate curve of a curvature surface f 0(x, y) on D, we have the +following facts (1) and (2): +(1) Any x-curve with y ̸= 0 has a cusp of type (2, 3, 4) at x = |y|. +(2) Any y-curve has a cusp of type (2, 3, 4) at y = 0. +Proof. (1) Let f 0(x, y) be an x-curve with fixed y ̸= 0. +We study the curve only in a +small neighborhood of (|y|, y). Now, the first derivative of the curve is given by f 0 +x(x, y) = +((x2 − y2)/(xh(x, y))) X0 +α(x, y) and +� +(x2 − y2)/(xh(x, y)) +� +x = (x2h2(x, y))−1 � +(x2 + y2)h(x, y) + (x2 − y2)2(X′′ +0 − X′ +0/x) +� +, +f 0 +xx(x, y) = +� +(x2 − y2)/(xh(x, y)) +� +x X0 +α + +� +(x2 − y2)/(xh(x, y)) +� � +∇′ +∂/∂xX0 +α +� +(x, y), +f 0 +xxx(|y|, y) = +� +(x2 − y2)/(xh(x, y)) +� +xx (|y|, y)X0 +α(|y|, y) + (4/h(|y|, y)) +� +∇′ +∂/∂xX0 +α +� +(|y|, y). +We define the functions wi(x) (i = 1, 2, 3) of x by +w1(x) := ⟨f 0(x, y), X0 +α(|y|, y)⟩, +w2(x) := ⟨f 0(x, y), u2(|y|, y)⟩, +w3(x) := ⟨f 0(x, y), f(|y|, y)⟩. +Then, we directly have +(w1)x(|y|) = (w2)x(|y|) = (w3)x(|y|) = (w2)xx(|y|) = (w3)xx(|y|) = 0, +(w1)xx(|y|) ̸= 0. +Furthermore, we have (w2)xxx(|y|) ̸= 0 by ∇′ +∂/∂xX0 +α = a1φ − b1ξ − c1X0 +β and Lemma 4.2. For +w3, (w3)xxx(|y|) = 0 holds by ⟨∇′ +∂/∂xX0 +α, f⟩(|y|, y) = 0, and further we have (w3)xxxx(|y|) ̸= 0 +by ⟨(∇′ +∂/∂x)2X0 +α, f⟩(|y|, y) ̸= 0. Hence, for t := x − |y|, we have +w1(|y| + t) ≈ w1(|y|) + (t2/2)(w1)xx(|y|), +w2(|y| + t) ≈ w2(|y|) + (t3/6)(w2)xxx(|y|), +w3(y + t) ≈ w3(|y|) + (t4/24)(w3)xxxx(|y|), +as t tends to 0, which show that the x-curve f 0(x, y) = (w1, w2, w3)(x) has the cusp of type +(2, 3, 4) at the point (|y|, y). +(2) Let f 0(x, y) be a y-curve with fixed x. We study the curve only in a small neighborhood +of (x, 0). The first derivative of the curve is given by f 0 +y (x, y) = (2y/h(x, y))X0 +β(x, y), and +(2y/h(x, y))y = (2/h2(x, y)) +� +h(x, y) − 2y2(X′ +0/x) +� +, +f 0 +yy(x, y) = (2y/h(x, y))yX0 +β(x, y) + (2y/h(x, y))(∇′ +∂/∂yX0 +β)(x, y), +f 0 +yyy(x, 0) = (2y/h(x, y))yy(x, 0)X0 +β(x, 0) + (4/h(x, 0))(∇′ +∂/∂yX0 +β)(x, 0). +We define the functions ˜wi(y) (i = 1, 2, 3) by +˜w1(y) := ⟨f 0(x, y), X0 +β(x, 0)⟩, +˜w2(y) := ⟨f 0(x, y), φ(x, 0)⟩, +˜w3(y) := ⟨f 0(x, y), ˜f(x, 0)⟩. +Then, we obtain +( ˜w1)yy(0) ̸= 0, +( ˜w2)yyy(0) ̸= 0, +( ˜w3)yyyy(0) ̸= 0, +and that the lower derivatives of each ˜wi(y) vanish at y = 0, in the same way as in (1). Hence, +we have verified that the y-curve also has the cusp of type (2,3,4) at y = 0. +□ + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +23 +By Theorem 4.8, the curves f 0(|y|, y) of y and f 0(x, 0) of x are the cuspidal edges in a +curvature surface f 0(x, y) on D. The curve f 0(x, 0) of x is also the singular set of the hyperbolic +2-metric ˇgH. For these curves, we have the following corollary. +Corollary 4.9. (1) The two curves f 0(|y|, y) of y are the envelopes of the family of y-curves +f 0(x, y) with x. (2) The vector φ0 := φ(x, 0) does not depend on x. The curve f 0(x, 0) of x lies +on a 2-sphere S2 +y=0 of radius 1/2 in an affine hyperplane R3 +y=0 perpendicular to φ0. +Proof. (1) For the sake of simplicity, we assume y > 0. The derivative of the y-curve f 0(y, y) +is given by ((∂/∂x + ∂/∂y) f 0) (y, y) = (∂f 0/∂y)(y, y) by (4.8). Hence, the tangent vectors of +both y-curves f 0(x, y) and f 0(y, y) coincide at (y, y), which implies that the y-curve f 0(y, y) is +the envelope of the family of y-curves f 0(x, y). +(2) We have (∇′ +∂/∂xφ)(x, 0) = 0 by Lemma 4.2. The other statement is already proved in +Theorem 4.5. +□ +To visualize the curvature surface f 0(x, y) locally, we use the approximation f δ(x, y) that +will be defined in Section 6, and illustrate them via the projection π: when we determine the +coordinates (r1, r2, r3, r4) of R4 by the frame F δ(y0, y0) = Id, the projection π eliminates the +4-th coordinate of R4, that is, π ((r1, r2, r3, r4)) = (r1, r2, r3). +4.75 +5.25 +5.75 +6.25 +6.75 +7.25 +x +5.75 +6.25 +y +π◦f1/20 +−−−−→ +Figure 2. This shows the x-curves passing through the points +� +f 1/20(yk, yk) +� +on the cuspidal edge: the lines in the domain are given by yk = 5.75 + 0.025k +(k = 0, 1, . . . , 20) and yk − 1 ≤ x ≤ yk + 1. Each x-curve is drawn with black for +yk − 1 ≤ x < yk and with gray for yk < x ≤ yk + 1. +4.75 +5.25 +5.75 +6.25 +6.75 +7.25 +x +5.75 +6.25 +y +π◦f1/20 +−−−−→ +Figure 3. This shows the envelope made by the y-curves around the points +� +f 1/20(xk, xk) +� +from the same domain as Figure 2: the lines in the domain are +given by xk = 4.75 + 0.05k (k = 1, 2, . . . , 49), xk − 1 ≤ y ≤ xk + 1 and 5.75 ≤ +y ≤ 6.25. +Next, for a curvature surface f 0(x, y) on D, we study the relation with f 0|D1(x, y) and +f 0|D2(x, −y). +To the end, we translate the surface f 0(x, y) such that the obtained surface +f 0(x, y) satisfies f 0(p0) = 0 at some point p0 := (x0, 0). Then, we consider that the frame of + +24 +N. MATSUURA AND Y. SUYAMA +R4 containing the new f 0(x, y) is given by F 0(p0) = Id, for the sake of simplicity. That is, +[X0 +α, X0 +β, ξ](p0) is a frame of R3 +y=0 and the x-curve f 0(x, 0) belongs to R3 +y=0 by Corollary 4.9-(2). +Corollary 4.10. Let f 0(x, y) and F 0(x, y) be the curvature surface and the frame field, ex- +pressed by the above coordinate system of R4. Then, with the orthogonal matrix B = [bij] such +that b11 = −1, bii = 1 (2 ≤ i ≤ 4) and bij = 0 (i ̸= j), we have (B ◦ f 0)(x, y) = f 0(x, −y), +(B ◦ φ)(x, y) = −φ(x, −y) and (B ◦ ξ)(x, y) = ξ(x, −y) for y ≥ 0. Furthermore, (B ◦ ˜u)(x) = +˜u(x) and (B ◦ ˜A)(x) = ˜A(x) hold. +Proof. Let ¯y = −y for y ≥ 0. We put ¯φ(x, y) := −φ(x, ¯y), ¯X0 +α(x, y) := X0 +α(x, ¯y), ¯X0 +β(x, y) := +X0 +β(x, ¯y) and ¯ξ(x, y) := ξ(x, ¯y). We study the derivative in y ≥ 0 of ¯F 0 := [¯φ, ¯X0 +α, ¯X0 +β, ¯ξ] by +Lemma 4.2. Then, we have d ¯F 0(x, y) = ¯F 0(x, y)Ω(x, y) for y ≥ 0, where Ω(x, y) is the differen- +tial 1-form in (4.6) such that dF 0(x, y) = F 0(x, y)Ω(x, y). Thus, ¯F 0(x, y) is another solution to +(4.6) in y ≥ 0, and hence there is an orthogonal matrix B such that BF 0(x, y) = ¯F 0(x, y). Then, +B is determined at p0 from F 0(p0) = [φ, X0 +α, X0 +β, ξ](p0) = Id and ¯F 0(p0) = [−φ, X0 +α, X0 +β, ξ](p0) +as in the statement. +Furthermore, f 0(x, ±y) is the integral surface of (X0 +α, X0 +β)(x, ±y) in +(4.8), respectively, and (B ◦ f 0)(x, 0) = f 0(x, 0) holds by f 0(x, 0) ∈ R3 +y=0. Hence, we have +(B ◦f 0)(x, y) = f 0(x, −y). In consequence, we have verified the first statement in the corollary. +For (B◦ ˜u)(x) = ˜u(x): Any y-curve f 0(x, y) lies on a 2-sphere S2 +x in R3 +x perpendicular to ˜u(x) +and it is not a circle (for example, Figure 6 in the next section). Hence, we have B(R3 +x) = R3 +x +by (B ◦ f 0)(x, y) = f 0(x, −y), which shows (B ◦ ˜u)(x) = ˜u(x). +For (B ◦ ˜A)(x) = ˜A(x): We have f 0(x, y) = (B2 +2 + C2 +2)−1/2(x)˜f(x, y) + ˜A(x) and f 0(x, −y) = +(B2 +2 + C2 +2)−1/2(x)˜f(x, −y) + ˜A(x) for y ≥ 0. +Then, since (B ◦ f 0)(x, y) = f 0(x, −y) and +(B ◦ ˜f)(x, y) = ˜f(x, −y) by the definition of ˜f(x, y), we obtain (B ◦ ˜A)(x) = ˜A(x). Hence, we +have verified the corollary. +Here, we remark the relation with F 0(x, y) and F 0(x, −y) for y ≥ 0, explicitly. For φ(x, ±y), +the first coordinate elements are equal and the other elements have the different sign. For each +X0 +α(x, ±y), X0 +β(x, ±y) and ξ(x, ±y), the first elements have the different sign and the other +elements are equal. +□ +By Theorem 4.8 and Corollaries 4.9 and 4.10, we may regard any curvature surface f 0 : +(D, g0) ∋ (x, y) �→ f 0(x, y) ∈ R4 as a realization in R4 of the regular coordinate system +ˆι : (D, g0) ∋ (x, y) �→ ˆι(x, y) ∈ ( ˆD±, ˆg± +0 ) in (3.13), that is, an isometric map ¯f 0 : ( ˆD±, ˆg± +0 ) ∋ +(ˆx, ˆy) �→ ¯f 0(ˆx, ˆy) ∈ R4 is determined by ( ¯f 0 ◦ ˆι)(x, y) = f 0(x, y) for (x, y) ∈ D. The sign of the +determinant W = det [φ, ∂f 0/∂x, ∂f 0/∂y, ξ] changes for each domain Dij, as mentioned at (4.9) +under the condition (4.7). However, if we replace (∂f 0/∂x, ∂f 0/∂y) with (∂ ¯f 0/∂ˆx, ∂ ¯f 0/∂ˆy), +then the determinant is positive on the whole D from (∂ ¯f 0/∂ˆx, ∂ ¯f 0/∂ˆy) = h−1(X0 +α/x, X0 +α +X0 +β) +by (3.6) and (4.8). Here, note that (∂ ¯f 0/∂ˆx)(ˆι(x, y)) = (xh(x, y))−1X0 +α(x, y) diverges to ∞ as +D ∋ (x, y) → (0, y) for any (0, y) including y = 0. +5. Structure of the extended curvature surface +Let f 0(x, y) be a curvature surface on D defined in the previous section. In this section, +we study the limit of x-curves f 0(x, y) with fixed y as x tends to 0 and ∞, and the limit of +y-curves f 0(x, y) with fixed x as y tends to ±∞. In order to study x-curves f 0(x, y) as x tends +to 0, we change x for a new parameter u by x = e−u: the change is reasonable for the metric +g0 in (3.1) and the equations of Lemma 4.2. Then, any u-curve f 0(e−u, y) uniformly converges +to a circle S1(y) parametrized by u as u tends to ∞, and any y-curve converges to a point +p(x) as y tends to ±∞. Furthermore, the convergence of u-curves is uniform with respect to +y ∈ (−∞, ∞) and the convergence of y-curves is also uniform in the wider sense with respect +to x ∈ (0, ∞) (see Definition 5.2 below). Thus, S1(y) and p(x), respectively, are continuous for + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +25 +y and x. Through further study for these convergences, we can understand the structure on +f 0(x, y) in R4 in detail as mentioned in the introduction, and connect two curvature surfaces +defined on D = {(x, y) | x > 0} and D(−) := {(x, y) | x < 0} continuously at the origin +(0, 0) ∈ R2. +Now, let f 0(x, y) be an x-curve with fixed y in a curvature surface on D. +The x-curve +f 0(x, y) is included in an affine hyperplane R3 +y perpendicular to u(y) by Theorem 4.5, and +the orthonormal frame field of R3 +y along the x-curve is given by [X0 +α, f, u2](x, y) in (4.10). We +change the parameter x for u by x = e−u, and then, for a vector Z(x) of x, we denote by Z(e−u) +the vector ¯Z(u) := Z(e−u) of u. The vectors f(e−u, y) and u2(e−u, y) satisfy the equations +∇′ +∂/∂uf = −v3(x, y)X0 +α, +∇′ +∂/∂uu2 = v2(x, y)X0 +α, +by ∂/∂u = −x ∂/∂x and Lemma 4.2, where +v2(x, y) := (5 + 4y2)−1/2h−1(x, y) +� +(5 + 4y2)(X0 + +√ +5/2) + +√ +5(x2 − y2) +� +, +v3(x, y) := 2 +√ +5(x2 − y2)h−1(x, y) +� +(1 + y2)(5 + 4y2)−1. +Hence we have ∇′ +∂/∂uX0 +α = v3(x, y)f−v2(x, y)u2. For the functions vi(x, y) (i = 2, 3), vi(0, y) are +also well defined: we have v2(0, 0) = +√ +5/2, v2(0, y) > 0 for y ∈ R and v3(0, 0) = 0, v3(0, y) < 0 +for y ̸= 0. As x tends to 0, the functions v2(x, y) and v3(x, y), respectively, converge to v2(0, y) +and v3(0, y) uniformly with respect to y ∈ R. In fact, we have the following fact in the same +way as in Corollary 2.3-(2): there is a number t1 (0 < t1 < 1) such that +|v2(x, y) − v2(0, y)| < 3 +2 +5 + +√ +5 + 4y2 +� +5 + 4y2 +x2 +h(x, y) < 3 +2 +5 + +√ +5 + 4y2 +� +5 + 4y2 +x2 +5 + +√ +5 + y2, +(5.1) +|v3(x, y) − v3(0, y)| < 2 +√ +5 +7 +� +1 + y2 +5 + 4y2 +x2(7 + y2) +h(x, y) +< 2 +√ +5 +7 +� +1 + y2 +5 + 4y2x2 +(5.2) +hold for 0 < x < t1 and any y ∈ R. By the above equations, M(u, y) := [X0 +α, f, u2](e−u, y) +satisfies the following equation for (u, y) ∈ R2: +∂M +∂u (u, y) = M(u, y)V (u, y), +V (u, y) := +� +� +0 +−v3(x(u), y) +v2(x(u), y) +v3(x(u), y) +0 +0 +−v2(x(u), y) +0 +0 +� +� . +(5.3) +In V (u, y), we replace the functions v2(x, y) and v3(x, y) with v0 +2(y) := v2(0, y) and v0 +3(y) := +v3(0, y), respectively, and denote by V 0(y) the new V (u, y). Then, we define another matrix +N(u, y) = [b(u, y), a(u, y), c(u, y)], by the differential equation +∂N +∂u (u, y) = N(u, y)V 0(y), +V 0(y) := +� +� +0 +−v0 +3 +v0 +2 +v0 +3 +0 +0 +−v0 +2 +0 +0 +� +� (y). +(5.4) +Lemma 5.1. Let N(u, y) = [b, a, c] (u, y) be a solution to (5.4). +Then, v0(y) := (v0 +2a + +v0 +3c)(u, y) does not depend on u. The solution N(u, y) is a rotation with respect to the axis +v0(y) and the speed of its rotation is ∥v0(y)∥. +Proof. In this proof, we fix the parameter y arbitrarily. We have ∇′ +∂/∂u(v0 +2a + v0 +3c) = (−v0 +2v0 +3 + +v0 +3v0 +2)b = 0 by (5.4). Hence, v0(y) does not depend on u. Now, we set +B0(y) := +1 +∥v0(y)∥ +� +� +0 +∥v0∥ +0 +v0 +2 +0 +v0 +3 +v0 +3 +0 +−v0 +2 +� +� (y), +Φu(y) := +� +� +1 +0 +0 +0 +cos(u∥v0(y)∥) +− sin(u∥v0(y)∥) +0 +sin(u∥v0(y)∥) +cos(u∥v0(y)∥) +� +� , + +26 +N. MATSUURA AND Y. SUYAMA +and then B0(y) and Φu(y) belong to the special orthogonal group SO(3). Then, we have +� +B0Φ−1 +u (∂Φu/∂u) (B0)−1� +(y) = V 0(y) +(5.5) +by direct calculation. Next, with any orthogonal matrix N ∞(y) depending on y, the solution +N(u, y) to (5.4) will be given by +N(u, y) = N ∞(y) +� +B0(y)Φu(y)(B0)−1(y) +� +. +(5.6) +The equation implies that the lemma holds good. We can verify that N(x, y) in (5.6) is a +solution to (5.4) as follows: taking the derivative of N(u, y) by u, we obtain +∂N/∂u = N ∞B0Φu(B0)−1 � +B0Φ−1 +u (∂Φu/∂u)(B0)−1� += NV 0 +by (5.5). In consequence, the proof of Lemma 5.1 has been completed. +□ +Now, we return to the equation (5.3) for M on R2. In the following lemma, we use the +notations in the proof of Lemma 5.1, and for a square matrix A = [aij], we define the norm +∥A∥ by ∥A∥ := +��(aij)2. +Lemma and Definition 5.2. +(1) Let ¯ +M(u, y) := M(u, y) (B0Φ−u(B0)−1) (y) for (u, y) ∈ +R2. Then, there is a continuous orthogonal matrix N ∞(y) of y ∈ R such that ¯ +M(u, y) +with fixed y converges to N ∞(y) as u tends to ∞. Furthermore, the convergence for +u-curves ¯ +M(u, y) is uniform with respect to y ∈ R. +(2) Let N ∞(y) be the matrix of y in (1). Then, the frame field M(u, y) with fixed y uni- +formly converges to the rotation N(u, y) = N ∞(y)(B0Φu(B0)−1)(y) as u tends to ∞, +and further the convergence for u-curves M(u, y) is also uniform with respect to y ∈ R. +Here, as u tends to ∞, we say that M(u, y) with fixed y uniformly converges to N(u, y), if +there is a real number U for any ε > 0 such that ∥M(u, y) − N(u, y)∥ < ε holds for u > U. +Then, we say that the convergence for u-curves M(u, y) is uniform with respect to y ∈ R, if +there is a number U satisfying above independently of y ∈ R for any ε > 0. +Proof. (1) Taking the derivative of ¯ +M(u, y) by u, we have +∂ ¯ +M/∂u = (∂M/∂u)B0Φ−u(B0)−1 + MB0(∂Φ−u/∂u)(B0)−1 += MV +� +B0Φ−u(B0)−1� +− M +� +B0(∂Φ−u/∂(−u))Φu(B0)−1� � +B0Φ−u(B0)−1� +. +Then, we have B0(∂Φ−u/∂(−u))Φu(B0)−1 = V 0 by (5.5). Hence, we obtain +∂ ¯ +M/∂u = M(V − V 0) +� +B0Φ−u(B0)−1� +. +(5.7) +Now, we have +��∂ ¯ +M/∂u(log(1/x), y) +�� = ∥V (x, y) − V (0, y)∥ +(5.8) +by (5.7), and hence we have +∥ ¯ +M(u1, y) − ¯ +M(u2, y)∥ ≤ +���� +� u2 +u1 +∥V (x(u), y) − V (0, y)∥du +���� = +����� +� e−u2 +e−u1 +∥V (x, y) − V (0, y)∥dx +x +����� +by (5.7) and (5.8). Then, by (5.1)–(5.2) there is an orthogonal matrix N ∞(y) for each y ∈ R +such that +∥ ¯ +M(log(1/x), y) − N ∞(y)∥ ≤ Cx2 +(5.9) +holds as x ↘ 0. In particular, N ∞(y) is continuous for y ∈ R, since the continuous matrix +¯ +M(u, y) of y converges to N ∞(y) uniformly with respect to y ∈ R as x ↘ 0. +(2) We have ∥M(u, y)−N ∞(y) (B0Φu(B0)−1) (y)∥ = ∥M(u, y) (B0Φ−u(B0)−1) (y)−N ∞(y)∥ ≤ +Ce−2u by (5.9). Then, since C is independent of y, we have the assertion. +□ + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +27 +In Lemma 5.2-(2), we put N ∞(y) = [b∞, a∞, c∞](y) and N(u, y) = [b, a, c](u, y). +Since +N(u, y) is a matrix in (5.6) determined by N ∞(y), the vector v0(y) := (v0 +2a + v0 +3c)(u, y) does +not depend on u by Lemma 5.1. Furthermore, we have ∥v0(y)∥ = +√ +5/2 by direct calculation. +We define v1(y) := −(2/ +√ +5)v0(y) and another unit vector v2(u, y) by +v2(u, y) := (2/ +√ +5) +� +v0 +3(y)a(u, y) − v0 +2(y)c(u, y) +� +, +which is perpendicular to b(u, y) and v1(y). Then, we have +[b, v2] (u, y) = +� +b∞, 2 +√ +5 +� +v0 +3a∞ − v0 +2c∞�� +(y) +� +cos( +√ +5u/2) +− sin( +√ +5u/2) +sin( +√ +5u/2) +cos( +√ +5u/2) +� +(5.10) +by (5.6), and +a(u, y) = +2 +√ +5 (−v0 +2(y)v1(y) + v0 +3(y)v2(u, y)) , +c(u, y) = − 2 +√ +5 (v0 +3(y)v1(y) + v0 +2(y)v2(u, y)) +by the definitions of vi (i = 1, 2). Here, any u-curve a(u, y) with fixed y is a circle of center +−(2/ +√ +5)v0 +2(y)v1(y) and radius (2/ +√ +5)|v0 +3(y)| by (5.10). In particular, in the case y = 0, the +circle degenerates into one point −v1(0) by v0 +3(0) = 0. +Furthermore, we have (∇′ +∂/∂yb)(u, y) = 0. In fact, as u tends to ∞, X0 +α(e−u, y) and (∇′ +∂/∂yX0 +α)(e−u, y) +with fixed y, respectively, converge uniformly to b(u, y) and zero-vector, and these convergences +for u-curves are also uniform with respect to y ∈ R. Here, the first convergence follows from +Lemma 5.2 and the second one follows from X′′ +0 −X′ +0/x = O(x2) in the equation of Lemma 4.2. +Hence, the vectors b∞(y) and (2/ +√ +5)(v0 +3a∞ − v0 +2c∞)(y) are constant by (5.10): we write these +constant vectors as +b∞ := b∞(y), +−c∞ := −c∞(0) = (2/ +√ +5)((v0 +3a∞ − v0 +2c∞))(y) +(5.11) +by v0 +2(0) > 0 and v0 +3(0) = 0. In consequence, [b, v2](u, y) in (5.10) does not depend on y. Then, +since v1(y) is perpendicular to u(y), b∞ and c∞, the pair (u(y), v1(y)) is an orthonormal frame +field of the plane perpendicular to the vectors b∞ and c∞. In particular, u(y) moves on a circle +S1, of which fact we have mentioned in Remark 4.7-(2). +Now, we write [b, v2](u) for [b, v2](u, y), and then by (5.10) and (5.11) we have +b(u) = cos( +√ +5u/2)b∞ − sin( +√ +5u/2)c∞, +v2(u) = − sin( +√ +5u/2)b∞ − cos( +√ +5u/2)c∞, +a(u, y) = (2/ +√ +5) +� +−v0 +2(y)v1(y) + v0 +3(y)v2(u) +� +, +a(u, 0) = −v1(0). +(5.12) +In the expression of a(u, y), y is a parameter for the family S1(y) of circles and u is a rotation +parameter for each circle S1(y): v1(y) is perpendicular to the circle S1(y). +Next, f(e−u, y) converges uniformly to a(u, y) as u tends to ∞, by Lemma 5.2. For each +y ∈ R, let Γ(u, y) be the following circle in R4 parametrized by u: +Γ(u, y) := +� +(2 +√ +5)−1� +(5 + 4y2)(1 + y2)−1a(u, y) + A(y) +y ̸= 0, +−(1/2)v1(0) + A(0) +y = 0, +(5.13) +where A(y) is the R4-valued function in Theorem 4.5-(2). Each circle Γ(u, y) with y has the +radius of 2y2/( +√ +5 h(0, y)). +In the following theorem, we use the notations in (5.11)–(5.13). +Theorem 5.3. Let f 0(x, y) be a curvature surface defined on D and F 0(x, y) = [φ, X0 +α, X0 +β, ξ](x, y) +be the orthonormal frame field on D determining f 0(x, y). Let u(y) and f(x, y) be the analytic +vectors in Theorem 4.5. Then, there is an orthonormal pair (b∞, c∞) of constant vectors such +that it satisfies the following conditions (1), (2) and (3): +(1) The vector u(y) moves on the unit circle in a plane perpendicular to b∞ and c∞. Let +v1(y) be a vector determined by (∇′ +d/dyu)(y) = (y2/(1+y2))v1(y). Then, [u(y), v1(y), b∞, c∞] +is an orthonormal frame field of R4 depending on y. + +28 +N. MATSUURA AND Y. SUYAMA +(2) As u tends to ∞, all u-curves X0 +α(e−u, y) with y uniformly converge to a circle b(u) = +cos( +√ +5u/2)b∞ − sin( +√ +5u/2)c∞, and each u-curve f(e−u, y) with fixed y uniformly con- +verges to the following circle a(u, y), +a(u, y) = +� +(2/ +√ +5) (−v0 +2(y)v1(y) + v0 +3(y)v2(u)) +y ̸= 0, +−v1(0) +y = 0, +where v2(u) is a circle given by v2(u) = − sin( +√ +5u/2)b∞ − cos( +√ +5u/2)c∞. +These +convergences for u-curves are also uniform with respect to y ∈ R. In particular, the +circles a(u, y) deform continuously for y and the circle a(u, 0) degenerates to one point. +(3) Any u-curve f 0(e−u, y) with fixed y uniformly converges to the circle Γ(u, y) as u tends +to ∞. The convergence for u-curves is also uniform with respect to y ∈ R. In particular, +the circles Γ(u, y) deform continuously for y and the circle Γ(u, 0) degenerates to one +point. +Proof. Almost all facts have been verified in the argument above. Now, since the vector u(y) of +y is perpendicular to b∞ and c∞, we have (∇′ +d/dyu)(y) = ±(y2/(1+y2))v1(y) for (1) by Remark +4.7-(2): we shall determine the sign ± at the end of this proof. The convergences in (2) follow +from Lemma 5.2, (5.10), (5.11) and (5.12): the frame field M(u, y) = [X0 +α, f, u2](e−u, y) with +fixed y uniformly converges to the rotation N(u, y) = [b, a, c](u, y) as u → ∞, and the converge +for u-curves is also uniform with respect to y ∈ R. Then, a(u, y) is analytic for y and a(u, 0) +is one point. We obtain (3) by Theorem 4.5 and (2). +Now, we verify (∇′ +d/dyu)(y) = (y2/(1+y2))v1(y). Then, we use the fact (2): limx→0 f(x, 0) = +−v1(0). Firstly, we have +(∇′ +d/dyu)(y) = (1 + y2)−3/2 � +(1 + (1 + y2)a2)(X0 +β + yφ) − y(1 + y2)(b2ξ + c2X0 +α) +� +(5.14) +by Lemma 4.2. Next, we define p(x, 0) for each x by the limit p(x, 0) := limy→0(∇′ +d/dyu)(y)/y2. +Then, we have limx→0 p(x, 0) = − limx→0 f(x, 0) = v1(0) by (5.14) and the definition (1.6) +of X0. +In consequence, we obtain the equation desired for all y ∈ R by the continuity of +(∇′ +d/dyu)(y) and v1(y). +□ +Remark 5.4. For the vector ˜u(x) in Theorem 4.6, ⟨u(y), ˜u(x)⟩ = 0 holds as in Remark 4.7-(1). +Hence, ˜u(x) is expressed as a linear combination of X0 +α, f and u2: explicitly, we have +˜u(e−u) = +1 +� +B2 +2 + C2 +2(e−u) +� +−B2X0 +α + +C2 +� +5 + 4y2 +� +u2 − 2 +� +1 + y2 f +�� +(e−u, y), +where B2 and C2 are the functions in Lemma 4.1. Then, since limu→∞ ˜u(e−u) = limu→∞ X0 +α(e−u, y) +by B2(x) < 0 and C2(x) = O(x2), ˜u(e−u) uniformly converges to b(u) = cos( +√ +5u/2)b∞ − +sin( +√ +5u/2)c∞ as u tends to ∞. +For a fixed y, let S2 +y be the 2-sphere in Theorem 4.5 including an x-curve f 0(x, y). Then, +we have Γ(u, y) ⊂ S2 +y. The two points ±(2 +√ +5)−1� +(5 + 4y2)(1 + y2)−1v1(y) + A(y) are the +antipodal points of S2 +y to each other and the tangent spaces at these points are spanned by the +vectors b∞ and c∞. In fact, we have limu→∞ f(e−u, y) = a(u, y) and ⟨a(u, y), v1(y)⟩ < 0, and +further ⟨v1(y), v2(u)⟩ ≡ 0 holds for any u ∈ R. +Now, for each y ∈ R and 0 < ε < 1, let ly(x) be an x-curve given by ly(x) := f 0(x, y) for +x ∈ [ε, 1] and L(ly) be the length of ly(x) by the metric g0. As ε → 0, L(ly) diverges to ∞ if +y ̸= 0 and L(ly=0) converges to a finite value. Then, we have the following corollary: +Corollary 5.5. Let {(xn, yn)}∞ +n=1 be a sequence in D convergent to the origin (0, 0) with respect +to the Euclidean distance of R2. Then, the sequence f 0(xn, yn) converges to the point Γ(u, 0) = + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +29 +−(1/2)v1(0) + A(0). That is, when we define f 0(0, 0) := Γ(u, 0), f 0(x, y) on D extends to +D ∪ {(0, 0)} continuously in this sense. +Proof. Let (xn, yn) ∈ D be a convergent sequence to (0, 0). Let un = exp(1/xn) and S1 +y := +{Γ(u, y)|u ∈ R}, where S1 +0 is one point p0 := Γ(u, 0). We define a kind of distance ¯d(S1 +y, p0) +between S1 +y and p0 by ¯d(S1 +y, p0) := Max{d(p, p0) | p ∈ S1 +y} for the distance d of R4. Then, +there is a number N1 for any ε > 0 such that ¯d(S1 +yn, p0) < ε holds for n > N1, because +S1 +y degenerates to p0 continuously as y → 0. Furthermore, f 0(e−un, y) converges to Γ(un, y) +uniformly with respect to y ∈ R as n → ∞. Hence, there is a number N2 for any ε > 0 such +that ∥f 0(e−un, y) − Γ(un, y)∥ < ε holds for any y ∈ R if n > N2. In consequence, for any ε > 0 +and n > Max{N1, N2} we have +∥f 0(e−un, yn) − p0∥ < ∥f 0(e−un, yn) − Γ(un, yn)∥ + ¯d(S1 +yn, p0) < 2ε, +which shows that the corollary holds. +□ +Corollary 5.6. The vector u(y) determines an orthonormal pair (˜b +∞, ˜c∞) of constant vectors +uniquely such that it satisfies the following conditions (1) and (2): +(1) The system [b∞, c∞, ˜b +∞, ˜c∞] is an othonormal base of R4. +(2) u(y) and v1(y) satisfy the following equations: +u(y) = − sin(y − arctan y) ˜b +∞ + cos(y − arctan y) ˜c∞, +v1(y) = − cos(y − arctan y) ˜b +∞ − sin(y − arctan y) ˜c∞. +In particular, as y → ∞, u(y) and v1(y), respectively, converge uniformly to the following +circles ˜b(y) and ˜c(y): +˜b(y) := cos y ˜b +∞ + sin y ˜c∞, +˜c(y) := − sin y ˜b +∞ + cos y ˜c∞. +Proof. The unit vector u(y) belongs to the plane perpendicular to b∞ and c∞, and it satisfies +the equation (∇′ +d/dyu)(y) = (y2/(1 + y2))v1(y). +□ +Now, for the function b1(x, y) in Lemma 4.2, we have b1(x, y) → − +√ +5x/(2X0 − xX′ +0) as +x → ∞ for any fixed y. Here, x/(2X0 − xX′ +0) for x > 0 is a positive and oscillating function +satisfying (x/(2X0 − xX′ +0))(0) = 0 and x/(2X0 − xX′ +0) = O(1) as x → ∞, by Propositions 2.2 +and 2.4. Hence, the integral w(x) := +√ +5 +� x +0 [x/(2X0 − xX′ +0)]dx is an increasing function with +w(x) = O(x) as x → ∞. Then, we have the following theorem: +Theorem 5.7. Let n(x, y) := (1 + y2)−1/2(X0 +β + yφ)(x, y). Then, we have the following facts +as x → ∞: +(1) Each x-curve n(x, y) for x ∈ [1, ∞) with fixed y has infinite length and the curve +approaches the point −v1(y) while winding uniformly around the point. Then, the curve +never reaches −v1(y) at any finite x. +(2) There is a constant orthonormal base [ˆb, ˆc] of the plane spanned by b∞ and c∞ such that +all x-curves ξ(x, y) and X0 +α(x, y) with y, respectively, converge uniformly to the circles +expressed as ¯b(x) = cos w(x) ˆb − sin w(x) ˆc and ¯c(x) = sin w(x) ˆb + cos w(x) ˆc. +(3) Each x-curve f 0(x, y) converges uniformly to the following circle of S2 +y: +−(2 +√ +5)−1(1 + y2)−1/2 � +v1(x) + 2(1 + y2)1/2 ¯b(x) +� ++ A(y). +The convergence for the x-curves in (2) and (3) is uniform in the wider sense with respect to +y ∈ R. + +30 +N. MATSUURA AND Y. SUYAMA +Proof. We consider the x-curves n(x, y) as x → ∞. Before the proof, we note that the vector +n(x, y) is perpendicular to u(y) for any x. The Propositions 2.2 and 2.4 are important in +the proof, which give the property of the function X0(x). Firstly, we verify the lemma for an +arbitrarily fix y. +For (1), we firstly have the following equation for y ̸= 06, +∥v1(y) + n(x, y)∥ = O(1/x) +(5.15) +by (5.14) and (∇′ +d/dyu)(y) = (y2/(1+y2))v1(y). Hence, the x-curve n(x, y) converges uniformly +to the point −v1(y) as x → ∞. Then, the equation n(x, y) = −v1(y) does not holds at any +finite x, since we have b2(x, y) ̸= 0 for (x, y) ∈ D in (5.14). Note that the convergence of the +x-curve is also uniform with respect to y of any bounded interval [−y1, y1], since (5.15) holds +uniformly for y ∈ [−y1, y1]. Next, we have +∇′ +∂/∂xn = (1 + y2)1/2c1X0 +α, +∇′ +∂/∂xX0 +α = −(1 + y2)1/2c1(x)n − b1ξ, +∇′ +∂/∂xξ = b1X0 +α +(5.16) +by Lemma 4.2. The length L(x) on [1, x] of the x-curve n(x, y) diverges to ∞ as x → ∞, from +(∇′ +∂/∂xn)(x, y) = O(1/x) by Proposition 2.4. +Now, we regard n(x, y) as an analytic curve on the unit 2-sphere S2, and then X0 +α and ξ +are the unit tangent vector and the unit normal vector of n(x, y), respectively. The curvature +κ(x, y) = (1 + y2)−1/2(b1/c1) of n(x, y) diverges to ∞ as x → ∞, by κ(x, y) = O(x). +In +consequence, when we take the geodesic mx on S2 for each x connecting n(x, y) with −v1(y) +and put t1 := inf{x > t0 | n(x, y) ∈ mt0} for x = t0, the length L(t1) − L(t0) of the curve +n(x, y) (t0 ≤ x ≤ t1) converges to 0 and further the curvature of the curve n(x, y) (t0 ≤ x ≤ t1) +is almost constant κ(t0, y), which diverges to ∞ as t0 → ∞. +This fact implies that, as x +increases, the curve n(x, y) gradually approaches a smaller and smaller nearly circle of central +axis −v1(y), which shows (1). +For (2): Firstly, note that every tangent plane at n(x, y) converges to T−v1(y)S2 uniformly as +x → ∞, by (1). Next, we regard the unit normal ξ(x, y) and the unit tangent vector X0 +α(x, y) +of the curve n(x, y) as the vectors at −v1(x) by the parallel translation of R4. Then, these +vectors ξ(x, y) and X0 +α(x, y) converge uniformly to the curves parametrized by x on the unit +circle of T−v1(y)S2 by (1), of which curves we denote by ¯b(x, y) and ¯c(x, y), respectively. Then, +we have ∇′ +∂/∂x¯b = −w′(x)¯c and ∇′ +∂/∂x¯c = w′(x)¯b from the definition of w(x) and (5.16) by +c1 = O(1/x), which implies the fact (2) for each y. That is, there is an orthonormal pair +(ˆb(y), ˆc(y)) of vectors depending on y and perpendicular to u(y) and v1(y) such that +¯b(x, y) = cos w(x) ˆb(y) − sin w(x) ˆc(y), +¯c(x, y) = sin w(x) ˆb(y) + cos w(x) ˆc(y) +hold. The fact (3) follows from (1) and (2) directly by f(x, y) = (5 + 4y2)−1/2[n − 2(1 + +y2)1/2ξ](x, y) in Theorem 4.5. +Finally, we obtain that ˆb(y) and ˆc(y) are constant vectors. In fact, as x → ∞, ∇′ +∂/∂yξ and +∇′ +∂/∂yX0 +α converges to zero-vector uniformly with respect to y of any bounded interval [−y1, y1], +by Lemma 4.2 and Proposition 2.4. +In consequence, we have completed the proof. +□ +6For y = 0, see the proof of Theorem 5.3. + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +31 +Figure 4. These show the x-curve n1/20(x, 1) on [1/500, 100]: the right hand- +side is a bird’s-eye view of the left. +Figure 5. These show the x-curve ξ1/20(x, 1) on [1/500, 100]: the right hand- +side is a bird’s-eye view of the left. +Next, we study the y-curves f 0(x, y) with fixed x as y tends to ∞. +Along any y-curve +f 0(x, y), we have an orthonormal frame field [˜u(x), X0 +β(x, y), ˜f(x, y), φ(x, y)] as in (4.11): any +y-curve belongs to an affine hyperplane R3 +x perpendicular to ˜u(x), where (˜u(x), ˜f(x, y)) is +defined in Theorem 4.6. In particular, the vector ˜f = (B2 +2 + C2 +2)−1/2(C2X0 +α + B2ξ) is analytic +and (B2 +2 + C2 +2)(x) = (X′ +0/x)(2X0 − xX′ +0 − X′ +0/x + +√ +5) > 0 holds. By Lemma 4.2, we have +∇′ +∂/∂y ˜f =: −˜v3X0 +β, +∇′ +∂/∂yφ = −a2X0 +β =: ˜v2X0 +β, +∇′ +∂/∂yX0 +β = ˜v3˜f − ˜v2φ, +where ˜v2(x, y) = h−1(x, y)[2X0+ +√ +5−(x2+y2)(X′ +0/x)], ˜v3(x, y) = −2(y/h(x, y))(B2 +2+C2 +2)1/2(x). + +32 +N. MATSUURA AND Y. SUYAMA +Lemma 5.8. +(1) For an arbitrarily fixed x1 > 0, let (0, x1] be the bounded interval. Then, +there is a constant C > 0 independent of x ∈ (0, x1] such that |˜v2(x, y) + 1| < Cy−2 and +|˜v3 + q(x)/y| < Cy−3 hold for x ∈ (0, x1] as y tends to ∞, where +q(x) := 2(x/X′ +0)(B2 +2 + C2 +2)1/2(x). +(2) For arbitrarily fixed two numbers x1 > x0 > 0, let [x0, x1] be the bounded interval. Then, +there is a constant C > 0 independent of x ∈ [x0, x1] such that +∥(∇′ +∂/∂x˜u)(x) − T(x)˜f(x, y)∥ < C/y, +∥(∇′ +∂/∂x˜f)(x, y) + T(x)˜u(x)∥ < C/y +hold for x ∈ [x0, x1] as y tends to ∞, where +T(x) := (X′ +0 + 2 +√ +5x)(2X0 + +√ +5)/[4xX′ +0(2X0 − xX′ +0 − X′ +0/x + +√ +5)] +given in Remark 4.7. +Proof. We have h(x, y) = 2X0 + +√ +5 + (−x2 + y2)(X′ +0/x) = 2X0 − xX′ +0 + y2(X′ +0/x) + +√ +5 > +√ +5 +and B2 +2 + C2 +2 = (X′ +0/x)(2X0 − xX′ +0 − X′ +0/x + +√ +5) > 0. Here, X′ +0/x is a bounded positive +functions on R satisfying (X′ +0/x)(0) = 2, and 2X0 − xX′ +0 is also a positive function satisfying +(2X0 − xX′ +0)(0) = 5 and (2X0 − xX0)(x) = O(x) as x → ∞, by Propositions 2.2 and 2.4. Now, +we have +1/h(x, y) = (1/y2)(x/X′ +0) − (x/X′ +0)(2X0 − xX′ +0 + +√ +5)/(y2h(x, y)). +The fact (1) follows from the equation. For (2), we have +T(x) = [(B2 +2 + C2 +2)−1(−B′ +2C2 + B2C′ +2) + +√ +5/X′ +0](x). +Then, we obtain (2) from Lemma 4.2 by direct calculation. +□ +In the study the case of y-curves f 0(x, y) as y → ∞, we can not adopt the way of the proof +for Theorem 5.3 by the fact ˜v3(x) → q(x)/y in Lemma 5.8-(1). +Now, for the function T(x) in Lemma 5.8, we define its integral ˜w(x) by ˜w(x) := ( +√ +5/2) log x+ +� x +0 [T(x) − ( +√ +5/2)(1/x)]dx, since T(x) = ( +√ +5/2)(1/x) + O(x) as x → 0 and T(x) = O(1) as +x → ∞ by the definition of X0(x) and Propositions 2.2 and 2.4. +For the function ˜w(x) and the orthonormal frame [b∞, c∞, ˜b +∞, ˜c∞] in Theorem 5.3 and Corol- +lary 5.6, we have the following theorem: +Theorem 5.9. There is a unit vector ˜v1(x) of x ∈ (0, ∞) such that it satisfy the following +facts (1)–(3): +(1) The vector ˜v1(x) is uniquely determined by the function ˜w(x) and the equations (∇′ +d/dx˜u)(x) = +T(x)˜v1(x) and (∇′ +d/dx˜v1)(x) = −T(x)˜u(x): we have +˜u(x) = cos ˜w(x) b∞ + sin ˜w(x) c∞, +˜v1(x) = − sin ˜w(x) b∞ + cos ˜w(x) c∞. +(2) As y tends to ∞, each y-curve f 0(x, y) with fixed x converges to the point (B2 +2 + +C2 +2)−1/2(x)˜v1(x) + ˜A(x). +(3) As y tends to ∞, every y-curve X0 +β(x, y) (resp. every y-curve φ(x, y)) with x uniformly +converges to the circle ˜b(y) = cos y ˜b +∞ + sin y ˜c∞ (resp. −˜c(y) = sin y ˜b +∞ − cos y ˜c∞). +Furthermore, the convergence for these y-curves in (2) and (3) are uniform in the wider sense +with respect to x ∈ (0, ∞). +Proof. Let (0, x1] or [x0, x1] be an arbitrarily fixed bounded interval, as in Lemma 5.8. +For (1) and (2): There is a vector ˜v1(x) := limy→∞ ˜f(x, y) on [x0, x1] by Lemma 5.8-(2) +such that (∇′ +d/dx˜u)(x) = T(x)˜v1(x) and (∇′ +d/dx˜v1)(x) = −T(x)˜u(x) hold. Then, since [x0, x1] + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +33 +is any bounded interval in (0, ∞), these equations hold on (0, ∞). Hence, there is a constant +orthonormal base [b, c] of the plane spanned by b∞ and c∞ such that +˜u(x) = cos ˜w(x) b + sin ˜w(x) c, +˜v1(x) = − sin ˜w(x) b + cos ˜w(x) c +hold, since ˜u(x) and u(y) are perpendicular to each other for all (x, y) ∈ D as mentioned in Re- +mark 4.7-(1). Then, ˜u(x) converges uniformly to the circle cos(( +√ +5/2) log x)b+sin(( +√ +5/2) log x)c +as x → 0. On the other hand, the circle is expressed as cos( +√ +5u/2)b∞ − sin( +√ +5u/2)c∞ by Re- +mark 5.4. Hence, we obtain b = b∞ and c = c∞ by u = − log x, which shows (1). Furthermore, +since any y-curve f 0(x, y) is given by (B2 +2 + C2 +2)−1/2(x)˜f(x, y) + ˜A(x), we obtain (2). +For (3): For x ∈ [x0, x1], we have u(y) = (1 + y2)−1/2(yX0 +β − φ)(x, y) = X0 +β(x, y) + O(1/y) +and (∇′ +d/dyu)(y) = −φ(x, y) + O(1/y) as y → ∞. Furthermore, by Corollary 5.6, u(y) and +v1(y) converge uniformly to the following circles as y → ∞: +u(y) → ˜b(y) = cos y ˜b +∞ + sin y ˜c∞, +v1(y) → ˜c(y) = − sin y ˜b +∞ + cos y ˜c∞. +Hence, we obtain (3). In consequence, the proof has been completed. +□ +Corollary 5.10. We have limy→∞ f 0(x, y) = limy→−∞ f 0(x, y). +Proof. For the sake of simplicity, we translate the curvature surface f 0(x, y) such that the +obtained surface f 0(x, y) satisfies f 0(p0) = 0 for some point p0 = (x0, 0), which means 0 ∈ R3 +y=0, +and take a coordinate system of R4 satisfying the setting in Corollary 4.10 for the new f 0(x, y): +F 0(p0) = Id and [X0 +α, X0 +β, ξ](p0) is a frame of R3 +y=0. Then, for the reflection B in Corollar 4.10, we +have B[φ, X0 +α, X0 +β, ξ](x, −y) = [−φ, X0 +α, X0 +β, ξ](x, y), B(f 0(x, −y)) = f 0(x, y), B(˜u(x)) = ˜u(x) +and B( ˜A(x)) = ˜A(x). Furthermore, we have B(˜f(x, −y)) = ˜f(x, y) by the definition of ˜f(x, y). +Now, we denote by ˆf 0(x, y)(= f 0(x, y)) (y ≥ 0) the curvature surface determined by the +frame field [−φ, X0 +α, X0 +β, ξ](x, y). Then, we have limy→∞ ˆf 0(x, y) = limy→∞ f 0(x, y) = (B2 +2 + +C2 +2)−1/2(x)˜v1(x) + ˜A(x) by Theorem 5.9. In consequence, we obtain limy→∞ f 0(x, −y) = (B2 +2 + +C2 +2)−1/2(x)˜v1(x) + ˜A(x). +In fact, we have B( ˆf 0(x, y)) = f 0(x, −y), B( ˜A(x)) = +˜A(x) and +B(˜v1(x)) = ˜v1(x) by B(∇′ +d/dx˜u(x)) = ∇′ +d/dx˜u(x). +□ +By Theorem 5.7 and Corollary 5.6, for every x ∈ (0, ∞), the tangent space at limy→∞ f 0(x, y) +in every 2-sphere S2 +x, which contains the y-curve f 0(x, y), is spanned by the constant vectors +˜b +∞ and ˜c∞. +Now, for the R4-valued functions A(y) and ˜A(x) in Theorems 4.5 and 4.6, we have the +following facts: +Corollary 5.11. (1) The curve A(y) lies on a plane spanned by ˜b∞ and ˜c∞. +(2) The curve ˜A(x) lies on a plane spanned by b∞ and c∞. +Proof. (1) A curvature surface f 0(x, y) on D is expressed as +f 0(x, y) = (2 +√ +5(1 + y2))−1 � +X0 +β − 2(1 + y2)ξ + yφ +� ++ A(y), +(as a family of x-curves f 0(x, y) with y). The vector ˜u(x) = (B2 +2 +C2 +2)−1/2(−B2X0 +α +C2ξ)(x, y) +moves on the circle of a plane spanned by b∞ and c∞. Hence, we have only to show that +⟨A′(y), (−B2X0 +α + C2ξ)(x, y)⟩ = 0 holds for any (x, y). Now, by f 0 +y (x, y) = (2y/h(x, y))X0 +β, we +have A′(y) = (2 +√ +5(1 + y2))−1(b2ξ + c2X0 +α)(x, y) except for the terms of X0 +β(x, y) and φ(x, y). +Then, we have −c2B2 + b2C2 = 0 by Lemmata 4.1 and 4.2, which implies ⟨A′(y), (−B2X0 +α + +C2ξ)(x, y)⟩ = 0. +(2) A curvature surface f 0(x, y) and the vector u(y) are expressed as +f 0(x, y) = (B2 +2 + C2 +2)−1(x)(B2ξ + C2X0 +α)(x, y) + ˜A(x), +u(y) = (1 + y2)−1/2(yX0 +β − φ)(x, y), + +34 +N. MATSUURA AND Y. SUYAMA +and u(y) moves on the circle of a plane spanned by ˜b +∞ and ˜c∞. We have only to show that +⟨ ˜A′(x), (yX0 +β − φ)(x, y)⟩ = 0 holds for any (x, y). We have +˜A′(x) = C2(x)(B2 +2 + C2 +2)−1(x) +� +−a1φ + c1X0 +β +� +(x, y) +except for the terms of X0 +α(x, y) and ξ(x, y). Then, we have a1 + yc1 = 0 by Lemma 4.2, which +implies that the desired equation holds. +□ +(a) y-curve at x0 = 2. +(b) y-curve at x0 = 5. +Figure 6. These are the y-curves f 1/20(x0, y) on [−100, 100]. Each curve is in +a sphere S2, and has a cusp at y = 0. We change its color from gray to black at +y = 0. As for the asymptotic behaviors, each curve converges to one point in S2 +as y → ±∞. +Figure 7. This shows the x-curve f 1/20(x, 2) on [1/500, 100]: the figure on the +right hand-side is a side view of the figure on the left. This curve is in a sphere +S2, and has a cusp at x = 2. We change its color from black to gray at x = 2. +As for the asymptotic behaviors, the curve converges uniformly to parallel small +circles in S2 as x → 0 and x → ∞. + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +35 +Figure 8. This shows the x-curve f 1/20(x, 0) on [1/100, 100]: the figure on the +right hand-side is a side view of the figure on the left. The curve is in a sphere +S2, and converges to a point in S2 as x → 0, and uniformly to a small circle in +S2 as x → ∞. This curve has no cusp. +In consequence of the all results above, we have obtained the simple structure on the curvature +surface f 0(x, y) on D, as mentioned in the introduction: in a sense, f 0(0, y) is a curve on the +plane {˜b∞, ˜c∞}R and f 0(x, ∞) = f 0(x, −∞) is a curve on the plane {b∞, c∞}R; these planes +are orthogonal to each other. +At the end of this section, we study a curvature surface defined on D(−) = {(x, y) | x < 0} in +relation to a curvature surface f 0(x, y) on D = {(x, y) | x > 0}. The singular metric g0 in (3.1) +is defined on D ∪ {(0, 0)} ∪ D(−), and the map j : D ∋ (x, y) ↔ (−x, y) ∈ D(−) is an isometry +between the spaces (D, g0|D) and (D(−), g0|D(−)). Furthermore, the coordinate system (ˆx, ˆy) +in §3, ˆx = (1/3)x3 − xy2 and ˆy = y2, is regular not only for the metric g0|D but also for the +metric g0|D(−). Although ¯P in (2.2) is a negative function on D(−), the equations of Lemma +4.1 and Lemma 4.2 also hold on D(−) for Φ in (2.1) and Ψ in (2.2), and hence these equations +also determine a curvature surface ˆf 0(x, y) defined on D(−). +Now, the following lemma is verified in the same way as the proof of Corollary 4.10. +Lemma 5.12. Let F 0(x, y) = [φ, X0 +α, X0 +β, ξ](x, y) be a frame field on D satisfying the equations +of Lemma 4.2. Then, for x > 0, the frame field ˆF 0(−x, y) := F 0(x, y) also satisfies the equations +in Lemma 4.2 on D(−). +By Lemma 5.12, a curvature surface ˆf 0(x, y) on D(−) is obtained by ˆf 0(−x, y) := f 0(x, y) +from a curvature surface f 0(x, y) on D determined by F 0(x, y). We regard the surface ˆf 0(x, y) +on D(−) as the back side of the surface f 0(x, y) on D. Then, since f 0(0, 0) = −(1/2)v1(0)+A(0) +holds by the continuity of f 0(x, y) in the sense at Corollary 5.5, we have f 0(0, 0) = ˆf 0(0, 0) and +limx→0 f 0(x, y) = limx→0 ˆf 0(x, y) for any y. In consequence, we could say that the curvature +surface formed by both sides of f 0(x, y) is a natural realization in R4 of the singular space +(D ∪ {(0, 0)} ∪ D(−), g0). +Corollary 5.13. Let f 0(x, y) be a curvature surface on D. Then, the surface formed by both +sides of f 0(x, y) is a curvature surface on D ∪ {(0, 0)} ∪ D(−) with the metric g0. +6. Approximation of frame field determining extended curvature surface +Let F 0(x, y) = [φ, X0 +α, X0 +β, ξ](x, y) be an orthonormal frame field determining a curvature +surface f 0(x, y) on D. In this section, we construct an approximation of F 0(x, y) on a compact +square E := [x0, x0 + a] × [y0, y0 + a] ⊂ D. Then, we regard φ(x, y) as a (singular) surface +in the standard unit 3-sphere S3: X0 +α(x, y) and X0 +β(x, y) are the principal curvature directions + +36 +N. MATSUURA AND Y. SUYAMA +and ξ(x, y) is a normal vector field of φ. With an integer n > 0, we divide [x0, x0 + a] and +[y0, y0 + a] into n sub-intervals of equal length δn = a/n: let x0 < x1 < · · · < xn = x0 + a and +y0 < y1 < · · · < yn = y0 + a. On each edge [xi, xi+1] × {yj} or {xi} × [yj, yj+1], we approximate +each vector of F 0(x, y) by a certain rational curve. +Then, independently of the width δn, +the approximation F δn(x, y) on the lattice in E made by the divisions is also an orthonormal +frame field and the approximation of each coordinate line in f 0(x, y) determined from F δn(x, y) +also lies on a 2-sphere. Furthermore, under an additional consideration, such approximations +F δn(x, y) with n extend to each point in E and the sequence F δn(x, y) converges to F 0(x, y) +uniformly on E as n → ∞. +Now, any orthonormal frame field F 0(x, y) = [φ, X0 +α, X0 +β, ξ](x, y) satisfies the structure equa- +tion dF 0 = F 0Ω in (4.6) on D, where Ω is the differential 1-form determined by (4.5) and φ +is a surface in S3 (if a1a2 ̸= 0 in (6.1) below) with a curvature line coordinate system (x, y). +Then, we have the following equations for F 0(x, y): +dφ = −(a1dx)X0 +α − (a2dy)X0 +β, +dξ = (b1dx)X0 +α + (b2dy)X0 +β, +∇′ +∂/∂xX0 +β = c1X0 +α, +∇∂/∂yX0 +α = c2X0 +β, +(6.1) +where ai, bi, ci are the analytic functions on D in Lemma 4.2. We have the following equations +by the Gauss and the Codazzi equations for φ(x, y). +Lemma 6.1. The functions ai, bi and ci satisfy the following equations on D: +(a1)y = a2c1, +(a2)x = a1c2, +(b1)y = b2c1, +(b2)x = b1c2, +(c2)x + (c1)y + a1a2 + b1b2 = 0. +Proof. These equations are equivalent to the Maurer-Cartan equation dΩ + Ω ∧ Ω = 0. Here, +we give another proof. The first four equations follow from (6.1) by d(dφ) = d(dξ) = 0. For +the last equation, suppose firstly that φ is a surface in S3. Then, the Gauss curvature Kφ of φ +is given by +Kφ = −(a1a2)−1� +((a2)x/a1)x + ((a1)y/a2)y +� += 1 + λ1λ2, +where λ1 := b1/a1 and λ2 := b2/a2 are the principal curvatures of φ. Thus, the lemma holds +good if φ is a surface. Next, since the functions ai, bi, ci are analytic on D and the domain for +φ to be a surface is open, the last equation also holds on D. +□ +Now, for a, x0 > 0 and y0 ∈ R, let [x0, x0 + a] and [y0, y0 + a] be the closed intervals: the +domain E := [x0, x0 + a] × [y0, y0 + a] ⊂ D is compact. From now on, we write xe := x0 + a and +ye := y0 + a. For the domain E, let us fix an orthonormal frame F 0(x0, y0) at P0,0 arbitrarily. +For a point (xp, yp) ∈ E and an integer n > 0, we put Ep := [x0, xp] × [y0, yp] and δn := a/n. +Definition 6.2 (Division of the domain Ep and Path). (1) For a point (xp, yp) ∈ E, we divide +the intervals [x0, xp] and [x0, yp], respectively, into sub-intervals of equal length: +x0 < x1 < · · · < xs = xp, +y0 < y1 < · · · < yt = yp. +Here, we take the integers s and t such that 0 < s, t ≤ n and (xp −x0)/s ≤ δn, (yp −y0)/t ≤ δn. +Then, we denote ∆n +i,j := [xi, xi+1] × [yj, yj+1] (⊂ Ep). +(2) For two lattice points Pi,j := (xi, yj) and Pi+k,j+l := (xi+k, yj+l) in a division of Ep, where +k ≥ 0 and l ≥ 0, let m be a polygonal line in the division connecting the two points. We +express m as +m : Pi,j → · · · → Pa,b → Pc,d → · · · → Pi+k,j+l +by pointing to lattice points through which m passes, in order. Then, we call m a path from +Pij to Pi+k,j+l if c ≥ a and d ≥ b are satisfied. + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +37 +Now, let us fix a division of Ep for (xp, yp) ∈ E. From F 0(x0, y0), we construct an approxi- +mation F δn(x, y) of F 0(x, y) on any path from P0,0 to (xp, yp). At the beginning, we state the +program for the construction of F δn(x, y). +Step A: We take a sub-domain ∆n +i,j arbitrarily, and suppose that an orthonormal frame +F δn(xi, yj) at Pi,j is determined. +(1) We construct an orthonormal frame field F δn(x, yj) on [xi, xi+1] × {yj} from F δn(xi, yj). +(2) We construct an orthonormal frame field F δn(xi, y) on {xi} × [yj, yj+1] from F δn(xi, yj). +In the constructions of F δn(xi+1, yj) and F δn(xi, yj+1) in (1) and (2), suppose that we +start from the other orthonormal frame ¯F δn(xi, yj) at Pi,j. Then, we obtain distinct frames +¯F δn(xi+1, yj) and ¯F δn(xi, yj+1) from F δn(xi+1, yj) and F δn(xi, yj+1), respectively. In this case, +these frames will satisfy the equations +∥(F δn − ¯F δn)(xi, yj)∥ = ∥(F δn − ¯F δn)(xi+1, yj)∥ = ∥(F δn − ¯F δn)(xi, yj+1)∥, +(6.2) +where ∥A∥ = +��(aij)2 for a square matrix A = [aij], as in the previous section. +(3) From F δn(xi+1, yj) obtained by (1), we firstly have an orthonormal frame field F δn(xi+1, y) +on {xi+1}×[yj, yj+1] by (2): we denote by F δn(xi+1, yj+1) the frame at Pi+1,j+1 determined in this +way. Next, from F δn(xi, yj+1) obtained by (2), we have an orthonormal frame field F δn(x, yj+1) +on [xi, xi+1] × {yj+1} by (1): we denote by F δn(xi+1, yj+1) the frame at Pi+1,j+1 determined in +this way. That is, the two frames F δn(xi+1, yj+1) and F δn(xi+1, yj+1) are determined at Pi+1,j+1. +Then, although F δn(xi+1, yj+1) and F δn(xi+1, yj+1) do not coincide, we shall have the following +inequality, +∥(F δn − F δn)(xi+1, yj+1)∥ ≤ K(δn)3, +(6.3) +if n is sufficiently large, where K is a constant on E determined independently of n (see +Definition 6.11 below). In the proof of (6.3), we shall use Lemma 6.1 essentially. +Step B: Let Pi,j be a lattice point of Ep and m be a path from P0,0 to Pi,j. By Step A, we +have an orthonormal frame field F δn on the path m. +In the following proposition, we fix F 0(x0, y0) arbitrarily. +Proposition 6.3. Let P := (xp, yp) ∈ E. For an integer n, we take a division of the domain +Ep. Let mi (i = 1, 2) be two paths from P0,0 to P in the division, and F δn +i (xp, yp) (i = 1, 2) be +the frames at P determined from mi, respectively. Then, we have +∥(F δn +1 − F δn +2 )(xp, yp)∥ ≤ Ka2δn, +if n is sufficiently large. +Proof. In this proof, we suppose that (6.2) and (6.3) hold good and that n is sufficiently large. +Now, under the assumption that a frame F δn(xi, yj) is determined at a point Pi,j (i ≤ s−2, j ≤ +t − 1), we study the frame at Pi+2,j+1. In this case, there are three paths mi (i = 1, 2, 3) from +Pi,j to Pi+2,j+1. For each mi, we point to the lattice points only on the way, and then we have +m1 : Pi+1,j → Pi+2,j, +m2 : Pi+1,j → Pi+1,j+1, +m3 : Pi,j+1 → Pi+1,j+1. +Let F δn +i (xi+2, yj+1) be the frames at Pi+2,j+1 determined from mi, respectively. Then, we firstly +have ∥(F δn +2 − F δn +1 )(xi+2, yj+1)∥ ≤ Kδ3 +n and ∥(F δn +3 − F δn +2 )(xi+2, yj+1)∥ ≤ Kδ3 +n by (6.2) and (6.3). +Furthermore, the closed polygonal line m3 − m1 includes two sub-domains ∆n +i,j and ∆n +i+1,j, and +we have +∥(F δn +3 − F δn +1 )(xi+2, yj+1)∥ +≤ ∥(F δn +3 − F δn +2 )(xi+2, yj+1)∥ + ∥(F δn +2 − F δn +1 )(xi+2, yj+1)∥ ≤ 2Kδ3 +n. + +38 +N. MATSUURA AND Y. SUYAMA +Next, we arbitrarily take two paths mi from P0,0 to P = (xp, yp). Then, the closed polygonal +line m1 − m2 includes at most n2 sub-domains ∆n +k,l. In consequence, we obtain the lemma by +(6.2), (6.3) and the above fact. +□ +Step C: For a given integer n and a point (xp, yp) ∈ E, we take a division of Ep. For the +division, let m and m be two paths defined by +m :P0,0 = (x0, y0) → Ps,0 = (xp, y0) → Ps,t = (xp, yp), +(6.4) +m :P0,0 = (x0, y0) → P0,t = (x0, yp) → Ps,t = (xp, yp). +(6.5) +Let F δn(xp, yp) and F δn(xp, yp) be the frames at (xp, yp) determined from m and m, respectively. +Then, we shall verify that F δn(xp, yp) converges to F 0(xp, yp) uniformly for any (xp, yp) ∈ E +as n tends to ∞, where F 0(x, y) is the frame field with a given F 0(x0, y0). In consequence, +F δn(xp, yp) also converges to the frame F 0(xp, yp) uniformly for any (xp, yp) ∈ E as n tends to +∞, since ∥(F δn − F δn)(xp, yp)∥ ≤ Ka2δn holds by Proposition 6.3. +Now, we verify Steps A and C. In these proofs, we fix an integer n, and study only the case +of E = [x0, xe] × [y0, ye] for Ep, by Definition 6.2. Hence, we have s = t = n, and denote +δ := δn = a/n, ∆i,j := ∆n +i,j and F δ := F δn. +For Step A: Under the assumption that an orthonormal frame F δ(xi, yj) at Pi,j is determined, +we construct F δ on boundary of the sub-domain ∆i,j according to the ways (1), (2) and (3), +and verify (6.2) and (6.3). Let F δ =: [φδ, Xδ +α, Xδ +β, ξδ]. We simply write (0, 0) with (xi, yj) in +the following Steps A-(1) and A-(2). +Step A-(1): For a given F δ(0, 0), we express F δ(x, yj) for x ∈ [xi, xi+1] as +φδ(x, yj) − φδ(0, 0) := − x−xi +2 a1(0, 0) +� +Xδ +α(0, 0) + Xδ +α(x, yj) +� +, +(6.6) +ξδ(x, yj) − ξδ(0, 0) := x−xi +2 b1(0, 0) +� +Xδ +α(0, 0) + Xδ +α(x, yj) +� +, +(6.7) +Xδ +β(x, yj) − Xδ +β(0, 0) := x−xi +2 c1(0, 0) +� +Xδ +α(0, 0) + Xδ +α(x, yj) +� +, +(6.8) +and find out a suitable Xδ +α(x, yj) from these equations. Now, let sδ +1(x, yj) and tδ +1(x, yj) be the +functions on (x, yj) ∈ [xi, xi+1] × {yj} defined by +sδ +1(x, yj) := ⟨Xδ +α(x, yj), Xδ +α(0, 0)⟩, +tδ +1(x, yj) := +� +1 + sδ +1(x, yj) +� +/2, +respectively. +Lemma 6.4. Suppose that the frame field F δ(x, yj) in (6.6)–(6.8) is orthonormal at each point +(x, yj) ∈ [xi, xi+1] × {yj}. Then, for x ∈ [xi, xi+1] we have the following equations: +⟨Xδ +α(x, yj), φδ(0, 0)⟩ = −⟨φδ(x, yj), Xδ +α(0, 0)⟩ = a1(0, 0)(x − xi)tδ +1(x, yj), +⟨Xδ +α(x, yj), ξδ(0, 0)⟩ = −⟨ξδ(x, yj), Xδ +α(0, 0)⟩ = −b1(0, 0)(x − xi)tδ +1(x, yj), +⟨Xδ +α(x, yj), Xδ +β(0, 0)⟩ = −⟨Xδ +β(x, yj), Xδ +α(0, 0)⟩ = −c1(0, 0)(x − xi)tδ +1(x, yj), +sδ +1(x, yj) = 4 − (x − xi)2(a2 +1 + b2 +1 + c2 +1)(0, 0) +4 + (x − xi)2(a2 +1 + b2 +1 + c2 +1)(0, 0), +tδ +1(x, yj) = +4 +4 + (x − xi)2(a2 +1 + b2 +1 + c2 +1)(0, 0). +Proof. Let x ∈ [xi, xi+1]. Suppose that F δ(x, yj) in (6.6)–(6.8) is an orthonormal frame field +at each point of [xi, xi+1] × {yj}. Firstly, we verify the first equation under the assumption +a1(0, 0) ̸= 0. By (6.6)–(6.8), we have +− ((x − xi)/2) a1(0, 0) ⟨Xδ +α(x, yj), φδ(0, 0)⟩ += ⟨φδ(x, yj) − φδ(0, 0), φδ(0, 0)⟩ = ⟨φδ(x, yj), φδ(0, 0)⟩ − 1, + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +39 +and +− ((x − xi)/2) a1(0, 0) ⟨φδ(x, yj), Xδ +α(0, 0)⟩ += ⟨φδ(x, yj) − φδ(0, 0), φδ(x, yj)⟩ = 1 − ⟨φδ(x, yj), φδ(0, 0)⟩. +Hence, we have ⟨Xδ +α(x, yj), φδ(0, 0)⟩ = −⟨φδ(x, yj), Xδ +α(0, 0)⟩ by a1(0, 0) ̸= 0. +Furthermore, +since +−((x − xi)/2) a1(0, 0) (1 + sδ +1(x, yj)) = ⟨φδ(x, yj) − φδ(0, 0), Xδ +α(0, 0)⟩ = ⟨φδ(x, yj), Xδ +α(0, 0)⟩, +the first equation is obtained: +⟨Xδ +α(x, yj), φδ(0, 0)⟩ = −⟨φδ(x, yj), Xδ +α(0, 0)⟩ = x−xi +2 a1(0, 0) +� +1 + sδ +1(x, yj) +� +. +In the case a1(0, 0) = 0, the equation is also satisfied from (6.6)–(6.8) by φδ(x, yj) = φδ(0, 0). +In the same way, we can verify the equations for ⟨Xδ +α(x, yj), ξδ(0, 0)⟩ and ⟨Xδ +α(x, yj), Xδ +β(0, 0)⟩. +Next, from these equations verified above, we have +Xδ +α(x, yj) = sδ +1(x, yj)Xδ +α(0, 0) + x−xi +2 +� +1 + sδ +1(x, yj) +� � +a1φδ − b1ξδ − c1Xδ +β +� +(0, 0). +(6.9) +Taking the norm of both sides of the equation, we obtain the equation for sδ +1(x, yj). Then, the +equation for tδ +1(x, yj) follows directly from the definition. +In consequence, we have completed the proof. +□ +The following lemma follows from (6.9) and the definition of tδ +1(x, yj). +Lemma 6.5. Suppose that the frame field F δ(x, yj) in (6.6)–(6.8) is orthonormal at each point +(x, yj) ∈ [xi, xi+1] × {yj}. Then, for x ∈ [xi, xi+1], we have +Xδ +α(0, 0) + Xδ +α(x, yj) = tδ +1(x, yj) +� +2Xδ +α(0, 0) + (x − xi)(a1φδ − b1ξδ − c1Xδ +β)(0, 0) +� +. +By Lemma 6.5, if the frame field F δ(x, yj) in (6.6)–(6.8) is orthonormal at each point (x, yj) ∈ +[xi, xi+1] × {yj}, then we have +Xδ +α(x, yj) − Xδ +α(0, 0) = (x − xi)tδ +1(x, yj) +� +Y δ +1 (0, 0) − x−xi +2 +� +(a2 +1 + b2 +1 + c2 +1)Xδ +α +� +(0, 0) +� +, +(6.10) +where Y δ +1 (0, 0) := (a1φδ − b1ξδ − c1Xδ +β)(0, 0), and +φδ(x, yj) − φδ(0, 0) = −(x − xi)tδ +1(x, yj)a1(0, 0) +� +Xδ +α(0, 0) + x−xi +2 Y δ +1 (0, 0) +� +, +(6.11) +ξδ(x, yj) − ξδ(0, 0) = (x − xi)tδ +1(x, yj)b1(0, 0) +� +Xδ +α(0, 0) + x−xi +2 Y δ +1 (0, 0) +� +, +(6.12) +Xδ +β(x, yj) − Xδ +β(0, 0) = (x − xi)tδ +1(x, yj)c1(0, 0) +� +Xδ +α(0, 0) + x−xi +2 Y δ +1 (0, 0) +� +. +(6.13) +Namely, with Ω1(x, y) in (4.5), +F δ(x, yj) − F δ(0) = (x − xi)tδ +1(x, yj)F δ(0) +� +Ω1(0) + x−xi +2 (Ω1)2(0) +� +holds. Conversely, we have the following theorem. +Theorem 6.6. Let sδ +1(x, yj) and tδ +1(x, yj) be the functions on [xi, xi+1] × {yj} given in Lemma +6.4. Let F δ(xi, yj)(= F δ(0, 0)) be an orthonormal frame. Then, the frame field F δ(x, yj) defined +by (6.10)–(6.13) is orthonormal at each point (x, yj) ∈ [xi, xi+1] × {yj}. Furthermore, for a +transformation AF δ(xi, yj) of F δ(xi, yj) by a special orthogonal matrix A, F δ(x, yj) in (6.10)– +(6.13) changes into AF δ(x, yj). In particular, for the unit matrix Id, we have +∥(A − Id)F δ(xi, yj)∥ = ∥(A − Id)F δ(xi+1, yj)∥ = ∥A − Id∥. + +40 +N. MATSUURA AND Y. SUYAMA +Proof. Note that the equations in (6.10)–(6.13) are induced from (6.6)–(6.8) and (6.9) (or +Lemma 6.5). Now, let the frame F δ(0, 0)(= F δ(xi, yj)) be orthonormal. The theorem follows +from (6.6)–(6.8) and (6.9) by direct calculation as follows. We firstly take the norm of the both +sides in the equation of Lemma 6.5, and then we have +∥Xδ +α(0, 0) + Xδ +α(x, yj)∥2 = 4tδ +1(x, yj). +By the equation, we can show that all vector fields of F δ(x, yj) have unit norm: for example, +by (6.9) we have +∥Xδ +α(x, yj)∥2 = (sδ +1(x, yj))2 + (x − xi)2(tδ +1(x, yj))2(a2 +1 + b2 +1 + c2 +1)(0, 0) += (1 − ((x − xi)/2)2(a2 +1 + b2 +1 + c2 +1)(0, 0))2 + (x − xi)2(a2 +1 + b2 +1 + c2 +1)(0, 0) +(1 + ((x − xi)/2)2(a2 +1 + b2 +1 + c2 +1)(0, 0))2 += 1. +In the same way, we can verify that φδ(x, yj), ξδ(x, yj) and Xδ +β(x, yj) are also unit vectors, by +(6.6)–(6.8) and Lemma 6.5. Next, we have +⟨Xδ +α(0, 0) + Xδ +α(x, yj), (a1φδ − b1ξδ − c1Xδ +β)(0, 0)⟩ = (x − xi)tδ +1(x, yj)(a2 +1 + b2 +1 + c2 +1)(0, 0) +by Lemma 6.5. By the equation, we can show that all vector fields of F δ(x, yj) are orthogonal +to each other: for example, by (6.6)–(6.8) and (6.9) we have +⟨Xδ +α, φδ⟩(x, yj) = − ((x − xi)/2)sδ +1(x, yj)(1 + sδ +1(x, yj))a1(0, 0) ++ (x − xi)tδ +1(x, yj)a1(0, 0) − ((x − xi)3/2)(tδ +1(x, yj))2 � +(a2 +1 + b2 +1 + c2 +1)a1 +� +(0, 0) += − (x − xi)(tδ +1(x, yj))2 � +1 − ((x − xi)/2)2(a2 +1 + b2 +1 + c2 +1)(0, 0) +� +a1(0, 0) ++ (x − xi)(tδ +1(x, yj))2 � +1 + ((x − xi)/2)2(a2 +1 + b2 +1 + c2 +1)(0, 0) +� +a1(0, 0) +− ((x − xi)3/2)(tδ +1(x, yj))2 � +(a2 +1 + b2 +1 + c2 +1)a1 +� +(0, 0) = 0. +The other orthogonality for these vectors is also obtained by (6.6)–(6.8), (6.9) and Lemma 6.5 +in the same way. +The assertion for transformation AF δ(xi, yj) of the initial condition follows directly from +(6.10)–(6.13). In consequence, we have verified the theorem. +□ +Step A-(2): For a given F δ(0, 0), we express the frame field F δ(xi, y) on {xi} × [yj, yj+1] as +a similar form to in (6.6)–(6.8): +φδ(xi, y) − φδ(0, 0) := − y−yj +2 a2(0, 0) +� +Xδ +β(0, 0) + Xδ +β(xi, y) +� +, +(6.14) +ξδ(xi, y) − ξδ(0, 0) := y−yj +2 b2(0, 0) +� +Xδ +β(0, 0) + Xδ +β(xi, y) +� +, +(6.15) +Xδ +α(xi, y) − Xδ +α(0, 0) := y−yj +2 c2(0, 0) +� +Xδ +β(0, 0) + Xδ +α(xi, y) +� +. +(6.16) +Let sδ +2(xi, y) and tδ +2(xi, y) be the functions on {xi} × [yj, yj+1] defined by +sδ +2(xi, y) := ⟨Xδ +β(xi, y), Xδ +β(0, 0)⟩, +tδ +2(xi, y) := +� +1 + sδ +2(xi, y) +� +/2. +Then, we have the following lemma and theorem in same way as in Step A-(1). +Lemma 6.7. Suppose that the frame field F δ(xi, y) in (6.14)–(6.16) is orthonormal on {xi} × +[yj, yj+1]. Then, we have +sδ +2(xi, y) = 4 − (y − yj)2(a2 +2 + b2 +2 + c2 +2)(0, 0) +4 + (y − yj)2(a2 +2 + b2 +2 + c2 +2)(0, 0), +tδ +2(xi, y) = +4 +4 + (y − yj)2(a2 +2 + b2 +2 + c2 +2)(0, 0), +Xδ +β(0, 0) + Xδ +β(xi, y) = tδ +2(xi, y) +� +2Xδ +β(0, 0) + (y − yj)(a2φδ − b2ξδ − c2Xδ +α)(0, 0) +� +. + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +41 +By Lemma 6.7, if the frame field F δ(xi, y) in (6.14)–(6.16) is orthonormal at each point +(xi, y) ∈ {xi} × [yj, yj+1], then we have +Xδ +β(xi, y) − Xδ +β(0, 0) = (y − yj)tδ +2(xi, y) +� +Y δ +2 (0, 0) − y−yj +2 +� +(a2 +2 + b2 +2 + c2 +2)Xδ +β +� +(0, 0) +� +, +(6.17) +where Y δ +2 (0, 0) := (a2φδ − b2ξδ − c2Xδ +α)(0, 0), and +φδ(xi, y) − φδ(0, 0) = −(y − yj)tδ +2(xi, y)a2(0, 0) +� +Xδ +β(0, 0) + y−yj +2 Y δ +2 (0, 0) +� +, +(6.18) +ξδ(xi, y) − ξδ(0, 0) = (y − yj)tδ +2(xi, y)b2(0, 0) +� +Xδ +β(0, 0) + y−yj +2 Y δ +2 (0, 0) +� +, +(6.19) +Xδ +α(xi, y) − Xδ +α(0, 0) = (y − yj)tδ +2(xi, y)c2(0, 0) +� +Xδ +β(0, 0) + y−yj +2 Y δ +2 (0, 0) +� +. +(6.20) +Namely, with Ω2(x, y) in (4.5), +F δ(xi, y) − F δ(0) = (y − yj)tδ +2(xi, y)F δ(0) +� +Ω2(0) + y−yj +2 (Ω2)2(0) +� +holds. Conversely, we have the following theorem. +Theorem 6.8. Let sδ +2(xi, y) and tδ +2(xi, y) be the functions on {xi} × [yj, yj+1] given in Lemma +6.7. Let F δ(xi, yj)(= F δ(0, 0)) be an orthonormal frame. Then, the frame field F δ(xi, y) defined +by (6.17)–(6.20) is orthonormal at each point (xi, y) ∈ {xi} × [yj, yj+1]. Furthermore, for a +transformation AF δ(xi, yj) of F δ(xi, yj) by a special orthogonal matrix A, F δ(xi, y) in (6.17)– +(6.20) changes into AF δ(xi, y). In particular, we have +∥(A − Id)F δ(xi, yj)∥ = ∥(A − Id)F δ(xi, yj+1)∥ = ∥A − Id∥. +Step A-(3): For a sub-domain ∆i,j, the orthonormal frames F δ(xi+1, yj) and F δ(xi, yj+1) +have been determined from F δ(xi, yj). +Hence, we can construct the frames F δ(xi+1, y) on +{xi+1} × [yj, yj+1] and F δ(x, yj+1) on [xi, xi+1] × {yj+1} by (2) and (1) of Step A, respectively. +Thus, we have two frames F δ(xi+1, yj+1) at the point Pi+1,j+1, since F δ(xi+1, yj+1) is defined for +each path from Pi,j to Pi+1,j+1. We study the difference of these two frames and verify (6.3). +In this step, we simply write +0 := (xi, yj), +1 = (xi+1, yj), +2 := (xi, yj+1), +3 := (xi+1, yj+1) +and denote by F δ(3) and F δ(3), respectively, the frames determined by the paths m : 0 → 1 → 3 +and m : 0 → 2 → 3. Note that tδ +1 (resp. tδ +2) is defined on [xi, xi+1] × {yj} and [xi, xi+1] × {yj+1} +(resp. on {xi} × [yj, yj+1] and {xi+1} × [yj, yj+1]). Then, we have +tδ +1(1) = 1/ +� +1 + (δ/2)2(a2 +1 + b2 +1 + c2 +1)(0) +� +, +tδ +2(3) = 1/ +� +1 + (δ/2)2(a2 +2 + b2 +2 + c2 +2)(1) +� +, +tδ +2(2) = 1/ +� +1 + (δ/2)2(a2 +2 + b2 +2 + c2 +2)(0) +� +, +tδ +1(3) = 1/ +� +1 + (δ/2)2(a2 +1 + b2 +1 + c2 +1)(2) +� +, +and hence tδ +i(k) = 1 + O(δ2) hold for the integers i and k above. Furthermore, since ai, bi and +ci are analytic functions on E, we have tδ +1(3) − tδ +1(1) = O(δ3) and tδ +2(3) − tδ +2(2) = O(δ3). +Now, we have +F δ(3) − F δ(0) = (F δ(3) − F δ(1)) + (F δ(1) − F δ(0)), +F δ(3) − F δ(0) = (F δ(3) − F δ(2)) + (F δ(2) − F δ(0)). +We define Gx and Gy by +Gx := (F δ(3) − F δ(2)) − (F δ(1) − F δ(0)), +Gy := (F δ(3) − F δ(1)) − (F δ(2) − F δ(0)). +Then, we have only to verify that Gx = Gy+O(δ3) holds. Hence, we study the third degree for δ +of Gx−Gy: in the asymptotic expansion Gx−Gy ≈ �3 +k=0 pkδk for δ, we show pk = 0 (k = 0, 1, 2). +In the estimate of each Gx and Gy, the frames F δ(3) and F δ(3) are naturally distinguished by +(6.10)–(6.13) and (6.17)–(6.20), and hence we can write F δ(3) with them. +Now, for a function or a vector field k(x, y), we denote [k]1 +0 := k(1) − k(0) and so on. Let +Y δ +1 := a1φδ − b1ξδ − c1Xδ +β and Y δ +2 := a2φδ − b2ξδ − c2Xδ +α. + +42 +N. MATSUURA AND Y. SUYAMA +Lemma 6.9. With the second degree for δ of [Y δ +1 ]2 +0 and [Y δ +2 ]1 +0, we have +[Y δ +1 ]2 +0 ≈ δ +� +c1c2Xδ +α + (c2)xXδ +β +� +(0) + (δ2/2) +� +((c2)x − (c1)y)Y δ +2 − c1(a2 +2 + b2 +2 − c2 +2)Xδ +β +� +(0) ++ (δ2/2) +� +(a1)yyφδ − (b1)yyξδ − (c1)yyXδ +β +� +(0), +[Y δ +2 ]1 +0 ≈ δ +� +(c1)yXδ +α + c1c2Xδ +β +� +(0) + (δ2/2) +� +((c1)y − (c2)x)Y δ +1 − c2(a2 +1 + b2 +1 − c2 +1)Xδ +α +� +(0) ++ (δ2/2) +� +(a2)xxφδ − (b2)xxξδ − (c2)xxXδ +α +� +(0). +Proof. We only prove the equation for Y δ +1 , since the equation for Y δ +2 is obtained in the same +way. Now, for the element [a1φδ]2 +0 of [Y δ +1 ]2 +0, we have +[a1φδ]2 +0/δ = +� +a1(2)[φδ]2 +0 + (a1(2) − a1(0))φδ(0) +� +/δ +≈ −(a2(a1 + δ(a1)y))(0) +� +Xδ +β + (δ/2)Y δ +2 +� +(0) + ((a1)y + (δ/2)(a1)yy)(0)φδ(0) +≈ +� +−a1a2Xδ +β + a2c1φδ� +(0) + (δ/2) +� +−a1a2Y δ +2 − 2a2 +2c1Xδ +β + (a1)yyφδ� +(0), +by (a1)y = a2c1 in Lemma 6.1. In the same way, we have +[b1ξδ]2 +0/δ ≈ +� +b1b2Xδ +β + b2c1ξδ� +(0) + (δ/2) +� +b1b2Y δ +2 + 2b2 +2c1Xδ +β + (b1)yyξδ� +(0), +[c1Xδ +β]2 +0/δ ≈ +� +c1Y δ +2 + (c1)yXδ +β +� +(0) + (δ/2) +� +2(c1)yY δ +2 − c1(a2 +2 + b2 +2 + c2 +2)Xδ +β + (c1)yyXδ +β +� +(0). +Then, by (c2)x + (c1)y + a1a2 + b1b2 = 0 in Lemma 6.1, we obtain the equation for [Y δ +1 ]2 +1. +□ +Lemma 6.10. With the third degree for δ of Gx − Gy, we have +([φδ]3 +2 − [φδ]1 +0) − ([φδ]3 +1 − [φδ]2 +0) +≈ − (δ3/2) +� +c1((a2)y − a1c2)Xδ +α − c2((a1)x − a2c1)Xδ +β +� +(0), +([ξδ]3 +2 − [ξδ]1 +0) − ([ξδ]3 +1 − [ξδ]2 +0) +≈ (δ3/2) +� +c1((b2)y − b1c2)Xδ +α − c2((b1)x − b2c1)Xδ +β +� +(0), +([Xδ +α]3 +2 − [Xδ +α]1 +0) − ([Xδ +α]3 +1 − [Xδ +α]2 +0) +≈ − (δ3/2) +� +−c1((a2)y − a1c2)φδ + c1((b2)y − b1c2)ξδ� +(0) +− (δ3/2) +� +(c2)xx + (c1)yy + c1(a2 +2 + b2 +2) + c2(a2 +1 + b2 +1) +� +(0)Xδ +β(0), +([Xδ +β]3 +2 − [Xδ +β]1 +0) − ([Xδ +β]3 +1 − [Xδ +β]2 +0) +≈ (δ3/2) +� +−c2((a1)x − a2c1)φδ + c2((b1)x − b2c1)ξδ� +(0) ++ (δ3/2) +� +(c2)xx + (c1)yy + c1(a2 +2 + b2 +2) + c2(a2 +1 + b2 +1) +� +(0)Xδ +α(0). +Proof. For the first equation, we have +([φδ]3 +2 − [φδ]1 +0)/δ = − [tδ +1]3 +1 +� +a1(Xδ +α + (δ/2)Y δ +1 ) +� +(2) − tδ +1(1) +� +a1(Xδ +α + (δ/2)Y δ +1 ) +�2 +0 +≈ − +� +a1(Xδ +α + (δ/2)Y δ +1 ) +�2 +0 += − a1(2) +� +Xδ +α + (δ/2)Y δ +1 +�2 +0 − [a1]2 +0 +� +Xδ +α + (δ/2)Y δ +1 +� +(0) +≈ − δ(a1 + δ(a1)y)(0) +� +c2Xδ +β + (δ/2)(c2Y δ +2 + c1c2Xδ +α + (c2)xXδ +β) +� +(0) +− δ +� +((a1)y + (δ/2)(a1)yy) (Xδ +α + (δ/2)Y δ +1 ) +� +(0). +Hence, we have +([φδ]3 +2 − [φδ]1 +0)/δ2 +≈ − +� +a2c1Xδ +α + a1c2Xδ +β +� +(0) − (δ/2) +� +a1a2(c1 + c2)φδ − (a1b2c2 + a2b1c1)ξδ� +(0) +− (δ/2) +� +(a1c2(c1 − c2) + (a1)yy) Xδ +α + +� +a1(c2)x + 2a2c1c2 − a2c2 +1 +� +Xδ +β +� +(0). + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +43 +In the same way, we have +([φδ]3 +1 − [φδ]2 +0)/δ2 +≈ − +� +a2c1Xδ +α + a1c2Xδ +β +� +(0) − (δ/2) +� +a1a2(c1 + c2)φδ − (a1b2c2 + a2b1c1)ξδ� +(0) +− (δ/2) +�� +−a1c2 +2 + 2a1c1c2 + a2(c1)y +� +Xδ +α + +� +−a2c2 +1 + a2c1c2 + (a2)xx +� +Xδ +β +� +(0). +From these equations, we have +([φδ]3 +2 − [φδ]1 +0) − ([φδ]3 +1 − [φδ]2 +0) +≈ −(δ3/2) +� +((a1)yy − a2(c1)y − a1c1c2) Xδ +α − ((a2)xx − a1(c2)x − a2c1c2) Xδ +β +� +(0). +Then, by Lemma 6.1, we have +(a1)yy − a2(c1)y − a1c1c2 = c1 ((a2)y − a1c2) , +(a2)xx − a1(c2)x − a2c1c2 = c2 ((a1)x − a2c1) . +Hence, we obtain the first equation. The second equation is also obtained in the same way. +Next, we prove the third equation. We have +([Xδ +α]3 +2 − [Xδ +α]1 +0)/δ ≈ +� +Y δ +1 − (δ/2)(a2 +1 + b2 +1 + c2 +1)Xδ +α +�2 +0 +≈ [Y δ +1 ]2 +0 − (δ/2)(a2 +1 + b2 +1 + c2 +1)(2)[Xδ +α]2 +0 − (δ2/2)(a2 +1 + b2 +1 + c2 +1)y(0)Xδ +α(0) +≈ [Y δ +1 ]2 +0 − (δ/2)(a2 +1 + b2 +1 + c2 +1)(0)[Xδ +α]2 +0 − (δ2/2)(a2 +1 + b2 +1 + c2 +1)y(0)Xδ +α(0) +≈ δ +� +c1c2Xδ +α + (c2)xXδ +β +� +(0) + (δ2/2) +� +((c2)x − (c1)y)Y δ +2 − (a2 +1 + b2 +1 + c2 +1)yXδ +α +� +(0) ++ (δ2/2) +� +− +� +c1(a2 +2 + b2 +2 − c2 +2) + c2(a2 +1 + b2 +1 + c2 +1) +� +Xδ +β + (a1)yyφδ − (b1)yyξδ − (c1)yyXδ +β +� +(0). +In the same way, we have +([Xδ +α]3 +1 − [Xδ +α]2 +0)/δ ≈ δ +� +c1c2Xδ +α + (c2)xXδ +β +� +(0) ++ (δ2/2) +� +c1c2Y δ +1 + (c2)xY δ +2 + (c2(c1)y + 2c1(c2)x)Xδ +α + (c1c2 +2 + (c2)xx)Xδ +β +� +(0). +From these equations, we have +([Xδ +α]3 +2 − [Xδ +α]1 +0) − ([Xδ +α]3 +1 − [Xδ +α]2 +0) +≈ −(δ3/2) +� +c1c2(a1φδ − b1ξδ) + (c1)y(a2φδ − b2ξδ) − (a1)yyφδ + (b1)yyξδ + (c1)yyXδ +β +� +(0) +− (δ3/2) +�� +2c1(c2)x + (a2 +1 + b2 +1 + c2 +1)y +� +Xδ +α + +� +c1(a2 +2 + b2 +2) + c2(a2 +1 + b2 +1) + (c2)xx +� +Xδ +β +� +(0). +Then, by Lemma 6.1, we have +(b1)yy − b2(c1)y − b1c1c2 = c1 ((b2)y − b1c2) , +2c1(c2)x + 2 (a1(a1)y + b1(b1)y + c1(c1)y) = 2c1 ((c2)x + (c1)y + a1a2 + b1b2) = 0. +Hence, we obtain the third equation. The last equation is also obtained in the same way. +□ +Definition 6.11 (Choice of the constant K in (6.3)). (1) For the four equations of Lemma +6.10, we take norm of the coefficient vectors for δ3. Then, let K1 be the maximum of these four +norms on the domain E. +(2) For the matrices Ω1 and Ω2 in (4.5), let +K2 := Max +p∈E {∥Ωi(p)∥, ∥(Ω1)x(p)∥/2, ∥(Ω2)y(p)∥/2} . +Then, we define K by K := Max{K1, K2} + 1. Here, we add 1 to Max{K1, K2}, since (F δn − +F δn)(xi+1, yj+1) also includes the terms of degree δi +n (i > 3), in (1). +By Lemma 6.10 and the definition of K, we have (6.3): + +44 +N. MATSUURA AND Y. SUYAMA +Theorem 6.12. For an integer n and (xp, yp) ∈ E, we take a division of Ep. With a sub-domain +∆n +i,j, suppose that an orthonormal frame F δn(xi, yj) is determined. Then, we have +∥(F δn − F δn)(xi+1,j+1)∥ ≤ Kδ3 +n, +if n is sufficiently large. +By Theorems 6.6, 6.8 and 6.12, the proof of Proposition 6.3 also has been completed. +For Step C: As in the proofs of Step A, we fix an integer n and study only the case of +E = [x0, xe] × [y0, ye], where xe = x0 + a and ye = y0 + a. Hence, we have s = t = n and +denote δ := δn = a/n. Let F 0(x, y) be the solution to (4.6) on D under a given initial condition +F 0(x0, y0). Let m be the path from P0,0 to Pn,n such that m : P0,0 → · · · → Pn,0 → · · · → +Pn,n = (xe, ye), and F δ(x, y) := F δ(x, y) be the orthonormal frame field on m determined from +F 0(x0, y0) by Step A. Firstly, note that we have +� +m +F 0(x, y)Ω = +� +m +dF 0 = +� +F 0�(xn,xn) +(x0,y0) , +and +� +F δ�(xi+1,y0) +(xi,y0) += +� (xi+1,y0) +(xi,y0) +dF δ +dx (x, y0)dx, +� +F δ�(xn,yj+1) +(xn,yj) += +� (xn,yj+1) +(xn,yj) +dF δ +dy (xn, y)dy +on each sub-interval of m. +Now, we study the norm ∥(F 0 − F δn)(xn, yn)∥. For the second degree Taylor polynomials +of both frames F 0(x, y) and F δ(x, y) at each point (xi, y0) and (xn, yj), we have the following +lemma. +Lemma 6.13. For x ∈ [xi, xi+1], the first degree Taylor polynomial of d(F 0−F δ) +dx +(x, y0) at x = xi +is given by +d +dx(F 0 − F δ)(x,y0) ≈ (x − xi)F 0 +(xi,y0)(Ω1)x(xi, y0) ++ (F 0 − F δ)(xi,y0) +� +Ω1(xi, y0) + (x − xi)Ω2 +1(xi, y0) +� +. +For y ∈ [yj, yj+1], the first degree Taylor polynomial of d(F 0−F δ) +dy +(xn, y) at y = yj is given by +d +dy(F 0 − F δ)(xn,y) ≈ (y − yj)F 0 +(xn,yj)(Ω2)y(xn, yj) ++ (F 0 − F δ)(xn,yj) +� +Ω2(xn, yj) + (y − yj)Ω2 +2(xn, yj) +� +. +Proof. Since dF 0 = F 0Ω and ∂2F 0/∂x2 = F 0[Ω2 +1 + (Ω1)x], we have the polynomial at x = xi of +dF 0(x, y0)/dx: +d +dxF 0(x, y0) ≈ F 0(xi, y0) +� +(Ω1)(xi,y0) + (x − xi) +� +Ω2 +1 + (Ω1)x +� +(xi,y0) +� +for x ∈ [xi, xi+1]. +Next, with the function tδ +1(x, y0) on x ∈ [xi, xi+1], we have tδ +1(xi, y0) = +1 and (dtδ +1/dx)(xi, y0) = 0. +From these equations, we obtain the polynomial at x = xi of +d(F δ(x, y0))/dx: +d +dxF δ(x, y0) ≈ F δ(xi, y0) +� +(Ω1)(xi,y0) + (x − xi)Ω2 +1|(xi,y0) +� +for x ∈ [xi, xi+1]. In the same way, we obtain the polynomials at y = yj of d(F 0(xn, y))/dy and +d(F δ(xn, y))/dy for y ∈ [yj, yj+1]. In consequence, we have verified the lemma. +□ + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +45 +Now, we put T δ(x, y) := (F 0−F δ)(x, y). Under the condition T δ(x0, y0) = (F 0−F δ)(x0, y0) = +0, we integrate (dT δ/dx)(x, y0) from xi to xi+1 in order of i = 0, 1, . . . , n − 1. Then, by Lemma +6.13 and Max {∥Ωi(p)∥, ∥(Ω1)x(p)∥/2, ∥(Ω2)y(p)∥/2 | p ∈ E} + 1 ≤ K, we have +∥T δ(x1, y0)∥ ≤ Kδ2, +∥T δ(xi+1, y0) − T δ(xi, y0)∥ ≤ Kδ2 + ∥T δ(xi, y0)∥Kδ for i ≥ 1, +if n is sufficiently large. Hence, we have +∥T δ(xi+1, y0)∥ + δ ≤ (∥T δ(xi, y0)∥ + δ)(1 + Kδ) +for i ≥ 1 +by ∥T δ(xi+1, y0)∥ ≤ Kδ2 + ∥T δ(xi, y0)∥(1 + Kδ). Therefore, we have +∥T δ(xn, y0)∥ + δ ≤ (∥T δ(x1, y0)∥ + δ)(1 + Kδ)n−1 ≤ δ(1 + Kδ)n. +(6.21) +Here, we have +(1 + Kδ)n = +n +� +k=0 +nCk +�Ka +n +�k +< +n +� +k=0 +Kkak +k! +< exp(Ka). +(6.22) +In the same way, by the integral of (dT δ/dy)(xn, y) from yj to yj+1, we have +∥T δ(xn, yj+1) − T δ(xn, yj)∥ ≤ Kδ2 + ∥T δ(xn, yj)∥Kδ +for j ≥ 0, +if n is sufficiently large. In consequence, we have +∥T δ(xn, yn)∥ ≤ (∥T δ(xn, y0)∥ + δ)(1 + Kδ)n − δ ≤ δ (exp(2Ka) − 1) +(6.23) +by (6.21) and (6.22). The inequality (6.23) implies that F δn(xe, ye)(= F δn(xn, yn)) converges +to F 0(xe, ye) as n tends to ∞. +In the argument above, we can replace (xe, ye) with an arbitrary point (xp, yp) ∈ E, by +Definition 6.2. Thus, Step C has been verified by the argument above and Proposition 6.3: +Theorem 6.14. Let F 0(x0, y0) be an orthonormal frame at (x0, y0) given arbitrarily. +Let +F 0(x, y) be the orthonormal frame field satisfying (4.6) under the initial condition F 0(x0, y0). +For an integer n and (xp, yp) ∈ E, we take a division of Ep. Let m and m be the two paths +from (x0, y0) to (xp, yp) given by (6.4)–(6.5). Let F δn(xp, yp) and F δn(xp, yp) be the orthonor- +mal frame determined from m and m by Step A. Then, F δn(xp, yp) and F δn(xp, yp) converge to +F 0(xp, yp) uniformly for all (xp, yp) ∈ E, as n tends to ∞. +By the argument above, all Steps A, B and C have been verified: for an integer n and +(x, y) ∈ E, we have obtained two approximations F δn(x, y) and F δn(x, y) of F 0(x, y). +In +Theorem 6.14, note that any frame F δn(xp, yp) determined by a path from (x0, y0) to (xp, yp) +is also an approximation of F 0(xp, yp) if n is sufficiently large, by Proposition 6.3. +Next, we construct an approximation of several coordinate curves in the curvature surface +f 0(x, y) on D. Before the construction, we state a remark. For an x-curve f 0(x, y0) with fixed +y0 ̸= 0 on an interval I = [x0, xe] := [y0 − a, y0 + a], where a, y0 − a > 0, we fix a division of +I with equal length δn = a/n and make the frame field F δn(x, y0) on I under the condition +F δn(x0, y0) = Id, by Theorem 6.14. Then, for a vector +f δn(x, y0) := (1 + y2 +0)−1 � +Xδn +β − 2(1 + y2)ξδn + yφδn� +(x, y0) +on each sub-interval [xi, xi+1], one might guess from Theorem 4.5 if the curve +f δn(x, y0) := +1 +2 +√ +5 +� +� +f δn�(x,y0) +(xi,y0) + +i +� +k=1 +� +f δn�(xk,y0) +(xk−1,y0) +� +for xi < x ≤ xi+1, +(6.24) +is an approximation of the x-curve (where we adopt the non-unit vector f δn(x, y0) in Theorem +4.5). However, it is not an approximation of the x-curve. In fact, for (6.24) we have f δn(x, y0)− +f δn(y0, y0) ≡ 0 for y0 = xn ≤ x ≤ xn+1, since (y0, y0) is in the singularity D∩S1 of the metric g0 + +46 +N. MATSUURA AND Y. SUYAMA +(or by (6.10)–(6.13)). In consequence, it is necessary for the interval [y0, xe] that we make the +frame F δn(x, y0) and the vector f δn(x, y0) in the direction that x decreases, because f δn(y0, y0) +is determined as the limit of f δn(p) for p ∈ D \S1. The approximation F δn(x, y0) on [xi−1, xi] is +also defined from F δn(xi, y0) by (6.10)–(6.13) as (x − xi) ≤ 0, and then f δn(x, y0) on [xi−1, xi] +is determined by the frame F δn(x, y0) in the inverse direction. +6.1. Approximation of x-curve f 0(x, y0). Let f 0(x, y0) be an x-curve with y0 ̸= 0 on +[y0 − a, y0 + a]. We divide the interval [y0 − a, y0 + a] as above. On the interval [y0 − a, y0], +we make F δn +1 (x, y0) := F δn(x, y0) from F δn +1 (y0 − a, y0) = Id by (6.10)–(6.13) and determine +f δn +1 (x, y0) := f δn(x, y0) by (6.24), which satisfies f δn +1 (y0 − a, y0) = 0: we write (F δn +1 , f δn +1 ) +with (F δn +1 (y0, y0), f δn +1 (y0, y0)) obtained in this way. +Next, on the interval [y0, y0 + a], we +make F δn +2 (x, y0) := F δn(x, y0) from F δn +2 (y0 + a, y0) = Id by (6.10)–(6.13) and determine +f δn +2 (x, y0) := f δn(x, y0) and f δn +2 (x, y0) := f δn(x, y0) in the inverse direction, which satisfies +f δn +2 (y0+a, y0) = 0: we write (F δn +2 , f δn +2 ) with (F δn +2 (y0, y0), f δn +2 (y0, y0)) obtained in this way. Then, +the curve f δn +2 (x, y0)−f δn +2 satisfies f δn +2 (y0, y0)−f δn +2 += 0. For the matrix C such that CF δn +2 += F δn +1 , +we define a curve kδn(x, y0) for x ∈ [y0, y0 + a] by kδn(x, y0) := C(f δn +2 (x, y0) − f δn +2 ) + f δn +1 . +Now, two curves f δn +1 (x, y0) on [y0 − a, y0] and kδn(x, y0) on [y0, y0 + a] connect continuously +and have the same frame F δn +1 +at x = y0. The connected curve at f δn +1 +is an approximation +of the x-curve. Then, since a vector (yXδn +β − φδn)(x, y0) does not depend on x and ⟨yXδn +β − +φδn, f δn⟩(x, y0) = 0 holds, the curve is contained in a hyperplane R3 +y0(∋ 0) perpendicular to the +vector (yXδn +β − φδn)(x, y0) by f δn(y0 − a, y0) = 0, and in particular the curve lies on a 2-sphere +of radius (2 +√ +5)−1� +(5 + 4y2 +0)/(1 + y2 +0) in R3 +y0. +(a) x-curves when n = 5 +(b) x-curves when n = 20 +Figure 9. These show the differences between two x-curves f δn(x, 1) and +¯f δn(x, 1) on [2, 3] under the conditions f δn(2, 1) = ¯f δn(2, 1) and F δn(2, 1) = +¯F δn(2, 1): the x-curve f δn(x, 1) of black (resp. ¯f δn(x, 1) of gray) is constructed in +the direction that x increases (resp. decreases). Here F δn(x, 1) (resp. ¯F δn(x, 1)) +is the frame field determining f δn(x, 1) (resp. ¯f δn(x, 1)). +6.2. Approximation of y-curve f 0(x0, y) with fixed x0(> 0). Let f 0(x0, y) be a y-curve +for y ∈ I = [−a, a], where a > 0. We take a division of I with equal length δn := a/n. On the +interval [−a, 0], we make F δn(x0, y) from F δn(x0, −a) = Id and determine a vector ˜f δn(x0, y) +on each [yj, yj+1] by +˜f δn(x0, y) := (B2ξδn + C2Xδn +α )(x0, y). +Then, the approximation f δn(x0, y) of f 0(x0, y) (−a ≤ y ≤ 0) is given by +f δn(x0, y) := +1 +(B2 +2 + C2 +2)(x0) +�� +˜f δn�(x0,y) +(x0,yj) + +j +� +k=1 +� +˜f δn�(x0,yk) +(x0,yk−1) +� +for yj < y ≤ yj+1, + +EXTENSION AND APPROXIMATION OF CURVATURE SURFACES +47 +where yj+1 ≤ yn = 0. On the interval [0, a], we make F δn(x0, y) from F δn(x0, a) = Id and +determine ˜f δn(x0, y) and f δn(x0, y) in the inverse direction. Then, we obtain the approximation +of the y-curve by the connection of these two curves, in the same way as the case of x-curves. +Then, since (−B2Xδn +α +C2ξδn)(x0, y) does not depend on y and ⟨−B2Xδn +α +C2ξδn, f δn⟩(x0, y) = 0 +holds, the connected curve f δn(x0, y) is contained a hyperplane R3 +x0 perpendicular to the vector +(−B2Xδn +α + C2ξδn)(x0, y), and in particular it lies on a 2-sphere of radius 1/ +� +B2 +2 + C2 +2(x0) in +R3 +x0. Furthermore, we have (B ◦f δn)(x, −y) = f δn(x, y) for the reflection B defined in Corollary +4.10 and have limy→∞ f δn(x0, y) = limy→∞ f δn(x0, −y) as mentioned in Corollary 5.10. +6.3. Approximation of the curve f 0(y, y) for b ≥ y ≥ a(> 0). Let F 0(x, y) be a solution +to (4.6) with F 0(a, a) = Id. Let us take a division a = y0 < y1 < y2 < · · · < yn = b of equal +length δn = (b − a)/n and a path mn : (a, a) = (y0, y0) → (y0, y1) → (y1, y1) → (y1, y2) → +· · · → (yn−1, yn−1) → (yn−1, yn) → (yn, yn) = (b, b). Let F δn(x, y) be an orthonormal frame +on mn determined by F δn(a, a) = Id and the path mn. For F δn(x, y), the vectors ˜f δn(yi, y) +and f δn(x, yi+1) above are determined on each edge {yi} × [yi, yi+1] and [yi, yi+1] × {yi+1}, +respectively. Then, since each lattice point (yi, yi) for y-curve (yi, y) is not singular for the +metric g0, the curve f δn(x, y) determined from these vectors is an approximation of the curve +f 0(x, y) on mn. In consequence, we obtain a sequence f δn(yi, yi) (i = 0, 1, . . . , n) of points, +which approximates to the curve f 0(y, y) as n → ∞. +Next, we attach the x-curve f δn(x, yi) and y-curve f δn(yi, y) to each points (yi, yi). Then, +in order to obtain correctly the relation between these curves, we have to take the frames +F δn(x, y) that (6.3) holds for these curves. That is, we determine these frames F δn(x, y) in the +directions that x and y increase, respectively. Hence, F δn(x, yi) and F δn(yi, y) are determined +from F δn(yi, yi) above. Then, the x-curve f δn(x, yi) (x ≤ yi) and the y-curve f δn(yi, y) are +naturally determined. However, for the x-curve, we have f δn(x, yi) ≡ f δn(yi, yi) for x ∈ [yi, yi+1] +mentioned above, and hence it is not sure how we define the curve f δn(x, yi) for x ∈ [yi, yi+1]. +On the above problem, for x ∈ [yi, yi+1] we determine a frame Aδn(x, yi) := F δn(x, yi) from +Aδn(yi+1, yi) = Id by (6.10)–(6.13) and a curve kδn(x, yi) := f δn(x, yi) in the inverse direction. +Thus, we obtain the curve f δn(x, yi) desired for x ∈ [yi, yi+1] by +f δn(x, yi) := C +� +kδn(x, yi) − kδn(yi, yi) +� ++ f δn(yi, yi), +where CAδn(xi, yi) = F δn(yi, yi). Then, we also obtain the x-curve f δn(x, yi) for x ≥ yi+1 from +the point f δn(yi+1, yi) and the frame F δn(yi+1, yi) given above. +References +[1] Burstall F E, Hertrich-Jeromin U, Suyama Y. Curvilinear coordinates on generic conformally flat hyper- +surfaces and constant curvature 2-metrics. J Math Soc Japan, 2018, 70: 617–649. +[2] Canevari S, Tojeiro R. Hypersurfaces of two space forms and conformally flat hypersurfaces. Ann Mat Pura +Appl (4), 2018, 197: 1–20. +[3] Canevari S, Tojeiro R. The Ribaucour transformation for hypersurfaces of two space forms and conformally +flat hypersurfaces. Bull Braz Math Soc (N.S.), 2018, 49: 593–613. +[4] Cartan E. La d´eformation des hypersurfaces dans l’´espace conforme `a n ≥ 5 dimensions. Bull Soc Math +France, 1917, 45: 57–121. +[5] Hertrich-Jeromin U. On conformally flat hypersurfaces and Guichard’s nets. Beitr Alg Geom, 1994, 35: +315–331. +[6] Hertrich-Jeromin U. Introduction to M¨obius Differential Geometry. London Math Soc Lect Note Ser. 300, +Cambridge University Press, 2003. +[7] Hertrich-Jeromin U, Suyama Y. Conformally flat hypersurfaces with cyclic Guichard net. Int J Math, 2007, +18: 301–329. +[8] Hertrich-Jeromin U, Suyama Y. Conformally flat hypersurfaces with Bianchi-type Guichard net. Osaka J +Math, 2013, 50: 1–30. + +48 +N. MATSUURA AND Y. SUYAMA +[9] Hertrich-Jeromin U, Suyama Y. Ribaucour pairs corresponding to dual pairs of conformally flat hypersur- +faces. Progr Math, 2015, 308: 449–469. +[10] Hertrich-Jeromin U, Suyama Y, Umehara M, Yamada K. A duality for conformally flat hypersurfaces. Beitr +Alg Geom, 2015, 56: 655–676. +[11] Lafontaine J. Conformal geometry from Riemannian viewpoint. In Kulkarni R S and Pinkall U, eds. Con- +formal Geometry. Aspects of Math, Vol. E12, Max-Plank-Ins. f¨ur Math, 1988, 65–92. +[12] Lin Y, Wong R. Asymptotics of generalized hypergeometric functions. Frontiers in orthogonal polynomials +and q-series, Contemp Math Appl Monogr Expo Lect Notes, vol. 1, World Sci Publ, Hackensack, NJ, 2018: +497–521. +[13] Luke Y L. The special functions and their approximations, Vol. I. Mathematics in Science and Engineering, +Vol. 53, Academic Press, New York-London, 1969. +[14] Santos D, Paulo J, Tojeiro R. Cyclic conformally flat hypersurfaces revisited. Mat Contemp, 2022, 49: +188–211. +[15] Suyama Y. Conformally flat hypersurfaces in Euclidean 4-space. Nagoya Math J, 2000, 158: 1–42. +[16] Suyama Y. Conformally flat hypersurfaces in Euclidean 4-space II. Osaka J Math, 2005, 42: 573–598. +[17] Suyama Y. A classification and non-existence theorem for conformally flat hypersurfaces in Euclidean +4-space. Int J Math, 2005, 16: 53–85. +[18] Suyama Y. Generic conformally flat hypersurfaces and surfaces in 3-sphere. Sci China Math, 2020, 63: +2439–2474. +[19] Wang C P. M¨obius geometry for hypersurfaces in S4. Nagoya Math J, 1995, 139: 1–20. +(N. Matsuura) Kurume Institute of Technology, 2228-66 Kamitsu, Kurume 830-0052, Japan +Email address: nozomu@kurume-it.ac.jp +(Y. Suyama) Fukuoka University, 8-19-1 Nanakuma, Jonan-ku Fukuoka 814-0180, Japan +Email address: suyama@fukuoka-u.ac.jp + diff --git a/zNFLT4oBgHgl3EQfni-P/content/tmp_files/load_file.txt b/zNFLT4oBgHgl3EQfni-P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d39e49725da57558d07361700d535d46274ab946 --- /dev/null +++ b/zNFLT4oBgHgl3EQfni-P/content/tmp_files/load_file.txt @@ -0,0 +1,1977 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf,len=1976 +page_content='EXTENSION AND APPROXIMATION OF CURVATURE SURFACES IN GENERIC CONFORMALLY FLAT HYPERSURFACES NOZOMU MATSUURA AND YOSHIHIKO SUYAMA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We study generic conformally flat (analytic-)hypersurfaces in the Euclidean 4- space R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Such a local-hypersurface is obtained as an evolution of surfaces issuing from a certain surface in R4, and then, in consequence, the original surface is a (principal-)curvature surface of the hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The hyperbolic 2-metric ˇgH of the upper half plane leads to a 6-dimensional set of singular (analytic-)Riemannian 2-metrics g0 of R2: on a simply connected open set in the regular domain of g0, a curvature surface with the metric g0 is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this paper, we choose a suitable singular metric g0 for ˇgH and clarify the structure of the curvature surfaces: there would be an analytic extension of the surfaces beyond the regular set of g0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' for the surface, the singularities and the points of infinity of g0 could be caught explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this case, all principal curvature lines in the extended surface are expressed by a frame field of R4 induced on the surface from hypersurfaces and they lie on some standard 2-spheres S2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We also provide a general method constructing an approximation of such frame fields, and obtain the figures of those lines including the singular points of g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Key words: conformally flat hypersurface, principal curvature surface, integrability condi- tion, cuspidal edge, envelope, point of infinity, numerical solution for orthonormal frame field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: 53A07, 53C40, 68W25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Introduction We study (principal-)curvature surfaces of generic conformally flat (analytic-)hypersurfaces in the Euclidean 4-space R4, arising from the hyperbolic 2-metric ˇgH of the upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The 2-metric ˇgH gives rise to many generic conformally flat local-hypersurfaces, but there is no known explicit representation of such hypersurfaces or their curvature surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this paper, we clarify the property and the structure of the analytically extended curvature surfaces by using a frame field of R4 induced on the surface from hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we say that a hypersurface is generic if it has distinct three principal curvatures at each points, and that a surface is curvature if it is woven of the curvature lines for two principal curvatures of some generic conformally flat hypersurface, because such lines always make a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The principal curvature line of a curvature surface is also that of the hypersurface including the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For n-dimentional hypersurfaces with n > 3, there are no generic conformally flat hypersurfaces by the result due to Cartan[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Any generic conformally flat local-hypersurface is regarded as a one-parameter family of curvature surfaces, in other words, it is obtained by an evolution of surfaces issuing from a certain surface in R4, and then, in consequence, the original surface is a curvature surface of the hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, for generic conformally flat hypersurfaces, it would be important to study the structure of the surfaces to be curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' A certain analytic (local-)surface φ in the standard 3-sphere S3 leads to a curvature surface: φ gives rise to an orthonormal frame field of R4, and the frame field induces a curvature surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, any metric ˇg of a certain family Met0, consisting of orthogonal analytic Riemannian 2-metrics on simply connected open sets V ⊂ R2 with constant Gauss curvature −1, leads to a 6-dimensional set of Riemannian This work was partially supported by JSPS KAKENHI Grant Number JP19K03507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12128v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='DG] 28 Jan 2023 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA 2-metrics g0 on V such that, for each metric g0, a surface φ in S3 mentioned above is determined and the curvature surface obtained has the metric g0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' [1], [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we regard the hyperbolic 2-metric ˇgH := ((dx)2 + (dy)2)/y2 on the upper half plane as a (singular) metric on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the 2-metric ˇgH, a pair Φ = (ϕ(x, y), ϕz(x, y)) of func- tions on R2 is determined (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For a simply connected open set V satisfying (ϕz(ϕz)x(ϕz)y)(x, y) ̸= 0, the metric ˇgH on V belongs to the family Met0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Moreover, for the metric ˇgH on R2 and Φ, a 5-dimensional set of singular (analytic-)Riemannian 2-metrics g0 on R2 is determined [18, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We choose a suitable singular metric g0 on R2 among them to get nice curvature surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Our first aim in this paper is to study an analytic extension of curvature surfaces beyond the regular set of the metric g0 and further to clarify the structure of the extended surface including the points at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, all extended principal curvature lines in the surface are expressed by the frame field determining curvature surfaces and they lie on some standard 2-spheres, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The second aim is to construct generally an approx- imation of such frame fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the approximation of each principal curvature line also lies on a standard 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By the approximation, we give several figures for the extended surface: principal curvature lines, the image of the singular curves for g0 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, let f be a generic conformally flat hypersurface in R4 defined on a domain U of R3, and κi (i = 1, 2, 3) be the principal curvatures1 of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for f a principal curvature line coordinate system (x, y, z) and a function ϕ = ϕ(x, y, z) on U are determined such that the (non-degenerate) 3-metric g = cos2 ϕ(dx)2 + sin2 ϕ(dy)2 + (dz)2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) is conformally flat and the first fundamental form If of f is given by If := P 2g = P 2(cos2 ϕ(dx)2 + sin2 ϕ(dy)2 + (dz)2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) with a function P = P(x, y, z) ̸= 0 on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let the principal curvatures κi (i = 1, 2, 3) correspond to x, y and z-lines in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, κ1, κ2 are determined from (ϕ, P, κ3) as κ1 = P −1 tan ϕ + κ3, κ2 = −P −1 cot ϕ + κ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) The conformally flat metric 3-metric g of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) is called the (principal) Guichard net2 of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Note that the above coordinate system (x, y, z) and the metric g in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) are defined only on the domain where f is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Conversely, for any conformally flat 3-metric g of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1), there is a generic conformally flat hypersurface with the Guichard net g uniquely up to a conformal transformation, if U is simply connected (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' [5]–[7], [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, in order to realize the hypersurface in R4, it is necessary to find out a function P in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) from ϕ such that the Gauss and the Codazzi equations are satisfied (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the examples of Guichard nets and generic conformally flat hypersurfaces, see a series of papers ([7], [8], [10], [15]–[17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The other construction of such hypersurfaces, which do not use the Guichard net, is given by the papers ([2], [3], [14]), where [14] is another proof of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Our problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the function P in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2), we study the curvature surfaces including the case P ≤ 0 not only the case P > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we state the setting in this paper, and briefly review the results for generic conformally flat local-hypersurfaces in the papers [1] and [18], which make our problems clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ˇgH be the hyperbolic 2-metric on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From now on, we assume that the domain U, where hypersurfaces f(x, y, z) are defined, is given by U = V × I ⊂ R2 × R or U = V ′ × I ⊂ R2 × R for a simply connected open set V or V ′ in R2 = R2 (x,y) and a suitable open interval I ⊂ R = R(z) with 0 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 1In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3), we have assumed that κ3 is the middle principal curvature: κ1 > κ3 > κ2 or κ1 < κ3 < κ2, for the sake of simplicity for the description later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 2We call the canonical principal Guichard net of f only the Guichard net (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 3 (R1) For ˇgH, a pair Φ = (ϕ(x, y), ϕz(x, y)) of functions is determined as cos ϕ(x, y) = x2 − y2 x2 + y2, sin ϕ(x, y) = 2xy x2 + y2, ϕz(x, y) = y x2 + y2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) which are analytic functions on R2 with the pole at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Actually, a one-parameter family Φc(x, y) = (ϕ(x, y), cϕz(x, y)) with parameter c ̸= 0 is determined for ˇgH, where we have chosen Φ = Φ1 in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (R2) Let S1 be the set defined by S1 := � (x, y) ∈ R2 | (ϕz(ϕz)x(ϕz)y)(x, y) = 0 � = � (x, y) ∈ R2 | xy(x2 − y2) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) For a domain V such that V ⊂ R2 \\ S1, the pair Φ leads to an analytic function ϕ(x, y, z) on U = V ×I uniquely as an evolution in z-direction under the initial conditions ϕ(x, y, 0) = ϕ(x, y) and ϕz(x, y, 0) = ϕz(x, y)3, and then ϕ(x, y, z) defines a conformally flat metric g on V × I in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) (by replacing V × I with a sub-domain V ′ × I such that ϕzϕzxϕzy ̸= 0 holds on V ′ × I, if necessary)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, a generic conformally flat hypersurface fV (x, y, z) on V × I with the Guichard net g is determined from Φ|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Note that the Guichard net g is not defined on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (R3) As the solutions to a certain system of differential equations defined from Φ, a class Ψ = ( ¯P(x, y), ¯Pz(x, y), ¯κ3(x, y)) consisting of three functions on R2 are also determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In order to describe the class Ψ explicitly, we prepare a generalized hypergeometric function X0 on R of type 1F2: for a sequence {ak}∞ k=1 given by a1 = 1 and 2(k + 1)(4k2 + 5/4)ak+1 + (2k − 1)ak = 0 (k ≥ 1), X0(x) is defined by X0 = X0(x) := 5/2 + �∞ k=1 akx2k = 5/2 + x2 − (1/21)x4 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) The class Ψ is determined from X0 as follows: ¯P −1(x, y) = x x2 + y2h(x, y), ¯κ3(x, y) = − y x2 + y2 � 2(X0 − xX′ 0) + √ 5 � , ( ¯P −1)z(x, y) := − ¯Pz ¯P 2(x, y) = 1 (x2 + y2)2 � (x2 − y2) � X0 + √ 5 2 � + 2xy2X′ 0 � + √ 5, where h(x, y) = 2X0 − xX′ 0 + y2X′ 0/x + √ 5 (> √ 5) on R2 (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3 in §2) and that the functions ¯P −1(x, y), ( ¯P −1)z(x, y) and ¯κ3(x, y) are analytic on the whole R2 with the pole at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the class Ψ a singular Riemannian 2-metric g0 on R2, which is our object, is determined by g0 := ¯P 2 � cos2 ϕ(dx)2 + sin2 ϕ(dy)2� = 1 h2(x, y) �(x2 − y2)2 x2 (dx)2 + 4y2(dy)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) Then, S1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) coincides with the singular set of g0, and we have g0(x, y) = g0(−x, y) = g0(x, −y) on R2 \\ S1 by the property of h (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Actually, a 5-dimensional set of classes {( ¯P, ¯Pz, ¯κ3)} is determined for Φ, and we have chosen one class Ψ from that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we review the relation between the class Ψ and the generic conformally flat hyper- surfaces fV (x, y, z) mentioned in (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the class Ψ, we define two functions ¯κ1 and ¯κ2 on R2 \\ S1 by ¯κ1(x, y) := 2y x2 − y2 � X0 + √ 5 2 � , ¯κ2(x, y) := 1 y � x2 + y2 2 X′ 0 x − � X0 + √ 5 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, ¯κ1¯κ2 ̸= 0 holds on R2 \\ S, where S is the set defined by S := S1 ∪ S2 and S2 := � (x, y) �� (x2 + y2)X′ 0 = (2X0 + √ 5)x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For a domain V such that V ⊂ R2 \\ S, we have the following facts (R4) and (R5): 3We applied the Cauchy-Kovalevskaya theorem for analytic evolution equations (more precisely, see [1], [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 4We shall omit this remark from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA (R4) For the pair Φ and the class Ψ, an analytic surface φ(x, y) in S3 is determined on V such that (x, y) is a principal curvature line coordinates of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let us denote the surface φ(x, y) on V by the frame field F 0 V (x, y) := � φ, X0 α, X0 β, ξ � (x, y), where X0 α and X0 β are the unit principal directions corresponding to the coordinates (x, y) and ξ is a unit normal vector field of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, an analytic surface f 0 V (x, y) in R4 with the metric g0 is determined on V as a certain integral surface of (X0 α, X0 β) (see Theorem 1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (R5) There is a generic conformally flat hypersurface fV (x, y, z) in (R2) defined on V × I such that fV (x, y, z) satisfies the following conditions (1) and (2): (1) fV (x, y, 0) = f 0 V (x, y) holds on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) For fV (x, y, z), the conformal element P 2 of IfV in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) and the principal curvatures κi satisfy the equations P(x, y, 0) = ¯P(x, y), Pz(x, y, 0) = ¯Pz(x, y), κi(x, y, 0) = ¯κi(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let FV (x, y, z) := [N, Xα, Xβ, Xγ] (x, y, z) be the orthonormal frame field determined by fV (x, y, z), where N(x, y, z) is normal and (Xα, Xβ, Xγ)(x, y, z) are the principal directions corresponding to the coordinates (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Actually, fV (x, y, z) is determined as an evolution of surfaces in z issuing from the surface f 0 V (x, y) on z = 0 under the condition FV (x, y, 0) = F 0 V (x, y) on V , and then the condition ¯κ1¯κ2 ̸= 0 is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, note that the other classes {( ¯P, ¯Pz, ¯κ3)} for Φ determine the conformal transformations of fV (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, by (R4) and (R5), f 0 V (x, y) is an analytic curvature surface with the metric g0|V , and f 0 V (x, y) and F 0 V (x, y) are determined only by Φ and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, each coordinate line of f 0 V (x, y) is a principal curvature line of fV (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We emphasize that the curvature surfaces f 0 V (x, y) are defined only on each domain V ⊂ R2 \\S (or at most on V ⊂ R2 \\S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Our first aim is to study the existence and the structure of an extended curvature surface f 0(x, y) including these singularities of g0: for example, let us take an interval (0, 1] × {y} ⊂ R2 for each y, then the length by the metric g0 of the interval converges to a finite value if y = 0, but it diverges to ∞ if y ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' hence, it would be interesting to study the curvature surface as x → ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the hyperbolic 2-metric ˇgH, the singular Riemannian metric g0 on R2 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) and the curvature (local-)surfaces are determined by Φ and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let D := {(x, y) ∈ R2 | x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Sections 3 and 4, we study the singular Riemannian space (D, g0) and an extended curvature surface only on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Section 5, we shall study them on the whole R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, the function X0(x) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) determines the properties of the space (D, g0) and the curvature surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Section 2 we study the property of X0(x) for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we also review how the class Ψ is determined from Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Section 3, we define a regular coordinates (ˆx, ˆy) of the space (D, g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ˆx and ˆy be the functions on D defined by ˆx := x3/3 − xy2, ˆy := y2 respectively, and ι be an analytic map given by ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The determinant of the Jacobi matrix for ι is a prime polynomial defining the singular set S1 ∩D of g0, where S1∩D = {(x, y) | y(x2−y2) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From the property ι(x, y) = ι(x, −y), we define the sub-domains Di (i = 1, 2) of D by D1 := {(x, y) ∈ D | y ≥ 0} and D2 := {(x, y) ∈ D | y ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, there is a Riemannian space ( ˆD, ˆg0) without singularity and an analytic map ˆι : D → ˆD is induced from ι such that ( ˆD, ˆg0) and ˆι satisfy the following conditions (C1), (C2) and (C3): (C1) For i = 1, 2, the restricted maps ˆι : (Di \\ S1, g0) → ( ˆD \\ ˆι(S1), ˆg0) are isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (C2) At each point ˆι(x, ±x) = ι(x, ±x) or ˆι(x, 0) = ι(x, 0), the tangent space of ˆD is defined only as a half plane: any x-curve ˆι(x, y) with fixed y ̸= 0 has a cusp at x = |y| and any y-curve ˆι(x, y) reflects at y = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' the curve ˆι(|y|, y) of y is the envelope of the family of y-curves ˆι(x, y) with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let T + ˆι(x,±x) ˆD and T + ˆι(x,0) ˆD be those tangent half spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (C3) The metric ˆg0 is well defined even on the tangent spaces T + ˆι(x,±x) ˆD and T + ˆι(x,0) ˆD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 5 By the above property, it is natural to consider both sides of the space ( ˆD, ˆg0): we denote by ( ˆD+, ˆg+ 0 ) and ( ˆD−, ˆg− 0 ) the front and the back of ( ˆD, ˆg0), respectively, and by ( ˆD±, ˆg± 0 ) the space equipped with both sides of ( ˆD, ˆg0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, ( ˆD+, ˆg+ 0 ) and ( ˆD−, ˆg− 0 ) are isometric with ( ˆD, ˆg0) and have only the common points ˆι(x, 0) and the common tangent spaces T + ˆι(x,0) ˆD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, we regard the map ˆι : (D1, g0) → ( ˆD, ˆg0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˆι : (D2, g0) → ( ˆD, ˆg0)) as ˆι : (D1, g0) → ( ˆD+, ˆg+ 0 ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˆι : (D2, g0) → ( ˆD−, ˆg− 0 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we could call ( ˆD±, ˆg± 0 ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (ˆx, ˆy)) a regular Riemannian space associated to (D, g0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' a regular coordinate system for (D, g0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Section 4, we firstly verify that Φ and Ψ give rise to an analytic curvature surface f 0(x, y) on D with the metric g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we say that f 0(x, y) is a curvature surface on D if, for a domain V such that V ⊂ D \\ S, there is a generic conformally flat hypersurface fV (x, y, z) on V × I such that fV (x, y, 0) = f 0(x, y) holds on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) The functions Φ and Ψ lead to an analytic orthonormal frame field F 0(x, y) := � φ, X0 α, X0 β, ξ � (x, y) of R4 defined on D such that the structure equation of F 0(x, y) sat- isfies the integrability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) An analytic curvature surface f 0(x, y) on D is determined as an integral surface of (X0 α, X0 β) by df 0 = (xh(x, y))−1((x2 − y2)X0 α dx + 2xyX0 β dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Theorem 1, the field F 0(x, y) is uniquely determined up to a transformation AF 0(x, y) by a constant orthogonal matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The coordinate lines of f 0(x, y) satisfy the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The surface f 0(x, y) on D in Theorem 1 satisfies the following conditions (1), (2), (3) and (4): (1) The unit analytic vector u(y) := (1 + y2)−1/2 [yX0 β − φ](x, y) depends only on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) Any x-curve f 0(x, y) with fixed y lies on a standard 2-sphere S2 y of radius (2 √ 5)−1� (5 + 4y2)/(1 + y2) in an affine hyperplane perpendicular to u(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) The unit analytic vector ˜u(x) := (B2 2 +C2 2)−1/2(x)(−B2X0 α +C2ξ)(x, y) depends only on x, where B2(x) = −(1/2)(X′ 0(x)/x) − √ 5 and C2(x) = X′′ 0 (x) − X′ 0(x)/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4) Any y-curve f 0(x, y) with fixed x lies on a standard 2-sphere S2 x of radius (B2 2+C2 2)−1/2(x) in an affine hyperplane perpendicular to ˜u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Theorem 2, when we regard the surface f 0(x, y) as a one-parameter family of x-curves, the surface f 0(x, y) is expressed as f 0(x, y) = (2 √ 5)−1� (5 + 4y2)(1 + y2)−1 f(x, y) + A(y) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) with the unit analytic vector f(x, y) := ((1 + y2)(5 + 4y2))−1/2(X0 β − 2(1 + y2)ξ + yφ)(x, y), where A(y) is a R4-valued analytic function of y determined uniquely up to a parallel translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Similarly, when we regard the surface f 0(x, y) as a one-parameter family of y-curves, the surface f 0(x, y) is expressed as f 0(x, y) = (B2 2 + C2 2)−1/2(x)˜f(x, y) + ˜A(x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) with the unit analytic vector ˜f(x, y) := (B2 2 + C2 2)−1/2(x) (B2ξ + C2X0 α) (x, y), where ˜A(x) is a R4-valued analytic function of x determined uniquely up to a parallel translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, in the singular set of g0, the surface f 0(x, y) has the following features: (F1) Any x-curve f 0(x, y) with fixed y ̸= 0 has a cusp of type (2, 3, 4) at x = |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (F2) Any y-curve f 0(x, y) with fixed x has a cusp of type (2, 3, 4) at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (F3) The curves f 0(|y|, y) of y are the envelopes of the family of y-curves f 0(x, y) with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (F4) We translate the surface f 0(x, y) such that f 0(p0) = 0 holds at some point p0 := (x, 0), where 0 is the origin of R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, there is a reflection B of R4 such that (B ◦ f 0)(x, y) = f 0(x, −y), (B ◦ φ)(x, y) = −φ(x, −y) and (B ◦ ξ)(x, y) = ξ(x, −y) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA From these facts, we may recognize that the map ˆι : (D, g0) → ( ˆD±, ˆg± 0 ) is a plane model of the curvature surface f 0(x, y) in R4, that is, there is an isometric map ¯f 0 : ( ˆD±, ˆg± 0 ) ∋ (ˆx, ˆy) �→ ¯f 0(ˆx, ˆy) ∈ R4 such that f 0(x, y) = ( ¯f 0 ◦ ˆι)(x, y) holds on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Section 5, we study the limits of x-curves f 0(x, y) as x → 0 and y-curves f 0(x, y) as y → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we can clarify the structure of the surface f 0(x, y) (and also the space (D, g0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, in the case of x → 0, we replace x with u given by x = e−u and consider the case of u → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the vectors u(y) and ˜u(x) in Theorem 2 have the following properties (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 for the uniform convergence): Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' There is an orthonormal frame [b∞, c∞, ˜b∞, ˜c∞] of R4 (consisting of constant vectors) such that it satisfies the following conditions: (1) The vector u(y) moves on the unit circle in a plane spanned by ˜b∞ and ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let v1(y) be a unit vector of y determined by (∇′ d/dyu)(y) = (y2/(1 + y2))v1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, u(y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' v1(y)) uniformly converges to the circle ˜b(y) := cos y ˜b∞ + sin y ˜c∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˜c(y) := − sin y ˜b∞ + cos y ˜c∞) as y tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) The vector ˜u(x) moves on the unit circle in a plane spanned by b∞ and c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ˜v1(x) be a unit vector of x determined by (∇′ d/du˜u)(e−u) = −T(u)˜v1(e−u), where T(u) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, ˜u(e−u) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˜v1(e−u)) uniformly converges to the circle b(u) := cos √ 5u 2 b∞ − sin √ 5u 2 c∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' −v2(u) := sin √ 5u 2 b∞ + cos √ 5u 2 c∞) as u tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, let v0 2(y) and v0 3(y) be the functions of y defined by v0 2(y) := 1 h(0, y) � 5 + 4y2 �5 2(5 + √ 5) + (10 + √ 5)y2 � , v0 3(y) := −2 √ 5y2 h(0, y) � 1 + y2 5 + 4y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have (v0 2(y))2 + (v0 3(y))2 = 5/4, v0 2(y) > 0 and v0 3(y) < 0 if y ̸= 0, v0 3(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the vectors v1(y) and v2(y) in Theorem 3, we define the vector Γ(u, y) by Γ(u, y) := 1 2 √ 5 � 5 + 4y2 1 + y2 a(u, y) + A(y), a(u, y) := 2 √ 5 � −v0 2(y)v1(y) + v0 3(y)v2(u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for each y ̸= 0, a(u, y) is a circle with parameter u and a(u, 0) degenerates to one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Note that −(2/ √ 5)v0 2(y)v1(y) is the center of the circle a(u, y) and that v1(y) and v2(u) are always perpendicular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have the following facts (1)–(4): (1) Any u-curve f 0(e−u, y) with y uniformly converges to the circle Γ(u, y) as u tends to ∞: only the u-curve f 0(e−u, 0) converges to the point Γ(u, 0) = −(1/2)v1(0) + A(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) All u-curve X0 α(e−u, y) with y uniformly converge to the circle b(u) as u tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) Any y-curve f 0(x, y) with x converges to the point (B2 2 + C2 2)−1/2(x)˜v1(x) + ˜A(x) as y tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4) All y-curve X0 β(x, y) with x uniformly converge to the circle ˜b(y) as y tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In (1) and (2), the convergences for those u-curves are also uniform with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In (3) and (4), the convergences for those y-curves f 0(x, y) are also uniform in the wider sense with respect to x ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have some other results: (1) As x → 0 and x → ∞, each x-curve f 0(x, y) with y ̸= 0 converges uniformly to parallel small circles in S2 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) limy→∞ f 0(x, y) = limy→−∞ f 0(x, y) holds for any x ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) The curve A(y) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˜A(x) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9)) lies on a plane H spanned by ˜b ∞ and ˜c∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' on a plane ˜H spanned by b∞ and c∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, a curvature surface on D has the following structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The curves −(v0 2(y)/5) � (5 + 4y2)(1 + y2)−1v1(y) + EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 7 A(y) and (B2 2 +C2 2)−1/2(x)˜v1(x)+ ˜A(x) in Theorem 4 lie on those planes H and ˜H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the spheres S2 y in Theorem 2, two points ±(2 √ 5)−1� (5 + 4y2)(1 + y2)−1v1(y) + A(y) are the antipodal points of S2 y to each other: the tangent spaces of S2 y at these points are spanned by b∞ and c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Similarly, for the sphere S2 x, two points ±(B2 2 + C2 2)−1/2(x)˜v1(x) + ˜A(x) are the antipodal points of S2 x to each other: the tangent spaces of S2 x at these points are spanned by ˜b ∞ and ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We also give the figures of several curves in the curvature surface f 0(x, y) on D explicitly: some coordinate lines, the cusps and the enveloping curves and so on, by using the approxima- tion F δn(x, y) for some integer n of the frame field F 0(x, y), which will be defined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, a curvature surface ˆf 0(x, y) on D(−) = {(x, y) | x < 0} is also defined from Φ and Ψ, and we can determine it by ˆf 0(−x, y) := f 0(x, y) for f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We regard the surfaces ˆf 0(x, y) on D(−) as the back side for f(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, two surfaces on D and D(−) connect at (0, 0) continuously in a sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we could recognize that the curvature surface formed by both sides of f 0(x, y) on D is a natural realization in R4 of the space (D∪{(0, 0)}∪D(−), g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Section 6, we define an approximation F δn(x, y) := [φδn, Xδn α , Xδn β , ξδn](x, y) of the frame field F 0(x, y) = [φ, X0 α, X0 β, ξ](x, y) for each positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We construct F δn(x, y) on a compact square E ⊂ D, by regarding φ (or F 0(x, y)) as a (singular) surface in the standard 3- sphere S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the Gauss and the Codazzi equations for φ are important in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let E := [x0, x0+a]×[y0, y0+a], and x0 < x1 < · · · < xn = x0+a and y0 < y1 < · · · < yn = y0+a be the divisions of [x0, x0 + a] and [y0, y0 + a] of equal length δn = a/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for an initial orthogonal matrix at (x0, y0), an orthonormal frame field F δn(x, y), not to depend on the width δn, is determined on the lattice in E made from the divisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the approximation of every coordinate line in f 0(x, y) also lies on a 2-sphere S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The approximation will be constructed by a kind of polygonal line method: on each edge [xi, xi+1]×{yj} or {xi}×[yj, yj+1], we approximate F 0(x, y) by a rational curve (not by a line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We further define F δn(x, y) at all points (x, y) ∈ E by a little change of the divisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, F δn(x, y) converges to F 0(x, y) uniformly on E as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By using F δn(x, y), we can draw the curves given in Section 5, since each coordinate curve is expressed by the frame field F 0(x, y) as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Choice of a singular metric g0 determining curvature surfaces As mentioned in the introduction, for the hyperbolic metric ˇgH = (dx2 + dy2)/y2 on R2, we have chosen the following pair Φ = (ϕ, ϕz) of functions on R2, cos ϕ(x, y) = x2 − y2 x2 + y2, sin ϕ(x, y) = 2xy x2 + y2, ϕz(x, y) = y x2 + y2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) and the class Ψ = ( ¯P, ¯Pz, ¯κ3) of three functions on R2, ¯P −1 = x x2 + y2h(x, y), ¯κ3 = − y x2 + y2 � 2(X0 − xX′ 0) + √ 5 � , ( ¯P −1)z := − ¯Pz ¯P 2 = 1 (x2 + y2)2 � (x2 − y2) � X0 + √ 5 2 � + 2xy2X′ 0 � + √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) Here X0 = X0(x) is the hypergeometric function on R2 given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) and h(x, y) is the function defined by h(x, y) = 2X0 − xX′ 0 + y2(X′ 0/x) + √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) The class Ψ is selected from the 5-dimensional set of classes {( ¯P, ¯Pz, ¯κ3)} in [18, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2] determined by Φ and it leads to a singular metric g0 determining curvature surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this section, we firstly explain that the choice of the class Ψ is suitable, and next study the property 8 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA of X0(x) for the argument later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Choice of the class Ψ and the singular metric g0: We define two 1-variable functions X1 = X1(x) of x and Y = Y (y) of y by the equations xX′′′ 1 − X′′ 1 + (x + 9/(4x)) X′ 1 − X1 = cx2, Y ′′ + Y = cy2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) respectively, where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the solutions X1 and Y to the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4), we define the functions GX1(x) and HY (y) by GX1(x) := (X′′ 1 )2 + 4cX1 + � 1 + 9/(4x2) � (X′ 1)2 + ((2/x)X2 − 4cx) X′ 1, HY (y) := (Y ′ − 2cy)2 + (Y − cy2 + 2c)2 − 4c2, where X2 := −X′′ 1 − X1 + cx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, for a pair (X1, Y ) of solutions, let ¯X = ¯X(x), ¯Y = ¯Y (y) and A = A(x, y) be the functions defined by ¯X := xX′ 1 − X1, ¯Y := yY ′ − Y, A := ¯X + ¯Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The following proposition are verified in [18, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2] except for the equations for GX1(x) and HY (y) in (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we only prove these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the proposition, we say that a triplet ( ¯P, ¯Pz, ¯κ3) is a class for Φ, if curvature surfaces f 0 V (x, y) and generic conformally flat hypersurfaces fV (x, y, z) in R4 are determined by Φ and the class ( ¯P, ¯Pz, ¯κ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let (X1, Y ) be a pair of solutions to the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have the following facts (1) and (2): (1) The functions GX1(x) and HY (y), respectively, are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) For any pair (X1, Y ) such that GX1(x) + HY (y) = 0, a class ( ¯P, ¯Pz, ¯κ3) for Φ is deter- mined as follows: ¯P −1 = X′ 1 − 2x x2 + y2A, ¯κ3 = −Y ′ + 2y x2 + y2A, ( ¯P −1)z = − ¯Pz ¯P 2 = 1 x2 + y2 � xX′ 1 − A + 2y2 x2 + y2A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Conversely, all classes ( ¯P, ¯Pz, ¯κ3) for Φ are determined by the above forms from the pairs (X1, Y ) such that GX1(x) + HY (y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have the following facts (3) and (4): (3) X0(x) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) is a solution to the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) with c = 0, and GX0(x) = −5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4) Let us take c = √ 5 in two equations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then X1(x) = X0(x)+ √ 5 (x2 + 5/2) and Y (y) = √ 5 (y2 − 2), respectively, are solutions to the equations, and the pair satisfies GX1(x) + HY (y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, we have GX1(x) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Firstly, note that X1(x) = X0(x) + c(x2 + 5/2) and Y (x) = c(y2 − 2) are the solutions of the first and the second equations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we verify that GX1(x) = −5+5c2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From (X′′ 1 (0))2 = 4(1 + c)2, 4cX1(0) = 10c(1 + c), � (1 + 9/(4x2))(X′ 1)2� (0) = 9(1 + c)2, ((2X2/x − 4cx)X′ 1) (0) = � (2X2 − 4cx2)(X′ 1/x) � (0) = −18(1 + c)2 by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6), we have GX1(0) = −5 + 5c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, GX1(x) = −5 + 5c2 holds for any x, since the function GX1(x) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the same way, we have HY (y) = −4c2 for Y (x) = c(y2 − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we also have verified that the pair X1(x) = X0(x) + √ 5(x2 + 5/2) and Y = √ 5(y2 − 2) is a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) and satisfies GX1(x) + HY (y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 9 Let X1(x) and Y (y) be a pair of solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1-(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From the pair, our class Ψ = ( ¯P(x, y), ¯Pz(x, y), ¯κ3(x, y)) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) is determined by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1-(2), and then for the class Ψ, the singular metric g0 on R2 and the functions ¯κi (i = 1, 2) mentioned in §1 are also determined as follows: g0 := ¯P 2 � cos2 ϕ(dx)2 + sin2 ϕ(dy)2� = h−2(x, y) � x−2(x2 − y2)2(dx)2 + 4y2(dy)2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) ¯κ1 := ¯P −1 tan ϕ + ¯κ3 = 2y(x2 − y2)−1� X0 + √ 5/2 � , ¯κ2 := − ¯P −1 cot ϕ + ¯κ3 = y−1� ((x2 + y2)/2)(X′ 0/x) − � X0 + √ 5/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) Property of the function X0(x): Since X0(x) determines the properties of g0 and ¯κi, we study the property of X0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The function X0(x) on R in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) satisfies the following conditions: (1) X0(−x) = X0(x) ≥ 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) X′ 0(x)/x > 0 for x ∈ R and (X′ 0(x)/x)(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) (2X0 − xX′ 0)(x) > 0 for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' X0(−x) = X0(x) follows from the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For X0(x) ≥ 5/2 and (2), suppose that there is a point x0 > 0 such that X′ 0(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have GX0(x0) = (X′′ 0 )2(x0) = −5, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Moreover, we have (X′ 0/x)(0) = 2 by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have X′ 0(x) > 0 for x > 0 and (X′ 0/x)(x) > 0 for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, since X′ 0(x) > 0 for x > 0, X0(x) is an increasing function on [0, ∞) and X0(0) = 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have X0(x) ≥ 5/2 for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (3), we have GX0(x) = (X′′ 0 − X′ 0/x)2 + (1 + 5/(4x2))(X′ 0)2 − 2X0(X′ 0/x) = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have (X′ 0)2 − 2X0(X′ 0/x) = (X′ 0/x)(xX′ 0 − 2X0) < −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By X′ 0/x > 0 for x ∈ R, we have 2X0 − xX′ 0 > 0 for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the function h(x, y), we have the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) h(x, y) > √ 5 on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) There is a number t1 (0 < t1 < 1) such that 5 + √ 5 + y2 < h(x, y) < 6 + √ 5 + 2y2 and |h(x, y) − h(0, y)| ≤ x2(1 + 2y2)/7 hold for 0 ≤ x < t1 and any y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The fact (1) follows from the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) of h(x, y) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The fact (2) follows from the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) of X0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, as x tends to 0, for any y ∈ R we have 2X0 − xX′ 0 = 5 + (2/21)x4 + O(x6), X′ 0/x = 2 − (4/21)x2 + O(x4), |h(x, y) − h(0, y)| ≤ |2X0 − xX′ 0 − 5| + y2|X′ 0/x − 2| < (x2/7)(1 + 2y2), where l(x) = O(xk) for a function l(x) implies that c1 < limx→0(l(x)/xk) < c2 holds for some constants ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ The function h(x, y) also satisfies the following equations: h(x, y) = h(−x, y) = h(x, −y), h(0, y) = 5 + √ 5 + 2y2, h(x, 0) = (2X0 − xX′ 0) + √ 5, h(0, 0) = 5 + √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, X′ 0(x) is an oscillating function, since X0(x) is a generalized hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We study whether X′ 0(x) oscillates even at x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we say that X′ 0(x) oscillates at x = ∞, if there is a bounded interval J (not one point) satisfying the following condition: for any point p ∈ J, there is a sequence xn (xn → ∞) such that X′ 0(xn) converges to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The function X0(x) on R satisfies the following conditions (1)–(4): 10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA (1) The function τ(x) := (X0 + X′′ 0 )(x)/x is a decreasing function on (0, ∞) and τ(x) converges to a non-negative constant τ(∞) := limx→∞ τ(x) as x tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have τ(∞) > √ 5, which implies that X′ 0(x) oscillates even at x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 (2) We have τ(∞) − � τ 2(∞) − 5 ≤ lim x→∞ X′ 0(x) ≤ τ(∞) + � τ 2(∞) − 5, where limx→∞ X′ 0(x) implies the oscillation of X′ 0(x) at x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) For x ≥ 0, we have (X0(x) − τ(x)x)2 = (X′′ 0 (x))2 ≤ τ 2(x) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4) We have τ(∞) − � τ 2(∞) − 5 ≤ lim x→∞((2X0 − xX′ 0)/x) ≤ τ(∞) + � τ 2(∞) − 5, where limx→∞ ((2X0 − xX′ 0)/x) implies the oscillation at x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The function X0(x) is a solution to the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, X0(x) satisfies the following equations: xX′′′ 0 + (x + 9/(4x)) X′ 0 − (X′′ 0 + X0) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) GX0(x) = (X′′ 0 )2 + (1 + 9/(4x2))(X′ 0)2 − 2τ(x)X′ 0 = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) Now, for (1) and (2), we have τ ′(x) = −9/(4x3)X′ 0 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have τ ′(x) < 0 for x > 0 by X′(x) > 0 in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we have τ(x)X′ 0 > 5/2 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, there is the limit τ(∞) := limx→∞ τ(x) by τ(x) > 0 for x ≥ 0: τ(∞) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we have (X′ 0)2 − 2τX′ 0 + 5 < 0 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have τ(x) ≥ √ 5, τ(x) − � τ 2(x) − 5 < X′ 0(x) < τ(x) + � τ 2(x) − 5, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) since X′ 0(x) is a real-valued function and τ(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, τ(x) − � τ 2(x) − 5 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' τ(x) + � τ 2(x) − 5) is an increasing function (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' a decreasing function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, we have obtained (1) τ(∞) ≥ √ 5 and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We shall verify τ(∞) ̸= √ 5 after the proofs of (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, note that τ(∞) = √ 5 is equivalent to limx→∞ X′′ 0 (x) = 0 and limx→∞ X′ 0(x) = √ 5 by the argument above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, τ(∞) > √ 5 implies that both functions X′′ 0 (x) and X′ 0(x) oscillate even at x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (3), by X′′ 0 = τ(x)x − X0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8), we have (X0 − τ(x)x)2 = (X′′ 0 )2 = −(1 + 9/(4x2))(X′ 0)2 + 2τ(x)X′ 0 − 5 < −(X′ 0)2 + 2τ(x)X′ 0 − 5 = −(X′ 0 − τ(x))2 + τ 2(x) − 5 ≤ τ 2(x) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (4), we have limx→∞ ((2X0 − xX′ 0)/x) = limx→∞(2τ(x)−X′ 0(x)) by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we obtain (4) by the existence of τ(∞) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Finally, we show τ(∞) ̸= √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Firstly, we have � � 1 + 9 4x2 �−1/2 τ(x) �′ = 9 4x3 � 1 + 9 4x2 �−3/2 � τ(x) − � 1 + 9 4x2 � X′ 0 � = 9 4x3 � 1 + 9 4x2 �−3/2 X′′′ 0 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) and τ ′(x) = −9/(4x3)X′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The equation shows that (1 + 9/(4x2))−1/2 τ(x) is almost equal to τ(∞) for x ≥ 10, because X′′′ 0 (x) is an oscillating function taking small values around 0 and limx→∞ � (1 + 9/(4x2))−1/2τ(x) � = τ(∞) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We shall precisely verify it below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We integrate the equation on the interval [x, C], where C > x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' As C → ∞, we have ���� � ∞ x �� 1 + 9 4x2 �−1/2τ �′dx ���� = ���τ(∞) − � 1 + 9 4x2 �−1/2τ(x) ��� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) 5Actually, we can show τ(∞) = √ 5 coth( √ 5π/4) ≈ √ 5 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0614 (> √ 5), by using the asymptotic expansion formula for large x of generalized hypergeometric functions of type 1F2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' [12], [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 11 for any x ∈ (0, ∞), since τ(C) converges to τ(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' And we have � C x 9 4t3 � 1 + 9 4t2 �−3/2 X′′′ 0 (t)dt = � 9 4t3 � 1 + 9 4t2 �−3/2 X′′ 0 (t) �C x − � C x � 9 4t3 � 1 + 9 4t2 �−3/2 �′X′′ 0 (t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the equation, we have � 9 4t3 � 1 + 9 4t2 �−3/2�′ = − 27 4t4 � 1 + 9 4t2 �−5/2, (X′′ 0 (t))2 ≤ 4τ 2(t), where the second inequality follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) and the fact that τ(x) is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, by ����− � C x � 9 4t3 � 1 + 9 4t2 �−3/2�′X′′ 0 (t)dt ���� ≤ 27 2 ���� � C x τ(t) t4 dt ���� ≤ 9 2x3τ(x), we have ���� � ∞ x 9 4t3 � 1 + 9 4t2 �−3/2X′′′ 0 (t)dt ���� ≤ 9 2x3τ(x) + 9 2x3τ(x) = 9 x3τ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11), we obtain the inequality ��τ(∞) − (1 + 9/(4x2))−1/2τ(x) �� ≤ (9/x3)τ(x) for any x ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, for any x ∈ (0, ∞), the following inequalities are satisfied � (1 + 9/(4x2))−1/2 − 9/x3� τ(x) ≤ τ(∞) � (1 + 9/(4x2))−1/2 + 9/x3� τ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, if there is an x0 ∈ (0, ∞) such that � (1 + 9/(4x2 0))−1/2 − (9/x3 0) � τ(x0) > √ 5, then we have τ(∞) > √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Actually, for x0 = 10, the desired inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We can make sure the fact as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The function τ(x) is expressed as the following alternating power series: τ(x) = (9/2)x−1 + (9/4) �∞ k=1 bkx2k−1, bk = ak/(4k2 + 5/4), where ak is the coefficients of X0(x) given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' At x = 10, the sequence |bk|x2k−1 is strongly decreasing for k ≥ 4, and |(9/4)bk102k−1| ≤ (3/2) · 10−12 holds for k = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have |τ(x) − � (9/2)x−1 + (9/4) �20 k=1 bkx2k−1� | ≤ (3/2) · 10−12 at x = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, at x = 10 we have the inequality � (1 + 9/(4x2))−1/2 − 9/x3� � (9/2)x−1 + (9/4) �20 k=1 bkx2k−1� ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='35 (> √ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, the proof of the proposition has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4-(4) implies that the first term of h(x, y) = 2X0 − xX′ 0 + y2(X′ 0/x) + √ 5 satisfies 2X0 −xX′ 0 = O(x) as x tends to ∞, and that (2X0 −xX′ 0)/x oscillates even at x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the next section, we study the property of the singular metric g0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The singular set of g0 is given by S1 = {(x, y) ∈ R2 | xy(x2 − y2) = 0}: g0 diverges along the line x = 0 except for the origin (0, 0) and degenerates along the lines y = 0 and x2 − y2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' limx→±0 g0(x, 0) totally degenerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Regular coordinates of the singular metric g0 on D Let D := {(x, y) ∈ R2 | x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this section, we define a regular coordinate system of the singular Riemannian metric space (D, g0) and study the property of the metric g0|D by using the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the next section, we define a curvature surface on D from Φ and Ψ, and then the curvature surface will be recognized as a regularization in R4 of the space (D, g0), by the results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the metric g0 on D(−) := {(x, y) ∈ R2 | x < 0}, we shall study it in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 12 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Now, let g0 be the metric on D given by g0 = 1 h2(x, y) �(x2 − y2)2 x2 (dx)2 + 4y2(dy)2 � and D = {(x, y) | x > 0}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) where g0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' h(x, y)) has been defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) from Φ(x, y) and ¯P(x, y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) from X0(x)): h(x, y) > √ 5 holds on R2 by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The metric g0 degenerates on the set S1 ∩ D = {(x, y) | y(x2 − y2) = 0}, where S1 = {(x, y) | xy(x2 − y2) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The aim in this section is to verify that g0 leads to a positive-definite metric on some domain ˆD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We firstly divide D into several domains according to the singularity of g0: D1 := {(x, y) ∈ D | y ≥ 0}, D2 := {(x, y) ∈ D | y ≤ 0}, Di1 := {(x, y) ∈ Di | x2 ≤ y2}, Di2 := {(x, y) ∈ Di | x2 ≥ y2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) Then, the Riemannian spaces (D1, g0|D1) and (D2, g0|D2) are isometric by g0(x, y) = g0(x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let e1(x, y) and e2(x, y) be the orthonormal vector fields on D \\ S1 defined by e1(x, y) := 1 ¯P cos ϕ ∂ ∂x = xh(x, y) x2 − y2 ∂ ∂x, e2(x, y) := 1 ¯P sin ϕ ∂ ∂y = h(x, y) 2y ∂ ∂y, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) respectively: e1(x, 0) and e2(|y|, y) are also determined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we adopt x2 − y2 not |x2 − y2| (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y not |y|) in the definition of e1(x, y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' e2(x, y)): the functions ¯P cos ϕ and ¯P sin ϕ are analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We extend the orthonormal frame field (e1, e2)(x, y) on D \\ S1 to the singular set of g0 as follows: e1(|y|−, y) := lim Ker(Di1)∋(x,˜y)→(|y|,y) e1(x, ˜y), e1(|y|+, y) := lim Ker(Di2)∋(x,˜y)→(|y|,y) e1(x, ˜y), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) e2(x, +0) := lim Ker(D12)∋(˜x,y)→(x,0) e2(˜x, y), e2(x, −0) := lim Ker(D22)∋(˜x,y)→(x,0) e2(˜x, y), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) where, for Dij ⊂ R2, Ker(Dij) is the open kernel of Dij in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These vectors e1(|y|±, y) and e2(x, ±0) are well defined: for example, e1(|y|+, y) is a unit vector at (|y|, y) with the same direction as ∂/∂x and g0(e1(|y|+, y), e2(|y|, y)) = 0 holds, because e1(|y|+, y) is the limit of unit vector e1(x, y) on Di2 \\ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have e1(|y|−, y) = −e1(|y|+, y) on x = |y| and e2(x, +0) = −e1(x, −0) on y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3 below, we shall obtain an explicit expression of these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, on each curve (|y|, y) ⊂ Dij (i, j = 1, 2), we determine e1(|y|, y) as follows: e1(|y|, y) := e1(|y|−, y) if (|y|, y) ∈ Di1 and e1(|y|, y) := e1(|y|+, y) if (|y|, y) ∈ Di2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Similarly, on each curve {y = 0} ⊂ Di2 (i = 1, 2), we determine e2(x, 0) as follows: e2(x, 0) := e2(x, +0) if (x, 0) ∈ D12 and e2(x, 0) := e2(x, −0) if (x, 0) ∈ D22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, (e1, e2)(x, y) is an orthonormal frame field defined on each domain Dij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The following lemma follows directly from the sign of (x2 − y2) or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The frame field (e1, e2)(x, y) on each Dij satisfies the following facts (1) and (2): (1) If (x, y) ∈ Di1 (i = 1, 2), then the two vectors e1(x, y) and (∂/∂x)(x, y) have the inverse directions to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' If (x, y) ∈ Di2 (i = 1, 2), then the two vectors e1(x, y) and (∂/∂x)(x, y) have the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) If (x, y) ∈ D1j (j = 1, 2), then the two vectors e2(x, y) and (∂/∂y)(x, y) have the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' If (x, y) ∈ D2j (j = 1, 2), then the two vectors e2(x, y) and (∂/∂y)(x, y) have the inverse directions to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 13 Next, for the functions ˆx := (1/3)x3 − xy2 and ˆy := y2, we define an analytic map ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The Jacobi matrix of the map ι is given by J(x, y) = � ˆxx ˆxy ˆyx ˆyy � = � x2 − y2 −2xy 0 2y � , det J(x, y) = 2y(x2 − y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) The singular set S1 ∩ D of the metric g0 coincides with the vanishing points of det J(x, y), and hence det J(x, y) is an irreducible polynomial determining the singularities of g0|D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, the following lemma is obtained by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The function ˆx = (1/3)x3 − xy2 (x > 0) satisfies the following facts (1) and (2): (1) For a fixed y ̸= 0, ˆx has only one critical point x0 = |y|, and then ˆx decreases (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' increases) on the interval (0, x0] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' [x0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, we have limx↘0 ˆx = 0 and limx→∞ ˆx = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) For y = 0, ˆx is an increasing function on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, we have limx↘0 ˜x = 0 and limx→∞ ˜x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have ι(|y|, y) = (−(2/3)(ˆy)3/2, ˆy), ˆy(ι(x, 0)) = 0, ι(x, y) = ι(x, −y) and g0(x, y) = g0(x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let η(ˆy) := (−(2/3)(ˆy)3/2, ˆy) = ι(|y|, y), and ˆDi (i = 1, 2) be the domains defined by ˆD1 := {(ˆx, ˆy) ∈ R2 | ˆy ≥ 0, −(2/3)(ˆy)3/2 ≤ ˆx < 0}, ˆD2 := {(ˆx, ˆy) ∈ R2 \\ {(0, 0)} | ˆy ≥ 0, −(2/3)(ˆy)3/2 ≤ ˆx < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' x y y = x D1 x y y = x D11 ι11 ˆx ˆy η �D1 x y y = x D12 ι12 ˆx ˆy η �D2 1 Figure 1 We have the following facts (1) and (2) by the definitions of ˆDi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) At the points η(ˆy) ∈ ˆDi (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' only the tangent half planes T + η(ˆy) ˆDi are deter- mined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' and similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' at the points (ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0) ∈ ˆD2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' only the tangent half planes T + (ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0)) ˆD2 are determined: precisely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' let v = c1∂/∂ˆx + c2∂/∂ˆy be a tangent vector at η(ˆy) in R2 (ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='ˆy) with ∂/∂ˆx := (∂/∂ˆx)(η(ˆy)) and ∂/∂ˆy := (∂/∂ˆy)(η(ˆy)) (or a tangent vector at (ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0) in R2 (ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='ˆy) with ∂/∂ˆx := (∂/∂ˆx)(ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0) and ∂/∂ˆy := (∂/∂ˆy)(ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' then we have T + η(ˆy) ˆDi := {v | ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' n⟩ ≥ 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' T + (ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0) ˆD2 := {v | c2 ≥ 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' where n and ⟨v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' n⟩ are the inward unit normal vector at η(ˆy) for the domain ˆDi and the Euclidean inner product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) For the maps ιi2 := ι|Di2 : Di2 → ˆD2 (i = 1, 2), we have dι12(e2(x, +0)) = dι22(e2(x, −0)) by ι(x, y) = ι(x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 14 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Now, although ˆD1 ⊂ ˆD2 as the subsets in R2, we have to recognize that ˆD1 and ˆD2 are the distinct domains without intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, for i = 1, 2, the map ιi1 := ι|Di1 : Di1 ∋ (x, y) �→ (ˆx, ˆy) ∈ ˆD1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ιi2 : Di2 ∋ (x, y) �→ (ˆx, ˆy) ∈ ˆD2) gives a new coordinate system for the Riemannian space (Di1, g0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (Di2, g0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the metrics ˆg1 on ˆD1 and ˆg2 on ˆD2 determined by g0|Dij = (ιij)∗ˆgj (j = 1, 2), the spaces ( ˆD1, ˆg1) and ( ˆD2, ˆg2) are not isometric even on the common domain ˆD1, since (Di1, g0|Di1) and (Di2, g0|Di2) are not isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, with ˆx = (1/3)x3 − xˆy, the metrics ˆgj (j = 1, 2) are explicitly given by ˆgj := � xh(x, (ˆy)1/2) �−2 � (dˆx)2 + 2x(dˆx)(dˆy) + 2x2(dˆy)2� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) on ˆDj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The metrics ˆgj (j = 1, 2) are positive-definite on ˆDj by xh(x, (ˆy)1/2) > 0: they are defined even on the tangent half spaces T + η(ˆy) ˆDj (j = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The domain ˆD2 does not extend to ˆD2 ∪ {(0, 0)}, since the metric ˆg2(ˆι(x, y)) diverges as (x, y) → (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By considering the Riemannian spaces ( ˆDi, ˆgi), we have verified that ˆD1 and ˆD2 are the distinct domains without intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' However, in the following Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4, we shall return to the original definition of the map ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ ˆD1 ∪ ˆD2 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ R2 be the analytic map given as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have the following facts (1), (2) and (3): (1) The Riemannian metrics ˆg1 and ˆg2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) are positive-definite on ˆD1 and ˆD2, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, at each point η(ˆy), the metrics ˆg1 and ˆg2 are also well defined on the tangent space T + η(ˆy) := T + η(ˆy) ˆD1 = T + η(ˆy) ˆD2 ⊂ Tη(ˆy)R2 and ˆg1|T + η(ˆy) = ˆg2|T + η(ˆy) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' similarly, at each points (ˆx, 0), ˆg2 is also well defined on T + (ˆx,0) := T + (ˆx,0) ˆD2 ⊂ T(ˆx,0)R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) We have |y|h(|y|, y) · (∂/∂ˆx)(η(ˆy)) = dι(e1(|y|−, y)) = dι(e1(|y|+, y)) at each point η(ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, e1(|y|−, y) and e1(|y|+, y) are identified with the vector of T + η(ˆy) through the map ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) We have h(x, 0) · (∂/∂ˆy)(ˆx, 0) = dι (e1(x, 0) + e2(x, +0)) = dι (e1(x, 0) + e2(x, −0)) at each point (ˆx, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, e1(x, 0) + e2(x, +0) and e1(x, 0) + e2(x, −0) are identified with the vector of T + (ˆx,0) through the map ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The fact (1) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (2), by ˆx = (1/3)x3 − xy2 and ˆy = y2, we have dˆx = (x2 − y2)dx − 2xydy, dˆy = 2ydy on D, and hence we have ∂ ∂ˆx = 1 x2 − y2 ∂ ∂x, ∂ ∂ˆy = x x2 − y2 ∂ ∂x + 1 2y ∂ ∂y (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) on D \\ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since the map ι|Dij = ιij is given by ιij : Dij ∋ (x, y) �→ (ˆx, ˆy) ∈ ˆDj and (ˆx, ˆy) is a coordinate system of ˆDj, the equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) are equivalent to the equations ∂ ∂ˆx = dι � 1 x2 − y2 ∂ ∂x � , ∂ ∂ˆy = dι � x x2 − y2 ∂ ∂x + 1 2y ∂ ∂y � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) on ι(D \\ S1), and then the first equation also holds at ι(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, the equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) are also satisfied even at each point η(ˆy): these right hand sides at η(ˆy) are determined as the dual base of (dˆx, dˆy)(η(ˆy)) as follows, dι � 1 x2−y2 ∂ ∂x � (ι(|y|, y)) = lim Ker(Dij)∋(x,˜y)→(|y|,y) dι � 1 x2−y2 ∂ ∂x � (ι(x, ˜y)), dι � x x2−y2 ∂ ∂x + 1 2y ∂ ∂y � (ι(|y|, y)) = lim Ker(Dij)∋(x,˜y)→(|y|,y) dι � x x2−y2 ∂ ∂x + 1 2y ∂ ∂y � (ι(x, ˜y)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 15 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For example, we have lim Ker(Dij)∋(x,˜y)→(|y|,y) � (ι∗dˆx) � (x2 − y2)−1∂/∂x �� (x, ˜y) = lim Ker(Dij)∋(x,˜y)→(|y|,y) �� (x2 − y2)dx − 2xydy � � (x2 − y2)−1∂/∂x �� (x, ˜y) = 1, lim Ker(Dij)∋(x,˜y)→(|y|,y) � (ι∗dˆx) � x(x2 − y2)−1∂/∂x + (2y)−1∂/∂y �� (ι(x, ˜y)) = lim Ker(Dij)∋(x,˜y)→(|y|,y) �� (x2 − y2)dx − 2xydy � � x(x2 − y2)−1∂/∂x + (2y)−1∂/∂y �� (x, ˜y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The other equations are also verified in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9), the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) are equivalent to the following equations, |y|h(|y|, y) · (∂/∂ˆx)(ι(|y|, y)) = dι(e1(|y|−, y)) = dι(e1(|y|+, y)), h(|y|, y) · (∂/∂ˆy)(ι(|y|, y)) = dι (e1(|y|−, y) + e2(|y|, y)) = dι (e1(|y|+, y) + e2(|y|, y)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11) In particular, we have obtained the fact (2) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The fact (3) is obtained in the same way as in (2), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, the proof has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ The spaces ( ˆD1, ˆg1) and ( ˆD2, ˆg2) are not isomeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' However, in the two spaces ( ˆDi, ˆgi) (i = 1, 2), the curves η(ˆy) ⊂ ˆDi, the tangent half spaces T + η(ˆy) ˆDi and the metrics ˆgi on T + η(ˆy) ˆDi have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the orthonormal frame fields (dι(e1(x, y)), dι(e2(x, y))) on ˆDi (i = 1, 2) are uniquely determined from (e1, e2) on each Dij in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 as xh(x, y) · ∂ ∂ˆx(ι(x, y)) = dι(e1(x, y)), h(x, y) · ∂ ∂ˆy(ι(x, y)) = dι(e1(x, y) + e2(x, y)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3: in particular, these equations also hold on ι(S1 ∩ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ι : D ∋ (x, y) �→ (ˆx, ˆy) ∈ R2 be the map given as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the curve η(ˆy) is the envelope of the family of y-curves ι(x, y) with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this proof, we can assume y ≥ 0 by ι(x, y) = ι(x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, any y-curve ι(x, y) is a half line ˆx + xˆy = (1/3)x3 in the domain ˆy ≥ 0 of R2 (ˆx,ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The tangent vector of the curve µ(ˆy) = (−(2/3)(ˆy)3/2, ˆy) is given by dµ/dˆy = −(ˆy)1/2∂/∂ˆx + ∂/∂ˆy = −x ∂/∂ˆx + ∂/∂ˆy from (ˆy)1/2 = x on η(ˆy) = ι(y, y) = ι(x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have h(y, y)dµ/dˆy = −dι(e1) + dι(e1 + e2) = dι(e2) = (h(y, y)/(2y))dι (∂/∂y) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The equation shows that any y-curve ι(x, y) is tangent to the curve η(ˆy) at ι(y, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4, each y-curve ι(x, y) with x is included in ˆD1 if |y| ≥ x and included in ˆD2 if −x ≤ y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, let ˆD1 and ˆD2 be the distinct domains in R2 without intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On the direct sum ˆD1 ⊕ ˆD2, we define the Riemannian metric ˆg0 := ˆg1 ⊕ ˆg2 by ˆg0(X, Y ) := ˆg1(X, Y ) for X, Y ∈ Tp ˆD1, ˆg0(X, Y ) := ˆg2(X, Y ) for X, Y ∈ Tp ˆD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3, we can define the domain ˆD := ˆD1 ⊕ ˆD2/ ∼, where ∼ implies two identifications: the first one is the identification of two curves η(ˆy) in both domains ˆD1 and ˆD2, and the second one is the identification with T + η(ˆy) ˆD1 and T + η(ˆy) ˆD2 at each point of η(ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, we have obtained the Riemannian space ( ˆD, ˆg0) without singularity, that is, the space ( ˆD, ˆg0) 16 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, an analytic map ˆι : D → ˆD = ˆD1 ⊕ ˆD2/ ∼ is defined from ι : D → R2 by ˆι : Di1 ∋ (x, y) → ι(x, y) ∈ ˆD1, ˆι : Di2 ∋ (x, y) → ι(x, y) ∈ ˆD2, which satisfies g0 = ˆι∗ˆg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We summarize the above results as the following theorem, where D = D1 ∪ D2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2): Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let (D, g0) and ( ˆD, ˆg0) be the Riemannian spaces and ˆι : D → ˆD be the analytic map, given as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let e1(x, y) and e2(x, y) be the orthonormal frame field on each Dij in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, they are satisfied the following facts (1), (2), (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) The space ( ˆD, ˆg0) is an isometric regularization of the two singular spaces (Di, g0|Di) (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, ( ˆD, ˆg0) is a regular Riemannian space and g0 = (ˆι|Di)∗ˆg0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) The vector field dˆι(e1(x, y)) is determined on ˆD as xh(x, y) · ∂/∂ˆx(ˆι(x, y)), and in par- ticular, dˆι(e1(|y|+, y)) = dˆι(e1(|y|−, y)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Any x-curve ˆι(x, y) with y ̸= 0 has a cusp at the point ˆι(|y|, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) The vector field dˆι(e2(x, y)) is determined on ˆD as h(x, y) · (∂/∂ˆy − x∂/∂ˆx)(ˆι(x, y)), and in particular, dˆι(e2(x, +0)) = dˆι(e2(x, −0)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Any y-curve ˆι(x, y) reflects at the point ˆι(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4) The curve ˆι(|y|, y) = η(ˆy) is the envelope of the family of y-curves ˆι(x, y) with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Almost all facts have been verified in the argument above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we only prove the fact in (2) that any x-curve ˆι(x, y) with y ̸= 0 has a cusp at ˆι(|y|, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we assume y > 0 by ˆι(x, y) = ˆι(x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, the vector field dˆι(e1) has the same direction as ∂/∂ˆx at any point in ˆD by h(x, y) > 0, and in particular, the fact holds even on η(ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, dˆι(e1) is a standard vector on ˆD to know the direction of the other vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, for each x-curve ˆι(x, y) with y, two tangent vector fields dˆι(e1) and dˆι(∂/∂x) of the curve has the inverse directions on the interval (0, y) to each other but the same direction on the interval (y, ∞), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The fact means that the x-curve reverses the direction at ˆι(y, y) as the curve is passing though the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have ˆι(x, y) ⊂ ˆD1 for x ∈ (0, y] and ˆι(x, y) ⊂ ˆD2 for x ∈ [y, ∞), and ( ˆD1, ˆg0| ˆD1) and ( ˆD2, ˆg0| ˆD2) are not isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By these facts, we can recognize the point ˆι(y, y) as a cusp (not a reflection) of the x-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ By (1) and (3) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5, it is natural to consider both sides of the space ( ˆD, ˆg0), which are the front ( ˆD+, ˆg+ 0 ) and the back ( ˆD−, ˆg− 0 ): the spaces ( ˆD+, ˆg+ 0 ) and ( ˆD−, ˆg− 0 ) are isometric with ( ˆD, ˆg0), respectively, and ( ˆD+, ˆg+ 0 ) and ( ˆD−, ˆg− 0 ) have only the common points (ˆx, 0) and tangent spaces T(ˆx,0) ˆD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ( ˆD±, ˆg± 0 ) be the space equipped with both sides of ( ˆD, ˆg0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we can regard the analytic map ˆι as a bijection ˆι : (D, g0) → ( ˆD±, ˆg± 0 ) defined by ˆι : D1 ∋ (x, y) → ˆι(x, y) ∈ ˆD+, ˆι : D2 ∋ (x, y) → ˆι(x, y) ∈ ˆD−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 and the above argument, we could say that ( ˆD±, ˆg± 0 ) is a regular Riemannian space associated with the singular Riemannian space (D, g0) and the coordinate system (ˆx, ˆy) of ( ˆD, ˆg0) is a regular coordinate system of (D, g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the next section, we construct a curvature surface f 0(x, y) defined on D as a realization in R4 of the space (D, g0), by replacing ˆι(x, y) with f 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the vector fields df 0(e1) and df 0(e2) give an orthonormal frame field on the surface f 0(x, y) in place of dˆι(e1) and dˆι(e2) and the map f 0 satisfies a similar property to the map ˆι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Extended frame field and analytic curvature surface defined on D For the hyperbolic metric ˇgH on R2, a pair Φ = (ϕ, ϕz) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) and a class Ψ = ( ¯P, ¯Pz, ¯κ3) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) are determined, and these functions give rise to a singular metric g0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) on D = EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 17 {(x, y) | x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this section, we verify that an orthonormal frame field of R4 is determined on the whole space D from Φ and Ψ and that the frame field leads to an analytic curvature surface in R4 defined on D, which is a realization in R4 of the space (D, g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, as mentioned in (R4) and (R5) of the introduction, for a simply connected open set V such that V ⊂ D \\ S, a curvature surface f 0 V (x, y) on V and a generic conformally flat hyper- surface fV (x, y, z) on V ×I are determined from Φ and Ψ such that f 0 V (x, y) = fV (x, y, 0) holds on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the coordinates (x, y, z) of fV (x, y, z) and (x, y) of f 0 V (x, y) are principal curvature line coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let FV (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z) = [N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Xα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Xβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Xγ] (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z) be the orthonormal frame field on fV (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' where N is a normal vector field of fV (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z) and (Xα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Xβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Xγ)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z) are the principal curvature directions corresponding to x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' and z-lines: let ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z) be the function determined from Φ by ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0) = ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) and ϕz(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0) = ϕz(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' and P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z) be the function determined from Ψ by P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0) = ¯P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) and Pz(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 0) = ¯Pz(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' then the frame field (Xα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Xβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Xγ)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' z) is given from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) by Xα := (P cos ϕ)−1∂fV /∂x, Xβ := (P sin ϕ)−1∂fV /∂y, Xγ := P −1∂fV /∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, the differential dfV (x, y, z) of fV (x, y, z) is expressed as dfV (x, y, z) = P(x, y, z) ((cos ϕ dx)Xα + (sin ϕ dy)Xβ + dzXγ) (x, y, z), For the frame FV (x, y, z), we put φ(x, y) := N(x, y, 0), X0 α(x, y) := Xα(x, y, 0), X0 β(x, y) := Xβ(x, y, 0) and ξ(x, y) := Xγ(x, y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have X0 α := ( ¯P cos ϕ)−1∂f 0 V /∂x = e1(f 0 V ), X0 β := ( ¯P sin ϕ)−1∂f 0 V /∂y = e2(f 0 V ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) with the vector fields e1(x, y) and e2(x, y) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3), and then F 0 V (x, y) := � φ, X0 α, X0 β, ξ � (x, y) is an orthonormal frame field on f 0 V (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The vector fields X0 α(x, y) and X0 β(x, y) are the curvature directions on f 0 V (x, y): if we regard φ(x, y) as a surface in S3, then X0 α and X0 β are the curvature directions and ξ is a normal vector field of φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, the differential df 0 V (x, y) of f 0 V (x, y) is determined as df 0 V (x, y) = ¯P(x, y) � cos ϕ dxX0 α + sin ϕ dyX0 β � (x, y) = h−1(x, y) � ((x2 − y2)/x)dxX0 α + 2ydyX0 β � (x, y), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) and in particular f 0 V (x, y) has the metric g0|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the functions ¯κi(x, y) (i = 1, 2) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6), the principal curvatures κi(x, y, z) (i = 1, 2, 3) of fV (x, y, z) satisfy κi(x, y, 0) = ¯κi(x, y) (i = 1, 2, 3) on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The structure equation of fV (x, y, z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=', the equation for dF(x, y, z)) is determined from (ϕ(x, y, z), P(x, y, z), κi(x, y, z)), and then the structure equation of f 0(x, y) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='e, the equa- tion for dF 0 V (x, y)) is determined from Φ and Ψ, as the restriction to FV (x, y, 0) of those for FV (x, y, z) (see the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let Dij (i, j = 1, 2) be the sub-domains in D defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For each Dij, we arbitrarily fix a simply connected open set Vij such that Vij ⊂ Dij \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the canonical connection ∇′ of R4, the structure equation of f 0 Vij(x, y) is determined on each domain Vij as the following form: ∇′ X0αX0 α = ¯κ1φ − B1ξ − C1X0 β, ∇′ X0αφ = −¯κ1X0 α, ∇′ X0αξ = B1X0 α, ∇′ X0αX0 β = C1X0 α, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and ∇′ X0 βX0 β = ¯κ2φ − B2ξ − C2X0 α, ∇′ X0 βφ = −¯κ2X0 β, ∇′ X0 βξ = B2X0 β, ∇′ X0 βX0 α = C2X0 β, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) where Bk and Ck are functions on Vij and ¯κi (i = 1, 2) are the functions given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6): ¯κ1 = (2y/(x2 − y2)) � X0 + √ 5 2 � , ¯κ2 = y−1 � ((x2 + y2)/2)(X′ 0/x) − � X0 + √ 5 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 18 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA The equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) imply that X0 α and X0 β are the principal directions of f 0 Vij(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On each Vij given above, the functions Bk, Ck in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) are determined as follows: B1(x, y) = −(x2 − y2)−1� X0 + √ 5 2 � − √ 5, C1(x, y) = −2(x2 − y2)−1� X0 + √ 5 2 � , B2(x) = −(1/2) (X′ 0/x) − √ 5, C2(x) = X′′ 0 − X′ 0/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for these functions Bk and Ck, the equations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) extend to D \\ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The structure equation of the hypersurface fVij(x, y, z) is given at [18, Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by F 0 Vij(x, y) = FVij(x, y, 0), the derivative of F 0 Vij = [φ, X0 α, X0 β, ξ] is obtained from the equation by taking as ϕ(x, y, 0) = ϕ(x, y), ϕz(x, y, 0) = ϕz(x, y) and P(x, y, z) = ¯P(x, y), Pz(x, y, 0) = ¯Pz(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For B1: We have B1 = ¯P −2 cos ϕ( ¯P cos ϕ)z = − 1 cos ϕ � ( ¯P −1)z cos ϕ + ¯P −1ϕz sin ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we obtain B1 by ϕz = y/(x2 + y2), ( ¯P −1)z = √ 5 + � (x2 − y2) � X0 + √ 5 2 � + 2xy2X′ 0 � /(x2 + y2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For C1: We have C1 = ¯P −2 sin ϕ cos ϕ( ¯P cos ϕ)y = −1 sin ϕ cos ϕ � ( ¯P −1)y cos ϕ + ¯P −1ϕy sin ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we obtain C1 by ϕy = 2x/(x2 + y2), ( ¯P −1)y = 2xy(x2 + y2)−2� − 2X0 + 2xX′ 0 − √ 5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For C2: We have C2 = ¯P −2 sin ϕ cos ϕ( ¯P sin ϕ)x = −1 sin ϕ cos ϕ � ( ¯P −1 sin ϕ)x − 2 ¯P −1ϕx cos ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by ϕx = −2y/(x2 + y2), we have ( ¯P −1 sin ϕ)x = 2 ¯P −1ϕx cos ϕ + 2y(x2 − y2)(x2 + y2)−2 (X′ 0 − xX′′ 0 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we obtain C2 = X′′ 0 − X′ 0/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For B2: We have B2 = ¯P −2 sin ϕ( ¯P sin ϕ)z = − 1 sin ϕ � ( ¯P −1)z sin ϕ − ¯P −1ϕz cos ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, in the same way as above, we obtain B2 from the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The last statement follows from the fact that the functions Bk and Ck above are independent of choice of the domains Vij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Next, we obtain the following lemma directly from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On each Vij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' we have the following equations: ∇′ ∂/∂xφ = −a1X0 α := −2y xh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) � X0 + √ 5 2 � X0 α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ∇′ ∂/∂xX0 β = c1X0 α := −2 xh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) � X0 + √ 5 2 � X0 α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ∇′ ∂/∂xξ = b1X0 α := −1 xh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) � X0 + √ 5 2 + √ 5(x2 − y2) � X0 α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ∇′ ∂/∂yφ = −a2X0 β := 1 h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) � 2X0 + √ 5 − (x2 + y2)X′ 0 x � X0 β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ∇′ ∂/∂yX0 α = c2X0 β := 2y h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) � X′′ 0 − X′ 0 x � X0 β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ∇′ ∂/∂yξ = b2X0 β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' := −2y h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' y) �X′ 0 2x + √ 5 � X0 β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 19 Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' all equations above are independent of choice of the domains Vij and they are analytic equations defined on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We also have ∇′ ∂/∂xX0 α = a1φ − b1ξ − c1X0 β and ∇′ ∂/∂yX0 β = a2φ − b2ξ − c2X0 α by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the analytic functions (ai, bi, ci) on D in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, we define the matrix-valued functions Ω1 and Ω2 by Ω1(x, y) = � ��� 0 a1 0 0 −a1 0 c1 b1 0 −c1 0 0 0 −b1 0 0 � ��� (x, y), Ω2(x, y) = � ��� 0 0 a2 0 0 0 −c2 0 −a2 c2 0 b2 0 0 −b2 0 � ��� (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) For a frame field F 0(x, y) := [φ, X0 α, X0 β, ξ](x, y) and the matrix-valued differential 1-form Ω = Ω1dx + Ω2dy on D, the equations of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 are summarized as dF 0 = F 0Ω, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) which is an analytic equation on the whole domain D, and in particular, it is independent of choice of Vij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3 below, we shall verify that the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) has a solution F 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the solution F 0(x, y) is uniquely determined up to a transformation AF 0(x, y) by a constant orthogonal matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the sake of simplicity for the argument, we determine an initial condition of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) by F 0(x0, y0) = Id (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) for a while, where (x0, y0) is a point of V12 and Id is the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' It is possible to take such an initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, for the frame field FV12(x, y, z) determining the hypersur- face fV12(x, y, z), suppose FV12(x0, y0, 0) ̸= Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We take a constant orthogonal matrix A such that AFV12(x0, y0, 0) = Id holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the frame AFV12(x, y, z) determines the hypersurface AfV12(x, y, z) by AdfV12 = d(AfV12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' An analytic orthonormal frame field F 0(x, y) is uniquely determined on D such that it is a solution to the structure equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) under the initial condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the surface f 0 V12(x, y) on V12 determined above extends to an analytic surface f 0(x, y) with the metric g0 defined on the whole domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, for any simply connected domain V ′ ij satisfying V ′ ij ⊂ Dij \\ S, there is a generic conformally flat hypersurface fV ′ ij(x, y, z) on V ′ ij × I such that f 0(x, y) = fV ′ ij(x, y, 0) holds on V ′ ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The frame field FV12(x, y, z) on V12 × I is determined from the hypersurface fV12(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Since F 0 V12(x, y) = FV12(x, y, 0) and f 0 V12(x, y) = fV12(x, y, 0), the 1-form Ω(x, y) satisfies the Maurer-Cartan equation dΩ + Ω ∧ Ω = 0 on the open domain V12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since Ω is analytic on D, the Maurer-Cartan equation is satisfied on the whole domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, a frame field F 0(x, y) on D is uniquely determined under the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7), which is the extension of F 0 V12(x, y) on V12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, for the vector fields X0 α and X0 β of F 0(x, y) on D, an analytic surface f 0(x, y) on D is determined by df 0 = θ1X0 α + θ2X0 β, θ1 := (x2 − y2)/(xh(x, y))dx, θ2 := 2y/h(x, y)dy, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) since f 0 V12(x, y) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) on V12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, df 0(x, y) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) satisfies d(df 0) = 0 on D by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, f 0(x, y) is an extension of f 0 V12(x, y) to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, φ and ξ are the normal vector fields of f 0(x, y), which are distinguished by the condition for the hypersurface fV12(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the surface f 0(x, y) on D has the metric g0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Finally, the existence of the generic conformally flat hypersurface fV ′ ij(x, y, z) in the last statement follows from the fact that F 0(x, y) is determined by Φ and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ 20 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4 (Curvature surface defined on D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3, we may recognize that the analytic surface f 0(x, y) on D is an extended curvature surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Our aim is to study the property and the structure on f 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We call f 0(x, y) and F 0(x, y) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3, respectively, a curvature surface defined on D and a frame field determining the curvature surface f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we can arbitrarily give the initial condition of F 0(x, y) by F 0(x0, y0) = A not only (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7), where (x0, y0) and A are a point D and an orthogonal matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For a fixed frame field F 0(x, y), the curvature surface f 0(x, y) is determined uniquely up to a parallel translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we remark on a curvature surface f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Since we have fixed F 0(x, y) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7), det F 0(x, y) = 1 holds on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, with W(x, y) := det[φ, ∂f 0/∂x, ∂f 0/∂y, ξ](x, y), we have W ≤ 0 on D11, W ≥ 0 on D12, W ≥ 0 on D21, W ≤ 0 on D22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) Next, we study the coordinate lines of a curvature surface f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ∥v∥ be the Euclidean norm for a vector v ∈ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let f 0(x, y) be a curvature surface on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for an x-curve f 0(x, y) with fixed y, we have the following facts (1), (2) and (3): (1) Along any x-curve f 0(x, y), the vector (yX0 β − φ)(x, y) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, an analytic unit vector u(y) of y is determined by u(y) = (1 + y2)−1/2(yX0 β − φ)(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) Let f(x, y) be an analytic unit vector defined by f(x, y) := ((1 + y2)(5 + 4y2))−1/2 � X0 β − 2(1 + y2)ξ + yφ � (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' When we regard the surface f 0(x, y) as a one-parameter family of x-curves, it is ex- pressed as f 0(x, y) = (2 √ 5)−1� (5 + 4y2)(1 + y2)−1f(x, y) + A(y), where A(y) is a R4-valued analytic function of y determined uniquely up to a parallel translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) Any x-curve f 0(x, y) lies on a 2-sphere S2 y of radius (2 √ 5)−1� (5 + 4y2)(1 + y2)−1 in an affine hyperplane R3 y perpendicular to u(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We firstly verify (1)–(3) on the domain U := {(x, y) ∈ D12 | x > y ≥ 0}, and then we use the equations in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we have ∇′ X0α(yX0 β − φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we have ∇′ X0α(X0 β − 2ξ) = 2 √ 5X0 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The component ˆf(x, y) of (X0 β − 2ξ)(x, y) perpendicular to u(y) is given by ˆf(x, y) := X0 β − 2ξ − y(1 + y2)−1/2u = (1 + y2)−1 � X0 β − 2(1 + y2)ξ + yφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have ∇′ X0α ˆf = 2 √ 5X0 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since f(x, y) is the normalization of ˆf(x, y), we have (2) by X0 αf 0 = X0 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Finally, since X0 α(x, y) ⊥ u(y) (or f(x, y) ⊥ u(y)), we obtain (3) by the norm ∥ˆf(x, y)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By the above argument, we have verified the theorem for x-curves on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, all x-curves on D are also expressed as the form (2), since all our objects: the frame field F 0(x, y), the surface f 0(x, y) and the vector f(x, y), are analytic on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Actually, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, we have the following equation, ∇′ ∂/∂x � (2 √ 5)−1� (5 + 4y2)(1 + y2)−1 f 2 + A � = [(x2 − y2)/(xh(x, y))] X0 α on D, which coincides with ∂f 0/∂x on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We can verify directly by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 that all x-curves on D also satisfy (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, the proof has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let f 0(x, y) be a curvature surface on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for a y-curve f 0(x, y) with fixed x, we have the following facts (1), (2) and (3): EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 21 (1) Along any y-curve f 0(x, y), the vector (−B2X0 α + C2ξ)(x, y) is constant, where B2(x) = −(1/2)(X′ 0/x) − √ 5 and C2(x) = X′′ 0 − (X′ 0/x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, an analytic unit vector ˜u(x) of x is determined by ˜u(x) = (B2 2 + C2 2)−1/2(−B2X0 α + C2ξ)(x, y) and (B2 2 + C2 2)(x) = (X′ 0/x)(2X0 − xX′ 0 − X′ 0/x + √ 5) > 0 holds for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) Let ˜f(x, y) be an analytic unit vector defined by ˜f(x, y) := (B2 2 + C2 2)−1/2(x)(B2ξ + C2X0 α)(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' When we regard the surface f 0(x, y) as a one-parameter family of y-curves, it is ex- pressed as f 0(x, y) = (B2 2 + C2 2)−1/2(x)˜f(x, y) + ˜A(x), where ˜A(x) is a R4-valued analytic function of x determined uniquely up to a parallel translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) Any y-curve f 0(x, y) lies on a standard 2-sphere S2 x of radius (B2 2 + C2 2)−1/2(x) in an affine hyperplane R3 x perpendicular to ˜u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We firstly prove the theorem on the domain ˜U := {(x, y) ∈ D1| y > 0}, and then we use the equations in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we have ∇′ X0 β(−B2X0 α + C2ξ) = 0 and (B2 2 + C2 2)(x) > 0 on R by B2(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore (B2 2 + C2 2)(x) is expressed as the form in (1) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have obtained (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we have ∇′ X0 β(B2ξ+C2X0 α) = (B2 2 +C2 2)X0 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by (B2 2 +C2 2)(x) ̸= 0 and X0 βf 0 = X0 β, we have (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Finally, since X0 β ⊥ ˜u(x) (or ˜f(x, y) ⊥ ˜u(x)), we have (3) by ∥(B2 2 + C2 2)−1/2(x)˜f(x, y)∥ = (B2 2 + C2 2)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, we have verified the theorem for y-curves on ˜U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These results also hold for all y-curves on D by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 similarly to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, the proof has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, the vectors u(y) and ˜u(x) are perpendic- ular for all (x, y) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, by the definitions of u(y) and ˜u(x), we have ⟨u(y), ˜u(x)⟩ = C(x, y) ⟨yX0 β − φ, −B2X0 α + C2ξ⟩(x, y) = 0, where C(x, y) := ((1 + y2)(B2 2 + C2 2)(x))−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) The vector u(y) of y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˜u(x) of x) moves on a circle S1 in a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We shall give a simple proof of these facts in the next section (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Certainly, we can also make sure these facts by direct calculation, but it is very hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we only give the norms of the first derivatives of u(y) and ˜u(x): ∥∇′ ∂/∂yu(y)∥ = y2 1 + y2, ∥∇′ ∂/∂x˜u(x)∥ = � X′ 0 + 2 √ 5x � � 2X0 + √ 5 � 4xX′ 0 � 2X0 − xX′ 0 − X′ 0/x + √ 5 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These norms show that the length of the curve u(y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˜u(x)) diverges to ∞ as y tends to ±∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' as x tends to ∞): in the right hand side of the second equation, we have (2X0−xX′ 0)(x) = O(x) and X′ 0(x) is a positive bounded function, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the length of ˜u(x) on (ε, 1] also diverges to ∞ as ε(> 0) → 0, since we have ∥∇′ ∂/∂x˜u(x)∥ ≈ √ 5/(2x) by X0(0) = 5/2 and X′ 0(x) ≈ 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, any x-curve f 0(x, y) with fixed y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' any y-curve f 0(x, y) with fixed x) belongs to an affine hyperplane perpendicular to u(y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ˜u(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We can determine the following orthonormal frame fields along the curves: let f(x, y) and ˜f(x, y) be vectors in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' along each x-curve, the frame field is given by � u(y), X0 α(x, y), f(x, y), u2(x, y) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) 22 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA where u2(x, y) is defined by u2(x, y) := (5 + 4y2)−1/2(2X0 β + ξ + 2yφ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' along each y-curve, the frame field is given by �˜u(x), X0 β(x, y), ˜f(x, y), φ(x, y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11) Now, in the following theorem, we verify that each x-curve with y ̸= 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' each y-curve with x) of a curvature surface f 0(x, y) on D has a cusp at the point x = |y| (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' at the point y = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we say that a curve p(t) in R3 has a cusp of type (2, 3, 4) at t = 0, if p(t) is expressed as p(t) = (at2, bt3, ct4) around t = 0 with constants a, b and c (abc ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For any coordinate curve of a curvature surface f 0(x, y) on D, we have the following facts (1) and (2): (1) Any x-curve with y ̸= 0 has a cusp of type (2, 3, 4) at x = |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) Any y-curve has a cusp of type (2, 3, 4) at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) Let f 0(x, y) be an x-curve with fixed y ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We study the curve only in a small neighborhood of (|y|, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, the first derivative of the curve is given by f 0 x(x, y) = ((x2 − y2)/(xh(x, y))) X0 α(x, y) and � (x2 − y2)/(xh(x, y)) � x = (x2h2(x, y))−1 � (x2 + y2)h(x, y) + (x2 − y2)2(X′′ 0 − X′ 0/x) � , f 0 xx(x, y) = � (x2 − y2)/(xh(x, y)) � x X0 α + � (x2 − y2)/(xh(x, y)) � � ∇′ ∂/∂xX0 α � (x, y), f 0 xxx(|y|, y) = � (x2 − y2)/(xh(x, y)) � xx (|y|, y)X0 α(|y|, y) + (4/h(|y|, y)) � ∇′ ∂/∂xX0 α � (|y|, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We define the functions wi(x) (i = 1, 2, 3) of x by w1(x) := ⟨f 0(x, y), X0 α(|y|, y)⟩, w2(x) := ⟨f 0(x, y), u2(|y|, y)⟩, w3(x) := ⟨f 0(x, y), f(|y|, y)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we directly have (w1)x(|y|) = (w2)x(|y|) = (w3)x(|y|) = (w2)xx(|y|) = (w3)xx(|y|) = 0, (w1)xx(|y|) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have (w2)xxx(|y|) ̸= 0 by ∇′ ∂/∂xX0 α = a1φ − b1ξ − c1X0 β and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For w3, (w3)xxx(|y|) = 0 holds by ⟨∇′ ∂/∂xX0 α, f⟩(|y|, y) = 0, and further we have (w3)xxxx(|y|) ̸= 0 by ⟨(∇′ ∂/∂x)2X0 α, f⟩(|y|, y) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, for t := x − |y|, we have w1(|y| + t) ≈ w1(|y|) + (t2/2)(w1)xx(|y|), w2(|y| + t) ≈ w2(|y|) + (t3/6)(w2)xxx(|y|), w3(y + t) ≈ w3(|y|) + (t4/24)(w3)xxxx(|y|), as t tends to 0, which show that the x-curve f 0(x, y) = (w1, w2, w3)(x) has the cusp of type (2, 3, 4) at the point (|y|, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) Let f 0(x, y) be a y-curve with fixed x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We study the curve only in a small neighborhood of (x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The first derivative of the curve is given by f 0 y (x, y) = (2y/h(x, y))X0 β(x, y), and (2y/h(x, y))y = (2/h2(x, y)) � h(x, y) − 2y2(X′ 0/x) � , f 0 yy(x, y) = (2y/h(x, y))yX0 β(x, y) + (2y/h(x, y))(∇′ ∂/∂yX0 β)(x, y), f 0 yyy(x, 0) = (2y/h(x, y))yy(x, 0)X0 β(x, 0) + (4/h(x, 0))(∇′ ∂/∂yX0 β)(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We define the functions ˜wi(y) (i = 1, 2, 3) by ˜w1(y) := ⟨f 0(x, y), X0 β(x, 0)⟩, ˜w2(y) := ⟨f 0(x, y), φ(x, 0)⟩, ˜w3(y) := ⟨f 0(x, y), ˜f(x, 0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we obtain ( ˜w1)yy(0) ̸= 0, ( ˜w2)yyy(0) ̸= 0, ( ˜w3)yyyy(0) ̸= 0, and that the lower derivatives of each ˜wi(y) vanish at y = 0, in the same way as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have verified that the y-curve also has the cusp of type (2,3,4) at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 23 By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8, the curves f 0(|y|, y) of y and f 0(x, 0) of x are the cuspidal edges in a curvature surface f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The curve f 0(x, 0) of x is also the singular set of the hyperbolic 2-metric ˇgH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For these curves, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) The two curves f 0(|y|, y) of y are the envelopes of the family of y-curves f 0(x, y) with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) The vector φ0 := φ(x, 0) does not depend on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The curve f 0(x, 0) of x lies on a 2-sphere S2 y=0 of radius 1/2 in an affine hyperplane R3 y=0 perpendicular to φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) For the sake of simplicity, we assume y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The derivative of the y-curve f 0(y, y) is given by ((∂/∂x + ∂/∂y) f 0) (y, y) = (∂f 0/∂y)(y, y) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, the tangent vectors of both y-curves f 0(x, y) and f 0(y, y) coincide at (y, y), which implies that the y-curve f 0(y, y) is the envelope of the family of y-curves f 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) We have (∇′ ∂/∂xφ)(x, 0) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The other statement is already proved in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ To visualize the curvature surface f 0(x, y) locally, we use the approximation f δ(x, y) that will be defined in Section 6, and illustrate them via the projection π: when we determine the coordinates (r1, r2, r3, r4) of R4 by the frame F δ(y0, y0) = Id, the projection π eliminates the 4-th coordinate of R4, that is, π ((r1, r2, r3, r4)) = (r1, r2, r3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 y π◦f1/20 −−−−→ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' This shows the x-curves passing through the points � f 1/20(yk, yk) � on the cuspidal edge: the lines in the domain are given by yk = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='025k (k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' , 20) and yk − 1 ≤ x ≤ yk + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Each x-curve is drawn with black for yk − 1 ≤ x < yk and with gray for yk < x ≤ yk + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25 y π◦f1/20 −−−−→ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' This shows the envelope made by the y-curves around the points � f 1/20(xk, xk) � from the same domain as Figure 2: the lines in the domain are given by xk = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='05k (k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' , 49), xk − 1 ≤ y ≤ xk + 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='75 ≤ y ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, for a curvature surface f 0(x, y) on D, we study the relation with f 0|D1(x, y) and f 0|D2(x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' To the end, we translate the surface f 0(x, y) such that the obtained surface f 0(x, y) satisfies f 0(p0) = 0 at some point p0 := (x0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we consider that the frame of 24 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA R4 containing the new f 0(x, y) is given by F 0(p0) = Id, for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, [X0 α, X0 β, ξ](p0) is a frame of R3 y=0 and the x-curve f 0(x, 0) belongs to R3 y=0 by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9-(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let f 0(x, y) and F 0(x, y) be the curvature surface and the frame field, ex- pressed by the above coordinate system of R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, with the orthogonal matrix B = [bij] such that b11 = −1, bii = 1 (2 ≤ i ≤ 4) and bij = 0 (i ̸= j), we have (B ◦ f 0)(x, y) = f 0(x, −y), (B ◦ φ)(x, y) = −φ(x, −y) and (B ◦ ξ)(x, y) = ξ(x, −y) for y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, (B ◦ ˜u)(x) = ˜u(x) and (B ◦ ˜A)(x) = ˜A(x) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let ¯y = −y for y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We put ¯φ(x, y) := −φ(x, ¯y), ¯X0 α(x, y) := X0 α(x, ¯y), ¯X0 β(x, y) := X0 β(x, ¯y) and ¯ξ(x, y) := ξ(x, ¯y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We study the derivative in y ≥ 0 of ¯F 0 := [¯φ, ¯X0 α, ¯X0 β, ¯ξ] by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have d ¯F 0(x, y) = ¯F 0(x, y)Ω(x, y) for y ≥ 0, where Ω(x, y) is the differen- tial 1-form in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) such that dF 0(x, y) = F 0(x, y)Ω(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, ¯F 0(x, y) is another solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) in y ≥ 0, and hence there is an orthogonal matrix B such that BF 0(x, y) = ¯F 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, B is determined at p0 from F 0(p0) = [φ, X0 α, X0 β, ξ](p0) = Id and ¯F 0(p0) = [−φ, X0 α, X0 β, ξ](p0) as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, f 0(x, ±y) is the integral surface of (X0 α, X0 β)(x, ±y) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8), respectively, and (B ◦ f 0)(x, 0) = f 0(x, 0) holds by f 0(x, 0) ∈ R3 y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have (B ◦f 0)(x, y) = f 0(x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we have verified the first statement in the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (B◦ ˜u)(x) = ˜u(x): Any y-curve f 0(x, y) lies on a 2-sphere S2 x in R3 x perpendicular to ˜u(x) and it is not a circle (for example, Figure 6 in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have B(R3 x) = R3 x by (B ◦ f 0)(x, y) = f 0(x, −y), which shows (B ◦ ˜u)(x) = ˜u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (B ◦ ˜A)(x) = ˜A(x): We have f 0(x, y) = (B2 2 + C2 2)−1/2(x)˜f(x, y) + ˜A(x) and f 0(x, −y) = (B2 2 + C2 2)−1/2(x)˜f(x, −y) + ˜A(x) for y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since (B ◦ f 0)(x, y) = f 0(x, −y) and (B ◦ ˜f)(x, y) = ˜f(x, −y) by the definition of ˜f(x, y), we obtain (B ◦ ˜A)(x) = ˜A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have verified the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we remark the relation with F 0(x, y) and F 0(x, −y) for y ≥ 0, explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For φ(x, ±y), the first coordinate elements are equal and the other elements have the different sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For each X0 α(x, ±y), X0 β(x, ±y) and ξ(x, ±y), the first elements have the different sign and the other elements are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8 and Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10, we may regard any curvature surface f 0 : (D, g0) ∋ (x, y) �→ f 0(x, y) ∈ R4 as a realization in R4 of the regular coordinate system ˆι : (D, g0) ∋ (x, y) �→ ˆι(x, y) ∈ ( ˆD±, ˆg± 0 ) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13), that is, an isometric map ¯f 0 : ( ˆD±, ˆg± 0 ) ∋ (ˆx, ˆy) �→ ¯f 0(ˆx, ˆy) ∈ R4 is determined by ( ¯f 0 ◦ ˆι)(x, y) = f 0(x, y) for (x, y) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The sign of the determinant W = det [φ, ∂f 0/∂x, ∂f 0/∂y, ξ] changes for each domain Dij, as mentioned at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) under the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' However, if we replace (∂f 0/∂x, ∂f 0/∂y) with (∂ ¯f 0/∂ˆx, ∂ ¯f 0/∂ˆy), then the determinant is positive on the whole D from (∂ ¯f 0/∂ˆx, ∂ ¯f 0/∂ˆy) = h−1(X0 α/x, X0 α +X0 β) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, note that (∂ ¯f 0/∂ˆx)(ˆι(x, y)) = (xh(x, y))−1X0 α(x, y) diverges to ∞ as D ∋ (x, y) → (0, y) for any (0, y) including y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Structure of the extended curvature surface Let f 0(x, y) be a curvature surface on D defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this section, we study the limit of x-curves f 0(x, y) with fixed y as x tends to 0 and ∞, and the limit of y-curves f 0(x, y) with fixed x as y tends to ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In order to study x-curves f 0(x, y) as x tends to 0, we change x for a new parameter u by x = e−u: the change is reasonable for the metric g0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) and the equations of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, any u-curve f 0(e−u, y) uniformly converges to a circle S1(y) parametrized by u as u tends to ∞, and any y-curve converges to a point p(x) as y tends to ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the convergence of u-curves is uniform with respect to y ∈ (−∞, ∞) and the convergence of y-curves is also uniform in the wider sense with respect to x ∈ (0, ∞) (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, S1(y) and p(x), respectively, are continuous for EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 25 y and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Through further study for these convergences, we can understand the structure on f 0(x, y) in R4 in detail as mentioned in the introduction, and connect two curvature surfaces defined on D = {(x, y) | x > 0} and D(−) := {(x, y) | x < 0} continuously at the origin (0, 0) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, let f 0(x, y) be an x-curve with fixed y in a curvature surface on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The x-curve f 0(x, y) is included in an affine hyperplane R3 y perpendicular to u(y) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5, and the orthonormal frame field of R3 y along the x-curve is given by [X0 α, f, u2](x, y) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We change the parameter x for u by x = e−u, and then, for a vector Z(x) of x, we denote by Z(e−u) the vector ¯Z(u) := Z(e−u) of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The vectors f(e−u, y) and u2(e−u, y) satisfy the equations ∇′ ∂/∂uf = −v3(x, y)X0 α, ∇′ ∂/∂uu2 = v2(x, y)X0 α, by ∂/∂u = −x ∂/∂x and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, where v2(x, y) := (5 + 4y2)−1/2h−1(x, y) � (5 + 4y2)(X0 + √ 5/2) + √ 5(x2 − y2) � , v3(x, y) := 2 √ 5(x2 − y2)h−1(x, y) � (1 + y2)(5 + 4y2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence we have ∇′ ∂/∂uX0 α = v3(x, y)f−v2(x, y)u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the functions vi(x, y) (i = 2, 3), vi(0, y) are also well defined: we have v2(0, 0) = √ 5/2, v2(0, y) > 0 for y ∈ R and v3(0, 0) = 0, v3(0, y) < 0 for y ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' As x tends to 0, the functions v2(x, y) and v3(x, y), respectively, converge to v2(0, y) and v3(0, y) uniformly with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, we have the following fact in the same way as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3-(2): there is a number t1 (0 < t1 < 1) such that |v2(x, y) − v2(0, y)| < 3 2 5 + √ 5 + 4y2 � 5 + 4y2 x2 h(x, y) < 3 2 5 + √ 5 + 4y2 � 5 + 4y2 x2 5 + √ 5 + y2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) |v3(x, y) − v3(0, y)| < 2 √ 5 7 � 1 + y2 5 + 4y2 x2(7 + y2) h(x, y) < 2 √ 5 7 � 1 + y2 5 + 4y2x2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) hold for 0 < x < t1 and any y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By the above equations, M(u, y) := [X0 α, f, u2](e−u, y) satisfies the following equation for (u, y) ∈ R2: ∂M ∂u (u, y) = M(u, y)V (u, y), V (u, y) := � � 0 −v3(x(u), y) v2(x(u), y) v3(x(u), y) 0 0 −v2(x(u), y) 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) In V (u, y), we replace the functions v2(x, y) and v3(x, y) with v0 2(y) := v2(0, y) and v0 3(y) := v3(0, y), respectively, and denote by V 0(y) the new V (u, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we define another matrix N(u, y) = [b(u, y), a(u, y), c(u, y)], by the differential equation ∂N ∂u (u, y) = N(u, y)V 0(y), V 0(y) := � � 0 −v0 3 v0 2 v0 3 0 0 −v0 2 0 0 � � (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let N(u, y) = [b, a, c] (u, y) be a solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, v0(y) := (v0 2a + v0 3c)(u, y) does not depend on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The solution N(u, y) is a rotation with respect to the axis v0(y) and the speed of its rotation is ∥v0(y)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this proof, we fix the parameter y arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have ∇′ ∂/∂u(v0 2a + v0 3c) = (−v0 2v0 3 + v0 3v0 2)b = 0 by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, v0(y) does not depend on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we set B0(y) := 1 ∥v0(y)∥ � � 0 ∥v0∥ 0 v0 2 0 v0 3 v0 3 0 −v0 2 � � (y), Φu(y) := � � 1 0 0 0 cos(u∥v0(y)∥) − sin(u∥v0(y)∥) 0 sin(u∥v0(y)∥) cos(u∥v0(y)∥) � � , 26 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA and then B0(y) and Φu(y) belong to the special orthogonal group SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have � B0Φ−1 u (∂Φu/∂u) (B0)−1� (y) = V 0(y) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, with any orthogonal matrix N ∞(y) depending on y, the solution N(u, y) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) will be given by N(u, y) = N ∞(y) � B0(y)Φu(y)(B0)−1(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) The equation implies that the lemma holds good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We can verify that N(x, y) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) is a solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) as follows: taking the derivative of N(u, y) by u, we obtain ∂N/∂u = N ∞B0Φu(B0)−1 � B0Φ−1 u (∂Φu/∂u)(B0)−1� = NV 0 by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Now, we return to the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) for M on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the following lemma, we use the notations in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1, and for a square matrix A = [aij], we define the norm ∥A∥ by ∥A∥ := ��(aij)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma and Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) Let ¯ M(u, y) := M(u, y) (B0Φ−u(B0)−1) (y) for (u, y) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, there is a continuous orthogonal matrix N ∞(y) of y ∈ R such that ¯ M(u, y) with fixed y converges to N ∞(y) as u tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the convergence for u-curves ¯ M(u, y) is uniform with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) Let N ∞(y) be the matrix of y in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the frame field M(u, y) with fixed y uni- formly converges to the rotation N(u, y) = N ∞(y)(B0Φu(B0)−1)(y) as u tends to ∞, and further the convergence for u-curves M(u, y) is also uniform with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, as u tends to ∞, we say that M(u, y) with fixed y uniformly converges to N(u, y), if there is a real number U for any ε > 0 such that ∥M(u, y) − N(u, y)∥ < ε holds for u > U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we say that the convergence for u-curves M(u, y) is uniform with respect to y ∈ R, if there is a number U satisfying above independently of y ∈ R for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) Taking the derivative of ¯ M(u, y) by u, we have ∂ ¯ M/∂u = (∂M/∂u)B0Φ−u(B0)−1 + MB0(∂Φ−u/∂u)(B0)−1 = MV � B0Φ−u(B0)−1� − M � B0(∂Φ−u/∂(−u))Φu(B0)−1� � B0Φ−u(B0)−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have B0(∂Φ−u/∂(−u))Φu(B0)−1 = V 0 by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we obtain ∂ ¯ M/∂u = M(V − V 0) � B0Φ−u(B0)−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) Now, we have ��∂ ¯ M/∂u(log(1/x), y) �� = ∥V (x, y) − V (0, y)∥ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7), and hence we have ∥ ¯ M(u1, y) − ¯ M(u2, y)∥ ≤ ���� � u2 u1 ∥V (x(u), y) − V (0, y)∥du ���� = ����� � e−u2 e−u1 ∥V (x, y) − V (0, y)∥dx x ����� by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) there is an orthogonal matrix N ∞(y) for each y ∈ R such that ∥ ¯ M(log(1/x), y) − N ∞(y)∥ ≤ Cx2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) holds as x ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, N ∞(y) is continuous for y ∈ R, since the continuous matrix ¯ M(u, y) of y converges to N ∞(y) uniformly with respect to y ∈ R as x ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) We have ∥M(u, y)−N ∞(y) (B0Φu(B0)−1) (y)∥ = ∥M(u, y) (B0Φ−u(B0)−1) (y)−N ∞(y)∥ ≤ Ce−2u by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since C is independent of y, we have the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 27 In Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2-(2), we put N ∞(y) = [b∞, a∞, c∞](y) and N(u, y) = [b, a, c](u, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Since N(u, y) is a matrix in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) determined by N ∞(y), the vector v0(y) := (v0 2a + v0 3c)(u, y) does not depend on u by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have ∥v0(y)∥ = √ 5/2 by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We define v1(y) := −(2/ √ 5)v0(y) and another unit vector v2(u, y) by v2(u, y) := (2/ √ 5) � v0 3(y)a(u, y) − v0 2(y)c(u, y) � , which is perpendicular to b(u, y) and v1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have [b, v2] (u, y) = � b∞, 2 √ 5 � v0 3a∞ − v0 2c∞�� (y) � cos( √ 5u/2) − sin( √ 5u/2) sin( √ 5u/2) cos( √ 5u/2) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6), and a(u, y) = 2 √ 5 (−v0 2(y)v1(y) + v0 3(y)v2(u, y)) , c(u, y) = − 2 √ 5 (v0 3(y)v1(y) + v0 2(y)v2(u, y)) by the definitions of vi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, any u-curve a(u, y) with fixed y is a circle of center −(2/ √ 5)v0 2(y)v1(y) and radius (2/ √ 5)|v0 3(y)| by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, in the case y = 0, the circle degenerates into one point −v1(0) by v0 3(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have (∇′ ∂/∂yb)(u, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, as u tends to ∞, X0 α(e−u, y) and (∇′ ∂/∂yX0 α)(e−u, y) with fixed y, respectively, converge uniformly to b(u, y) and zero-vector, and these convergences for u-curves are also uniform with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, the first convergence follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 and the second one follows from X′′ 0 −X′ 0/x = O(x2) in the equation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, the vectors b∞(y) and (2/ √ 5)(v0 3a∞ − v0 2c∞)(y) are constant by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10): we write these constant vectors as b∞ := b∞(y), −c∞ := −c∞(0) = (2/ √ 5)((v0 3a∞ − v0 2c∞))(y) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11) by v0 2(0) > 0 and v0 3(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, [b, v2](u, y) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) does not depend on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since v1(y) is perpendicular to u(y), b∞ and c∞, the pair (u(y), v1(y)) is an orthonormal frame field of the plane perpendicular to the vectors b∞ and c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, u(y) moves on a circle S1, of which fact we have mentioned in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7-(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we write [b, v2](u) for [b, v2](u, y), and then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11) we have b(u) = cos( √ 5u/2)b∞ − sin( √ 5u/2)c∞, v2(u) = − sin( √ 5u/2)b∞ − cos( √ 5u/2)c∞, a(u, y) = (2/ √ 5) � −v0 2(y)v1(y) + v0 3(y)v2(u) � , a(u, 0) = −v1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12) In the expression of a(u, y), y is a parameter for the family S1(y) of circles and u is a rotation parameter for each circle S1(y): v1(y) is perpendicular to the circle S1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, f(e−u, y) converges uniformly to a(u, y) as u tends to ∞, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For each y ∈ R, let Γ(u, y) be the following circle in R4 parametrized by u: Γ(u, y) := � (2 √ 5)−1� (5 + 4y2)(1 + y2)−1a(u, y) + A(y) y ̸= 0, −(1/2)v1(0) + A(0) y = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) where A(y) is the R4-valued function in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5-(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Each circle Γ(u, y) with y has the radius of 2y2/( √ 5 h(0, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the following theorem, we use the notations in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let f 0(x, y) be a curvature surface defined on D and F 0(x, y) = [φ, X0 α, X0 β, ξ](x, y) be the orthonormal frame field on D determining f 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let u(y) and f(x, y) be the analytic vectors in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, there is an orthonormal pair (b∞, c∞) of constant vectors such that it satisfies the following conditions (1), (2) and (3): (1) The vector u(y) moves on the unit circle in a plane perpendicular to b∞ and c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let v1(y) be a vector determined by (∇′ d/dyu)(y) = (y2/(1+y2))v1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, [u(y), v1(y), b∞, c∞] is an orthonormal frame field of R4 depending on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 28 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA (2) As u tends to ∞, all u-curves X0 α(e−u, y) with y uniformly converge to a circle b(u) = cos( √ 5u/2)b∞ − sin( √ 5u/2)c∞, and each u-curve f(e−u, y) with fixed y uniformly con- verges to the following circle a(u, y), a(u, y) = � (2/ √ 5) (−v0 2(y)v1(y) + v0 3(y)v2(u)) y ̸= 0, −v1(0) y = 0, where v2(u) is a circle given by v2(u) = − sin( √ 5u/2)b∞ − cos( √ 5u/2)c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These convergences for u-curves are also uniform with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, the circles a(u, y) deform continuously for y and the circle a(u, 0) degenerates to one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) Any u-curve f 0(e−u, y) with fixed y uniformly converges to the circle Γ(u, y) as u tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The convergence for u-curves is also uniform with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, the circles Γ(u, y) deform continuously for y and the circle Γ(u, 0) degenerates to one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Almost all facts have been verified in the argument above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, since the vector u(y) of y is perpendicular to b∞ and c∞, we have (∇′ d/dyu)(y) = ±(y2/(1+y2))v1(y) for (1) by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7-(2): we shall determine the sign ± at the end of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The convergences in (2) follow from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12): the frame field M(u, y) = [X0 α, f, u2](e−u, y) with fixed y uniformly converges to the rotation N(u, y) = [b, a, c](u, y) as u → ∞, and the converge for u-curves is also uniform with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, a(u, y) is analytic for y and a(u, 0) is one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We obtain (3) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we verify (∇′ d/dyu)(y) = (y2/(1+y2))v1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we use the fact (2): limx→0 f(x, 0) = −v1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Firstly, we have (∇′ d/dyu)(y) = (1 + y2)−3/2 � (1 + (1 + y2)a2)(X0 β + yφ) − y(1 + y2)(b2ξ + c2X0 α) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we define p(x, 0) for each x by the limit p(x, 0) := limy→0(∇′ d/dyu)(y)/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have limx→0 p(x, 0) = − limx→0 f(x, 0) = v1(0) by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14) and the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we obtain the equation desired for all y ∈ R by the continuity of (∇′ d/dyu)(y) and v1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the vector ˜u(x) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, ⟨u(y), ˜u(x)⟩ = 0 holds as in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7-(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, ˜u(x) is expressed as a linear combination of X0 α, f and u2: explicitly, we have ˜u(e−u) = 1 � B2 2 + C2 2(e−u) � −B2X0 α + C2 � 5 + 4y2 � u2 − 2 � 1 + y2 f �� (e−u, y), where B2 and C2 are the functions in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since limu→∞ ˜u(e−u) = limu→∞ X0 α(e−u, y) by B2(x) < 0 and C2(x) = O(x2), ˜u(e−u) uniformly converges to b(u) = cos( √ 5u/2)b∞ − sin( √ 5u/2)c∞ as u tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For a fixed y, let S2 y be the 2-sphere in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 including an x-curve f 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have Γ(u, y) ⊂ S2 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The two points ±(2 √ 5)−1� (5 + 4y2)(1 + y2)−1v1(y) + A(y) are the antipodal points of S2 y to each other and the tangent spaces at these points are spanned by the vectors b∞ and c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, we have limu→∞ f(e−u, y) = a(u, y) and ⟨a(u, y), v1(y)⟩ < 0, and further ⟨v1(y), v2(u)⟩ ≡ 0 holds for any u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, for each y ∈ R and 0 < ε < 1, let ly(x) be an x-curve given by ly(x) := f 0(x, y) for x ∈ [ε, 1] and L(ly) be the length of ly(x) by the metric g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' As ε → 0, L(ly) diverges to ∞ if y ̸= 0 and L(ly=0) converges to a finite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have the following corollary: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let {(xn, yn)}∞ n=1 be a sequence in D convergent to the origin (0, 0) with respect to the Euclidean distance of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the sequence f 0(xn, yn) converges to the point Γ(u, 0) = EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 29 −(1/2)v1(0) + A(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, when we define f 0(0, 0) := Γ(u, 0), f 0(x, y) on D extends to D ∪ {(0, 0)} continuously in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let (xn, yn) ∈ D be a convergent sequence to (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let un = exp(1/xn) and S1 y := {Γ(u, y)|u ∈ R}, where S1 0 is one point p0 := Γ(u, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We define a kind of distance ¯d(S1 y, p0) between S1 y and p0 by ¯d(S1 y, p0) := Max{d(p, p0) | p ∈ S1 y} for the distance d of R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, there is a number N1 for any ε > 0 such that ¯d(S1 yn, p0) < ε holds for n > N1, because S1 y degenerates to p0 continuously as y → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, f 0(e−un, y) converges to Γ(un, y) uniformly with respect to y ∈ R as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, there is a number N2 for any ε > 0 such that ∥f 0(e−un, y) − Γ(un, y)∥ < ε holds for any y ∈ R if n > N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, for any ε > 0 and n > Max{N1, N2} we have ∥f 0(e−un, yn) − p0∥ < ∥f 0(e−un, yn) − Γ(un, yn)∥ + ¯d(S1 yn, p0) < 2ε, which shows that the corollary holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The vector u(y) determines an orthonormal pair (˜b ∞, ˜c∞) of constant vectors uniquely such that it satisfies the following conditions (1) and (2): (1) The system [b∞, c∞, ˜b ∞, ˜c∞] is an othonormal base of R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) u(y) and v1(y) satisfy the following equations: u(y) = − sin(y − arctan y) ˜b ∞ + cos(y − arctan y) ˜c∞, v1(y) = − cos(y − arctan y) ˜b ∞ − sin(y − arctan y) ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, as y → ∞, u(y) and v1(y), respectively, converge uniformly to the following circles ˜b(y) and ˜c(y): ˜b(y) := cos y ˜b ∞ + sin y ˜c∞, ˜c(y) := − sin y ˜b ∞ + cos y ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The unit vector u(y) belongs to the plane perpendicular to b∞ and c∞, and it satisfies the equation (∇′ d/dyu)(y) = (y2/(1 + y2))v1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Now, for the function b1(x, y) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, we have b1(x, y) → − √ 5x/(2X0 − xX′ 0) as x → ∞ for any fixed y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, x/(2X0 − xX′ 0) for x > 0 is a positive and oscillating function satisfying (x/(2X0 − xX′ 0))(0) = 0 and x/(2X0 − xX′ 0) = O(1) as x → ∞, by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, the integral w(x) := √ 5 � x 0 [x/(2X0 − xX′ 0)]dx is an increasing function with w(x) = O(x) as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have the following theorem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let n(x, y) := (1 + y2)−1/2(X0 β + yφ)(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have the following facts as x → ∞: (1) Each x-curve n(x, y) for x ∈ [1, ∞) with fixed y has infinite length and the curve approaches the point −v1(y) while winding uniformly around the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the curve never reaches −v1(y) at any finite x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) There is a constant orthonormal base [ˆb, ˆc] of the plane spanned by b∞ and c∞ such that all x-curves ξ(x, y) and X0 α(x, y) with y, respectively, converge uniformly to the circles expressed as ¯b(x) = cos w(x) ˆb − sin w(x) ˆc and ¯c(x) = sin w(x) ˆb + cos w(x) ˆc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) Each x-curve f 0(x, y) converges uniformly to the following circle of S2 y: −(2 √ 5)−1(1 + y2)−1/2 � v1(x) + 2(1 + y2)1/2 ¯b(x) � + A(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The convergence for the x-curves in (2) and (3) is uniform in the wider sense with respect to y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 30 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We consider the x-curves n(x, y) as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Before the proof, we note that the vector n(x, y) is perpendicular to u(y) for any x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4 are important in the proof, which give the property of the function X0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Firstly, we verify the lemma for an arbitrarily fix y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (1), we firstly have the following equation for y ̸= 06, ∥v1(y) + n(x, y)∥ = O(1/x) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='15) by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14) and (∇′ d/dyu)(y) = (y2/(1+y2))v1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, the x-curve n(x, y) converges uniformly to the point −v1(y) as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the equation n(x, y) = −v1(y) does not holds at any finite x, since we have b2(x, y) ̸= 0 for (x, y) ∈ D in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Note that the convergence of the x-curve is also uniform with respect to y of any bounded interval [−y1, y1], since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='15) holds uniformly for y ∈ [−y1, y1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we have ∇′ ∂/∂xn = (1 + y2)1/2c1X0 α, ∇′ ∂/∂xX0 α = −(1 + y2)1/2c1(x)n − b1ξ, ∇′ ∂/∂xξ = b1X0 α (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='16) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The length L(x) on [1, x] of the x-curve n(x, y) diverges to ∞ as x → ∞, from (∇′ ∂/∂xn)(x, y) = O(1/x) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we regard n(x, y) as an analytic curve on the unit 2-sphere S2, and then X0 α and ξ are the unit tangent vector and the unit normal vector of n(x, y), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The curvature κ(x, y) = (1 + y2)−1/2(b1/c1) of n(x, y) diverges to ∞ as x → ∞, by κ(x, y) = O(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, when we take the geodesic mx on S2 for each x connecting n(x, y) with −v1(y) and put t1 := inf{x > t0 | n(x, y) ∈ mt0} for x = t0, the length L(t1) − L(t0) of the curve n(x, y) (t0 ≤ x ≤ t1) converges to 0 and further the curvature of the curve n(x, y) (t0 ≤ x ≤ t1) is almost constant κ(t0, y), which diverges to ∞ as t0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' This fact implies that, as x increases, the curve n(x, y) gradually approaches a smaller and smaller nearly circle of central axis −v1(y), which shows (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (2): Firstly, note that every tangent plane at n(x, y) converges to T−v1(y)S2 uniformly as x → ∞, by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we regard the unit normal ξ(x, y) and the unit tangent vector X0 α(x, y) of the curve n(x, y) as the vectors at −v1(x) by the parallel translation of R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, these vectors ξ(x, y) and X0 α(x, y) converge uniformly to the curves parametrized by x on the unit circle of T−v1(y)S2 by (1), of which curves we denote by ¯b(x, y) and ¯c(x, y), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have ∇′ ∂/∂x¯b = −w′(x)¯c and ∇′ ∂/∂x¯c = w′(x)¯b from the definition of w(x) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='16) by c1 = O(1/x), which implies the fact (2) for each y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, there is an orthonormal pair (ˆb(y), ˆc(y)) of vectors depending on y and perpendicular to u(y) and v1(y) such that ¯b(x, y) = cos w(x) ˆb(y) − sin w(x) ˆc(y), ¯c(x, y) = sin w(x) ˆb(y) + cos w(x) ˆc(y) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The fact (3) follows from (1) and (2) directly by f(x, y) = (5 + 4y2)−1/2[n − 2(1 + y2)1/2ξ](x, y) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Finally, we obtain that ˆb(y) and ˆc(y) are constant vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, as x → ∞, ∇′ ∂/∂yξ and ∇′ ∂/∂yX0 α converges to zero-vector uniformly with respect to y of any bounded interval [−y1, y1], by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we have completed the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ 6For y = 0, see the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 31 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These show the x-curve n1/20(x, 1) on [1/500, 100]: the right hand- side is a bird’s-eye view of the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These show the x-curve ξ1/20(x, 1) on [1/500, 100]: the right hand- side is a bird’s-eye view of the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we study the y-curves f 0(x, y) with fixed x as y tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Along any y-curve f 0(x, y), we have an orthonormal frame field [˜u(x), X0 β(x, y), ˜f(x, y), φ(x, y)] as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11): any y-curve belongs to an affine hyperplane R3 x perpendicular to ˜u(x), where (˜u(x), ˜f(x, y)) is defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, the vector ˜f = (B2 2 + C2 2)−1/2(C2X0 α + B2ξ) is analytic and (B2 2 + C2 2)(x) = (X′ 0/x)(2X0 − xX′ 0 − X′ 0/x + √ 5) > 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, we have ∇′ ∂/∂y ˜f =: −˜v3X0 β, ∇′ ∂/∂yφ = −a2X0 β =: ˜v2X0 β, ∇′ ∂/∂yX0 β = ˜v3˜f − ˜v2φ, where ˜v2(x, y) = h−1(x, y)[2X0+ √ 5−(x2+y2)(X′ 0/x)], ˜v3(x, y) = −2(y/h(x, y))(B2 2+C2 2)1/2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 32 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) For an arbitrarily fixed x1 > 0, let (0, x1] be the bounded interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, there is a constant C > 0 independent of x ∈ (0, x1] such that |˜v2(x, y) + 1| < Cy−2 and |˜v3 + q(x)/y| < Cy−3 hold for x ∈ (0, x1] as y tends to ∞, where q(x) := 2(x/X′ 0)(B2 2 + C2 2)1/2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) For arbitrarily fixed two numbers x1 > x0 > 0, let [x0, x1] be the bounded interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, there is a constant C > 0 independent of x ∈ [x0, x1] such that ∥(∇′ ∂/∂x˜u)(x) − T(x)˜f(x, y)∥ < C/y, ∥(∇′ ∂/∂x˜f)(x, y) + T(x)˜u(x)∥ < C/y hold for x ∈ [x0, x1] as y tends to ∞, where T(x) := (X′ 0 + 2 √ 5x)(2X0 + √ 5)/[4xX′ 0(2X0 − xX′ 0 − X′ 0/x + √ 5)] given in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have h(x, y) = 2X0 + √ 5 + (−x2 + y2)(X′ 0/x) = 2X0 − xX′ 0 + y2(X′ 0/x) + √ 5 > √ 5 and B2 2 + C2 2 = (X′ 0/x)(2X0 − xX′ 0 − X′ 0/x + √ 5) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, X′ 0/x is a bounded positive functions on R satisfying (X′ 0/x)(0) = 2, and 2X0 − xX′ 0 is also a positive function satisfying (2X0 − xX′ 0)(0) = 5 and (2X0 − xX0)(x) = O(x) as x → ∞, by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we have 1/h(x, y) = (1/y2)(x/X′ 0) − (x/X′ 0)(2X0 − xX′ 0 + √ 5)/(y2h(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The fact (1) follows from the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (2), we have T(x) = [(B2 2 + C2 2)−1(−B′ 2C2 + B2C′ 2) + √ 5/X′ 0](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we obtain (2) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ In the study the case of y-curves f 0(x, y) as y → ∞, we can not adopt the way of the proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3 by the fact ˜v3(x) → q(x)/y in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8-(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, for the function T(x) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8, we define its integral ˜w(x) by ˜w(x) := ( √ 5/2) log x+ � x 0 [T(x) − ( √ 5/2)(1/x)]dx, since T(x) = ( √ 5/2)(1/x) + O(x) as x → 0 and T(x) = O(1) as x → ∞ by the definition of X0(x) and Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the function ˜w(x) and the orthonormal frame [b∞, c∞, ˜b ∞, ˜c∞] in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3 and Corol- lary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, we have the following theorem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' There is a unit vector ˜v1(x) of x ∈ (0, ∞) such that it satisfy the following facts (1)–(3): (1) The vector ˜v1(x) is uniquely determined by the function ˜w(x) and the equations (∇′ d/dx˜u)(x) = T(x)˜v1(x) and (∇′ d/dx˜v1)(x) = −T(x)˜u(x): we have ˜u(x) = cos ˜w(x) b∞ + sin ˜w(x) c∞, ˜v1(x) = − sin ˜w(x) b∞ + cos ˜w(x) c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) As y tends to ∞, each y-curve f 0(x, y) with fixed x converges to the point (B2 2 + C2 2)−1/2(x)˜v1(x) + ˜A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) As y tends to ∞, every y-curve X0 β(x, y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' every y-curve φ(x, y)) with x uniformly converges to the circle ˜b(y) = cos y ˜b ∞ + sin y ˜c∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' −˜c(y) = sin y ˜b ∞ − cos y ˜c∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the convergence for these y-curves in (2) and (3) are uniform in the wider sense with respect to x ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let (0, x1] or [x0, x1] be an arbitrarily fixed bounded interval, as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (1) and (2): There is a vector ˜v1(x) := limy→∞ ˜f(x, y) on [x0, x1] by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8-(2) such that (∇′ d/dx˜u)(x) = T(x)˜v1(x) and (∇′ d/dx˜v1)(x) = −T(x)˜u(x) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since [x0, x1] EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 33 is any bounded interval in (0, ∞), these equations hold on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, there is a constant orthonormal base [b, c] of the plane spanned by b∞ and c∞ such that ˜u(x) = cos ˜w(x) b + sin ˜w(x) c, ˜v1(x) = − sin ˜w(x) b + cos ˜w(x) c hold, since ˜u(x) and u(y) are perpendicular to each other for all (x, y) ∈ D as mentioned in Re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7-(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, ˜u(x) converges uniformly to the circle cos(( √ 5/2) log x)b+sin(( √ 5/2) log x)c as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On the other hand, the circle is expressed as cos( √ 5u/2)b∞ − sin( √ 5u/2)c∞ by Re- mark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we obtain b = b∞ and c = c∞ by u = − log x, which shows (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, since any y-curve f 0(x, y) is given by (B2 2 + C2 2)−1/2(x)˜f(x, y) + ˜A(x), we obtain (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For (3): For x ∈ [x0, x1], we have u(y) = (1 + y2)−1/2(yX0 β − φ)(x, y) = X0 β(x, y) + O(1/y) and (∇′ d/dyu)(y) = −φ(x, y) + O(1/y) as y → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, u(y) and v1(y) converge uniformly to the following circles as y → ∞: u(y) → ˜b(y) = cos y ˜b ∞ + sin y ˜c∞, v1(y) → ˜c(y) = − sin y ˜b ∞ + cos y ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we obtain (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, the proof has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have limy→∞ f 0(x, y) = limy→−∞ f 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the sake of simplicity, we translate the curvature surface f 0(x, y) such that the obtained surface f 0(x, y) satisfies f 0(p0) = 0 for some point p0 = (x0, 0), which means 0 ∈ R3 y=0, and take a coordinate system of R4 satisfying the setting in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10 for the new f 0(x, y): F 0(p0) = Id and [X0 α, X0 β, ξ](p0) is a frame of R3 y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for the reflection B in Corollar 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10, we have B[φ, X0 α, X0 β, ξ](x, −y) = [−φ, X0 α, X0 β, ξ](x, y), B(f 0(x, −y)) = f 0(x, y), B(˜u(x)) = ˜u(x) and B( ˜A(x)) = ˜A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have B(˜f(x, −y)) = ˜f(x, y) by the definition of ˜f(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we denote by ˆf 0(x, y)(= f 0(x, y)) (y ≥ 0) the curvature surface determined by the frame field [−φ, X0 α, X0 β, ξ](x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have limy→∞ ˆf 0(x, y) = limy→∞ f 0(x, y) = (B2 2 + C2 2)−1/2(x)˜v1(x) + ˜A(x) by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we obtain limy→∞ f 0(x, −y) = (B2 2 + C2 2)−1/2(x)˜v1(x) + ˜A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, we have B( ˆf 0(x, y)) = f 0(x, −y), B( ˜A(x)) = ˜A(x) and B(˜v1(x)) = ˜v1(x) by B(∇′ d/dx˜u(x)) = ∇′ d/dx˜u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, for every x ∈ (0, ∞), the tangent space at limy→∞ f 0(x, y) in every 2-sphere S2 x, which contains the y-curve f 0(x, y), is spanned by the constant vectors ˜b ∞ and ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, for the R4-valued functions A(y) and ˜A(x) in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, we have the following facts: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) The curve A(y) lies on a plane spanned by ˜b∞ and ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) The curve ˜A(x) lies on a plane spanned by b∞ and c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) A curvature surface f 0(x, y) on D is expressed as f 0(x, y) = (2 √ 5(1 + y2))−1 � X0 β − 2(1 + y2)ξ + yφ � + A(y), (as a family of x-curves f 0(x, y) with y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The vector ˜u(x) = (B2 2 +C2 2)−1/2(−B2X0 α +C2ξ)(x, y) moves on the circle of a plane spanned by b∞ and c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have only to show that ⟨A′(y), (−B2X0 α + C2ξ)(x, y)⟩ = 0 holds for any (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, by f 0 y (x, y) = (2y/h(x, y))X0 β, we have A′(y) = (2 √ 5(1 + y2))−1(b2ξ + c2X0 α)(x, y) except for the terms of X0 β(x, y) and φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have −c2B2 + b2C2 = 0 by Lemmata 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, which implies ⟨A′(y), (−B2X0 α + C2ξ)(x, y)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) A curvature surface f 0(x, y) and the vector u(y) are expressed as f 0(x, y) = (B2 2 + C2 2)−1(x)(B2ξ + C2X0 α)(x, y) + ˜A(x), u(y) = (1 + y2)−1/2(yX0 β − φ)(x, y), 34 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA and u(y) moves on the circle of a plane spanned by ˜b ∞ and ˜c∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have only to show that ⟨ ˜A′(x), (yX0 β − φ)(x, y)⟩ = 0 holds for any (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have ˜A′(x) = C2(x)(B2 2 + C2 2)−1(x) � −a1φ + c1X0 β � (x, y) except for the terms of X0 α(x, y) and ξ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have a1 + yc1 = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2, which implies that the desired equation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ (a) y-curve at x0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (b) y-curve at x0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These are the y-curves f 1/20(x0, y) on [−100, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Each curve is in a sphere S2, and has a cusp at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We change its color from gray to black at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' As for the asymptotic behaviors, each curve converges to one point in S2 as y → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' This shows the x-curve f 1/20(x, 2) on [1/500, 100]: the figure on the right hand-side is a side view of the figure on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' This curve is in a sphere S2, and has a cusp at x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We change its color from black to gray at x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' As for the asymptotic behaviors, the curve converges uniformly to parallel small circles in S2 as x → 0 and x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 35 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' This shows the x-curve f 1/20(x, 0) on [1/100, 100]: the figure on the right hand-side is a side view of the figure on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The curve is in a sphere S2, and converges to a point in S2 as x → 0, and uniformly to a small circle in S2 as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' This curve has no cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence of the all results above, we have obtained the simple structure on the curvature surface f 0(x, y) on D, as mentioned in the introduction: in a sense, f 0(0, y) is a curve on the plane {˜b∞, ˜c∞}R and f 0(x, ∞) = f 0(x, −∞) is a curve on the plane {b∞, c∞}R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' these planes are orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' At the end of this section, we study a curvature surface defined on D(−) = {(x, y) | x < 0} in relation to a curvature surface f 0(x, y) on D = {(x, y) | x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The singular metric g0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) is defined on D ∪ {(0, 0)} ∪ D(−), and the map j : D ∋ (x, y) ↔ (−x, y) ∈ D(−) is an isometry between the spaces (D, g0|D) and (D(−), g0|D(−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the coordinate system (ˆx, ˆy) in §3, ˆx = (1/3)x3 − xy2 and ˆy = y2, is regular not only for the metric g0|D but also for the metric g0|D(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Although ¯P in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) is a negative function on D(−), the equations of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 also hold on D(−) for Φ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) and Ψ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2), and hence these equations also determine a curvature surface ˆf 0(x, y) defined on D(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, the following lemma is verified in the same way as the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F 0(x, y) = [φ, X0 α, X0 β, ξ](x, y) be a frame field on D satisfying the equations of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for x > 0, the frame field ˆF 0(−x, y) := F 0(x, y) also satisfies the equations in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 on D(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12, a curvature surface ˆf 0(x, y) on D(−) is obtained by ˆf 0(−x, y) := f 0(x, y) from a curvature surface f 0(x, y) on D determined by F 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We regard the surface ˆf 0(x, y) on D(−) as the back side of the surface f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since f 0(0, 0) = −(1/2)v1(0)+A(0) holds by the continuity of f 0(x, y) in the sense at Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5, we have f 0(0, 0) = ˆf 0(0, 0) and limx→0 f 0(x, y) = limx→0 ˆf 0(x, y) for any y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we could say that the curvature surface formed by both sides of f 0(x, y) is a natural realization in R4 of the singular space (D ∪ {(0, 0)} ∪ D(−), g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let f 0(x, y) be a curvature surface on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the surface formed by both sides of f 0(x, y) is a curvature surface on D ∪ {(0, 0)} ∪ D(−) with the metric g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Approximation of frame field determining extended curvature surface Let F 0(x, y) = [φ, X0 α, X0 β, ξ](x, y) be an orthonormal frame field determining a curvature surface f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this section, we construct an approximation of F 0(x, y) on a compact square E := [x0, x0 + a] × [y0, y0 + a] ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we regard φ(x, y) as a (singular) surface in the standard unit 3-sphere S3: X0 α(x, y) and X0 β(x, y) are the principal curvature directions 36 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA and ξ(x, y) is a normal vector field of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' With an integer n > 0, we divide [x0, x0 + a] and [y0, y0 + a] into n sub-intervals of equal length δn = a/n: let x0 < x1 < · · · < xn = x0 + a and y0 < y1 < · · · < yn = y0 + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On each edge [xi, xi+1] × {yj} or {xi} × [yj, yj+1], we approximate each vector of F 0(x, y) by a certain rational curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, independently of the width δn, the approximation F δn(x, y) on the lattice in E made by the divisions is also an orthonormal frame field and the approximation of each coordinate line in f 0(x, y) determined from F δn(x, y) also lies on a 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, under an additional consideration, such approximations F δn(x, y) with n extend to each point in E and the sequence F δn(x, y) converges to F 0(x, y) uniformly on E as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, any orthonormal frame field F 0(x, y) = [φ, X0 α, X0 β, ξ](x, y) satisfies the structure equa- tion dF 0 = F 0Ω in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) on D, where Ω is the differential 1-form determined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) and φ is a surface in S3 (if a1a2 ̸= 0 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) below) with a curvature line coordinate system (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have the following equations for F 0(x, y): dφ = −(a1dx)X0 α − (a2dy)X0 β, dξ = (b1dx)X0 α + (b2dy)X0 β, ∇′ ∂/∂xX0 β = c1X0 α, ∇∂/∂yX0 α = c2X0 β, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) where ai, bi, ci are the analytic functions on D in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have the following equations by the Gauss and the Codazzi equations for φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The functions ai, bi and ci satisfy the following equations on D: (a1)y = a2c1, (a2)x = a1c2, (b1)y = b2c1, (b2)x = b1c2, (c2)x + (c1)y + a1a2 + b1b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These equations are equivalent to the Maurer-Cartan equation dΩ + Ω ∧ Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we give another proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The first four equations follow from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) by d(dφ) = d(dξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the last equation, suppose firstly that φ is a surface in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the Gauss curvature Kφ of φ is given by Kφ = −(a1a2)−1� ((a2)x/a1)x + ((a1)y/a2)y � = 1 + λ1λ2, where λ1 := b1/a1 and λ2 := b2/a2 are the principal curvatures of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, the lemma holds good if φ is a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, since the functions ai, bi, ci are analytic on D and the domain for φ to be a surface is open, the last equation also holds on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Now, for a, x0 > 0 and y0 ∈ R, let [x0, x0 + a] and [y0, y0 + a] be the closed intervals: the domain E := [x0, x0 + a] × [y0, y0 + a] ⊂ D is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From now on, we write xe := x0 + a and ye := y0 + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the domain E, let us fix an orthonormal frame F 0(x0, y0) at P0,0 arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For a point (xp, yp) ∈ E and an integer n > 0, we put Ep := [x0, xp] × [y0, yp] and δn := a/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 (Division of the domain Ep and Path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) For a point (xp, yp) ∈ E, we divide the intervals [x0, xp] and [x0, yp], respectively, into sub-intervals of equal length: x0 < x1 < · · · < xs = xp, y0 < y1 < · · · < yt = yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we take the integers s and t such that 0 < s, t ≤ n and (xp −x0)/s ≤ δn, (yp −y0)/t ≤ δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we denote ∆n i,j := [xi, xi+1] × [yj, yj+1] (⊂ Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) For two lattice points Pi,j := (xi, yj) and Pi+k,j+l := (xi+k, yj+l) in a division of Ep, where k ≥ 0 and l ≥ 0, let m be a polygonal line in the division connecting the two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We express m as m : Pi,j → · · · → Pa,b → Pc,d → · · · → Pi+k,j+l by pointing to lattice points through which m passes, in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we call m a path from Pij to Pi+k,j+l if c ≥ a and d ≥ b are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 37 Now, let us fix a division of Ep for (xp, yp) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From F 0(x0, y0), we construct an approxi- mation F δn(x, y) of F 0(x, y) on any path from P0,0 to (xp, yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' At the beginning, we state the program for the construction of F δn(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Step A: We take a sub-domain ∆n i,j arbitrarily, and suppose that an orthonormal frame F δn(xi, yj) at Pi,j is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) We construct an orthonormal frame field F δn(x, yj) on [xi, xi+1] × {yj} from F δn(xi, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) We construct an orthonormal frame field F δn(xi, y) on {xi} × [yj, yj+1] from F δn(xi, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the constructions of F δn(xi+1, yj) and F δn(xi, yj+1) in (1) and (2), suppose that we start from the other orthonormal frame ¯F δn(xi, yj) at Pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we obtain distinct frames ¯F δn(xi+1, yj) and ¯F δn(xi, yj+1) from F δn(xi+1, yj) and F δn(xi, yj+1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this case, these frames will satisfy the equations ∥(F δn − ¯F δn)(xi, yj)∥ = ∥(F δn − ¯F δn)(xi+1, yj)∥ = ∥(F δn − ¯F δn)(xi, yj+1)∥, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) where ∥A∥ = ��(aij)2 for a square matrix A = [aij], as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (3) From F δn(xi+1, yj) obtained by (1), we firstly have an orthonormal frame field F δn(xi+1, y) on {xi+1}×[yj, yj+1] by (2): we denote by F δn(xi+1, yj+1) the frame at Pi+1,j+1 determined in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, from F δn(xi, yj+1) obtained by (2), we have an orthonormal frame field F δn(x, yj+1) on [xi, xi+1] × {yj+1} by (1): we denote by F δn(xi+1, yj+1) the frame at Pi+1,j+1 determined in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, the two frames F δn(xi+1, yj+1) and F δn(xi+1, yj+1) are determined at Pi+1,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, although F δn(xi+1, yj+1) and F δn(xi+1, yj+1) do not coincide, we shall have the following inequality, ∥(F δn − F δn)(xi+1, yj+1)∥ ≤ K(δn)3, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) if n is sufficiently large, where K is a constant on E determined independently of n (see Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3), we shall use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 essentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Step B: Let Pi,j be a lattice point of Ep and m be a path from P0,0 to Pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Step A, we have an orthonormal frame field F δn on the path m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the following proposition, we fix F 0(x0, y0) arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let P := (xp, yp) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For an integer n, we take a division of the domain Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let mi (i = 1, 2) be two paths from P0,0 to P in the division, and F δn i (xp, yp) (i = 1, 2) be the frames at P determined from mi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have ∥(F δn 1 − F δn 2 )(xp, yp)∥ ≤ Ka2δn, if n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this proof, we suppose that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) hold good and that n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, under the assumption that a frame F δn(xi, yj) is determined at a point Pi,j (i ≤ s−2, j ≤ t − 1), we study the frame at Pi+2,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this case, there are three paths mi (i = 1, 2, 3) from Pi,j to Pi+2,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For each mi, we point to the lattice points only on the way, and then we have m1 : Pi+1,j → Pi+2,j, m2 : Pi+1,j → Pi+1,j+1, m3 : Pi,j+1 → Pi+1,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F δn i (xi+2, yj+1) be the frames at Pi+2,j+1 determined from mi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we firstly have ∥(F δn 2 − F δn 1 )(xi+2, yj+1)∥ ≤ Kδ3 n and ∥(F δn 3 − F δn 2 )(xi+2, yj+1)∥ ≤ Kδ3 n by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, the closed polygonal line m3 − m1 includes two sub-domains ∆n i,j and ∆n i+1,j, and we have ∥(F δn 3 − F δn 1 )(xi+2, yj+1)∥ ≤ ∥(F δn 3 − F δn 2 )(xi+2, yj+1)∥ + ∥(F δn 2 − F δn 1 )(xi+2, yj+1)∥ ≤ 2Kδ3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 38 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Next, we arbitrarily take two paths mi from P0,0 to P = (xp, yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the closed polygonal line m1 − m2 includes at most n2 sub-domains ∆n k,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we obtain the lemma by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) and the above fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Step C: For a given integer n and a point (xp, yp) ∈ E, we take a division of Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the division, let m and m be two paths defined by m :P0,0 = (x0, y0) → Ps,0 = (xp, y0) → Ps,t = (xp, yp), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4) m :P0,0 = (x0, y0) → P0,t = (x0, yp) → Ps,t = (xp, yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5) Let F δn(xp, yp) and F δn(xp, yp) be the frames at (xp, yp) determined from m and m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we shall verify that F δn(xp, yp) converges to F 0(xp, yp) uniformly for any (xp, yp) ∈ E as n tends to ∞, where F 0(x, y) is the frame field with a given F 0(x0, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, F δn(xp, yp) also converges to the frame F 0(xp, yp) uniformly for any (xp, yp) ∈ E as n tends to ∞, since ∥(F δn − F δn)(xp, yp)∥ ≤ Ka2δn holds by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we verify Steps A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In these proofs, we fix an integer n, and study only the case of E = [x0, xe] × [y0, ye] for Ep, by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have s = t = n, and denote δ := δn = a/n, ∆i,j := ∆n i,j and F δ := F δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For Step A: Under the assumption that an orthonormal frame F δ(xi, yj) at Pi,j is determined, we construct F δ on boundary of the sub-domain ∆i,j according to the ways (1), (2) and (3), and verify (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F δ =: [φδ, Xδ α, Xδ β, ξδ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We simply write (0, 0) with (xi, yj) in the following Steps A-(1) and A-(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Step A-(1): For a given F δ(0, 0), we express F δ(x, yj) for x ∈ [xi, xi+1] as φδ(x, yj) − φδ(0, 0) := − x−xi 2 a1(0, 0) � Xδ α(0, 0) + Xδ α(x, yj) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) ξδ(x, yj) − ξδ(0, 0) := x−xi 2 b1(0, 0) � Xδ α(0, 0) + Xδ α(x, yj) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7) Xδ β(x, yj) − Xδ β(0, 0) := x−xi 2 c1(0, 0) � Xδ α(0, 0) + Xδ α(x, yj) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) and find out a suitable Xδ α(x, yj) from these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, let sδ 1(x, yj) and tδ 1(x, yj) be the functions on (x, yj) ∈ [xi, xi+1] × {yj} defined by sδ 1(x, yj) := ⟨Xδ α(x, yj), Xδ α(0, 0)⟩, tδ 1(x, yj) := � 1 + sδ 1(x, yj) � /2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Suppose that the frame field F δ(x, yj) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) is orthonormal at each point (x, yj) ∈ [xi, xi+1] × {yj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for x ∈ [xi, xi+1] we have the following equations: ⟨Xδ α(x, yj), φδ(0, 0)⟩ = −⟨φδ(x, yj), Xδ α(0, 0)⟩ = a1(0, 0)(x − xi)tδ 1(x, yj), ⟨Xδ α(x, yj), ξδ(0, 0)⟩ = −⟨ξδ(x, yj), Xδ α(0, 0)⟩ = −b1(0, 0)(x − xi)tδ 1(x, yj), ⟨Xδ α(x, yj), Xδ β(0, 0)⟩ = −⟨Xδ β(x, yj), Xδ α(0, 0)⟩ = −c1(0, 0)(x − xi)tδ 1(x, yj), sδ 1(x, yj) = 4 − (x − xi)2(a2 1 + b2 1 + c2 1)(0, 0) 4 + (x − xi)2(a2 1 + b2 1 + c2 1)(0, 0), tδ 1(x, yj) = 4 4 + (x − xi)2(a2 1 + b2 1 + c2 1)(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let x ∈ [xi, xi+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Suppose that F δ(x, yj) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) is an orthonormal frame field at each point of [xi, xi+1] × {yj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Firstly, we verify the first equation under the assumption a1(0, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8), we have − ((x − xi)/2) a1(0, 0) ⟨Xδ α(x, yj), φδ(0, 0)⟩ = ⟨φδ(x, yj) − φδ(0, 0), φδ(0, 0)⟩ = ⟨φδ(x, yj), φδ(0, 0)⟩ − 1, EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 39 and − ((x − xi)/2) a1(0, 0) ⟨φδ(x, yj), Xδ α(0, 0)⟩ = ⟨φδ(x, yj) − φδ(0, 0), φδ(x, yj)⟩ = 1 − ⟨φδ(x, yj), φδ(0, 0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have ⟨Xδ α(x, yj), φδ(0, 0)⟩ = −⟨φδ(x, yj), Xδ α(0, 0)⟩ by a1(0, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, since −((x − xi)/2) a1(0, 0) (1 + sδ 1(x, yj)) = ⟨φδ(x, yj) − φδ(0, 0), Xδ α(0, 0)⟩ = ⟨φδ(x, yj), Xδ α(0, 0)⟩, the first equation is obtained: ⟨Xδ α(x, yj), φδ(0, 0)⟩ = −⟨φδ(x, yj), Xδ α(0, 0)⟩ = x−xi 2 a1(0, 0) � 1 + sδ 1(x, yj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the case a1(0, 0) = 0, the equation is also satisfied from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) by φδ(x, yj) = φδ(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the same way, we can verify the equations for ⟨Xδ α(x, yj), ξδ(0, 0)⟩ and ⟨Xδ α(x, yj), Xδ β(0, 0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, from these equations verified above, we have Xδ α(x, yj) = sδ 1(x, yj)Xδ α(0, 0) + x−xi 2 � 1 + sδ 1(x, yj) � � a1φδ − b1ξδ − c1Xδ β � (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) Taking the norm of both sides of the equation, we obtain the equation for sδ 1(x, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the equation for tδ 1(x, yj) follows directly from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we have completed the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ The following lemma follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) and the definition of tδ 1(x, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Suppose that the frame field F δ(x, yj) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) is orthonormal at each point (x, yj) ∈ [xi, xi+1] × {yj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for x ∈ [xi, xi+1], we have Xδ α(0, 0) + Xδ α(x, yj) = tδ 1(x, yj) � 2Xδ α(0, 0) + (x − xi)(a1φδ − b1ξδ − c1Xδ β)(0, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5, if the frame field F δ(x, yj) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) is orthonormal at each point (x, yj) ∈ [xi, xi+1] × {yj}, then we have Xδ α(x, yj) − Xδ α(0, 0) = (x − xi)tδ 1(x, yj) � Y δ 1 (0, 0) − x−xi 2 � (a2 1 + b2 1 + c2 1)Xδ α � (0, 0) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10) where Y δ 1 (0, 0) := (a1φδ − b1ξδ − c1Xδ β)(0, 0), and φδ(x, yj) − φδ(0, 0) = −(x − xi)tδ 1(x, yj)a1(0, 0) � Xδ α(0, 0) + x−xi 2 Y δ 1 (0, 0) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11) ξδ(x, yj) − ξδ(0, 0) = (x − xi)tδ 1(x, yj)b1(0, 0) � Xδ α(0, 0) + x−xi 2 Y δ 1 (0, 0) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12) Xδ β(x, yj) − Xδ β(0, 0) = (x − xi)tδ 1(x, yj)c1(0, 0) � Xδ α(0, 0) + x−xi 2 Y δ 1 (0, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) Namely, with Ω1(x, y) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5), F δ(x, yj) − F δ(0) = (x − xi)tδ 1(x, yj)F δ(0) � Ω1(0) + x−xi 2 (Ω1)2(0) � holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Conversely, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let sδ 1(x, yj) and tδ 1(x, yj) be the functions on [xi, xi+1] × {yj} given in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F δ(xi, yj)(= F δ(0, 0)) be an orthonormal frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the frame field F δ(x, yj) defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) is orthonormal at each point (x, yj) ∈ [xi, xi+1] × {yj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, for a transformation AF δ(xi, yj) of F δ(xi, yj) by a special orthogonal matrix A, F δ(x, yj) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)– (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) changes into AF δ(x, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, for the unit matrix Id, we have ∥(A − Id)F δ(xi, yj)∥ = ∥(A − Id)F δ(xi+1, yj)∥ = ∥A − Id∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 40 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Note that the equations in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) are induced from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) (or Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, let the frame F δ(0, 0)(= F δ(xi, yj)) be orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The theorem follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) by direct calculation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We firstly take the norm of the both sides in the equation of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5, and then we have ∥Xδ α(0, 0) + Xδ α(x, yj)∥2 = 4tδ 1(x, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By the equation, we can show that all vector fields of F δ(x, yj) have unit norm: for example, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) we have ∥Xδ α(x, yj)∥2 = (sδ 1(x, yj))2 + (x − xi)2(tδ 1(x, yj))2(a2 1 + b2 1 + c2 1)(0, 0) = (1 − ((x − xi)/2)2(a2 1 + b2 1 + c2 1)(0, 0))2 + (x − xi)2(a2 1 + b2 1 + c2 1)(0, 0) (1 + ((x − xi)/2)2(a2 1 + b2 1 + c2 1)(0, 0))2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the same way, we can verify that φδ(x, yj), ξδ(x, yj) and Xδ β(x, yj) are also unit vectors, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we have ⟨Xδ α(0, 0) + Xδ α(x, yj), (a1φδ − b1ξδ − c1Xδ β)(0, 0)⟩ = (x − xi)tδ 1(x, yj)(a2 1 + b2 1 + c2 1)(0, 0) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By the equation, we can show that all vector fields of F δ(x, yj) are orthogonal to each other: for example, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) we have ⟨Xδ α, φδ⟩(x, yj) = − ((x − xi)/2)sδ 1(x, yj)(1 + sδ 1(x, yj))a1(0, 0) + (x − xi)tδ 1(x, yj)a1(0, 0) − ((x − xi)3/2)(tδ 1(x, yj))2 � (a2 1 + b2 1 + c2 1)a1 � (0, 0) = − (x − xi)(tδ 1(x, yj))2 � 1 − ((x − xi)/2)2(a2 1 + b2 1 + c2 1)(0, 0) � a1(0, 0) + (x − xi)(tδ 1(x, yj))2 � 1 + ((x − xi)/2)2(a2 1 + b2 1 + c2 1)(0, 0) � a1(0, 0) − ((x − xi)3/2)(tδ 1(x, yj))2 � (a2 1 + b2 1 + c2 1)a1 � (0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The other orthogonality for these vectors is also obtained by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The assertion for transformation AF δ(xi, yj) of the initial condition follows directly from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we have verified the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Step A-(2): For a given F δ(0, 0), we express the frame field F δ(xi, y) on {xi} × [yj, yj+1] as a similar form to in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8): φδ(xi, y) − φδ(0, 0) := − y−yj 2 a2(0, 0) � Xδ β(0, 0) + Xδ β(xi, y) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14) ξδ(xi, y) − ξδ(0, 0) := y−yj 2 b2(0, 0) � Xδ β(0, 0) + Xδ β(xi, y) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='15) Xδ α(xi, y) − Xδ α(0, 0) := y−yj 2 c2(0, 0) � Xδ β(0, 0) + Xδ α(xi, y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='16) Let sδ 2(xi, y) and tδ 2(xi, y) be the functions on {xi} × [yj, yj+1] defined by sδ 2(xi, y) := ⟨Xδ β(xi, y), Xδ β(0, 0)⟩, tδ 2(xi, y) := � 1 + sδ 2(xi, y) � /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have the following lemma and theorem in same way as in Step A-(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Suppose that the frame field F δ(xi, y) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='16) is orthonormal on {xi} × [yj, yj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have sδ 2(xi, y) = 4 − (y − yj)2(a2 2 + b2 2 + c2 2)(0, 0) 4 + (y − yj)2(a2 2 + b2 2 + c2 2)(0, 0), tδ 2(xi, y) = 4 4 + (y − yj)2(a2 2 + b2 2 + c2 2)(0, 0), Xδ β(0, 0) + Xδ β(xi, y) = tδ 2(xi, y) � 2Xδ β(0, 0) + (y − yj)(a2φδ − b2ξδ − c2Xδ α)(0, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 41 By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7, if the frame field F δ(xi, y) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='16) is orthonormal at each point (xi, y) ∈ {xi} × [yj, yj+1], then we have Xδ β(xi, y) − Xδ β(0, 0) = (y − yj)tδ 2(xi, y) � Y δ 2 (0, 0) − y−yj 2 � (a2 2 + b2 2 + c2 2)Xδ β � (0, 0) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='17) where Y δ 2 (0, 0) := (a2φδ − b2ξδ − c2Xδ α)(0, 0), and φδ(xi, y) − φδ(0, 0) = −(y − yj)tδ 2(xi, y)a2(0, 0) � Xδ β(0, 0) + y−yj 2 Y δ 2 (0, 0) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='18) ξδ(xi, y) − ξδ(0, 0) = (y − yj)tδ 2(xi, y)b2(0, 0) � Xδ β(0, 0) + y−yj 2 Y δ 2 (0, 0) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='19) Xδ α(xi, y) − Xδ α(0, 0) = (y − yj)tδ 2(xi, y)c2(0, 0) � Xδ β(0, 0) + y−yj 2 Y δ 2 (0, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='20) Namely, with Ω2(x, y) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5), F δ(xi, y) − F δ(0) = (y − yj)tδ 2(xi, y)F δ(0) � Ω2(0) + y−yj 2 (Ω2)2(0) � holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Conversely, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let sδ 2(xi, y) and tδ 2(xi, y) be the functions on {xi} × [yj, yj+1] given in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F δ(xi, yj)(= F δ(0, 0)) be an orthonormal frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the frame field F δ(xi, y) defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='17)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='20) is orthonormal at each point (xi, y) ∈ {xi} × [yj, yj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, for a transformation AF δ(xi, yj) of F δ(xi, yj) by a special orthogonal matrix A, F δ(xi, y) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='17)– (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='20) changes into AF δ(xi, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In particular, we have ∥(A − Id)F δ(xi, yj)∥ = ∥(A − Id)F δ(xi, yj+1)∥ = ∥A − Id∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Step A-(3): For a sub-domain ∆i,j, the orthonormal frames F δ(xi+1, yj) and F δ(xi, yj+1) have been determined from F δ(xi, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we can construct the frames F δ(xi+1, y) on {xi+1} × [yj, yj+1] and F δ(x, yj+1) on [xi, xi+1] × {yj+1} by (2) and (1) of Step A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, we have two frames F δ(xi+1, yj+1) at the point Pi+1,j+1, since F δ(xi+1, yj+1) is defined for each path from Pi,j to Pi+1,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We study the difference of these two frames and verify (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In this step, we simply write 0 := (xi, yj), 1 = (xi+1, yj), 2 := (xi, yj+1), 3 := (xi+1, yj+1) and denote by F δ(3) and F δ(3), respectively, the frames determined by the paths m : 0 → 1 → 3 and m : 0 → 2 → 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Note that tδ 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' tδ 2) is defined on [xi, xi+1] × {yj} and [xi, xi+1] × {yj+1} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' on {xi} × [yj, yj+1] and {xi+1} × [yj, yj+1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have tδ 1(1) = 1/ � 1 + (δ/2)2(a2 1 + b2 1 + c2 1)(0) � , tδ 2(3) = 1/ � 1 + (δ/2)2(a2 2 + b2 2 + c2 2)(1) � , tδ 2(2) = 1/ � 1 + (δ/2)2(a2 2 + b2 2 + c2 2)(0) � , tδ 1(3) = 1/ � 1 + (δ/2)2(a2 1 + b2 1 + c2 1)(2) � , and hence tδ i(k) = 1 + O(δ2) hold for the integers i and k above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, since ai, bi and ci are analytic functions on E, we have tδ 1(3) − tδ 1(1) = O(δ3) and tδ 2(3) − tδ 2(2) = O(δ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we have F δ(3) − F δ(0) = (F δ(3) − F δ(1)) + (F δ(1) − F δ(0)), F δ(3) − F δ(0) = (F δ(3) − F δ(2)) + (F δ(2) − F δ(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We define Gx and Gy by Gx := (F δ(3) − F δ(2)) − (F δ(1) − F δ(0)), Gy := (F δ(3) − F δ(1)) − (F δ(2) − F δ(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have only to verify that Gx = Gy+O(δ3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we study the third degree for δ of Gx−Gy: in the asymptotic expansion Gx−Gy ≈ �3 k=0 pkδk for δ, we show pk = 0 (k = 0, 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the estimate of each Gx and Gy, the frames F δ(3) and F δ(3) are naturally distinguished by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='17)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='20), and hence we can write F δ(3) with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, for a function or a vector field k(x, y), we denote [k]1 0 := k(1) − k(0) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let Y δ 1 := a1φδ − b1ξδ − c1Xδ β and Y δ 2 := a2φδ − b2ξδ − c2Xδ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 42 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' With the second degree for δ of [Y δ 1 ]2 0 and [Y δ 2 ]1 0, we have [Y δ 1 ]2 0 ≈ δ � c1c2Xδ α + (c2)xXδ β � (0) + (δ2/2) � ((c2)x − (c1)y)Y δ 2 − c1(a2 2 + b2 2 − c2 2)Xδ β � (0) + (δ2/2) � (a1)yyφδ − (b1)yyξδ − (c1)yyXδ β � (0), [Y δ 2 ]1 0 ≈ δ � (c1)yXδ α + c1c2Xδ β � (0) + (δ2/2) � ((c1)y − (c2)x)Y δ 1 − c2(a2 1 + b2 1 − c2 1)Xδ α � (0) + (δ2/2) � (a2)xxφδ − (b2)xxξδ − (c2)xxXδ α � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We only prove the equation for Y δ 1 , since the equation for Y δ 2 is obtained in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, for the element [a1φδ]2 0 of [Y δ 1 ]2 0, we have [a1φδ]2 0/δ = � a1(2)[φδ]2 0 + (a1(2) − a1(0))φδ(0) � /δ ≈ −(a2(a1 + δ(a1)y))(0) � Xδ β + (δ/2)Y δ 2 � (0) + ((a1)y + (δ/2)(a1)yy)(0)φδ(0) ≈ � −a1a2Xδ β + a2c1φδ� (0) + (δ/2) � −a1a2Y δ 2 − 2a2 2c1Xδ β + (a1)yyφδ� (0), by (a1)y = a2c1 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the same way, we have [b1ξδ]2 0/δ ≈ � b1b2Xδ β + b2c1ξδ� (0) + (δ/2) � b1b2Y δ 2 + 2b2 2c1Xδ β + (b1)yyξδ� (0), [c1Xδ β]2 0/δ ≈ � c1Y δ 2 + (c1)yXδ β � (0) + (δ/2) � 2(c1)yY δ 2 − c1(a2 2 + b2 2 + c2 2)Xδ β + (c1)yyXδ β � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by (c2)x + (c1)y + a1a2 + b1b2 = 0 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1, we obtain the equation for [Y δ 1 ]2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' With the third degree for δ of Gx − Gy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' we have ([φδ]3 2 − [φδ]1 0) − ([φδ]3 1 − [φδ]2 0) ≈ − (δ3/2) � c1((a2)y − a1c2)Xδ α − c2((a1)x − a2c1)Xδ β � (0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ([ξδ]3 2 − [ξδ]1 0) − ([ξδ]3 1 − [ξδ]2 0) ≈ (δ3/2) � c1((b2)y − b1c2)Xδ α − c2((b1)x − b2c1)Xδ β � (0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ([Xδ α]3 2 − [Xδ α]1 0) − ([Xδ α]3 1 − [Xδ α]2 0) ≈ − (δ3/2) � −c1((a2)y − a1c2)φδ + c1((b2)y − b1c2)ξδ� (0) − (δ3/2) � (c2)xx + (c1)yy + c1(a2 2 + b2 2) + c2(a2 1 + b2 1) � (0)Xδ β(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ([Xδ β]3 2 − [Xδ β]1 0) − ([Xδ β]3 1 − [Xδ β]2 0) ≈ (δ3/2) � −c2((a1)x − a2c1)φδ + c2((b1)x − b2c1)ξδ� (0) + (δ3/2) � (c2)xx + (c1)yy + c1(a2 2 + b2 2) + c2(a2 1 + b2 1) � (0)Xδ α(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the first equation, we have ([φδ]3 2 − [φδ]1 0)/δ = − [tδ 1]3 1 � a1(Xδ α + (δ/2)Y δ 1 ) � (2) − tδ 1(1) � a1(Xδ α + (δ/2)Y δ 1 ) �2 0 ≈ − � a1(Xδ α + (δ/2)Y δ 1 ) �2 0 = − a1(2) � Xδ α + (δ/2)Y δ 1 �2 0 − [a1]2 0 � Xδ α + (δ/2)Y δ 1 � (0) ≈ − δ(a1 + δ(a1)y)(0) � c2Xδ β + (δ/2)(c2Y δ 2 + c1c2Xδ α + (c2)xXδ β) � (0) − δ � ((a1)y + (δ/2)(a1)yy) (Xδ α + (δ/2)Y δ 1 ) � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have ([φδ]3 2 − [φδ]1 0)/δ2 ≈ − � a2c1Xδ α + a1c2Xδ β � (0) − (δ/2) � a1a2(c1 + c2)φδ − (a1b2c2 + a2b1c1)ξδ� (0) − (δ/2) � (a1c2(c1 − c2) + (a1)yy) Xδ α + � a1(c2)x + 2a2c1c2 − a2c2 1 � Xδ β � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 43 In the same way, we have ([φδ]3 1 − [φδ]2 0)/δ2 ≈ − � a2c1Xδ α + a1c2Xδ β � (0) − (δ/2) � a1a2(c1 + c2)φδ − (a1b2c2 + a2b1c1)ξδ� (0) − (δ/2) �� −a1c2 2 + 2a1c1c2 + a2(c1)y � Xδ α + � −a2c2 1 + a2c1c2 + (a2)xx � Xδ β � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From these equations, we have ([φδ]3 2 − [φδ]1 0) − ([φδ]3 1 − [φδ]2 0) ≈ −(δ3/2) � ((a1)yy − a2(c1)y − a1c1c2) Xδ α − ((a2)xx − a1(c2)x − a2c1c2) Xδ β � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1, we have (a1)yy − a2(c1)y − a1c1c2 = c1 ((a2)y − a1c2) , (a2)xx − a1(c2)x − a2c1c2 = c2 ((a1)x − a2c1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we obtain the first equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The second equation is also obtained in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we prove the third equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='([Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 − [Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α]1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0)/δ ≈ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='Y δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 − (δ/2)(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='≈ [Y δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 ]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0 − (δ/2)(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)(2)[Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0 − (δ2/2)(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)y(0)Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='≈ [Y δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 ]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0 − (δ/2)(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)(0)[Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='0 − (δ2/2)(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)y(0)Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='≈ δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='c1c2Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α + (c2)xXδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='(0) + (δ2/2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='((c2)x − (c1)y)Y δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 − (a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1)yXδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='+ (δ2/2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='c1(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2 − c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2) + c2(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='Xδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='β + (a1)yyφδ − (b1)yyξδ − (c1)yyXδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the same way, we have ([Xδ α]3 1 − [Xδ α]2 0)/δ ≈ δ � c1c2Xδ α + (c2)xXδ β � (0) + (δ2/2) � c1c2Y δ 1 + (c2)xY δ 2 + (c2(c1)y + 2c1(c2)x)Xδ α + (c1c2 2 + (c2)xx)Xδ β � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From these equations, we have ([Xδ α]3 2 − [Xδ α]1 0) − ([Xδ α]3 1 − [Xδ α]2 0) ≈ −(δ3/2) � c1c2(a1φδ − b1ξδ) + (c1)y(a2φδ − b2ξδ) − (a1)yyφδ + (b1)yyξδ + (c1)yyXδ β � (0) − (δ3/2) �� 2c1(c2)x + (a2 1 + b2 1 + c2 1)y � Xδ α + � c1(a2 2 + b2 2) + c2(a2 1 + b2 1) + (c2)xx � Xδ β � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1, we have (b1)yy − b2(c1)y − b1c1c2 = c1 ((b2)y − b1c2) , 2c1(c2)x + 2 (a1(a1)y + b1(b1)y + c1(c1)y) = 2c1 ((c2)x + (c1)y + a1a2 + b1b2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we obtain the third equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The last equation is also obtained in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='11 (Choice of the constant K in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (1) For the four equations of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10, we take norm of the coefficient vectors for δ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, let K1 be the maximum of these four norms on the domain E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (2) For the matrices Ω1 and Ω2 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5), let K2 := Max p∈E {∥Ωi(p)∥, ∥(Ω1)x(p)∥/2, ∥(Ω2)y(p)∥/2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we define K by K := Max{K1, K2} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here, we add 1 to Max{K1, K2}, since (F δn − F δn)(xi+1, yj+1) also includes the terms of degree δi n (i > 3), in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10 and the definition of K, we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3): 44 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For an integer n and (xp, yp) ∈ E, we take a division of Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' With a sub-domain ∆n i,j, suppose that an orthonormal frame F δn(xi, yj) is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we have ∥(F δn − F δn)(xi+1,j+1)∥ ≤ Kδ3 n, if n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='12, the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3 also has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For Step C: As in the proofs of Step A, we fix an integer n and study only the case of E = [x0, xe] × [y0, ye], where xe = x0 + a and ye = y0 + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have s = t = n and denote δ := δn = a/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F 0(x, y) be the solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) on D under a given initial condition F 0(x0, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let m be the path from P0,0 to Pn,n such that m : P0,0 → · · · → Pn,0 → · · · → Pn,n = (xe, ye), and F δ(x, y) := F δ(x, y) be the orthonormal frame field on m determined from F 0(x0, y0) by Step A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Firstly, note that we have � m F 0(x, y)Ω = � m dF 0 = � F 0�(xn,xn) (x0,y0) , and � F δ�(xi+1,y0) (xi,y0) = � (xi+1,y0) (xi,y0) dF δ dx (x, y0)dx, � F δ�(xn,yj+1) (xn,yj) = � (xn,yj+1) (xn,yj) dF δ dy (xn, y)dy on each sub-interval of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, we study the norm ∥(F 0 − F δn)(xn, yn)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the second degree Taylor polynomials of both frames F 0(x, y) and F δ(x, y) at each point (xi, y0) and (xn, yj), we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For x ∈ [xi, xi+1], the first degree Taylor polynomial of d(F 0−F δ) dx (x, y0) at x = xi is given by d dx(F 0 − F δ)(x,y0) ≈ (x − xi)F 0 (xi,y0)(Ω1)x(xi, y0) + (F 0 − F δ)(xi,y0) � Ω1(xi, y0) + (x − xi)Ω2 1(xi, y0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For y ∈ [yj, yj+1], the first degree Taylor polynomial of d(F 0−F δ) dy (xn, y) at y = yj is given by d dy(F 0 − F δ)(xn,y) ≈ (y − yj)F 0 (xn,yj)(Ω2)y(xn, yj) + (F 0 − F δ)(xn,yj) � Ω2(xn, yj) + (y − yj)Ω2 2(xn, yj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Since dF 0 = F 0Ω and ∂2F 0/∂x2 = F 0[Ω2 1 + (Ω1)x], we have the polynomial at x = xi of dF 0(x, y0)/dx: d dxF 0(x, y0) ≈ F 0(xi, y0) � (Ω1)(xi,y0) + (x − xi) � Ω2 1 + (Ω1)x � (xi,y0) � for x ∈ [xi, xi+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, with the function tδ 1(x, y0) on x ∈ [xi, xi+1], we have tδ 1(xi, y0) = 1 and (dtδ 1/dx)(xi, y0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' From these equations, we obtain the polynomial at x = xi of d(F δ(x, y0))/dx: d dxF δ(x, y0) ≈ F δ(xi, y0) � (Ω1)(xi,y0) + (x − xi)Ω2 1|(xi,y0) � for x ∈ [xi, xi+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the same way, we obtain the polynomials at y = yj of d(F 0(xn, y))/dy and d(F δ(xn, y))/dy for y ∈ [yj, yj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we have verified the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' □ EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 45 Now, we put T δ(x, y) := (F 0−F δ)(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Under the condition T δ(x0, y0) = (F 0−F δ)(x0, y0) = 0, we integrate (dT δ/dx)(x, y0) from xi to xi+1 in order of i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13 and Max {∥Ωi(p)∥, ∥(Ω1)x(p)∥/2, ∥(Ω2)y(p)∥/2 | p ∈ E} + 1 ≤ K, we have ∥T δ(x1, y0)∥ ≤ Kδ2, ∥T δ(xi+1, y0) − T δ(xi, y0)∥ ≤ Kδ2 + ∥T δ(xi, y0)∥Kδ for i ≥ 1, if n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, we have ∥T δ(xi+1, y0)∥ + δ ≤ (∥T δ(xi, y0)∥ + δ)(1 + Kδ) for i ≥ 1 by ∥T δ(xi+1, y0)∥ ≤ Kδ2 + ∥T δ(xi, y0)∥(1 + Kδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Therefore, we have ∥T δ(xn, y0)∥ + δ ≤ (∥T δ(x1, y0)∥ + δ)(1 + Kδ)n−1 ≤ δ(1 + Kδ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='21) Here, we have (1 + Kδ)n = n � k=0 nCk �Ka n �k < n � k=0 Kkak k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' < exp(Ka).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='22) In the same way, by the integral of (dT δ/dy)(xn, y) from yj to yj+1, we have ∥T δ(xn, yj+1) − T δ(xn, yj)∥ ≤ Kδ2 + ∥T δ(xn, yj)∥Kδ for j ≥ 0, if n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we have ∥T δ(xn, yn)∥ ≤ (∥T δ(xn, y0)∥ + δ)(1 + Kδ)n − δ ≤ δ (exp(2Ka) − 1) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='23) by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='21) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='23) implies that F δn(xe, ye)(= F δn(xn, yn)) converges to F 0(xe, ye) as n tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In the argument above, we can replace (xe, ye) with an arbitrary point (xp, yp) ∈ E, by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, Step C has been verified by the argument above and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F 0(x0, y0) be an orthonormal frame at (x0, y0) given arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F 0(x, y) be the orthonormal frame field satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) under the initial condition F 0(x0, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For an integer n and (xp, yp) ∈ E, we take a division of Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let m and m be the two paths from (x0, y0) to (xp, yp) given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='4)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F δn(xp, yp) and F δn(xp, yp) be the orthonor- mal frame determined from m and m by Step A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, F δn(xp, yp) and F δn(xp, yp) converge to F 0(xp, yp) uniformly for all (xp, yp) ∈ E, as n tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' By the argument above, all Steps A, B and C have been verified: for an integer n and (x, y) ∈ E, we have obtained two approximations F δn(x, y) and F δn(x, y) of F 0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14, note that any frame F δn(xp, yp) determined by a path from (x0, y0) to (xp, yp) is also an approximation of F 0(xp, yp) if n is sufficiently large, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we construct an approximation of several coordinate curves in the curvature surface f 0(x, y) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Before the construction, we state a remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For an x-curve f 0(x, y0) with fixed y0 ̸= 0 on an interval I = [x0, xe] := [y0 − a, y0 + a], where a, y0 − a > 0, we fix a division of I with equal length δn = a/n and make the frame field F δn(x, y0) on I under the condition F δn(x0, y0) = Id, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, for a vector f δn(x, y0) := (1 + y2 0)−1 � Xδn β − 2(1 + y2)ξδn + yφδn� (x, y0) on each sub-interval [xi, xi+1], one might guess from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5 if the curve f δn(x, y0) := 1 2 √ 5 � � f δn�(x,y0) (xi,y0) + i � k=1 � f δn�(xk,y0) (xk−1,y0) � for xi < x ≤ xi+1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='24) is an approximation of the x-curve (where we adopt the non-unit vector f δn(x, y0) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' However, it is not an approximation of the x-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In fact, for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='24) we have f δn(x, y0)− f δn(y0, y0) ≡ 0 for y0 = xn ≤ x ≤ xn+1, since (y0, y0) is in the singularity D∩S1 of the metric g0 46 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' MATSUURA AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' SUYAMA (or by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, it is necessary for the interval [y0, xe] that we make the frame F δn(x, y0) and the vector f δn(x, y0) in the direction that x decreases, because f δn(y0, y0) is determined as the limit of f δn(p) for p ∈ D \\S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The approximation F δn(x, y0) on [xi−1, xi] is also defined from F δn(xi, y0) by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) as (x − xi) ≤ 0, and then f δn(x, y0) on [xi−1, xi] is determined by the frame F δn(x, y0) in the inverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Approximation of x-curve f 0(x, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let f 0(x, y0) be an x-curve with y0 ̸= 0 on [y0 − a, y0 + a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We divide the interval [y0 − a, y0 + a] as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On the interval [y0 − a, y0], we make F δn 1 (x, y0) := F δn(x, y0) from F δn 1 (y0 − a, y0) = Id by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) and determine f δn 1 (x, y0) := f δn(x, y0) by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='24), which satisfies f δn 1 (y0 − a, y0) = 0: we write (F δn 1 , f δn 1 ) with (F δn 1 (y0, y0), f δn 1 (y0, y0)) obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, on the interval [y0, y0 + a], we make F δn 2 (x, y0) := F δn(x, y0) from F δn 2 (y0 + a, y0) = Id by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) and determine f δn 2 (x, y0) := f δn(x, y0) and f δn 2 (x, y0) := f δn(x, y0) in the inverse direction, which satisfies f δn 2 (y0+a, y0) = 0: we write (F δn 2 , f δn 2 ) with (F δn 2 (y0, y0), f δn 2 (y0, y0)) obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the curve f δn 2 (x, y0)−f δn 2 satisfies f δn 2 (y0, y0)−f δn 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For the matrix C such that CF δn 2 = F δn 1 , we define a curve kδn(x, y0) for x ∈ [y0, y0 + a] by kδn(x, y0) := C(f δn 2 (x, y0) − f δn 2 ) + f δn 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Now, two curves f δn 1 (x, y0) on [y0 − a, y0] and kδn(x, y0) on [y0, y0 + a] connect continuously and have the same frame F δn 1 at x = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' The connected curve at f δn 1 is an approximation of the x-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since a vector (yXδn β − φδn)(x, y0) does not depend on x and ⟨yXδn β − φδn, f δn⟩(x, y0) = 0 holds, the curve is contained in a hyperplane R3 y0(∋ 0) perpendicular to the vector (yXδn β − φδn)(x, y0) by f δn(y0 − a, y0) = 0, and in particular the curve lies on a 2-sphere of radius (2 √ 5)−1� (5 + 4y2 0)/(1 + y2 0) in R3 y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' (a) x-curves when n = 5 (b) x-curves when n = 20 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' These show the differences between two x-curves f δn(x, 1) and ¯f δn(x, 1) on [2, 3] under the conditions f δn(2, 1) = ¯f δn(2, 1) and F δn(2, 1) = ¯F δn(2, 1): the x-curve f δn(x, 1) of black (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ¯f δn(x, 1) of gray) is constructed in the direction that x increases (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' decreases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Here F δn(x, 1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ¯F δn(x, 1)) is the frame field determining f δn(x, 1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' ¯f δn(x, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Approximation of y-curve f 0(x0, y) with fixed x0(> 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let f 0(x0, y) be a y-curve for y ∈ I = [−a, a], where a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' We take a division of I with equal length δn := a/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On the interval [−a, 0], we make F δn(x0, y) from F δn(x0, −a) = Id and determine a vector ˜f δn(x0, y) on each [yj, yj+1] by ˜f δn(x0, y) := (B2ξδn + C2Xδn α )(x0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the approximation f δn(x0, y) of f 0(x0, y) (−a ≤ y ≤ 0) is given by f δn(x0, y) := 1 (B2 2 + C2 2)(x0) �� ˜f δn�(x0,y) (x0,yj) + j � k=1 � ˜f δn�(x0,yk) (x0,yk−1) � for yj < y ≤ yj+1, EXTENSION AND APPROXIMATION OF CURVATURE SURFACES 47 where yj+1 ≤ yn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On the interval [0, a], we make F δn(x0, y) from F δn(x0, a) = Id and determine ˜f δn(x0, y) and f δn(x0, y) in the inverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we obtain the approximation of the y-curve by the connection of these two curves, in the same way as the case of x-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since (−B2Xδn α +C2ξδn)(x0, y) does not depend on y and ⟨−B2Xδn α +C2ξδn, f δn⟩(x0, y) = 0 holds, the connected curve f δn(x0, y) is contained a hyperplane R3 x0 perpendicular to the vector (−B2Xδn α + C2ξδn)(x0, y), and in particular it lies on a 2-sphere of radius 1/ � B2 2 + C2 2(x0) in R3 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Furthermore, we have (B ◦f δn)(x, −y) = f δn(x, y) for the reflection B defined in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10 and have limy→∞ f δn(x0, y) = limy→∞ f δn(x0, −y) as mentioned in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Approximation of the curve f 0(y, y) for b ≥ y ≥ a(> 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F 0(x, y) be a solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='6) with F 0(a, a) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let us take a division a = y0 < y1 < y2 < · · · < yn = b of equal length δn = (b − a)/n and a path mn : (a, a) = (y0, y0) → (y0, y1) → (y1, y1) → (y1, y2) → · · → (yn−1, yn−1) → (yn−1, yn) → (yn, yn) = (b, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Let F δn(x, y) be an orthonormal frame on mn determined by F δn(a, a) = Id and the path mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' For F δn(x, y), the vectors ˜f δn(yi, y) and f δn(x, yi+1) above are determined on each edge {yi} × [yi, yi+1] and [yi, yi+1] × {yi+1}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, since each lattice point (yi, yi) for y-curve (yi, y) is not singular for the metric g0, the curve f δn(x, y) determined from these vectors is an approximation of the curve f 0(x, y) on mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' In consequence, we obtain a sequence f δn(yi, yi) (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' , n) of points, which approximates to the curve f 0(y, y) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Next, we attach the x-curve f δn(x, yi) and y-curve f δn(yi, y) to each points (yi, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, in order to obtain correctly the relation between these curves, we have to take the frames F δn(x, y) that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='3) holds for these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' That is, we determine these frames F δn(x, y) in the directions that x and y increase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Hence, F δn(x, yi) and F δn(yi, y) are determined from F δn(yi, yi) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, the x-curve f δn(x, yi) (x ≤ yi) and the y-curve f δn(yi, y) are naturally determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' However, for the x-curve, we have f δn(x, yi) ≡ f δn(yi, yi) for x ∈ [yi, yi+1] mentioned above, and hence it is not sure how we define the curve f δn(x, yi) for x ∈ [yi, yi+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' On the above problem, for x ∈ [yi, yi+1] we determine a frame Aδn(x, yi) := F δn(x, yi) from Aδn(yi+1, yi) = Id by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='10)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content='13) and a curve kδn(x, yi) := f δn(x, yi) in the inverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Thus, we obtain the curve f δn(x, yi) desired for x ∈ [yi, yi+1] by f δn(x, yi) := C � kδn(x, yi) − kδn(yi, yi) � + f δn(yi, yi), where CAδn(xi, yi) = F δn(yi, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Then, we also obtain the x-curve f δn(x, yi) for x ≥ yi+1 from the point f δn(yi+1, yi) and the frame F δn(yi+1, yi) given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' References [1] Burstall F E, Hertrich-Jeromin U, Suyama Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' Curvilinear coordinates on generic conformally flat hyper- surfaces and constant curvature 2-metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} +page_content=' J Math Soc Japan, 2018, 70: 617–649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf'} 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